AnAlySiS of STRuCTuRAl DynAmiC ABSTRACT

AnAlySiS of STRuCTuRAl DynAmiC ABSTRACT
Furthermore, vibration in milling has also been
studied in relation to milling tool holders with a
long overhang.A basic investigation concerning the
spatial dynamic properties of the tool holders of
milling machines, both when not cutting and during
cutting, has been carried out. Also, active control
of milling tool holder vibration has been investigated and a first prototype of an active milling tool
holder was implemented and tested.The challenge
of transferring electrical power while maintaining
good signal quality to and from a rotating object is
addressed and a solution to this is proposed.
Finally, vibration is also a problem for the hydroelectric power industry. In Sweden, hydroelectric power plants stand for approximately half of
Sweden’s electrical power production and are
also considered to be a so-called green source of
energy. When renovating water turbines in smallscale hydroelectric power plants and modifying
them to optimize efficiency, it is not uncommon
that disturbing vibrations occur in the power
plant. These vibrations have a negative influence
on the production capacity and will wear various
components quickly. Occasionally, these vibrations
may cause severe damage to the power plant. To
identify this vibration problem, experimental modal analysis and operating deflection shape analysis
were utilized. To reduce the vibration problem, active control using inertial mass actuators was investigated. Preliminary results indicate a significant
attenuation of the vibrations.
Henrik Åkesson
ISSN 1653-2090
ISBN 978-91-7295-172-3
2009:05
2009:05
Analysis of Structural Dynamic Properties
Vibration in metal cutting is a common problem in
the manufacturing industry, especially when long
and slender tool holders or boring bars are involved in the manufacturing process. Vibration has a
detrimental effect on machining. In particular the
surface finish is likely to suffer, but tool life is also
most likely to be reduced. Tool vibration also results in loud noise that may disturb the working
environment.
The first part of this thesis describes the development of a robust and manually adjustable
analog controller capable of actively controlling
boring bar vibrations related to internal turning.
This controller is compared with an adaptive
digital feedback filtered-x LMS controller and it
displays similar performance with a vibration attenuation of up to 50 dB.
A thorough experimental investigation of the
influence of the clamping properties on the dynamic properties of clamped boring bars is also
carried out in second part of the thesis. In relation to this, it is demonstrated that the number
of clamping screws, the clamping screw diameter
size, the screw tightening torque and the order
the screws are tightened, have a significant influence on a clamped boring bar’s eigenfrequencies
as well as on its mode shape orientation in the
cutting speed - cutting depth plane. Also, an initial
investigation of nonlinear dynamic properties of
clamped boring bars was carried out.
and Active Vibration Control Concerning Machine Tools and a Turbine Application
ABSTRACT
Analysis of Structural Dynamic
Properties and Active Vibration
Control Concerning Machine
Tools and a Turbine Application
Henrik Åkesson
Blekinge Institute of Technology
Doctoral Dissertation Series No. 2009:05
School of Engineering
Analysis of Structural Dynamic Properties and
Active Vibration Control Concerning Machine Tools
and a Turbine Application
Henrik Åkesson
Blekinge Institute of Technology Doctoral Dissertation Series
No 2009:05
Analysis of Structural Dynamic Properties
and Active Vibration Control Concerning
Machine Tools and a Turbine Application
Henrik Åkesson
Department of Electrical Engineering
School of Engineering
Blekinge Institute of Technology
SWEDEN
© 2009 Henrik Åkesson
Department of Electrical Engineering
School of Engineering
Publisher: Blekinge Institute of Technology
Printed by Printfabriken, Karlskrona, Sweden 2009
ISBN 978-91-7295-172-3
Blekinge Institute of Technology Doctoral Dissertation Series
ISSN 1653-2090
urn:nbn:se:bth-00452
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Abstract
Vibration in metal cutting is a common problem in the manufacturing industry, especially when long and slender tool holders or boring bars are involved
in the manufacturing process. Vibration has a detrimental effect on machining.
In particular the surface finish is likely to suffer, but tool life is also most likely
to be reduced. Tool vibration also results in loud noise that may disturb the
working environment.
The first part of this thesis describes the development of a robust and manually adjustable analog controller capable of actively controlling boring bar vibrations related to internal turning. This controller is compared with an adaptive
digital feedback filtered-x LMS controller and it displays similar performance
with a vibration attenuation of up to 50 dB.
A thorough experimental investigation of the influence of the clamping properties on the dynamic properties of clamped boring bars is also carried out in
second part of the thesis. In relation to this, it is demonstrated that the number of clamping screws, the clamping screw diameter size, the screw tightening
torque and the order the screws are tightened, have a significant influence on
a clamped boring bar’s eigenfrequencies as well as on its mode shape orientation in the cutting speed - cutting depth plane. Also, an initial investigation of
nonlinear dynamic properties of clamped boring bars was carried out.
Furthermore, vibration in milling has also been studied in relation to milling
tool holders with a long overhang. A basic investigation concerning the spatial
dynamic properties of the tool holders of milling machines, both when not cutting and during cutting, has been carried out. Also, active control of milling
tool holder vibration has been investigated and a first prototype of an active
milling tool holder was implemented and tested. The challenge of transferring
electrical power while maintaining good signal quality to and from a rotating
object is addressed and a solution to this is proposed.
Finally, vibration is also a problem for the hydroelectric power industry.
In Sweden, hydroelectric power plants stand for approximately half of Sweden’s
electrical power production and are also considered to be a so-called green source
of energy. When renovating water turbines in small-scale hydroelectric power
plants and modifying them to optimize efficiency, it is not uncommon that disturbing vibrations occur in the power plant. These vibrations have a negative
influence on the production capacity and will wear various components quickly.
Occasionally, these vibrations may cause severe damage to the power plant.
To identify this vibration problem, experimental modal analysis and operating
deflection shape analysis were utilized. To reduce the vibration problem, active control using inertial mass actuators was investigated. Preliminary results
indicate a significant attenuation of the vibrations.
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Preface
This thesis summarizes my work at the Department of Electrical Engineering at
Blekinge Institute of Technology. The thesis is comprised by an introduction followed
by six parts:
Part
I
On the Development of a Simple and Robust Active Control System for Boring
Bar Vibration in Industry.
II
Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping
Conditions.
III
Estimation and Simulation of Nonlinear Dynamic Properties of a Boring bar.
IV
Investigation of the Dynamic Properties of a Milling Tool Holder.
V
Preliminary Investigation of Active Control of a Milling Tool Holder.
VI
Noise Source Identification and Active Control in a Water Turbine Application.
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Acknowledgments
Firstly, I would like to express my sincere gratitude to professor Ingvar Claesson for
giving me the opportunity to begin a PhD candidacy. Special thanks to my supervisor
and friend professor Lars Håkansson, for his profound knowledge within the field of
both applied signal processing and mechanical engineering, but also for all his help
in improving this thesis. I would also like to thank former and present colleagues at
the department, especially my co-worker and friend Tatiana for all the constructive
discussions, her help and support.
I would like to express my sincere gratitude to Dr. Thomas L Lagö, Adjunct
Professor, who provided an industrial touch to my candidacy by introducing me to
different vibration problems in industry. Special thanks to the president of Acticut
International AB, CEO Rolf Zimmergren, for believing in my abilities. Furthermore,
thanks go to the colleagues at Acticut International AB for their help, discussions
and good collaboration.
I am indebted to my parents for providing me with a good foundation, giving me
the freedom to follow my interest, letting me form my own ideas and supporting me
in the pursuit of knowledge. My warmest gratitude is directed to my sisters for their
support.
I would also like to thank my wife Lisa for her love, support and patience throughout the years and for being the person she is. Finally, my greatest thanks and love go
to the most important people in the world, Pontus and Ebba for their unconditional
love.
Henrik Åkesson
Ronneby, November 2009
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Publication list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Part
I
On the Development of a Simple and Robust Active Control System for
Boring Bar Vibration in Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
II
Analysis of Dynamic Properties of Boring Bars Concerning Different
Clamping Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
III
Estimation and Simulation of Nonlinear Dynamic Properties of a
Boring bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
IV
Investigation of the Dynamic Properties of a Milling Tool Holder . . . . . . . . . 130
V
Preliminary Investigation of Active Control of a Milling Tool Holder . . . . . . 134
VI
Noise Source Identification and Active Control in a Water Turbine
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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Publication List
Part I is published as:
H. Åkesson, T. Smirnova, I. Claesson and L. Håkansson, On the Development of
a Simple and Robust Active Control System for Boring Bar Vibration in Industry,
IJAV-International Journal of Acoustics and Vibration, 12(4), (pp 139-152), 2007.
Part II is published as:
H. Åkesson, T. Smirnova, and L. Håkansson, ”Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping Conditions ”, Journal of Mechanical Systems
& Signal Processing, 23(8), pp. 2629-2647, 2009.
Part III is published as:
H. Åkesson, T. Smirnova, I. Claesson, T. Lagö and L. Håkansson, ”Estimation and
Simulation of Nonlinear Dynamic Properties of a Boring bar”, submitted to publication in International Journal of Acoustics and Vibration, August, 2009.
Part IV is published as:
H. Åkesson, T. Smirnova, T. Lagö, L. Håkansson and I. Claesson, Investigation of the
Dynamic Properties of a Milling Tool Holder, Research Report No 2009:14, Blekinge
Institute of Technology, ISSN: 1103-1581, November, 2009.
Part V is based on the publication:
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö Preliminary investigation of active control of a milling tool holder, ACTIVE 2009 - The 2009 International Symposium on Active Control of Sound and Vibration, Ottawa, Canada,
20-22 August, 2009.
Part VI is based on the publication:
H. Åkesson, A. Sigfridsson, T. Lagö, I. Andersson and L. Håkansson, Noise Source
Identification and Active Control in a Water Turbine Application, In Proceedings of
The Sixth International Conference on Condition Monitoring and Machinery Failure
Prevention Technologies, Dublin, Ireland, 23-25 June, 2009.
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Other Publications
Invited: I. Gustavsson, J. Zackrisson, H. Åkesson ans L. Håkansson, "A Remotely
Operated Traditional Electronics Laboratory", International Journal of Online Engineering, Vol. 2, No. 1, 2006.
H. Åkesson "Active control of vibration and analysis of dynamic properties concerning machine tools", Technology Licentiate Dissertion, ISSN 1650-2140, 2007.
H. Åkesson, T. Smirnova, T. Lagö and L. Håkansson "Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping Conditions", Research report
No 2007:06, ISSN 1101-1581, 2007.
T. Smirnova, H. Åkesson and L. Håkansson "Modeling of an Active Boring Bar",
Research report No 2007:14, ISSN 1103-1581, 2007.
B. Sällberg, H. Åkesson, N. Westerlund, M. Dahl, I. Claesson, "Analog Circuit Implementation for Speech Enhancement Purposes" published at the 38th ASILOMAR
conference, Pacific Grove, California, USA,
November 2004
I. Gustavsson, H. Åkesson, "A Remote Laboratory providing Teacher-defined Sessions", Proceedings of the ICEE 2004 Conference in Gainesville, USA, October 17 21, 2004.
H. Åkesson, A. Brandt, T. Lagö, L. Håkansson, I. Claesson, "Operational Modal
Analysis of a Boring Bar During Cutting" published at the 1st IOMAC conference,
April 26-27, 2005, Copenhagen, Denmark
I. Gustavsson, T. Olsson, H. Åkesson, J. Zackrisson, L. Håkansson, "A Remote Electronics Laboratory for Physical Experiments using Virtual Breadboards", Proceedings
of the 2005 ASEE Annaual Conference, Portland, USA, June 12 - 15, 2005.
B. Sällberg, H. Åkesson, M. Dahl and I. Claesson, "A Mixed Analog - Digital Hybrid
for Speech Enhancement Purposes" published at ISCAS 2005.
H. Åkesson, I. Gustavsson, L. Håkansson, I. Claesson, "Remote Experimental Vibration Analysis of Mechanical Structures over the Internet", Proceedings of the 2005
ASEE Annaual Conference, Portland, USA, June 12 - 15, 2005.
I. Gustavsson, J. Zackrisson, H. Åkesson and L. Håkansson, ”A Flexible Remote Electronics Laboratory”, In proceedings of the International Remote Engineering Virtual
Instrumentation Symposium, REV2005, Bukarest, Romania, 30 June-1 July, 2005.
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö, I. Claesson, "Analog and Digital Approaches of Attenuation Boring Bar Vibrations During Metal Cutting Operations", published at the 12th ICSV conference, Lisabon, Portugal, July 11 - 14, 2005.
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H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Analog versus
Digital Control of Boring Bar Vibration", Accepted for publication In proceedings of
the SAE World Aerospace Congress, WAC, Dallas, Texas, USA, October 3-6, 2005.
I. Gustavsson, J. Zackrisson, H. Åkesson ans L. Håkansson, "Experimentera hemma
med utrustningen i universitetens övningslaboratorier", 4:e Pedagogiska Inspirationskonferensen 2006, Lund
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson, A. Sigfridsson, T. Svensson
and T. Lagö, "Active Boring Bar Prototype Tested in Industry ," Accepted Paper,
Adaptronic Congress 2006, 03-04 May, Göttingen, Germany.
H. Åkesson, I. Gustavsson, L. Håkansson, I. Claesson and T. Lagö, "Vibration Analysis of Mechanical Structures over the Internet Integrated into Engineering Education",
In proceedings of the 2006 ASEE Annual Conference & Exposition, Chicago IL, June
18 - 21, 2006.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson, A. Sigfridsson, T. Svensson
and T. Lagö, "A First Prototype of an Active Boring Bar Tested in Industry", In
Proceedings of the Twelfth International Congress on Sound and Vibration, ICSV12,
Lisbon, Portugal, 11 - 14 July, 2006.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Identification
of Spatial Dynamic Properties of the Boring Bar by means of Finite Element Model:
Comparison with Experimental Modal Analysis and Euler-Bernoulli Model", In Proceedings of the Twelfth International Congress on Sound and Vibration, ICSV12,
Lisbon, Portugal, 11 - 14 July, 2006.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Comparison of
different controllers in the active control of tool vibration; including abrupt changes
in the engagement of metal cutting", Sixth International Symposium on Active Noise
and Vibration Control, ACTIVE, Adelaide, Australia, 18-20 September, 2006.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Building an Accurate Finite Element Model of the Boring Bar with "Free-Free" Boundary Conditions:
Correlation to Experimental Modal Analysis and Euler-Bernoulli Model", Noise and
Vibration Engineering Conference, ISMA, Leuven, Belgium, 18 - 20 September 2006.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Vibration in
Turning and the Active Control of Tool Vibration", Second World Congress on Engineering Asset Management and the Fourth International Conference on Condition
Monitoring, Harrogate, UK, 11-14 June, 2007.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Investigation
of the Dynamic Properties of a Boring Bar Concerning Different Boundary Conditions", ICSV14 - The Fourteenth International Congress on Sound and Vibration,
Cairns, Australia, 9-12 July, 2007.
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T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Initial Experiments
With a Finite Element Model of an Active Boring Bar",ICSV14 - The Fourteenth International Congress on Sound and Vibration, Cairns, Australia, 9-12 July, 2007.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson, A. Sigfridsson, T. Svensson
and T. Lagö, "Investigation of the Dynamic Properties of a Passive Damped Boring
Bar", In Proceedings of Adaptronic Congress, Berlin, Germany, 20 - 21 May, 2008.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Advantages and
Drawbacks Using Different Sensors in Feedback Control in Active Boring Bar Applications", ICSV15 - The 15th International Congress on Sound and Vibration, Daejon,
Korea, 6 - 10 July, 2008.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Investigation Concerning Actuator Position in an Active Boring Bar Regarding it´s Performance by
Means of "3-D" Finite Element Models", ICSV15 - The 15th International Congress
on Sound and Vibration, Daejon, Korea, 6 - 10 July, 2008.
T. Lagö, L. Håkansson and H. Åkesson, "Classification of metal cutting vibrations, is
it all chatter?", In Proceedings of Fifth International Conference on Condition Monitoring and Machinery Failure Prevention Technologies, Edinburgh, Scotland, UK, 15
- 18 July, 2008.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Dynamic Properties of Tooling Structure: Hydraulic Clamping versus Standard Screw Clamping in
a Lathe Application", Ninth International Conference on Computational Structures
Technology, Athens, Greece, 2-5 September, 2008.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Estimation of an
active boring bar´s control path FRF:s by means of its 3-D FE-model with Coulomb
friction", The Ninth International Conference on Computational Structures Technology, Athens, Greece, 2-5 September, 2008.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Developments
Steps of an Active Boring Bar for Industrial Application", Internoise, Shanghai,
China, 26-29 October, 2008.
M. Winberg, H. Åkesson, T. Lagö, and I. Claesson, "Acoustical Measurements Using
a Virtual Microphone Technique", Internoise, Shanghai, China, 26-29 October, 2008.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Investigation
of the Dynamic Properties of a Milling Structure; Using a Tool Holder with Moderate
Overhang", IMAC XXVII, Orlando, USA, 9-12 February, 2009.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Accurate FEmodeling of a Boring Bar Correlated with Experimental Modal Analysis", IMAC
XXVII, Orlando, USA, 9-12 February, 2009.
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H. Åkesson, A. Sigfridsson, T. Lagö, I. Andersson and L. Håkansson, "Noise Source
Identification and Active Control in a Water Turbine Application", In Proceedings of
The Sixth International Conference on Condition Monitoring and Machinery Failure
Prevention Technologies, Dublin, Ireland, 23-25 June, 2009.
T. Smirnova, H. Åkesson, I. Claesson, L. Håkansson and T. Lagö, "Modeling a
Clamped Boring Bar using Euler-Bernoulli Beam Models with Various Boundary
Conditions",3rd Conference on Mathematical Modeling of Wave Phenomena/20th
Nordic Conference on Radio Science and Communications, v1106, p149-156, 2009.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Investigation
Concerning Dynamic Properties of an Active Boring Bar Regarding its Perfomance
by Means of "1-D" Finite Element Models",ICSV16 - The Sixteenth International
Congress on Sound and Vibration, Krakow, Poland, 5-9 July, 2009.
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö, "Preliminary investigation of active control of a milling tool holder", ACTIVE 2009 - The 2009 International Symposium on Active Control of Sound and Vibration, Ottawa, Canada,
20-22 August, 2009.
T. Smirnova, H. Åkesson, L. Håkansson, I. Claesson and T. Lagö, "Simulation of
Active Suppression of Boring Bar Vibrations by Means of Boring Bar’s "1-D" Finite
Element Model", ACTIVE 2009 - The 2009 International Symposium on Active Control of Sound and Vibration, Ottawa, Canada, 20-22 August, 2009.
T. Smirnova, H. Åkesson, and L. Håkansson, ”Dynamic Modeling of a Boring Bar Using Theoretical and Experimental Engineering Methods Part 1: Distributed-Parameter
System Modeling and Experimental Modal Analysis”, IJAV-International Journal of
Acoustics and Vibration, 14(3), pp. 124-133, 2009.
T. Smirnova, H. Åkesson, and L. Håkansson, ”Dynamic Modeling of a Boring Bar
Using Theoretical and Experimental Engineering Methods Part 2: Finite Element
Modeling and Sensitivity Analysis”, IJAV-International Journal of Acoustics and Vibration, 14(3), pp. 134-142, 2009.
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Introduction
1
Introduction
Vibration concerns the repetitive motion of an object or objects relative to a stationary
frame referred to as the equilibrium of the vibration. Vibrations may be measured
in terms of displacement, velocity or acceleration. Vibrations exist everywhere and
may have a great impact on the surrounding environment. One general phenomenon
of vibration is ”self-oscillation” or resonance [1], meaning that a system exposed to
even a weak force that excites a resonance, may causes a substantial vibration level
that eventually results in damage to or failure of the system. Thus, it is of great
importance in engineering design to consider the dynamic properties of the system
from a vibration point of view. In Fig. 1, an example of the simplest possible vibrating
system, a single degree of freedom (SDOF) system, is presented in conjunction with
a diagram describing the displacement x(t) of the mass. An equation according to
D a m p e r
c
k
M a ss
m
A
D is p la c e m e n t
S p rin g
E q u ilib riu m
x (t)
112
f
A e
0
- z 2 p f0t
t
T im e
Figure 1: A single degree of freedom system, consisting of a rigid object with a mass
m (kg), suspended to a supporting rigid structure via a spring, having a stiffness
coefficient of k (N/m), and a damper with a damping coefficient of c (Ns/m). The
figure also describes the system’s oscillation behavior for the initial displacement A
where f0 is the natural frequency and ζ is the relative damping ratio.
Newton’s second law for the forces acting on the mass m in a SDOF-system in the x
direction, yields the equation of motion as [1]
dx(t)
d2 x(t)
+c
+ kx(t) = f (t)
(1)
2
dt
dt
where f (t) is an external force acting on the mass. In general, the dynamic response of
a structure cannot be described adequately by a SDOF model. The response usually
includes time variations in the displacement shape as well as its amplitude. Thus,
a multiple degree of freedom system or a distributed parameter system is usually
required to describe the motion or response of a structure [1]. Generalizing to include
several masses connected to each other and to the ground by springs and dampers,
results in multiple degree of freedom (MDOF) system whose equation of motion may
in matrix form be written as [1]
m
d2 {x(t)}
d {x(t)}
+ [C]
+ [K] {x(t)} = {f (t)}
(2)
dt2
dt
where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {x(t)} is the displacement vector for all masses and {f (t)} is the force vector
representing all external forces applied to each mass.
[M]
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This way of representing systems is referred to as lumped parameter representation, since each mass in the system is modeled as a concentrated mass (single point).
Also, distributed parameter system theory may be utilized for the modeling of structures, resulting in a model which represents an infinite dimensional continuous system
in both space and time [1]. The two latest methods have the ability to model real
structures, including their spatial dynamic properties, mainly in terms of natural
frequencies fr , relative damping coefficients ζr and mode shapes {ψr }.
An application where vibration is a frequent problem may be found in the manufacturing industry, particularly in the workshop, where metal cutting operations such
as external and internal turning, boring, milling etc., take place. Vibrations affect
the surface finish of the workpiece, tool life and the noise level in the working environment. In order to increase productivity and tool life, and improve the tolerance of
machined workpieces, it is necessary to develop methods which increase stability and
restrain tool vibration in metal cutting.
In many cases, the tooling structure may be considered a bottleneck with regard to
the achievable accuracy imposed by static deflections and cutting regimes, but also in
respect of surface finish due to forced and self-excited vibrations [2,3]. A long overhang
cantilever tooling structure (e.g. boring bars) is often the critical component [2, 3].
Fig. 2 presents a typical configuration for internal turning using a boring bar with
a long overhang, in which a tube is clamped to one side in the chuck by three jaws
while the boring bar is clamped to the clamping house.
Ja w s
C la m p in g h o u s e
W o rk p ie c e
B o rin g b a r
Figure 2: Typical configuration for internal turning, illustrating the long boring bar
overhang required to turn deep holes.
There are several possible sources of boring bar vibrations, including transient
excitations due to rapid movements or to the engagement phase of cutting; periodic
excitation related to residual rotor mass unbalance in the spindle-chuck-workpiece
system or random excitation from the material deformation process [4]. The two
most widely used theories for explaining self-excited chatter or tool vibration are the
regenerative effect and the mode coupling effect [5–8]. These theories generally assume
that the dynamic interaction of the cutting process and the machine tool structure
constitute the basic causes of chatter [5–8].
During cutting, the cutting force Fr (t) is generated between the tool and the
workpiece, see Fig. 3 b). The cutting force applied by the material deformation
process during turning will strain the tool-boring bar structure and may introduce a
relative displacement of the tool and the workpiece, changing the tool and workpiece
engagement. This relation between cutting force and tool displacement is commonly
described by the feedback system pictured in Fig. 3 a). The causes of instability
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Introduction
3
are generally considered to derive from mechanisms providing energy, along with the
regenerative effect and the mode coupling effect. The regenerative effect is considered
to be the most frequent cause of instability and chatter, and may appear when the
tool removes an undulation on the workpiece surface that was cut during the previous
revolution of the workpiece. Fig. 3 b) illustrates this scenario, where h0 (t) is the
desired cutting depth or chip thickness, h(t) the actual chip thickness, y(t) is the
displacement of the tool at the time t and y(t − T ) is the displacement of the tool
during the preceding revolution of the workpiece.
y (t - T )
y (t)
M a c h in e to o l
s tru c tu re
C u ttin g
sp e e d
y (t)
F r(t)
h (t)
T o o l
C u ttin g p ro c e s s
W o rk p ie c e
F r(t)
h 0(t)
a)
b)
Figure 3: a) Block diagram describing the cutting process-machine tool structure
feedback system and b) the principle for regenerative chatter where the h0 (t) is the
desired cutting depth or chip thickness, h(t) the actual chip thickness, y(t − T ) the
previous cut shape and y(t) the present cut shape.
The classic self-excited chatter models fail to explain some types of vibration which
might exist in the system [9]. For instance, chips produced by the chip formation
process - which is a part of the material deformation process - may indicate a presence
of narrow band excitation [10] (see Fig. 4 where a saw-toothed continuous fragmentary
chip process is presented). The shape of the chips depends on many factors such as
C h ip
T o o l
T o o l
W o rk p ie c e
a)
b)
Figure 4: a) A model of the chip formation of a saw-toothed continuous fragmentary
chip and b) a photo of the chip formation, during continuous cutting [10].
tool geometry, the workpiece material, the cutting speed, and the cutting depth.
Another model considers the material deformation process as broadband vibration
excitation of the boring bar. This model is referred to as ”self-excited vibration with
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white noise excitation” [9]. The irregularities of the workpiece surface, the chemical
composition, the inhomogeneities, the microstructure and the spatial stochastic variation of the hardening [11] result in a cutting force, which may be considered as a
stochastic process [12, 13]. Fig. 5 illustrates the material structure of one common
work material: chromium molybdenum nickel steel SS 2541-03 (AISI 3239).
0 .1 m m
Figure 5: The material structure of chromium molybdenum nickel steel SS 2541-03
(AISI 3239).
A number of methods have been proposed to reduce harmful tool vibration. Three
of these methods are as follows:
• ”trial and error” - the operator tries to adapt the cutting data in an iterative
fashion;
• passive control - constructional enhancement of the dynamic stiffness, can be
achieved by increasing the structural damping and/or stiffness of the boring
bar;
• active control - selective increase of the dynamic stiffness of a fundamental
boring bar’s natural frequency.
The ”trial and error” approach requires continuous supervision and control of the
machining process by a skilled operator. Passive control is frequently tuned to increase
the dynamic stiffness at a certain eigenfrequency, (for example, that of a particular
boring bar), making it an inflexible solution [14, 15].
Active control, which is based on an adaptive feedback controller and a boring bar
with an integrated actuator and vibration sensor, can easily be adapted to various
configurations. Thus, active control provides a more flexible, and therefore preferable,
solution. Active control of a boring bar can be implemented using either a digital
or an analog approach. The use of a digital controller based on a feedback filteredx LMS algorithm [16] results in substantial attenuation of vibrations, and exhibits
stable behavior. However, it should be noted that an analog controller with the
corresponding vibration attenuation performance is able to avoid unnecessary delay
in control authority and eventual tool failure in the engagement phase of the tool. A
digital controller always introduces delay associated with controller processing time,
A/D-and D/A- conversion processes and anti-aliasing and reconstruction filtering.
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Other benefits with an analog controller may include, low complexity, reduced cost
and flexible bandwidth.
Both the analog and digital domains may be utilized for the implementation of
feedback controllers [17–19]. Implementation of an active control solution requires
a modification of the boring bar structure, and thus of the bar’s dynamic properties. Modifications should be carried out with care to avoid undesired problems that
may result from making the boring bar too flexible, or moving boring bar resonance
frequencies so that they coincide with other structural resonance frequencies of the
machine tool system.
PART I - On the Development of a Simple and Robust Active
Control System for Boring Bar Vibration in Industry
The application of the active control of boring bar vibration in industry requires
reliable, robust adaptive feedback controllers or manually tuned feedback controllers,
that are easy to adjust on the workshop floor by the lathe operator. This part of the
thesis presents the development of a simple adjustable robust analog controller, based
on a digitally controlled, analog design that is suitable for the control of boring bar
vibration in industry. Fig. 6 presents a block diagram of a feedback control system.
D is tu rb a n c e
C o n tro l
s ig n a l
D y n a m ic S y s te m
S e c o n d a ry
v ib ra tio n
C o n tro lle r
F e e d b a c k
s ig n a l
+
Figure 6: Block diagram of a feedback control system.
Initially, a digitally controlled, analog and manually adjustable lead compensator
was developed. This manually adjustable controller approach was further developed to
provide controller responses appropriate for the active control of boring bar vibration.
The controller relies on a lead-lag compensator and enables manual tuning by the
lathe operator. The emphasis has been on designing a controller that enables simple
adjustment of its gain and phase. This should provide the robust control appropriate
for industry application.
Furthermore, this part of the thesis features a comparative evaluation of the two
analog controllers and a digital controller based on the feedback filtered-x LMS algorithm [18, 20] capable of providing good performance and robustness in the active
control of boring bar vibration. This evaluation includes the consideration of a number of different dynamic properties of the boring bar, produced using a set of different
clamping conditions likely to occur in industry. Both the developed analog controller
and the adaptive digital controller are able to reduce the boring bar vibration level
by up to approximately 50 dB.
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PART II - Analysis of the Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
Successful implementation of active control, such as in the method presented in Part
I, requires substantial knowledge concerning the dynamic properties of the tooling
system. Furthermore, the interface between the boring bar and the lathe (i.e. the
clamping house) has a significant influence on the dynamic properties of the clamped
boring bar. Part II of this thesis presents the dynamic properties of boring bars
for different clamping conditions, based on experimental and analytical results. The
different cases reflect on the variations that may be introduced in the clamping conditions of a boring bar when the operator mounts the boring bar in the clamping house
and tightens the clamp screws, as illustrated in Fig. 7. Thus, this section focuses
on those dynamic properties of a boring bar which arise due to different clamping
conditions of the boring bar introduced by a clamping house. In connection with this,
Euler-Bernoulli modeling of a clamped boring bar with emphasis on the modeling of
the clamping conditions was considered.
Figure 7: A lathe operator, mounting a boring bar in a clamping house using a
standard wrench, accomplishing an arbitrary tightening torque. The boring bar may
be expected to exhibit different properties when clamped or mounted in the clamping
house by different operators.
PART III - The Estimation and Simulation of the Nonlinear
Dynamic Properties of a Boring bar
The third part addresses the nonlinear behavior, frequently observed, in the dynamic
properties of a clamped boring bar [21,22]. Two nonlinear SDOF models with different
softening spring nonlinearity were introduced for modeling the nonlinear dynamic
behavior of the fundamental bending mode in the cutting speed direction of a boring
bar. Also, two different methods for the simulation of nonlinear models were used.
PART IV - Investigation of the Dynamic Properties of a Milling
Tool Holder
In the previous parts, vibration in turning operations was discussed. This part addresses the vibrations that appear in the metal cutting operation called milling. An
introduction to the milling operation and the basic terminology are presented. In
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Fig. 8, central parts and components of a milling machine during milling and a milling
operation are pictured. Furthermore, an extensive experimental investigation, involvS p in d le h e a d
W o rk p ie c e
S p in d le
T o o l h o ld e r
T o o l
T o o l h o ld e r
T a b le
T o o l
In se rt
C la m p
a)
W o rk p ie c e
b)
Figure 8: a) An overview of a milling setup presenting central machine parts and
components. In b) a closer view of the place in the milling machine where the actual
cutting in the workpiece take place.
ing an analysis of the dynamic properties of a certain milling tool holder, both during
cutting operation and when not cutting, is conducted. The angular vibrations of the
rotating tool, the vibrations of the machine tool structure, and the vibration of the
workpiece were examined during cutting. The focus was on identifying the source or
sources of the dominant milling vibrations and on determining which of these vibrations that are related to the structural dynamic properties of the milling tool holder.
Also, basic distributed parameter system models of the milling told holder were developed in order to simplify, for example, the measurement configuration and the sensor
setup.
PART V - Preliminary Investigation of the Active Control of a
Milling Tool Holder
A thorough investigation of milling dynamics is conducted in Part IV. Large vibration of both the spindle frame structure, the milling tool holder and the workpiece
was observed during experimentation and this is presented in this study. In order to
reduce the vibration levels observed on the workpiece and spindle frame, an active
control system was proposed and implemented. This section analyzes and investigates the challenges of implementing active control of milling tool holder vibrations.
Basically, the strategy is the same as for the active boring bar application, that is,
to increase the dynamic stiffness of the milling tool holder by introducing secondary
vibrations with the opposite phase for a certain mode of interest. However, moving
from an application using a non-rotating tool to an application using a rotating tool,
implies several challenges that must be resolved in order to successfully implement
the proposed solution.
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PART VI - Noise Source Identification and Active Control in a
Water Turbine Application
Another application with vibration problems is presented in Part VI. This part discusses vibration generated by a turbine in the production of electrical power in a
hydroelectric power plant. See Fig. 9 for an overview of the principle of a hydroelectric power plant.
G e n e ra to r
P o w e r h o u se
R e s e rv o ir
P o w e r lin e s
P e n s to c k
G e n e ra to r s h a ft
R o to r
G e n e ra to r
In ta k e
S ta to r
T u rb in e
R iv e r
W a te r
flo w
W ic k e t
g a te s
T u rb in e b la d e s
a)
b)
Figure 9: a) Sketch of a hydroelectric power station. b) The generator zoomed in
presenting how the water makes the generator shaft rotate by applying force on turbine
blades.
In the considered hydroelectrical power plant, the vibrations become severe even
when the operating condition is characterized by a moderate load. This operating
condition results in a power production far from the maximum production capacity
of the power plant. These vibrations thus have a negative influence on the production capacity, but also on the wear of various components such as bearings, hydraulic
connections, etc. In order to reduce the vibration problem, active control is proposed
also in relation to this application. Active control using inertial mass actuators was
investigated and preliminary results indicate a significant attenuation of the vibrations.
References
[1] D.J. Inman. Engineering Vibration. Prentice-Hall, second edition, 2001.
[2] E. I. Revin. Tooling structure: Interface between cutting edge and machine tool.
Annals of the CIRP, 49(2):591–634, 2000.
[3] L. Håkansson, S. Johansson, and I. Claesson. Chapter 81– Machine Tool Noise,
Vibration and Chatter Prediction and Control in Handbook of Noise and Vibration Control. John Wiley & Sons, first edition, 2007.
[4] J. Tlusty. Analysis of the state of research in cutting dynamics. In Annals of the
CIRP, volume 27, pages 583–589. CIRP, 1978.
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[5] S.A. Tobias. Machine-Tool Vibration. Blackie & Son, 1965.
[6] F. Koenigsberger and J. Tlusty. Machine Tool Structures, Volume 1. Pergemon
Press, 1970.
[7] G. Boothroyd and W.A. Knight. Fundamentals of machining and machine toolsS.
Marcel Dekker, second edition, 1989.
[8] H.E. Merritt. Theory of self-excited machine-tool chatter, contribution to
machine-tool chatter, research –1. Journal of Engineering for Industry, Transactions of the ASME, pages 447–454, November 1965.
[9] S.M. Pandit, T.L. Subramanian, and S.M. Wu. Modeling machine tool chatter
by time series. Journal of Engineering for Industry, Transactions of the ASME,
97:211–215, February 1975.
[10] Viktor P. Astakhov. Tribology of Metal Cutting. Elsevier, first edition edition,
2006.
[11] D.R. Askeland. The Science and Engineering of Materials. CHAPMAN & HALL,
second edition, 1990.
[12] L. Andrén, L. Håkansson, A. Brandt, and I. Claesson. Identification of dynamic
properties of boring bar vibrations in a continuous boring operation. Journal of
Mechanical Systems & Signal Processing, 18(4):869–901, 2004.
[13] P-O. H. Sturesson, L. Håkansson, and I. Claesson. Identification of the statistical properties of the cutting tool vibration in a continuous turning operation correlation to structural properties. Journal of Mechanical Systems and Signal
Processing, Academic Press, 11(3), July 1997.
[14] D.G. Lee. Manufacturing and testing of chatter free boring bars. Annals of the
CIRP, 37/1:365–368, 1988.
[15] F. Kuster. Cutting dynamics and stability of boring bars. Annals of the CIRP,
39/1:361–366, 1990.
[16] Sen M. Kuo and Dennis R. Morgan. Active noise control systems, 1996.
[17] S.J. Elliott. Signal Processing for Active Control. Academic Press, London, first
edition, 2001.
[18] L. Håkansson, S. Johansson, M. Dahl, P. Sjösten, and I. Claesson. Chapter 12 Noise canceling headsets for speech communication in CRC Press Handbook on
Noise Reduction in Speech Applications, Gillian M. Davis (ed.). CRC Press, first
edition, 2002.
[19] G.F Franklin, J.D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic
Systems. Prentice Hall, fifth edition, 2006.
[20] I. Claesson and L. Håkansson. Adaptive active control of machine-tool vibration
in a lathe. IJAV-International Journal of Acoustics and Vibration, 3(4):155–162,
1998.
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[21] L. Andrén, L. Håkansson, A. Brandt, and I. Claesson. Identification of motion of
cutting tool vibration in a continuous boring operation - correlation to structural
properties. Journal of Mechanical Systems & Signal Processing, 18(4):903–927,
2004.
[22] H. Åkesson, T. Smirnova, and L. Håkansson. Analysis of dynamic properties
of boring bars concerning different clamping conditions. Journal of Mechanical
Systems & Signal Processing, 23(8):2629–2647, November 2009.
Part I
On the Development of a Simple
and Robust Active Control System
for Boring Bar Vibration in
Industry
This part is published as:
H. Åkesson, T. Smirnova, I. Claesson and L. Håkansson, On the Development of
a Simple and Robust Active Control System for Boring Bar Vibration in Industry,
IJAV-International Journal of Acoustics and Vibration, 12(4), (pp 139-152), 2007.
On the Development of a Simple and
Robust Active Control System for Boring
Bar Vibration in Industry
H. Åkesson, T. Smirnova, I. Claesson and L. Håkansson
Blekinge Institute of Technology
Department of Signal Processing
372 25 Ronneby
Sweden
Abstract
Vibration in internal turning is a problem in the manufacturing industry. A
digital adaptive controller for the active control of boring bar vibration may not
be a sufficient solution to the problem. The inherent delay in a digital adaptive
controller delays control authority and may result in tool failure when the load
applied by the workpiece on the tool changes abruptly, e.g. in the engagement
phase of the cutting edge. A robust analog controller, based on a lead-lag
compensator, with simple adjustable gain and phase, suitable for the industry
application, has been developed. Also, the basic principle of an active boring
bar with embedded actuator is addressed. The performance and robustness of
the developed controller has been investigated and compared with an adaptive
digital controller based on the feedback filtered-x algorithm. In addition, this
paper takes into account those variations in boring bar dynamics which are
likely to occur in industry; for example, when the boring bars is clamped in a
lathe. Both the analog and the digital controller manage to reduce the boring
bar vibration level by up to approximately 50 dB.
1
Introduction
Degrading vibrations in metal cutting e.g. turning, milling, boring and grinding are a
common problem in the manufacturing industry. In particular, vibration in internal
turning operations is a pronounced problem. To obtain the required tolerances of the
workpiece shape, and adequate tool-life, the influence of vibration in the process of
machining a workpiece must be kept to a minimum. This requires that extra care be
taken in production planning and preparation. Vibration problems in internal turning
have a considerable influence on important factors such as productivity, production
costs, working environment, etc. In internal turning the dimensions of the workpiece
hole will generally determine the length and limit the diameter or cross-sectional size
of the boring bar. As a result, boring bars are frequently long and slender - longoverhang cantilever tooling- and thus sensitive to excitation forces introduced by the
14
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Part I
material deformation process in the turning operation [1–3]. The vibrations of a boring bar are often directly related to its low-order bending modes [4–6].
The vibration problems in internal turning can be addressed using both passive and
active methods [1, 2, 7]. Common methods used to increase dynamic stiffness of cantilever tooling involve making them (in high Young’s modulus) non-ductile materials,
such as sintered tungsten carbide and machinable sintered tungsten, and/or utilizing
passive Tuned Vibration Absorbers (TVA) [1,2,8]. These passive methods are known
to enhance the dynamic stiffness and stability (chatter-resistance) of long cutting tools
and thus, enable the allowable overhang to be increased [1, 2, 8]. The passive methods offer solutions with a fix enhancement of the dynamic stiffness frequently tuned
for a narrow frequency range comprising a certain bending mode frequency that in
some cases may be manually adjusted [1, 2, 8]. On the other hand, the active control of tool vibration enables a flexible solution that selectively increases the dynamic
stiffness at the actual frequency of the dominating bending modes until the level of
the chatter component in the feedback signal is negligible [2, 7, 9]. An active control
approach was reported by Tewani et al. [10, 11] concerning active dynamic absorbers
in boring bars controlled by a digital state feedback controller. It was claimed to
provide a substantial improvement in the stability of the cutting process. Browning
et al. [12] reported an active clamp for boring bars controlled by a feedback version
of the filtered-x LMS algorithm. They assert that the method enables to extend the
operable length of boring bars. Claesson and Håkansson [9] controlled tool vibration
by using the feedback filtered-x LMS algorithm to control tool shank vibration in the
cutting speed direction without applying the traditional regenerative chatter theory.
Two important constraints concerning the active control of tool vibration involve the
difficult environment in a lathe and industry demands. It is necessary to protect the
actuator and sensors from the metal chips and cutting fluid. Also, the active control
system should be applicable to a general lathe. Pettersson et al. [13] reported an
adaptive active feedback control system based on a tool holder shank with embedded
actuators and vibration sensors. This control strategy was later applied to boring
bars by Pettersson et al. [6]. Åkesson et al. [14] reported successful application of
active adaptive control of boring bar vibration in industry using an active boring bar
with embedded actuators and vibration sensors.
During the process of machining a workpiece in a lathe, the boundary conditions applied by the workpiece on the cutting tool may exhibit large and abrupt variation,
particularly in the engagement phase between the cutting tool and workpiece. These
abrupt changes of load applied by the workpiece on the tool may result in tool failure.
However when utilizing active adaptive digital control of tool vibration, the problem
of tool failure in the engagement phase may remain. For instance, the time required
for the adaptive tuning of the controller, the inherent delay in, controller processing
time, A/D and D/A- conversion processes, and analog anti-aliasing and reconstruction
filtering might impede an active adaptive digital control system to produce control
authority sufficiently fast to avoid tool failure. To provide means to address the issue
of delay in control authority in the active control of boring bar vibration in industry
an analog controller approach is suggested.
This article focuses on the development of a simple adjustable robust analog controller,
based on digitally controlled analog design, that is suitable for the control of boring
bar vibration in industry. Initially a digitally controlled analog manually adjustable
lead compensator was developed. To provide more appropriate controller responses a
manually adjustable bandpass lead-lag compensator was developed. Gain and phase
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
15
of the controller response may, at a selectable frequency, be independently adjusted
on the two developed controller prototypes. Also, the performance and robustness
using the two analog controllers was evaluated and compared with a digital adaptive
controller based on the feedback filtered-x LMS-algorithm [15,16] in the active control
of boring bar vibration.
2
2.1
Materials and Methods
Experimental Setup
Experiments concerning active control of tool vibration have been carried out in
a Mazak SUPER QUICK TURN - 250M CNC turning centre. The CNC lathe,
presented by the photo in Fig. 1, shows the room in the lathe in which the machining
is carried out. In this photo (Fig. 1(b)), the turret configured with a boring bar
clamped in a clamping house, and a workpiece clamped in the chuck are observable.
Y
Z
X
a)
b)
Figure 1: a) Mazak SUPER QUICK TURN - 250M CNC lathe and b) the room in
the lathe where the machining is carried out.
A coordinate system was defined: z was in the feed direction, y in the reversed
cutting speed direction and x in the cutting depth direction (see upper left corner of
Fig. 1 b)).
2.1.1
Work Material - Cutting Data - Tool Geometry
The cutting experiments used the work material chromium molybdenum nickel steel
SS 2541-03 (AISI 3239). The material deformation process of this material during
turning excites the boring bar with a narrow bandwidth and has a susceptibility
to induce severe boring bar vibration levels [4, 5], resulting in poor surface finish,
tool breakage, and severe acoustic noise levels. The workpiece used in the cutting
experiments had a diameter of 225 mm and a length of 230 mm. To enable supervision
of the metal-cutting process during continuous turning, the cutting operation was
performed externally, see Fig. 1 b). An active boring bar was firmly clamped in a
clamping house rigidly attached to the lathe turret. Only one side of the workpiece
shaft’s end was firmly clamped into the chuck of the lathe, see Fig. 1 b). As a
cutting tool, a standard 55◦ diagonal insert with geometry DNMG 150608-SL and
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carbide grade TN7015 for medium roughing was used. The following cutting data
was selected: Cutting speed v = 60 m/min, Depth of cut a = 1.5 mm, and Feed
s = 0.2 mm/rev.
2.1.2
Measurement Equipment and Setup
A block diagram of the experimental setup for the active control of boring bar vibrations is presented in Fig. 2.
B o r in g b a r
T o o l tip
A c c e le r o m e te r
A c tu a to r
S ig n a l a n a ly z e r
C h 1
H P V X I M a in fr a m e
C h 1
C h 2
C la m p in g h o u s in g
C h a r g e
a m p lifie r
A c tu a to r
a m p lifie r
S ig n a l
c o n d itio n in g
S ig n a l
c o n d itio n in g
O s c illo s c o p e
C h 2
C h 1
C o n tr o lle r
Figure 2: A block diagram describing the experimental setup for the active vibration
control system.
The control experiments used an active boring bar equipped with an accelerometer
and an embedded piezoceramic stack actuator. The actuator was powered with an
actuator amplifier, custom designed for capacitive loads, and the accelerometer was
connected to a charge amplifier. A floating point signal processor with Successive
Approximation Register AD- and DA- converters were used. Two commercial signal
conditioning filters were used in the control experiments. A VXI Mainframe E8408A
with two 16-channel 51.2kSa/s cards were used for data collection.
2.2
Active Boring Bar
The active boring bar used in this experiment is based on the standard WIDAX
S40T PDUNR15 boring bar with an accelerometer and an embedded piezoceramic
stack actuator. By embedding accelerometers and piezoceramic stack actuators in
conventional boring bars, a solution for the introduction of control force to the boring
bar with physical features and properties that fit the general lathe application may
be obtained.
2.2.1
Active Boring Bar - Simple Model
A Euler-Bernoulli beam may be used as a simple model to illustrate the structural
dynamic properties of a boring bar [4, 5]. The Euler-Bernoulli differential equation
describing the transversal motion in the y direction of the boring bar may be written
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
17
as: [17–19].
∂ 2 u(z, t)
∂ 2 u(z, t)
∂me (z, t)
∂2
ρA(z)
=
+ 2 EI(z)
∂t2
∂z
∂z 2
∂z
(1)
where ρ is the density of the boring bar, A(z) the cross-sectional area, u(z, t) the
deflection in y direction, E the Young’s elastic modulus, I(z) the cross-sectional area
moment of inertia, and me (z, t) the space- and time-dependent external moment
load per unit length. The boundary conditions of the boring bar depend on the
suspension of the boring bar ends, and a clamped-free model is suggested [4, 5, 20].
Fig. 3 illustrates a boring bar with a piezoelectric stack actuator embedded in a milled
space in the underside of the boring bar.
A c tiv e b o rin g b a r
P ie c o c e ra m ic s ta c k a c tu a to r
a
z
S e n so r
- m e(t)
fa(t)
z2(t)
y
+ m e(t)
z1(t)
- fa(t)
Figure 3: The configuration of the piezoceramic stack actuator in the active boring
bar.
Assuming that the actuator operates well below its resonance frequency, thus neglecting inertial effects of the actuator. Then, in the frequency domain, the force
the actuator exerts on the boring bar, Fa (f ) (the Fourier transform of fa (t) and
f is frequency), may approximately be related to the constraint expansion or motion of the actuator, Z(f ) = Z2 (f ) − Z1 (f ), as Fa (f ) = Ka (∆La (f ) − Z(f )) [17].
Where Z1 (f ) and Z2 (f ) are the Fourier transform of the displacements z1 (t) and
z2 (t), ∆La (f ) is the Fourier transform of the free expansion of an unloaded piezoelectric stack actuator [17] and Ka is the actuator equivalent spring constant. If the
point receptance at the respective actuator end are summed to form the receptance
HB (f ) = Z1 (f )/Fa (f ) + Z2 (f )/Fa (f ), the relative displacement Z(f ) may be expressed Z(f ) = HB (f )Fa (f ). With the aid of Newton’s second law, an expression
for the actuator force applied on the boring bar as a function of the actuator voltage
V (f ) may now be written as [17];
Fa (f ) = Hf v (f )V (f )
(2)
where Hf v (f ) is the electro-mechanic frequency function between input actuator voltage V (f ) and output actuator force Fa (f ). The distance between the actuator - boring
bar interface center and the natural surface of the active boring bar in the y-direction
is α (see Fig. 3). Thus, the external moment per unit length applied on the boring
bar by the actuator force Fa (f ) may be approximated as:
me (z, f ) = αFa (f )(δ(z − z1 ) − δ(z − z2 )))
(3)
where δ(z) is the Dirac delta function and z1 respective z2 are the z-coordinates for the
actuator - boring bar interfaces. Based on the method of eigenfunction expansion [21],
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if z1 = 0 the generalized load of mode r is Fload,r (f ) = αFa (f )ψr0 (z2 ), ψr0 is the
derivative of the normal mode r. Expressing the generalized load with the aid of Eq.
2 and relying on the method of eigenfunction expansion [21], the frequency-domain
dynamic response of the boring bar in the y direction may be written as:
u(z, f ) =
∞
X
ψr (z)Hr (f )αHf v (f )V (f )ψr0 (z2 )
(4)
r=1
where Hr (f ) is the frequency response function and ψr is the normal mode, for mode
r.
2.3
System Identification
The design of feedback controllers usually rely on detailed knowledge of the dynamic
properties of the system to be controlled, e.g. a dynamic model of the system to
be controlled [22, 23]. Non-parametric spectrum estimation may be utilized to produce non-parametric linear least-squares estimates of dynamic systems [24,25]. Welch
spectrum estimator [25] was used and Table 1 column A gives the spectral density
estimation parameters used in the production of plant frequency function estimates,
with various clamping conditions for the boring bar. Also, the spectral density estimation parameters used in the production of plant frequency function estimates during
continuous metal cutting can be found in Table 1 column B. In Table 1 column C,
the spectral density estimation parameters used in the production of frequency function estimates for the controller responses are given, and in Table 1 column D, the
spectrum estimation parameters used for the production of power spectral density
estimates for boring bar vibration with and without active control are presented.
Parameter
Excitation signal
Sampling frequency f s
Block length N
Freq. resolution ∆f
Number of averages L
Burst length
Freq. range of burst
Window w(n)
Overlap
A
Burst rand.
8192 Hz
16384
0.5 Hz
160
90%
0-4000 Hz
Rectangular
0%
B
True rand.
10240 Hz
20480
0.5 Hz
265-635
Hanning
50%
C
True rand.
10240 Hz
20480
0.5 Hz
160
Hanning
50%
D
Cutting proc.
10240 Hz
10240
1 Hz
100
Hanning
50%
Table 1: Spectral density estimation parameters used for the production of boring
bar vibration spectra, with and without active control.
2.4
Controllers
The application of active control of boring bar vibration in industry requires reliable
robust adaptive feedback control or manually tuned feedback control, which is simple
to adjust at the shop floor by the lathe operator. A controller suitable for active
control of boring bar vibrations might be implemented using different approaches.
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
19
However, the abrupt changes that occur in the turning operations, i.e. in the engagement phase, suggest that an analog control approach might be suitable with respect
to, for example, the controller delay and thus delay in control authority. Simple and
effective compensators or controllers that may be implemented in the analog domain
are the lag and lead compensators that approximate the PI controller and the PD
controller, respectively. Moreover, the PID controller may be approximated in the
analog domain by combining a lag compensator with a lead compensator [26].
A block diagram of the active boring bar vibration feedback control system, a feedback control system for disturbance rejection, is presented in Fig. 4 where −W is the
controller, y(t) is the controller output signal, C is the plant or control path, yc (t) is
the plant output vibration (secondary vibration), d(t) is the undesired tool vibration,
and e(t) is the error signal.
y (t)
C
y c(t)
d (t)
å
- W
e (t)
Figure 4: Block diagram of the active tool vibration control system.
If the plant and the controller are linear time invariant stable systems, their dynamic properties may be described by the frequency response functions C(f ) and
W (f ), respectively. Then, the feedback control systems’ open-loop frequency function can be written as Hol (f ) = W (f )C(f ) and the closed-loop frequency function
for the active boring bar vibration feedback control system is given by Hcl (f ) =
1/(1 + W (f )C(f )).
2.4.1
Controller performance and robustness
Principally, there are two important aspects of the behavior of feedback controllers in
active control systems, their performance and their robustness; that is, the ability of
the controller to reject disturbance and to remain stable under varying conditions [22].
Basically, good performance in feedback control requires high loop gain; while on the
other hand, robust stability usually implies a more retained loop gain. Usually, the
discussion concerning the performance and robust stability of feedback controllers is
based on the sensitivity function S(f ) = 1/(1 + W (f )C(f )) and the complementary
sensitivity function T (f ) = W (f )C(f )/(1 + W (f )C(f )) [22,23]. The sensitivity function S(f ) gives a measure on disturbance reduction of a feedback control system. In
the determination of the stability properties of the system, the complementary sensitivity function T (f ) has a vital role. It also governs the performance of the control
system regarding the reduction of noise from the sensor detecting the error e(t). Generally, the design of controllers rely on a model of the system to be controlled or the
so called plant. However, a model of a physical system is an approximation of the
true dynamics of the system and it is therefore likely to affect the performance of
the control system [22, 23]. To incorporate the plant uncertainty in the design procedure of a controller, a model of the plant uncertainty is usually included. A common
way to model the plant uncertainty is with a multiplicative perturbation, yielding a
White
Part I
20
frequency function model of the plant as [22, 23, 27];
Cactual (f ) = Cnominal (f )(1 + ∆C (f ))
(5)
where ∆C (f ) is an unstructured perturbation given by;
∆C (f ) =
Cactual (f )
−1
Cnominal (f )
(6)
and Cnominal (f ) is a nominal plant model of the plant. Thus, assuming that a control
system design based on the nominal plant model results in a theoretically stable
control system, then the denominator 1 + W (f )Cnominal (f ) has no zeros in the right
half complex plane. However, the actual control system’s frequency response function
will have the denominator 1 + W (f )Cactual (f ). Thus, if |1 + W (f )Cnominal (f )| >
|W (f )Cnominal (f )∆C (f )|, ∀f is fulfilled, the actual control system is stable [22,28,29].
Hence, for stable control the unstructured perturbation is upper limited as |∆C (f )| <
1/|Tnominal (f )|, ∀f where Tnominal (f ) = W (f )Cnominal (f )/(1 + W (f )Cnominal (f )).
If ∆C (f ) is bounded as |∆C (f )| 6 β(f ), ∀f . The condition for robust stability of a
control system is given by [28]
β(f ) <
1
|Tnominal (f )|
, ∀f
(7)
The performance and robustness of an active feedback control system may also be
visualized by a polar plot of its open-loop frequency response function in a Nyquist
diagram [22, 28, 29]. If the closed loop system is to be stable, the polar plot of the
open loop frequency response for the feedback control system W (f )C(f ) must not
enclose the polar coordinate (-1, 0) in the Nyquist diagram. The larger the distance
between the polar plot and the (-1, 0) point, the more robust the feedback control
system becomes, with respect to variation in plant response.
2.4.2
Compensators
A lead compensators purpose is to advance the phase of the open loop frequency
response W (f )C(f ) for a feedback control system, usually by adding maximal positive
phase shift in the frequency range where the loop gain equals 0 dB, i.e. at the
crossover frequency [26,29]. This will increase the phase margin and generally increase
the bandwidth of a feedback control system [26, 29]. The characteristic equation or
frequency function for a lead compensator may be written as [30]:
WLead (f ) = Klead
1 j2πf + zlead
τlead j2πf + 1
= Klead
αlead j2πf + plead
αlead τlead j2πf + 1
(8)
Where zlead > 0 and zlead ∈ R (−zlead is the compensator zero), plead > 0 and
plead ∈ R (−plead is the compensator pole), αlead = zlead /plead < 1 is the inverse lead
ratio for a lead compensator, Klead is the compensator gain, and τlead = 1/zlead . By
utilizing a lag compensator, the low frequency loop gain of a feedback control system
may be increased as the phase-lag filter attenuates the high frequency gain. In this
way, the gain margin of the open loop frequency response for the feedback control
system can be improved, and the phase shift added by the compensation filter can
be minimized [26, 28, 29]. The characteristic equation or frequency function for a
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
21
lag compensator is fairly similar to the lead compensator characteristic equation and
may be expressed as [30]:
WLag (f ) = Klag
τlag j2πf + 1
1 j2πf + zlag
= Klag
αlag j2πf + plag
αlag τlag j2πf + 1
(9)
Here zlag > 0 and zlag ∈ R (−zlag is the compensator zero), plag > 0 and plag ∈ R
(−plag is the compensator pole), αlag = zlag /plag > 1 is the inverse lag ratio for a lag
compensator, Klag is the compensator gain, and τlag = 1/zlag .
A lead-lag compensator is obtained by connecting a lead compensator and a lag
compensator in series, thus by combining Eqs. (8) and (9). The lag compensator
may be adjusted to provide a suitable low frequency loop gain of the feedback control
system. Subsequently, the lead compensator may be adjusted to provide an additional
positive phase shift in the frequency range, where the loop gain equals 0 dB, i.e. at
the crossover frequency [26, 29].
2.4.3
Digitally Controlled Analog Controller
The intention is to develop an analog controller with response properties that can
be easily adjusted manually, without necessitating the replacement of discrete components, i.e. resistors and capacitances. If a lead compensator is considered, its
frequency response function is given by Eq. (8) [26, 29]. A lead compensator may
be designed according to the circuit diagram shown in Fig. 5, where Rd,2 and Rd,f
are adjustable resistors, Rd,1 is a fixed resistor and Cd is a fixed capacitor. The
Rd,f
,
parameters in Eq. (8) are related to the discrete components as Klead = Rd,2 +R
d,1
Rd,1
Rd,2 +Rd,1 .
τlead = Cd Rd,2 , and αlead =
R
R
V
in
d
R
d
R
d ,2
R
d , f
d ,1
C
d
V
o u t
Figure 5: Circuit diagram of a lead compensator.
If Rd,2 and Rd,f are implemented by digitally controlled potentiometers, the phase
and gain of the compensator response may be adjusted independently at one selectable
frequency, in discrete steps, to successively increase or decrease phase and gain respectively. For instance, by using two knobs for the compensator tuning (one for phase
adjustment and one for gain adjustment) a function of the two knob angles may be
produced according to:
[a, g] = AGlead (gain knob angle, phase knob angle)
(10)
This function produces integers a ∈ {1, 2, . . . , La } and g ∈ {1, 2, . . . , Lg }, selecting
the appropriate analog compensator frequency response function in the set of La × Lg
White
Part I
22
different analog compensator frequency response functions:
τa,g j2πf + 1
, a ∈ {1, 2, . . . , La } and g ∈ {1, 2, . . . , Lg }
αa,g τa,g jω + 1
(11)
The micro-controller realized the AG(gain knob angle, phase knob angle) function,
by controlling the adjustable resistors Rd,2 and Rd,f , (the so called digital potentiometers that have a digital control interface and an analog signal path). Such a
micro-controller will allow the implementation of an analog lead-circuit which enabling orthogonal adjustment of the phase function and magnitude function at one
selectable frequency of the compensator response. If the frequency for orthogonal
adjustment of the phase function and magnitude function is set to 500 Hz, the magnitude and phase functions of the frequency response function realized by this circuit
may be adjusted with the phase knob according to the 3-D plots in Fig. 6 a) and b),
respectively. Observe, the gain knob only adjusts the level of the magnitude function
surface and has no influence on its shape or the shape of the phase function surface.
80
15
Phase [Degree]
Magnitude [dB]
Wlead;a,g (f ) = Ka,g
10
5
0
−5
60
40
20
0
100
−10
100
Tu 50
ni
ng
[%
]
0
500
0
1000
2000
1500
Tu
n
50
in
g
Frequency [Hz]
a)
[%
]
0
500
0
1500
1000
2000
Frequency [Hz]
b)
Figure 6: a) Magnitude function and b) phase function of lead compensator frequency
response function as a function of phase knob adjustment range in % and frequency.
The gain knob only adjusts the level of the magnitude function surface.
A lag compensator with orthogonal adjustment of the phase function and the
magnitude function at one selectable frequency of the compensator response may be
designed similar to the adjustable analog lead compensator. Thus, an adjustable lag
compensator may be designed according to the circuit diagram shown in Fig. 7.
R
V
R
in
g
R
g
R
R
g ,1
C
g , f
g ,3
R
g
g ,2
Figure 7: Circuit diagram of a lag compensator.
V
o u t
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
23
15
10
5
0
−5
−10
0
]
[%
100
50
500
1000
1500
Frequency [Hz]
a)
2000
0
Tu
ng
ni
Phase [Degree]
Magnitude [dB]
In Fig. 7, Rg,1 and Rg,f are adjustable resistors and Rg,2 is a fixed resistor and Cg is
a fixed capacitor. The frequency response function for the lag compensator is given by
Eq. (9) [26,29]. The parameters in the equation are related to the discrete components
Rg,f
R R +Rg,1 Rg,3 +Rg,2 Rg,3
according to, Klag = Rg,1 +R
, τlag = Cg Rg,2 , and αlag = g,1 Rg,2
.
g,3
g,1 Rg,2 +Rg,2 Rg,3
In much the same way as the orthogonally adjustable lead compensator, a microcontroller realizes an AGlag (gain knob angle, phase knob angle) function suitable for
steering the digital potentiometers implementing the adjustable resistors Rg,1 and
Rg,f for the lag compensator. If the frequency for orthogonal adjustment of the phase
and magnitude functions is selected to 500 Hz, the magnitude and phase functions
of the frequency response function realized by this circuit may be adjusted with the
phase knob, according to the 3-D plots in Fig. 8 a) and b), respectively.
0
−10
−20
−30
−40
−50
−60
50
0
]
[%
100
−70
500
1000
1500
Frequency [Hz]
b)
2000
0
ng
ni
u
T
Figure 8: Magnitude function and b) phase function of lag compensator frequency
response function as a function of phase knob adjustment range in % and frequency.
The gain knob only adjusts the level of the magnitude function surface.
By connecting the adjustable lead compensator in series with the adjustable lag
compensator, a lead-lag compensator with orthogonal adjustment of the phase function and the magnitude function at one selectable frequency of the response may be
realized. The magnitude and phase functions of the lead-lag compensator may be
adjusted with the phase knob e.g. according to the 3-D plots in Fig. 9 a) and b)
respectively.
Also, to utilize the capacity of the actuator amplifier, to limit the active control
frequency range, a suitable high-pass filter, followed by a suitable low-pass filter was
connected in series with the lead-lag compensator. The block diagram of the obtained
analog band-pass lead-lag controller is shown in Fig. 10.
Selecting the frequency for orthogonal adjustment of the phase and magnitude
functions to 500 Hz, this controller’s frequency response magnitude and phase functions are adjustable with the phase knob, according to the 3-D plots in Fig. 11 a) and
b), respectively.
The implemented analog lead-lag controller consists of several blocks: high-pass
filter, low-pass filter, lead compensator and lag compensator. Each block can be used
separately or arbitrary combinations of the blocks can be used to enable controller
flexibility.
5
0
g[
%]
100
−5
100
50
0
100
−50
50
−10
0
500
1000
1500
Frequency [Hz]
a)
0
2000
Tu
nin
g[
%]
10
Tu
nin
Magnitude [dB]
15
Phase [Degree]
White
Part I
24
50
−100
0
500
1000
1500
Frequency [Hz]
b)
2000
0
Figure 9: a) Magnitude function and b) phase function of lead-lag compensator frequency response function, as a function of phase knob adjustment range in % and
frequency. The gain knob only adjusts the level of the magnitude function surface.
A
H ig h p a s s
filte r
W
L o w p a ss
filte r
W
le a d
la g
100
0
−100
−200
100
50
500
1000
1500
Frequency [Hz]
a)
2000
0
100
−300
Tu
nin
g[
%]
0
[%
]
Phase [Degree]
15
10
5
0
−5
−10
−15
−20
−25
Tu
nin
g
Magnitude [dB]
Figure 10: Block diagram of the implemented lead-lag circuit.
50
−400
0
500
1000
1500
Frequency [Hz]
b)
2000
0
Figure 11: a) Magnitude function and b) phase function of band-pass lead-lag compensator frequency response function, as a function of phase knob adjustment range
in % and frequency. The gain knob only adjusts the level of the magnitude function
surface.
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
2.4.4
25
Feedback Filtered-x LMS Algorithm
The feedback filtered-x LMS algorithm is an adaptive digital feedback controller suitable for narrow-band applications [15,16,31]. This algorithm is based on the method
of steepest descent and the objective of the control is to minimize the disturbance
signal or desired signal in the mean square sense [15, 16, 31]. A block diagram of the
feedback filtered-x LMS algorithm is shown in Fig. 12.
F IR filte r
w (n )
x (n )
E s tim a te o f th e
fo rw a rd p a th Cˆ
x
Cˆ
(n )
e (n - 1 )
y (n )
F o rw a rd p a th
C
y C (n )
A
å
e (n )
A d a p tiv e
a lg o rith m
Z
d (n )
- 1
Figure 12: Block diagram of the feedback filtered-x LMS algorithm.
The feedback filtered-x LMS algorithm with leakage coefficient is defined by the
following equations:
y(n) = wT (n)x(n)
e(n)
w(n + 1)
xĈ (n) =
"I−1
X
i=0
(12)
= d(n) + yC (n)
= γw(n) − µxĈ (n)e(n)
ĉi x(n − i), . . . ,
I−1
X
i=0
(13)
(14)
#T
ĉi x(n − i − M + 1)
(15)
where µ is the adaptation step size and γ is the leakage coefficient 0 < γ < 1, usually
selected close to unity. By selecting γ = 1 the feedback filtered-x LMS algorithm is
obtained. Furthermore, xĈ (n) is the filtered reference signal vector, which usually
is produced by filtering the reference signal x(n) with an I-coefficients FIR-filter
estimate, ĉi , i ∈ 0, 1, . . . , I − 1, of the control path or plant. Furthermore, w(n) is the
adaptive FIR filter coefficient vector, y(n) is the output signal from the adaptive FIR
filter, e(n) is the error signal, yC (n) the secondary vibration (the output signal from
the plant), Ĉ is an estimate of the forward path and d(n) is the primary disturbance.
The reference signal vector x(n) = [x(n), x(n − 1), . . . , x(n − M + 1)]T is related to
the error signal as x(n) = e(n − 1) [15, 16, 31]. In order to select a step size µ to
enable the feedback filtered-x LMS algorithm to converge, the inequality 0 < µ <
2/(E[x2Ĉ (n)](M + δ)) may be used [15, 16, 31], where δ is the overall delay in the
forward path, M is the length of the adaptive FIR filter and E[x2Ĉ (n)] is the mean
square value of the filtered reference signal to the algorithm. Incorporating a leakage
factor γ in the feedback filtered-x LMS-algorithm causes the loop gain of the control
system to be reduced, yielding a more robust behavior [15, 16].
White
Part I
26
3
Results
3.1
The Plant
The system to be controlled, C, is comprised of several parts: a signal conditioning
filter, an actuator amplifier, an actuator, the structural path between the force applied by actuator on the boring bar, and the boring bar response (measured by an
accelerometer mounted close to the tool-tip). In order to clamp the boring bar, it
is first inserted into the cylindrical space of the clamping house. It is then clamped
by means of four/six clamping screws; two/three on the tool side and two/three on
the opposite side of the boring bar. The two standard versions of the clamping house
are distinguishable only by the fact that one supports four clamping screws while the
other supports six. It is obvious that the boundary conditions applied by the fourscrew version of the clamping house will differ from the boundary conditions applied
by the six-screw version of the clamping house. Also, to enable the boring bar to
be inserted in the clamping house, the diameter of the clamping house’s cylindrical
clamping space is slightly larger than the diameter of the boring bar. Thus, the exact
spatial position of the clamped boring bar end in the clamping space of the clamping
house is difficult to pinpoint. Furthermore, the tightening torque of the clamping
screws, i.e. the clamping force, is likely to vary between the screws each time the
boring bar is clamped and each screw is tightened. Thus, each time the boring bar is
clamped it is likely that the clamped boring bar will have different dynamic properties.
Plant frequency function estimates were produced when the boring bar was not in
contact with the workpiece, i.e. off-line. The control path was estimated off-line for
10 different possible clamping conditions with respect to the tightening torque of the
screws and the spatial position of the boring bar in the clamping space of the clamping house. Two spatial positions within clamping space were selected. The first was
that in which the upper side of the boring bar’s end (the tool side) was clamped in
contact with the upper section of the clamping space surface. The second was that in
which the opposite, underside of the boring bar’s end, was clamped in contact with
the lower section of the clamping space surface. For each of these two spatial position configurations, five various tightening torques were used. The different off-line
clamping conditions are presented by Table 2.
Notation
Torq. [Nm]
Contact
BC1
10
BC2
BC3
BC4
BC5
20
30
40
50
Boring bar in contact with
upper side to clamping housing
BC6
10
BC7
BC8
BC9
BC10
20
30
40
50
Boring bar in contact with
under side to clamping housing
Table 2: The ten different clamping conditions of the boring bar used for the production of plant estimates when the boring bar is not in contact with the workpiece
(off-line). Four clamping screws were used, two on the tool side and two on the
opposite side.
The spectrum estimation parameters and identification signal used in the production of off-line frequency function estimates are given in Table 1 column A. Fig. 13
shows frequency function estimates of the plant for five different clamping screw tightening torques (BC6 - BC10 ). These plant frequency function estimates are shown in
the frequency range of the fundamental resonance frequency of the boring bar in
Fig. 14.
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
27
|Ĥ(f )| [dB rel 1m/s2 /V]
40
30
20
Phase ∠Ĥ(f ) [Degree]
50
50
BC
6
BC7
0
BC8
BC
9
−50
BC10
−100
10
−150
0
BC
6
BC7
−10
BC8
BC
−20
9
−200
−250
BC10
−30
0
500
1000
1500
2000
Frequency [Hz]
a)
2500
3000
−300
0
500
1000
1500
2000
Frequency [Hz]
b)
2500
3000
Figure 13: Frequency function estimates of the plant, for the five different tightening
torques of the clamping screws, when the underside, of the boring bar end, is clamped
in contact with the lower part of the clamping space surface.
|Ĥ(f )| [dB rel 1m/s2 /V]
BC6
BC
7
BC
8
BC
45
9
BC10
BC
6
BC7
BC
8
−50
BC
9
BC10
−100
40
−150
35
30
400
Phase ∠Ĥ(f ) [Degree]
0
50
−200
450
500
550
Frequency [Hz]
a)
600
400
450
500
550
Frequency [Hz]
b)
600
Figure 14: Frequency function estimates of the plant in the frequency range of the
fundamental resonance frequency of the boring bar, for the five different tightening
torques clamping screws, when the underside, of the boring bar end is clamped in
contact with the lower part of the clamping space surface.
White
Part I
28
The five other plant estimates shows similar differences but with the resonance
frequency peak between 450 to 470 Hz. As opposed to a situation in which the boring
bar is not in contact with the workpiece, contact with the workpiece during a continuous cutting operation will cause the boundary conditions on the cutting tool to
change [4, 5]. Hence, the dynamic properties of the plant will be different when the
boring bar is not in contact with the workpiece and during continuous turning. Also,
different cutting data and work material are likely to affect the dynamic properties
of the plant during continuous turning [4,5]. Plant frequency function estimates were
produced during continuous turning for a variety of different cutting data. The clamping conditions with respect to the tightening torque of the clamping screws and the
spatial position of the boring bar in the clamping space of the clamping house were
fixed and are given by clamping condition BC10 in Table 2. The spectrum estimation parameters and identification signal used in the production of on-line frequency
function estimates are given in Table 1 column B.
Notation
On-line 1
On-line 2
Cutting Parameters
Cutting Depth Cutting Speed
Feed Rate
1.2 mm
80 m/min
0.2 mm/rev
1 mm
150 m/min
0.2 mm/rev
Table 3: Cutting data used for plant frequency response function estimation during
continuous turning (on-line).
Fig. 15 presents two different plant frequency function estimates produced during
continuous turning (on-line) with different cutting data (see Table 3). This diagram
also present a plant frequency function estimate produced when the boring bar is
not in contact with the workpiece (off-line). The off-line frequency function estimate
was produced using the spectrum estimation parameters and identification signal
according to Table 1 column A.
The coherence functions corresponding to both the on-line control path frequency
response function estimates and the off-line estimates are shown in Fig. 16 a). Estimates of the random error for the on-line and off-line frequency response function
estimates in Fig. 15 are shown in Fig. 16 b).
3.2
Active Boring Bar Vibration Control Results
The cutting experiments utilized three different feedback controllers in the active
control of boring bar vibration: first, an analog manually adjustable controller based
on lead compensation; secondly, a manually adjustable analog stand alone controller,
based on a lead-lag compensation; and finally an adaptive digital controller based on
the feedback filtered-x LMS algorithm. To illustrate the results of the active control
of boring bar vibration using the three different controllers, power spectral densities
of boring bar vibration with and without active vibration control are presented in the
same diagram. The spectrum estimation parameters used in the production of boring
bar vibration power spectral density estimates are shown in Table 1 column D.
Initially, a simple analog manually adjustable lead compensator, based on digitally controlled analog design was developed. The adjustable lead compensator was
tuned manually and it was possible to provide an attenuation of the boring bar vibration level by up to approximately 35 dB. However, using the manually adjustable
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
50
45
40
35
Off−line
On−line 1
On−line 2
−50
−100
30
−150
25
20
−200
15
10
460
0
Phase ∠Ĉ(f ) [Degree]
|Ĉ(f )| [dB rel 1m/s2 /V]
Off−line
On−line 1
On−line 2
29
470
480
490
500
510
520
530
Frequency [Hz]
a)
540
550
560
570
−250
460
470
480
490
500
510
520
530
Frequency [Hz]
b)
540
550
560
570
Figure 15: Frequency function estimates of the plant during a continuous cutting
operation (on-line) and when the boring bar is not in contact with the workpiece
(off-line). The on-line estimation of the plant was produced using workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015. On-line 1: feed rate
s=0.2mm/rev, cutting depth a=1.2mm, cutting speed v=80m/min and On-line 2:
feed rate s=0.2mm/rev, cutting depth a=1mm, cutting speed v=150m/min.
1
Random error εr (|Ĥ(f )|)
1
2
(f )
Coherence γ̂yx
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
460
Off−line
On−line 1
On−line 2
470
480
490
500
510
520
530
Frequency [Hz]
a)
540
550
560
570
Off−line
On−line 1
On−line 2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
460
470
480
490
500
510
520
530
Frequency [Hz]
b)
540
550
560
570
Figure 16: a) Coherence function estimates between input and output signal of the
plant during continuous cutting (on-line) and when the boring bar is not in contact
with the workpiece (off-line). b) Estimate of the random error for the on-line and
off-line frequency response function estimates.
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Part I
30
60
Without Control
With Control
40
20
0
−20
−40
−60
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]
a)
PSD [dB rel 1(m/s2 )2 /Hz
PSD [dB rel 1(m/s2 )2 /Hz
lead compensator in the active control of boring bar vibration frequently resulted in
stability problems. By using the manually adjustable band-pass lead-lag controller in
the active control of boring bar vibration, the vibration level was reduced by up to
approximately 50 dB after a simple manual tuning of the controller (see Fig. 17). Furthermore in numerous cutting experiments, the active control of boring bar vibration
based on the band-pass lead-lag controller after initial manual tuning has provided
stable control with significant vibration attenuation.
Without Control
With Control
60
50
40
30
20
10
0
−10
−20
0
200
400
600
800
1000
1200
Frequency [Hz]
b)
1400
1600
Figure 17: a) Power spectral densities of boring bar vibration in the cutting speed
direction with active control using the adjustable lead-lag controller (solid line) and
without active control (dashed line). b) The corresponding spectra zoomed in to
the three first resonance peaks. Workpiece material SS2541-03, cutting tool DNMG
150806-SL, grade TN7015, feed rate s=0.24 mm/rev, cutting depth a=2 mm, cutting
speed v=60 m/min.
By utilizing the feedback filtered-x LMS algorithm as controller in the active
control of boring bar vibration, the adaptive controller will tune the adaptive FIR filter
to de-correlate the error signal with the filtered reference signal vector. Therefore,
high loop gain is provided in the frequency range of the resonance frequency that
dominates the boring bar vibration. Frequently, at high boring bar vibration levels,
the feedback filtered-x LMS algorithm in the active control of boring bar vibration
yields an attenuation of the vibration by more than 50 dB at the dominating resonance
frequency. However, feedback filtered-x LMS algorithm requires leakage to provide
stable and robust control [15, 31] and the cost for improved robustness is a somewhat
reduced vibration attenuation performance.
3.3
Stability and Robustness of the Controllers
The stability of a feedback control system requires that its open loop frequency response Hol (f ) does not violate the closed loop stability requirements, i.e. the Nyquist
stability criterion [22,28,29]. A closed loop system is said to be stable if the polar plot
of the open loop frequency response Hol (f ) for the feedback control system does not
enclose the (-1, 0) point in the Nyquist diagram. The greater the shortest distance
between the polar plot and the (-1, 0) point, the more robust the feedback control
system is with respect to variation in plant response and controller response. The
system fulfills the conditions for robust stability [22, 28, 29] if there is no phase function present which (in combination with maximal magnitude of the possible plant
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
31
5
BC1
BC5
4
BC6
Imag (Ĥol (f ))
BC10
3
2
1
0
−1
−3
−2
−1
0
1
2
Real (Ĥol (f ))
3
4
Figure 18: Nyquist diagram for a boring bar vibration control system based on a
manually adjustable lead compensator for the four different plant frequency response
function estimates corresponding to the clamping conditions: BC1 , BC5 , BC6 and
BC10 .
uncertainties at each frequency) can result in a feedback control system open loop
frequency response that encloses the (-1,0) point in the Nyquist diagram.
An estimate of the open loop frequency function for a feedback control system
may be produced based on the controller frequency response function and the plant
frequency response function. The analog controller frequency response function was
estimated after manually tuning for active control of boring bar vibration. In the
case of the adaptive digital controller, the controller frequency response function was
estimated after convergence with the step size set to zero. All the controllers were estimated with the spectrum estimation parameters and the identification signal shown
in Table 1 column C. The open loop frequency responses for the active boring bar
vibration control system were produced for the manually adjustable lead compensator
and the off-line control path frequency function estimate for each of the 10 different
clamping conditions of the boring bar (see Table 2). Also the open loop frequency
responses for the manually adjustable lead compensator and the two different on-line
plant frequency function estimates were produced. Observe that in order to facilitate interpretation of the Nyquist diagrams the number of the open loop frequency
response functions plotted in the same diagram were limited to four. These open
loop frequency response functions were selected in order to avoid redundancy in the
Nyquist diagrams, and to form the open loop frequency response functions. The plant
frequency response function estimates corresponding to the clamping conditions (BC1 ,
BC5 , BC6 and BC10 ) were selected (see Table 2). The Nyquist diagram in Fig. 18
shows polar plots of the selected open loop frequency responses. The corresponding
Bode plot is shown in Fig. 19.
Observe, in Fig. 18, that the polar plots of the open loop frequency responses for
the feedback control system based on the manually adjustable lead compensator are
close to, or enclose, the (-1, 0) point in the Nyquist diagram.
Fig. 19 demonstrates that open loop frequency responses for the boring bar vibration control system based on the manually adjustable lead compensator provide
substantial loop gain at resonance frequencies above 2000 Hz. Open loop frequency
responses for the boring bar vibration control system were produced based on the
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32
20
BC1
Phase ∠Ĥol (f ) [Degree]
Magnitude |Ĥol (f )| [dB]
−100
10
BC5
−200
0
BC
6
BC
−300
−10
10
−400
−20
−500
−30
−600
−40
BC1
−50
BC5
BC
−60
6
−700
−800
BC
10
−70
0
500
1000
1500
2000
2500
Frequency [Hz]
a)
−900
0
3000
500
1000
1500
2000
Frequency [Hz]
b)
2500
3000
Figure 19: Open loop frequency response function estimates for a boring bar vibration
control system, based on a manually adjustable lead compensator for the four different
plant frequency response function estimates corresponding to the clamping conditions:
BC1 , BC5 , BC6 and BC10 .
manually adjustable band-pass lead-lag compensator and the off-line plant frequency
function estimate for each of the ten different boring bar clamping conditions (see
Table 2).
Also, open loop frequency responses were produced for the manually adjustable
lead compensator and the two different on-line control path frequency function estimates. The selected open loop frequency functions for the active boring bar control
system based on band-pass lead-lag compensator are shown in the Nyquist diagram in
Fig. 20 (these estimates were based on the clamping conditions: (BC1 , BC5 , BC6 and
BC10 ). The corresponding Bode plot of the open loop frequency response functions
for the active boring bar control system are shown in Fig. 21.
BC
4
1
BC
5
3
BC
Imag (Ĥol (f ))
6
BC
2
10
1
0
−1
−2
−3
−4
0
2
4
6
Real (Ĥol (f ))
8
10
Figure 20: Nyquist diagram for a boring bar vibration control system, based on a
manually adjustable band-pass lead-lag compensator for the four different plant frequency response function estimates corresponding to the clamping conditions: BC1 ,
BC5 , BC6 and BC10 .
Observe the distance between the polar plots of the open loop frequency response
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
BC
Magnitude |Ĥol (f )| [dB]
1
BC
10
5
BC6
BC10
0
−10
BC1
BC5
−200
BC
6
BC
−300
10
−400
−20
−500
−30
−600
−40
−700
−50
−60
0
−100
Phase ∠Ĥol (f ) [Degree]
20
33
−800
500
1000
1500
2000
Frequency [Hz]
a)
2500
3000
−900
0
500
1000
1500
2000
Frequency [Hz]
b)
2500
3000
Figure 21: Open loop frequency response function estimates, for a boring bar vibration
control system, based on a manually adjustable band-pass lead-lag compensator for
the four different plant frequency response function estimates corresponding to the
clamping conditions: BC1 , BC5 , BC6 and BC10 .
function estimates for the boring bar vibration control system based on the manually
adjustable band-pass lead-lag compensator and the (-1, 0) point in the Nyquist diagram in Fig. 20. In addition,the Bode plot (see Fig. 19) demonstrates low loop gain
above 1000 Hz.
The adaptive digital control of boring bar vibration was carried out with and without a leakage factor in the feedback filtered-x LMS algorithm. The Nyquist diagram
in Fig. 22 shows the polar plots of the four open loop frequency responses, which
is based on the feedback filtered-x LMS algorithm. The four open loop frequency
functions for the active boring bar control system based on the feedback filtered-x
LMS algorithm are shown in the Nyquist diagram in Fig. 22.
The corresponding Bode plots of the open loop frequency response functions based
on the feedback filtered-x LMS algorithm are shown in Fig. 23.
The feedback filtered-x LMS algorithm will automatically tune the adaptive FIR
filter to de-correlate the error signal with the filtered reference signal vector. Thus, a
high loop gain will be provided in the frequency range of the resonance frequency that
dominates the boring bar vibration (see the Bode plot in Fig. 23). Finally, the leaky
feedback filtered-x LMS controller yields a significantly lower loop gain compared to
the case of no leakage.
If, for example, the plant frequency function estimate corresponding to the clamping condition BC10 is selected as a nominal control path or plant model of the plant
then an estimate of the upper bound β(f ) for the multiplicative perturbation modeling the plant uncertainty may be produced based on Eq. (6) using all the other plant
frequency function estimates (not the plant frequency function estimate corresponding to the clamping condition BC10 ). In Fig. 24 the magnitude of the inverse of the
nominal complementary function for each of the controllers is plotted in the same
diagram as the estimated upper bound β̂(f ) for the multiplicative perturbation.
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Part I
34
BC
1
3
BC
Imag (Ĥol (f ))
5
BC
6
2
BC10
1
0
−1
−2
−3
−4
−5
−2
0
2
4
Real (Ĥol (f ))
6
8
Figure 22: Nyquist diagram for a boring bar vibration control system, based on the
feedback filtered-x LMS algorithm for the four different plant frequency response
function estimates, corresponding to the clamping conditions: BC1 , BC5 , BC6 and
BC10 . Number of adaptive filter coefficients M=20, step size µ = −0.5, sampling
frequency of the DSP Fs=8 kHz.
10
0
−10
Phase ∠Ĥol (f ) [Degree]
Magnitude |Ĥol (f )| [dB]
20
500
BC1
0
BC5
BC6
−500
BC10
−1000
−20
−1500
−30
−2000
−40
−2500
−50
BC1
−60
BC
−70
BC6
5
−3000
−3500
BC
10
−80
0
500
1000
1500
2000
Frequency [Hz]
a)
2500
3000
−4000
0
500
1000
1500
2000
Frequency [Hz]
b)
2500
3000
Figure 23: Open loop frequency response function estimates for a boring bar vibration
control system, based on the feedback filtered-x LMS algorithm for the four different
plant frequency response function estimates, corresponding to the clamping conditions: BC1 , BC5 , BC6 and BC10 . Number of adaptive filter coefficients M=20, step
size µ = −0.5, sampling frequency of the DSP Fs=8 kHz.
1/Tnorm(f ) and β̂(f ) in [dB]
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
35
120
100
80
β̂(f)
Filtered-x, γ = 1
Filtered-x, γ = 0.9999
Lead
Lead-Lag
60
40
20
0
−20
−40
0
500
1000
1500
2000
Frequency [Hz]
2500
3000
Figure 24: Magnitude of the inverse of the nominal complementary function for each
of the controllers and the estimated upper bound for the multiplicative perturbation.
4
Discussion and Conclusions
The above results demonstrate that boring bar vibration in internal turning may
be reduced by utilizing active control based on active boring bars with embedded
actuator and sensor and a suitable feedback controller such as an analog manually
adjustable band-pass lead-lag controller or an adaptive digital controller based on the
feedback filtered-x LMS algorithm. It has been established that for each time the
boring bar is clamped, it is likely that the clamped boring bar will have different
dynamic properties (see Figs 13 and 14). Also, the dynamic properties of the
clamped boring bar will differ between these instances when the boring bar is not in
contact with the workpiece and during continuous turning (see Fig. 15). In addition,
different cutting data and work materials will likely affect the dynamic properties
of the clamped boring bar during continuous turning [4]. Thus, the plant in the
active boring bar vibration system may display significant variations in its dynamic
properties. Hence, a robust controller that performs well for substantial variations in
the dynamics of the plant is required for the active control of boring bar vibration.
The development of a simple adjustable analog controller, based on digitally controlled analog design, started initially with a lead compensator. However, the vibration to be controlled is related to the fundamental bending modes of the boring
bar and not its higher order modes [4, 5]. Thus, the high loop gain (provided by a
controller above the fundamental bending modes eigenfrequencies) will likely be an
issue concerning the stability and robustness of the active boring bar vibration control
system. However if the manually adjustable lead control is utilized for active boring
bar vibration control, it will (when stable) perform a broad-band attenuation of the
tool-vibration. Therefore, the vibration level is reduce by over 35 dB at 460 Hz and
harmonics of the 460 Hz boring bar resonance frequency are attenuated. The reduction of the harmonics of the dominating fundamental resonance frequency is probably
a consequence of a linearization of the boring bars dynamic response imposed by the
active vibration control. Polar plots of the open loop frequency responses (for the
feedback control system based on the manually adjustable lead compensator), which
approach or enclose the (-1, 0) point of the Nyquist diagram, also demonstrate problems with robustness and stability (see Fig. 18). According to the Bode plots (Fig. 19),
the manually adjustable lead compensators provides (as expected) a substantial loop
36
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gain at the boring bar resonance frequencies above 2000 Hz. Thus, to provide high
boring bar vibration attenuation, high loop gain is only required in the frequency
range of the boring bar’s fundamental bending modes eigenfrequencies. Basically, a
manually adjustable controller should be able to provide adjustable band-pass gain
and adjustable phase enabling to control the plant in the frequency range of the boring bar’s fundamental bending modes eigenfrequencies. This will produce adequate
anti-vibration canceling the original vibration excited by the material deformation
process.
The adaptive digital feedback control based on feedback filtered-x LMS algorithm
(with and without leakage) performs a broad-band attenuation of tool-vibration and
is able to reduce vibration levels by over 50 dB at 460 Hz, as well as to attenuate
the harmonics of the 460 Hz boring bar resonance frequency. However, slightly lower
vibration attenuation might be observed for the leaky feedback filtered-x LMS algorithm. The introduction of a leakage factor or a ”forgetting factor” in the feedback
filtered-x LMS algorithm’s recursive coefficient adjustment algorithm will provide a
restraining influence on the loop gain of the control system and may thereby cause a
somewhat reduced attenuation of the boring bar vibration.
The active boring bar vibration control system based on the manually adjustable
band-pass lead-lag control performs a broad-band attenuation of tool-vibration and
is also able to reduce the vibration level by over 50 dB at 460 Hz, as well as to
attenuate the harmonics of the 460 Hz boring bar resonance frequency (see Fig. 17).
Thus, the band-pass lead-lag controller provides attenuation performance comparable
to that of the adaptive controller by tuning the adaptive FIR filter to de-correlate
the error signal with the filtered reference signal vector. It thereby provides high loop
gain in the frequency range of the resonance frequency that dominates the boring
bar vibration (see Fig. 23). By comparing the loop gains provided by the adaptive
digital controller with the loop gains provided by the band-pass lead-lag controller
(see Fig. 19), it follows that the analog lead-lag controller is tuned to provide high loop
gain over a broader frequency range as compared to the loop gain resulting from the
adaptive digital control. On the other hand, by examining the Nyquist plots for the
open loop frequency response functions concerning the band-pass lead-lag controller
(see Fig. 20) and the feedback filtered-x LMS controller (see Fig. 22), it follows that
the shortest distance between the polar plots and the (-1, 0) point is greater for the
analog controller as compared with the feedback filtered-x LMS controller.
If robust stability is considered (see Fig. 24), it follows that the active boring bar
vibration control system based on the leaky feedback filtered-x LMS controller is the
only system that fulfills the conditions for robust stability. It is also the controller
that results in the greatest shortest distance between the polar plots and the (-1, 0)
point. However, the active boring bar vibration control system based on the bandpass lead-lag controller, tuned initially, remained stable during the course of numerous
trials with varied clamping conditions and cutting data.
Acknowledgments
The present project is sponsored by the Foundation for Knowledge and Competence
Development and the company Acticut International AB.
On the Development of a Simple and Robust Active Control System
for Boring Bar Vibration in Industry
37
References
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[15] I. Claesson and L. Håkansson. Adaptive active control of machine-tool vibration
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Part II
Analysis of Dynamic Properties of
Boring Bars Concerning Different
Clamping Conditions
This part is published as:
H. Åkesson, T. Smirnova, and L. Håkansson, ”Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping Conditions ”, Journal of Mechanical Systems
& Signal Processing, 23(8), pp. 2629-2647, 2009.
Analysis of Dynamic Properties of Boring
Bars Concerning Different Clamping
Conditions
H. Åkesson, T. Smirnova, and L. Håkansson
Blekinge Institute of Technology,
Department of Signal Processing
372 25 Ronneby
Sweden
Abstract
Boring bars are frequently used in the manufacturing industry to turn deep
cavities in workpieces and are usually associated with vibration problems. This
paper focuses on the clamping properties’ influence on the dynamic properties
of clamped boring bars. A standard clamping housing of the variety commonly
used in industry today has been used. Both a standard boring bar and a modified boring bar have been considered. Two methods have been used: EulerBernoulli beam modeling and experimental modal analysis. It is demonstrated
that the number of clamping screws, the clamping screw diameter sizes, the
screw tightening torques, the order the screws are tightened has a significant
influence on a clamped boring bars eigenfrequencies and its mode shapes orientation in the cutting speed-cutting depth plane. Also, the damping of the
modes is influenced. The results indicate that multi-span Euler-Bernoulli beam
models with pinned boundary condition or elastic boundary condition modeling
the clamping are preferable as compared to a fixed-free Euler-Bernoulli beam for
modeling dynamic properties of a clamped boring bar. It is also demonstrated
that a standard clamping housing clamping a boring bar with clamping screws
imposes non-linear dynamic boring bar behavior.
1
Introduction
In industry where metal cutting operations such as turning, milling, boring and grinding take place, degrading vibrations are a common problem. In internal turning operations vibration is a pronounced problem, as long and slender boring bars are usually
required to perform the internal machining of workpieces. Tool vibration in internal
turning frequently has a degrading influence on surface quality, tool life and production efficiency, and will also result in severe environmental issues such as high noise
levels. Boring bar vibrations are usually directly related to the lower order bending
modes of the clamped boring bar [1, 2]. Also, the dynamic properties of a boring bar
installed in a lathe are influenced by the boundary conditions imposed by the clamping of the bar [2,3]. A number of theories concerning the machine tool chatter and the
behavior of the dynamic system has been developed explaining tool vibration during
42
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turning operations [4–7]. In 1946 the principles of the traditional theory of chatter
in simple machine-tool systems was worked out by Arnold [4] based on experiments
carried out in a rigid lathe, using a stiff workpiece but a flexible tool holder. In this
way he was able to investigate chatter under controlled conditions. Later in 1965
Tobias [5] presented an extensive summary of results from various researchers concerning the dynamic behavior of the lathe, the chatter theory, and further developed
the chatter phenomena considering the chip-thickness variation and the phase lag of
the undulation of the surface. Also, in the same year, Merritt et al. [6] discussed the
stability of structures with n-degrees of freedom, assuming no dynamics in the cutting
process; they also proposed a simple stability criterion.
Parker et al. [8] investigated the stability behavior of a slender boring bar by
representing it with a two-degree-of-freedom mass-spring-damper system and experimenting with regenerative cutting. They also investigated how the behavior of the
vibration was affected by coupling between modes, by using different cutting speeds,
feed rates and angles of the boring bar head relative to the two planes of vibrations. Pandit et al. [9] developed a procedure for modeling chatter from time series
by including unknown factors of random disturbances present in the cutting process,
they formulated self-excited random vibrations with white noise as a forcing function. Kato et al. [10] investigated regenerative chatter vibration due to deflection
of the workpiece, and introduced a differential equation describing chatter vibration
based on experimental data. Furthermore, various analytical models/analysis methods relating to the boring bar/or cutting process have been continuously developed,
assuming various conditions. For example, Zhang et al. [11] who’s model is derived
from a two-degree-of-freedom model of a clamped boring bar and four cutting force
components. In addition, Rao et al. [12] produced a continuous system model of
boring dynamics based on a dynamic boring force model, including variation of chip
cross-sectional area, and a uniform Euler-Bernoulli cantilever beam, while Kuster et
al. [13] developed a computer simulation based on a three-dimensional model of regenerative chatter. A time series approach was used by Andrén et al. [1] to investigate
boring bar chatter and the results were compared with an analytical Euler-Bernoulli
model. Also, Euler-Bernoulli beam modeling, experimental modal analysis (EMA)
and operating deflection shape analysis were used by Andrén et al. [2] to investigate
the dynamic properties of a clamped boring bar. The issue of modeling the boundary
conditions of a clamped boring bar is discussed briefly. Results obtained demonstrate
observable differences concerning the fundamental bending modes. Scheuer et al. [3]
made a preliminary investigation of the dynamic properties of a boring bar, for two
different clamping housings based on experimental modal analysis. Their study indicates that different clamping conditions using a clamping housing with clamp screws
may affect the fundamental bar bending modes slightly.
Following the literature review, it appears that little work has been done concerning the modeling of the clamping in the dynamic models of clamped boring bars.
Also, when it comes to experimental investigations concerning the clamping properties’ influence on the dynamic properties of a clamped boring bar, it appears that
little work has been done. Thus, it is of importance to investigate the clamping properties’ influence on the dynamic properties of the clamped boring bar in order to gain
further understanding of the dynamic behavior of clamped boring bars in the metal
cutting process. Also, the modeling of the boring bar clamping has to be addressed
in order to further improve dynamic models of clamped boring bars.
This paper discusses Euler-Bernoulli modeling of a clamped boring bar with em-
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
43
phasis on the modeling of the clamping conditions, i.e. the boundary conditions. Also
the variation in the dynamic properties of a clamped boring bar imposed by controlled
discrete variations in the clamping conditions produced by a standard clamping housing commonly used in industry today is investigated experimentally. The focus is on
the bars first two fundamental bending modes. Firstly, the Euler-Bernoulli theory
for the modeling of a clamped boring bar using ”fixed-free” boundary conditions is
considered. In order to incorporate clamping flexibility in the distributed-parameter
models, three-span Euler-Bernoulli boring bar model with free-pinned-pinned-free
and free-spring-spring-free are considered. The three-span model is used to model
clamping conditions of the boring bar when screws are clamping the bar at two positions along the bar. A four-span model with free-pinned-pinned-pinned-free and
free-spring-spring-spring-free are considered to model the clamping conditions when
screws are clamping the bar at three positions along the bar. Furthermore, derivations
of the spring coefficients used in the elastic support models are presented for various
screw sizes. An experimental investigation, based on experimental modal analysis,
of dynamic properties of boring bars for a comprehensive set of different clamping
conditions has been carried out. The investigation has included two standard clamping housings with different number of clamping screws, two different clamping screw
diameter sizes, a number of different screw tightening torques, the order in which the
screws were tightened and a ”linearized” clamping setup. Both a standard boring bar
and a modified boring bar have been considered. Experimental modal analysis of the
boring bars has been carried out for the different clamping conditions using a number
of different excitation force levels. The results from the Euler-Bernoulli modeling of
a clamped boring bar and from the experimental investigation of dynamic properties
of boring bars for a comprehensive set of different clamping conditions have been
compared and discussed.
2
2.1
Materials and Methods
Experimental Setup
The experimental setup and subsequent measurements were carried out in a Mazak
SUPER QUICK TURN-250M CNC turning center. The CNC lathe has 18.5 kW
spindle power and a maximal machining diameter of 300 mm, with 1005 mm between
the centers, a maximal spindle speed of 400 revolutions per minute (r.p.m.) and a
flexible turret with a tool capacity of 12 tools. Fig. 1 a) illustrates some of the basic
structural parts of internal turning, i.e. a workpiece clamped in a chuck and a boring
bar clamped in a clamping housing. The room in the Mazak SUPER QUICK TURN250M CNC lathe where machining is carried out presented by the photo in Fig. 1
b).
Initially, a right-hand cartesian coordinate system was defined. Subsequently a
sign convention was defined for use throughout this paper. The coordinate system
and sign convention are based on the right-hand definition where the directions of
displacements and forces in positive directions of the coordinate axes are considered
positive. Moreover, moment about an axis in the clockwise direction (when viewing
from the origin in the positive direction of the axis) is considered positive.
The boring bar was positioned in its operational position in all setups, that is, mounted
in a clamping housing attached to a turret with screws, during all measurements. The
turret may be controlled to move in the cutting depth direction, x-direction, and in
the feed direction, z-direction, as well as to rotate about the z-axis for tool change.
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Ja w s
C la m p in g h o u s e
W o rk p ie c e
B o rin g b a r
y
z
a)
x
b)
Figure 1: a) An internal turning setup with a workpiece clamped in a chuck to the
left and a boring bar clamped in a clamping housing to the right. b) The room in the
Mazak SUPER QUICK TURN - 250M CNC lathe where machining is carried out.
The turret, etc. is supported by a slide which in turn is mounted onto the lathe bed.
Even though the turret is a movable component, it is relative rigid, rendering the
dynamic properties of the boring bars observable.
2.1.1
Measurement Equipment and Setup
The following equipment was used in the experimental setup;
•
•
•
•
•
•
•
•
•
12 PCB Piezotronics, Inc. 333A32 accelerometers.
2 Brüel & Kjǽr 8001 impedance head.
1 Brüel & Kjǽr NEXUS 2 channel conditioning amplifier 2692.
OSC audio power amplifier, USA 850.
Ling dynamic systems shaker v201.
Gearing & Watson electronics shaker v4.
Hewlett Packard VXI mainframe E8408A.
Hewlett Packard E1432A 4-16 channel 51.2 kSa/s digitizer.
PC with I-DEAS 10 NX Series.
Twelve accelerometers and two cement studs for the impedance heads were attached onto the boring bars with X60 glue. The sensors were evenly distributed along
the centerline, on the under-side and on the backside of the boring bar; six accelerometers and one stud on the respective side (see Fig. 2 a). To excite all the lower order
bending modes, two shakers were attached via stinger rods to the impedance heads,
one in the cutting speed direction (y-) and the other in the cutting depth direction
(x-). The shakers were positioned relatively close to the cutting tool.
2.1.2
Boring Bars
Two different boring bars were used in the experiment. The first boring bar used
in the modal analysis was a standard ”non-modified” boring bar, WIDAX S40T
PDUNR15F3 D6G.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
C e m e n t stu d s
y
F r o n t V ie w
l4
B
B
1
B
2
3
45
3 7 .5 m m
Ø 4 0 m m
z
C
C
3 7 m m
l7
l1
l2
- A c c e le r o m e te r s
l8
l3
l8
y
M
C
1
x
l5
a)
b)
Figure 2: a) Drawings of the boring bar including clamp screws, cement studs and
sensors. The sensors are attached along the underside and the backside of the boring
bar. The threaded holes denoted B1 , B2 and B3 are screw positions for clamping the
boring bar from top and bottom. The dimension are in mm, where l1 = 10.7, l2 = 18,
l3 = 101, l4 = 250, l5 = 35, l6 = 100, l7 = 18.5, l8 = 25. b) The cross section of the
boring bar where CC is the center of the circle and MC is the mass center.
The second boring bar used in this experiment was a modified boring bar, based
on the standard WIDAX S40T PDUNR15 boring bar, with an accelerometer and an
embedded piezo-stack actuator, see Fig. 3. This boring bar is designed for active
control of boring bar vibration [14]. The accelerometer was mounted 25 mm from the
tool tip to measure the vibrations in the cutting speed direction (y-). This position
was as close as possible to the tool tip, but at a sufficient distance to prevent metalchips from the material removal process from damaging the accelerometer.
T o o l
y
B o rin g B a r
z
A c c e le ro m e te r
A c tu a to r
x
Figure 3: The modified boring bar with an accelerometer close to the tool tip and an
embedded piezo-stack actuator in a milled space below the centerline.
The standard WIDAX S40T PDUNR15 boring bar is manufactured in the material
30CrNiMo8, (AISI 4330) which is a heat treatable steel alloy (for high strength).
2.1.3
Clamping Housing
The clamping housing is a basic 8437-0 40 mm Mazak holder, presented to the right
in Fig. 4. The clamping housing has a circular cavity that the boring bar fits easily
into; the clamping is then carried out by means of screws on the tool side and on
the opposite side of the boring bar by means of either four or six screws of size M8:
two/three from the top and two/three from bottom. The basic holder itself is mounted
onto the turret with four screws. Furthermore, one clamping housing was rethreaded
in order to be able to use screws of size M10.
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2.1.4
Clamping Conditions
Six different setups were considered using the different boring bars described in section 2.1.2 in conjunction with the different clamping housings.
In the first setup, the standard boring bar was clamped using four M8 class 8.8
screws. The clamp screws were first tightened from the top and then from the bottom
using the following five different torques (10 Nm, 15 Nm, 20 Nm, 25 Nm, 30 Nm).
The recommended torque for this screw class is 26.6 Nm, but from evaluations of the
screws used, the threads stayed intact and the screws did not break for 30 Nm. As
only four clamp screws were used, the clamping housing center screw positions were
not used.
The second setup involved the same five torques as for the previous setup, but
with six screws of size M10 class 8.8, which were, again, tightened first from the top
and then from the underside. The use of six screws involved the use of all clamping
housing screw positions.
The third setup involved the same five torques as for the previous setup, six
screws of size M10 class 8.8, which were, however, tightened first from the underside
and then from the top.
Setup four and five are configured in the same way as setup one and two,
respectively, with the difference that the modified boring bar was used instead of the
standard boring bar.
The last setup (setup 6) used a modification of the standard clamping in
order to accomplish a more rigid clamping condition. A boring bar WIDAX S40T
PDUNR15F3 D6G, the same model as the standard boring bar, was used together
with three steel wedges produced of the material SS 1650 (AISI 1148). The steel
wedges were glued with epoxy on the flat surfaces of the boring bar along the clamping
length of the bar end. The steel wedges were shaped geometrically to render a circular
cross-section on the boring bar along its clamped end. After the epoxy was set; the
boring bar end with circular cross-section was pressed into the clamping housing and
glued to it with epoxy to make the clamping rigid, see Fig 4. Table 1 summarizes the
setup configurations used in the experimental modal analysis.
C la m p in g h o u s e
W e d g e
B o rin g b a r
y
z
W e d g e s
Figure 4: The linearized boring bar-clamping housing setup.
x
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
Setup
number
1
2
3
4
5
6
Boring bar
Standard
Standard
Standard
Modified
Modified
Linearized
Configuration
Number of Screws Screw Size
four
M8
six
M10
six
M10
four
M8
six
M10
-
47
Tighten first from
top
top
bottom
top
top
-
Table 1: The configuration of the different setups from the experimental modal analysis.
2.2
Analytical Models of the Boring Bars
To approximately model a boring bars lower order bending modes, Euler-Bernoulli
beam theory may be utilized [1, 2]. The Euler-Bernoulli beam theory is generally
considered for slender beams having a diameter to length ratio above 10 as it ignores
the effects of shear deformation and rotary inertia [15]. As a result, it tends to slightly
overestimate the eigenfrequencies; this problem increases for the eigenfrequencies of
higher modes [15]. The Euler-Bernoulli differential equation describing the transversal motion of the boring bar in the cutting speed direction or y-direction is given
by [15]:
∂ 2 u(z, t)
∂ 2 u(z, t)
∂2
ρA(z)dz
EI
(z)
= f (z, t)
(1)
+
x
∂t2
∂z 2
∂z 2
where A(z) is the boring bars cross section area, E is Young’s elastic modulus for the
bar and Ix (z) is the cross-sectional area moment of inertia about the ”x axis.”, ρ is
the density, t is the time, u(z, t) is the deflection in the y-direction and f (z, t) is the
external force per unit length.
It is assumed that both the cross-sectional area A(z) and the flexural stiffness
EIx (z) are constant along the boring bar. The boring bar dimensions provided by
Fig. 2 b) result in a cross-sectional area A and a moment of inertia Ix , Iy presented
by Table 2.
Variable
A
Ix
Iy
Value
1.19330295 · 10−3
1.13858895 · 10−7
1.13787080 · 10−7
Unit
m2
m4
m4
Table 2: Cross-sectional properties of the boring bar, illustrated in Fig. 2 b), where A
is the cross-sectional area and Ix , Iy are the moments of inertia around the denoted
axis.
2.2.1
Single Span Boring Bar Model
The simplest and most straightforward model of a boring bar is the Euler-Bernoulli
model, which consists of a homogenous single span beam with constant cross-sectional
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area A(z) = A and constant cross-sectional moment of inertia Ix (z) = Ix . The beam
has four boundary conditions, two at each end. One end is fixed and the other is
free [15], see Fig. 5.
z
E , r ,I , A
l
Figure 5: Model of a Fixed - Free beam, where E is the elasticity modulus (Young’s
coefficient), ρ the density, A the cross-sectional area, I the moment of inertia and the
length of the beam is l = 200.
2.2.2
Multi-span Boring Bar with Elastic Foundation Model
The boring bar may be clamped with either two screws on the top and two on the
underside or three screws on the top and three on the underside. If the clamping
housing is consider to be a rigid body, and the screws to be rigid in the transverse
direction, a number of boundary conditions are yielded, i.e. approximated as pinned.
The pinned boundary condition assumes an infinitely stiff spring in the transverse direction but no rotational stiffness. Letting the screws assume more realistic properties
as deformable bodies will yield ”elastic supports” [16] boundary conditions, instead
of the pinned condition. The elastic support can be seen as two springs applied to
one point, one spring acting in the transverse direction applying transverse stiffness
resistance and one spring acting in the rotational direction applying rotational stiffness resistance. The configurations of the ”elastic support” condition are presented
in Fig. 6.
Two types of boundary conditions may be categorized from the models presented
in Fig. 6, where zpos denotes the position of the boundary condition along the beam.
One is the ”free” boundary condition where there is no bending or shear forces
present [15]. The other boundary conditions derive from the ”elastic support” condition and may be expressed as [16]
∂ 3 u(z, t) ∂ 2 u(z, t) ∂u(z, t)
,
EI
EI
=
−k
= kT u(z, t)
(2)
R
∂z 2 z=zpos
∂z
∂z 3 z=zpos
where the transverse spring produces a transverse force proportional to the displacement, and the rotational spring produces a bending moment proportional to the beam
slope. However, if we let the rotational spring coefficient equal zero kR = 0, and the
transverse spring coefficient go to infinity kT = ∞, we will have a third boundary
condition termed ”pinned” [15].
The stiffness coefficients of the screws were calculated by modeling the screws as a
beam rigidly clamped at one end, and free at the other. The beam was considered to
be homogeneous, having a constant cross-sectional area A, a constant cross-sectional
moment of inertia I and a length of l. When a screw is threaded in the clamping
housing and is clamping the boring bar, a part of the screw’s tip will not be in
contact with the clamping housing; thus yielding both transverse, lateral and bending
elasticity. This is due to the fact that the inside of the clamping housing is circular
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
k
a )
l1
l
k
l1
2
k
k
T
4
l
k
R
l
R
E , r ,I , A
T
k
b )
k
R
49
k
l
T
T
k
R
3
R
4
E , r ,I , A
k
l
3
T
Figure 6: a) A model of a three span beam with elastic support, b) a model of a
four span beam with elastic support, where E is the elasticity modulus (Young’s
coefficient), ρ the density, A the cross-sectional area, I the moment of inertia, kT
the transverse spring coefficient, kR the rotational spring coefficient the length of the
different spans in mm are l1 = 35, l2 = 50, l3 = 215 and l4 = 25.
with a radius of 40 mm plus tolerance and the boring bar has a thickness of 37 mm,
plus tolerance where the boring bar is clamped.
The screw clamping model is presented in Fig. 7, where a) shows the clamping
configuration, b) illustrates the beam model of transverse vibrations and the transverse spring coefficient, and c) illustrates the beam model of the rotational spring
coefficient.
C la m p in g h o u s in g
T ra n s v e rs e S tiffn e s s
l
E A
Þ
k
T
B e n d in g S tiffn e s s
E I
l
Þ
k
R
M
F
B o rin g b a r
a)
b)
c)
Figure 7: a) Sketch illustrating screw clamping of the boring bar, via the clamping
housing, b) the transverse stiffness model, and c) the rotational stiffness model
The transverse spring constant kT and rotational spring constant kR were calculated as [15, 17]
kT =
EA
,
l
kR =
EI
l
(3)
The calculated spring coefficients and the spring parameters used in the spring
models are presented in Table 3 together with the dimensions and elasticity modulus
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and are based on dimensions and properties defined by MC6S norm ”DIN 912, ISO
4762” .
Size
M8
M10
A [m2 ]
I [m4 ]
−5
3.661 · 10
5.799 · 10−5
E [N/m2 ]
−10
1.067 · 10
2.676 · 10−10
200 · 109
l [m]
kT [N/m]
kR [Nm]
1.5 · 10−3
4.881 · 109
7.732 · 109
1.422 · 104
3.568 · 104
Table 3: The longitudinal and rotational spring coefficients and the spring parameters
used in the spring models.
2.3
Experimental Modal Analysis
The primary goal of experimental modal analysis is to identify the dynamic properties of the system under examination; i.e. to determine the natural frequencies, mode
shapes, and damping ratios from experimental vibration measurements. The procedure of modal analysis may be divided into two parts: the acquisition of data and
the parameter estimation or parameter identification from these data, also known as
curve fitting [18].
2.4
Spectral Properties
Non-parametric spectrum estimation may be utilized to produce non-parametric linear least-squares estimates of dynamic systems [19]. By using the Welch spectrum
estimator [20], the cross-power spectral density P̂yx (fk ) between the input signal
x(n) and the output signal y(n) and the power spectral density P̂xx (fk ) for the ink
F s is the discrete frequency,
put signal x(n) may be produced [2, 19], where fk = N
k = 0, . . . , N − 1, N is the length of the data segments used to produce the periodograms and F s is the sampling frequency.
In the case of a multiple-input-multiple-output system (MIMO system) with P
number of responses and Q number of references, an estimate of the cross-spectrum
matrix [P̂xx (fk )] between all the inputs is produced, where the diagonal elements is
power spectral densities for the respective input signal and the of-diagonal are crossspectral densities. Also a cross-spectrum matrix [P̂yx (fk )] between all the inputs and
outputs may be estimated in the same way.
The least-square estimate for a (MIMO system) may be written as [19]
h
i
Ĥ(fk )
=
h
ih
i−1
P̂yx (fk ) P̂xx (fk )
(4)
In the case of a multiple inputs, the multiple coherence is of interest as a quality
of the MIMO system estimates [19].
2.4.1
Parameter Estimation
There are several different methods for identification of the modal parameters [18,21].
There are two basic curve fitting methods: curve fitting in frequency domain using
measured frequency response function (FRF) data and a parametric model of the
FRF; or curve fitting toward the measured impulse response function (IRF) data
using a parametrical model of the IRF [18]. Many methods use both domains, depending on which parameter is estimated [18].A parametric model of the FRF matrix,
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
51
h
i
Ĥ(f ) , expressed as the receptance between the reference points, input signals, and
the responses, output signals, may be written as [18],
h
i
Ĥ(f )
=
N
X
Qr {ψ}r {ψ}T
r
r=1
j2πf − λr
+
Q∗r {ψ}∗r {ψ}H
r
j2πf − λ∗r
(5)
where r is the mode number, N the number of modes used in the model, Qr the
scaling factor of mode r, {ψ}r the mode shape vector of mode r, and λr is the pole
belonging to mode r.
Due to the fact that two sources (references) were used during data acquisition,
a method capable of handling multi-references is required. One such method is the
Polyreference least square complex exponential method developed by Vold [22, 23].
This method is defined for identification of MIMO-systems with the purpose of obtaining a global least-squares estimates of the modal parameters. This method was used
in this work; however, the mode shapes were estimated using the frequency domain
polyreference method [24]. The used modal scaling was unity modal mass [21, 25].
As quality assessment of the estimated parameters the FRF’s were synthesized
using the estimated parameters and overlayed with the non-parametric estimates of
the FRF’s. Furthermore the modal assurance criterion (MAC) [18] defined by Eq. 6.
M ACkl
2
{ψ}H
k {ψ}l
=
H
{ψ}H
k {ψ}k {ψ}l {ψ}l
(6)
was used as a measure of correlation between mode shape {ψ}k belonging to mode
k, and mode shape {ψ}l belonging to mode l, where H is the Hermitian transpose
operator.
2.4.2
Excitation Signal
For the experimental modal analysis, burst random was used as the excitation signal.
Based on initial experiments concerning suitable burst length and frequency resolution
(data segment time or data block length time), a burst length of 90% of the data block
length time was selected, see Table 4. Basically, the frequency resolution was tuned
to provide high overall coherence in the analysis bandwidth and the burst length was
tuned to provide high coherence at resonance frequencies.
Parameter
Excitation signal
Sampling Frequency f s
Block Length N
Frequency Resolution ∆f
Number of averages L
Window
Overlap
Frequency Range of Burst
Burst Length
Value
Burst Random
10240 Hz
20480
0.5 Hz
200
Rectangular
0%
0-4000 Hz
90%
Table 4: Spectral density estimation parameters.
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Furthermore, four different excitation levels with the proportion {1, 2, 3, 4} were
applied for each of the clamping conditions of the boring bars. By using a number of
different excitation levels and carrying out system identification for each of the excitation levels, differences between the estimates of the system may indicate nonlinear
behavior of the system and might provide information concerning the structure of the
nonlinearity or the nonlinearities involved.
3
Results
The results presented constitute a small part of an extensive investigation of the
dynamic properties of boring bars for the different setups; however they represent the
essence of the experimental results.
3.1
Results of the Analytical Models
This section presents results from the different Euler-Bernoulli models of the boring
bar. The first model assumed rigid clamping of the boring bar by the clamping housing. The second model assumes that boring bar clamping is pinned at the positions
where the actual clamp screws clamp the boring bar inside the clamping housing.
Finally, the multi-span boring bar models with flexible boundary conditions (corresponding to the standard boring bar clamped using four clamp screws or six clamp
screws) are considered.
The simplest model is the single span model with rigid clamping (fixed) at one end
and no clamping (free) at other. The first three resonance frequencies in the cutting
speed direction and in the cutting depth direction are presented in Table 5, and the
three first mode shapes in Fig. 8 a).
Two multi-span Euler-Bernoulli boring bar models with pinned boundary conditions were considered: one corresponded to the boring bar clamped with four screws
in the clamping housing, and one corresponding to the boring bar clamped with six
screws in the clamping housing. The eigenfrequencies and mode shapes (eigenfunctions) for the two models were calculated. The first three resonance frequencies in
both the cutting speed direction and the cutting depth direction for the two different
models are presented in Table 5. Thus, when the fixed clamping model is changed
to the pinned model, the first resonance drops by approximately 170 Hz for the
four-screw-clamped boring bar, and approximately 140 Hz for the six-screw-clamped
boring bar. The first three mode shapes for the two models are presented in Fig. 8
b) and c).
Two multi-span models with elastic foundation were calculated in the same way
as for the multi-span models with pinned boundary condition, but now for the elastic
boundary condition, using the stiffness coefficients in Table 3. The length of the clamp
screw overhang was selected to 1.5 mm. Both eigenfrequencies and mode shapes were
calculated for the two multi-span boring bar models with flexible clamping boundary
conditions. The calculated eigenfrequencies are presented in Table 5, and mode shapes
are shown in Fig. 8 d) and e).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
1
Mode 1
Mode 2
Mode 3
0.8
Normalized mode shape
53
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
Normalized mode shape
0.6
0.1
0.8
0.2
0
−0.2
0.05
0.1
0.15
0.2
0.25
0.2
0
Feed direction (z+) [m]
b)
−1
0
0.3
1
Mode 1
Mode 2
Mode 3
0.8
Normalized mode shape
Normalized mode shape
0.4
−0.8
0.4
0.2
0
−0.2
0.6
0.05
0.1
0.15
0.2
0.25
0.3
0.15
0.2
0.25
0.3
Feed direction (z+) [m]
c)
Mode 1
Mode 2
Mode 3
0.4
0.2
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−1
0
0.6
Mode 1
Mode 2
Mode 3
−0.6
−0.8
0.6
0.3
−0.4
−0.6
1
0.25
−0.2
−0.4
0.8
0.2
1
Mode 1
Mode 2
Mode 3
0.4
−1
0
0.15
Feed direction (z+) [m]
a)
Normalized mode shape
1
0.8
0.05
−0.8
0.05
0.1
0.15
0.2
0.25
Feed direction (z+) [m]
d)
0.3
−1
0
0.05
0.1
Feed direction (z+) [m]
e)
Figure 8: The first three mode shapes for the Euler-Bernoulli boring bar model
with boundary conditions a) fixed-free b) free-pinned-pinned-free (four clamp screws),
c) free-pinned-pinned-pinned-free (six clamp screws), d) free-spring-spring-free (four
clamp screws) and e) free-spring-spring-spring-free (six clamp screws).
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Model
Fixed-free
Pinned, four screws
Pinned, six screws
Spring, four M8 screws
Spring, six M8 screws
Spring, four M10 screws
Spring, six M10 screws
f1 [Hz]
698.33
527.47
566.92
519.43
532.09
525.24
541.52
f2 [Hz]
4376.36
3390.18
3575.59
3303.79
3335.17
3346.84
3398.74
f3 [Hz]
12253.94
9539.40
10058.76
9257.16
9278.08
9404.65
9484.36
Table 5: The first three resonance frequencies in cutting speed direction for the EulerBernoulli models
3.2
Experimental Modal Analysis
Shaker excitation was used for the experimental modal analysis (EMA) of the boring
bars. The utilized spectrum estimation parameters and excitation signals’ properties
are given in Table 4. A frequency range covering the significant part near the resonance frequencies was selected, i.e. ±100 to ±200 Hz around the resonance peaks.
The coherence values for the involved transfer paths at each eigenfrequency were
greater or equal to 0.996. A number of different phenomena were observed during the
experimental modal analysis of the boring bars for various configurations and setups.
For instance, large variations were observed in the first resonance frequencies of the
boring bar for different tightening torques of the clamp screws. Also, the order in
which the clamp screws were tightened (first from the upper side of the boring bar or
first from the under side of the boring bar) had a significant impact on, for example,
the fundamental bending resonance frequencies.
3.2.1
Standard Boring Bar
When clamping the standard boring bar so that the bottom side of the boring bar is
clamped against the clamping housing (i.e. the screws are tightened from the topside
first and subsequently from the bottom side) the fundamental boring bar resonance
frequencies increase with increasing tightening, see Fig. 9 a). In this setup, screws
of size M8 were used; the spectrum estimation parameters and excitation signal are
presented in Table 4.
By changing the excitation levels, nonlinearities in the dynamic properties of the
boring bar might be observable via changes in frequency response function estimates
for the same input and output locations at the boring bar. Four different excitation
levels were used with the proportion {1,2,3,4} for each of the torque configuration
presented in section 2.1.4. As can be seen in Fig. 9 a) the fundamental boring bar
resonance frequencies increases with increasing torque. And in Fig. 9 b) it can be
seen that the fundamental boring bar resonance frequencies decreases slightly with
increased excitation level. The estimated resonance frequencies and relative damping
from all 20 measurements are presented in Table 6.
The clamp screws were replaced with M10 screws and the number of clamp screws
was increased to six. Using these clamping conditions, experiments were performed
which were identical to those carried out using a clamping housing with four M8
screws.
When clamping the standard boring bar so that the bottom side of the boring
32
30
28
55
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
26
24
Level 1, 10Nm
Level 2, 10Nm
Level 3, 10Nm
Level 4, 10Nm
Level 1, 30Nm
Level 2, 30Nm
Level 3, 30Nm
Level 4, 30Nm
24
22
22
20
20
18
18
16
14
490
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
16
500
510
520
530
540
550
Frequency [Hz]
a)
560
570
580
14
490
500
510
520
530
540
550
Frequency [Hz]
b)
560
570
580
Figure 9: The driving point accelerance magnitude function in cutting speed direction
(y-) of the boring bar response using the standard boring bar, four screws of size
M8 and when clamp screws were tightened firstly from the upper-side, a) using five
different tightening torques and b) using two different tightening torques and four
different excitation levels.
Setup 1
Resonance Frequency, Mode 1 [Hz]
Torque
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
Level 1
509.52
518.13
523.84
526.64
526.72
Torque
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
Level 1
0.99
0.97
0.88
0.87
0.86
Level 2
507.87
516.50
522.97
526.05
526.23
Level 3
506.61
515.23
522.13
525.50
525.79
Level 4
505.51
514.18
521.55
525.10
525.45
Relative Damping of Mode 1 [%]
Level 2
1.04
1.00
0.91
0.88
0.88
Level 3
1.08
1.03
0.94
0.90
0.90
Level 4
1.14
1.08
0.96
0.92
0.93
Resonance Frequency, Mode 2 [Hz]
Level 1
540.86
546.50
553.01
556.07
555.67
Level 2
540.15
546.31
552.86
555.84
555.68
Level 3
539.33
545.73
552.49
555.66
555.55
Level 4
540.07
544.60
552.13
555.42
555.35
Relative Damping of Mode 2 [%]
Level 1
1.31
1.26
1.04
0.97
0.97
Level 2
1.33
1.32
1.04
0.93
0.95
Level 3
1.46
1.28
1.01
0.91
0.92
Level 4
0.26
1.23
0.99
0.90
0.90
Table 6: Estimates of the fundamental boring bar resonance frequencies and its relative damping based on all the measurements using the setup with standard boring
bar, clamped with four screws first tightened from the upper-side of the boring bar.
The grey columns of mode 1 and mode 2 correspond to frequency response functions in Fig. 9 a). The grey rows of mode 1 and mode 2 correspond to the boring
bar frequency response functions Fig. 9 b) produced for the four different excitation
levels.
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Part II
56
32
30
28
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
26
24
Level 1, 10Nm
Level 2, 10Nm
Level 3, 10Nm
Level 4, 10Nm
Level 1, 30Nm
Level 2, 30Nm
Level 3, 30Nm
Level 4, 30Nm
24
22
22
20
20
18
18
16
14
490
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
bar is clamped against the clamping housing, the fundamental boring bar resonance
frequencies increases with increasing tightening, see Fig. 10 a). As can be seen in
Fig. 10 b), the fundamental boring bar resonance frequencies decreases slightly with
increasing excitation level.
16
500
510
520
530
540
550
Frequency [Hz]
a)
560
570
580
14
490
500
510
520
530
540
550
Frequency [Hz]
b)
560
570
580
Figure 10: The driving point accelerance magnitude function in cutting speed direction (y-) of the boring bar response using the standard boring bar, six screws of size
M10 and when clamp screws were tightened firstly from the upper-side of the boring
bar. a) Using five different tightening torques and b) using two different tightening
torques and four different excitation levels.
Clamping by first tightening the clamp screws from the boring bar’s underside
changes the frequency response functions significantly as compared with the case
where the clamp screws were tighten first from the upper side. This might be observed
by comparing Fig. 11 with Fig. 10.
3.2.2
Modified Boring Bar
The modified boring bar has a cavity, a milled space, onto which an embedded actuator
was placed. This space constitutes a change in the dynamic properties of the boring
bar in comparison to the standard boring bar. This is obvious since the material,
steel, is removed from the boring bar and replaced partly with an actuator with a
lower Young’s module, etc. The actuator was kept passive during the experiments,
thus, no control authority was applied. The same experiments were conducted with
the modified boring bar as were performed with the standard boring bar.
From the results presented in Fig. 12 it is clear that the dynamic properties of the
modified boring bar have changed significantly, mostly with regard to cutting speed
direction (compare with the results from the standard boring bar Fig. 9). However,
we can observe the same phenomenon that occurred in results obtained with the
standard boring bar; i.e. increasing resonance frequency with increasing torque and
decreasing resonance frequency with increasing excitation force. The estimated resonance frequencies and relative damping from all the 20 measurements are presented
in Table 7.
When the modified boring bar is clamped with six, size M10 screws, results obtained resemble those derived from clamping the same bar with size M8 screws, see
Fig. 12.
32
57
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
30
28
26
26
24
24
22
22
20
Level 1, 10Nm
Level 2, 10Nm
Level 3, 10Nm
Level 4, 10Nm
Level 1, 30Nm
Level 2, 30Nm
Level 3, 30Nm
Level 4, 30Nm
20
18
18
16
14
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
16
420
440
460
480
500
Frequency [Hz]
a)
520
540
14
420
440
460
480
500
Frequency [Hz]
b)
520
540
32
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
Level 1, 10Nm
Level 2, 10Nm
Level 3, 10Nm
Level 4, 10Nm
Level 1, 30Nm
Level 2, 30Nm
Level 3, 30Nm
Level 4, 30Nm
30
28
26
24
24
22
22
20
20
18
18
16
14
440
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Figure 11: The accelerance magnitude function of the boring bar response in the
driving point in cutting speed direction (y-)using the standard boring bar, six screws
of size M10 and when clamp screws were tightened firstly from the underside of the
boring bar. a) Using five different tightening torques and b) using two different
tightening torques and four different excitation levels.
16
450
460
470
480
490
500
Frequency [Hz]
a)
510
520
530
14
440
450
460
470
480
490
500
Frequency [Hz]
b)
510
520
530
Figure 12: The driving point accelerance magnitude function in cutting speed direction (y-) of the boring bar response using the modified boring bar, four screws of size
M8 and when clamp screws were tightened firstly from the upper-side of the boring
bar. a) Using five different tightening torques and b) using two different tightening
torques and four different excitation levels.
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58
Setup 4
Resonance Frequency, Mode 1 [Hz]
Torque
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
Level 1
449.83
466.62
478.76
482.29
484.39
Torque
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
Level 1
1.40
1.22
1.17
1.16
1.36
Level 2
447.10
464.55
477.87
481.62
483.63
Level 3
445.31
463.10
476.51
480.81
482.87
Level 4
444.34
462.37
475.90
480.36
482.88
Relative Damping of Mode 1 [%]
Level 2
1.41
1.20
1.14
1.22
1.35
Level 3
1.43
1.25
1.06
1.23
1.42
Level 4
1.51
1.35
1.15
1.27
1.51
Resonance Frequency, Mode 2 [Hz]
Level 1
473.12
478.55
501.49
510.13
515.08
Level 2
472.57
477.36
499.88
508.39
513.91
Level 3
472.50
476.75
499.29
507.35
513.11
Level 4
471.76
475.98
497.79
506.96
512.79
Relative Damping of Mode 2 [%]
Level 1
1.61
1.25
1.67
1.18
0.99
Level 2
1.72
1.24
1.73
1.23
1.02
Level 3
1.85
1.25
1.86
1.25
1.03
Level 4
1.89
1.34
1.62
1.33
1.05
Table 7: Estimates of the fundamental boring bar resonance frequencies and its relative damping based on all measurements, using the setup in which the modified boring
bar is clamped with four screws first tightened from the upper-side of the boring bar.
The grey columns of mode 1 and mode 2 correspond to frequency response functions
in Figs. 12 a). The grey rows of mode 1 and mode 2 correspond to the boring bar
frequency response functions in Figs. 12 b), produced for the four different excitation
levels.
3.2.3
Linearized Boring Bar
Finally, the results from the boring bar with a so-called ”linearized” clamping condition are presented. Since no screws were used in this setup, only the excitation
levels were changed. The results are presented in Fig. 13 and Table 8, which are the
driving point frequency response functions in both the cutting speed direction and
the cutting depth direction. Thus, only a slight variation in the boring bar’s resonance frequencies and damping might be observed. Unfortunately, both resonance
frequencies coincide with periodic disturbances originating from the engines in the
lathe producing the hydraulic pressure. One disturbance was at approximately 591
Hz and the other disturbance at approximately 600 Hz. These disturbances will have
different influences on the estimates, depending on the excitation level, this may be
observed near the peak in Fig. 13.
Setup 6
Excitation
Mode 1, Frequency [Hz]
Mode 2, Frequency [Hz]
Mode 1, Relative Damping [%]
Mode 2, Relative Damping [%]
Level 1
583.82
602.25
2.12
0.76
Level 2
584.15
602.07
2.04
0.75
Level 3
583.13
601.92
2.16
0.75
Level 4
582.52
601.79
2.16
0.74
Table 8: Estimates of resonance frequency and the relative damping for the fundamental bending modes of the linearized boring bar.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
59
36
Accelerance [dB rel 1(m/s2 )/N]
Level 1
Level 2
Level 3
Level 4
34
32
30
28
26
24
22
20
560
570
580
590
600
610
620
Frequency [Hz]
630
Figure 13: The accelerance magnitude function of the boring bar response using the
linearized setup and with four different excitation levels, in the driving point in cutting
speed direction (Y-).
3.2.4
Mode shapes
This section presents all the mode shapes estimated from the three different setups:
the standard boring bar, the modified boring bar and the linearized boring bar. The
mode shapes were estimated in I-DEAS using the frequency poly-reference method.
First, results are presented from the standard boring bar, with size M8 screws, tightening clamp screws firstly from the upper-side. The shapes are presented in zy-plane
and xy-plane in Fig. 14, a) and b) respectively. The angle of rotation around z-axis
(relative the cutting depth direction for each measurement) is presented in Table 9.
The mode shapes in xy-plane illustrated in Fig. 14 b) and the corresponding values
in Table 9 show an average rotation of approximately 20 degrees.
Setup 1
Angle of Mode 1, [Degree]
Torque
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
Level 1
-17.55
-21.42
-22.09
-21.31
-22.54
Level 2
-17.10
-20.84
-21.81
-20.90
-22.27
Level 3
-16.33
-20.44
-21.33
-20.66
-22.04
Level 4
-17.81
-20.13
-21.08
-20.48
-21.84
Angle of Mode 2, [Degree]
Level 1
-107.13
-110.30
-110.96
-109.90
-110.92
Level 2
-106.89
-109.80
-110.30
-109.65
-110.68
Level 3
-106.68
-109.45
-109.92
-109.42
-110.45
Level 4
-106.43
-109.22
-109.76
-109.32
-110.35
Table 9: Angle of mode shapes for the standard boring bar, relative to cutting depth
direction axis.
Measurements derived from the modified boring bar differ somewhat from those
obtained with the standard boring bar. Mode shapes are presented in Fig. 15 and
the values of the angle of rotation in Table 10. The shapes are almost identical in
the yz-plane, but in the xy-plane the shapes rotate around the z-axis. Table 10,
demonstrates a trend of clockwise rotation with increasing torque, as well as counterclockwise rotation with increasing excitation level; this applies to both modes. The
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60
Negative Cutting Speed Direction (y+)
0.4
Mode 1
Mode 2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
0.25
Feed Direction (z+)
a)
Negative Cutting Speed Direction (y+)
0.4
0.3
Mode 1
Mode 2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Cutting Depth Direction (x+)
b)
0.8
1
1.2
Figure 14: The two first mode shapes of the standard boring bar clamped with four
M8 screws, when the clamp screws were tightened firstly from the upper-side, for five
different tightening torques and four different excitation levels. a) in the zy-plane and
b) in the xy-plane.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
61
angles lies between 20 and 73 degrees for the first mode, thus the first mode shifts
from being most significant in cutting depth direction, to being most significant in
cutting speed direction.
Setup 4
Angle of Mode 1, [Degree]
Torque
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
Level 1
-20.98
-33.78
-60.47
-69.59
-72.25
Level 2
-19.20
-29.55
-56.97
-68.21
-71.41
Level 3
-18.28
-28.04
-55.28
-67.65
-70.93
Angle of Mode 2, [Degree]
Level 4
-18.50
-27.58
-55.76
-67.69
-70.50
Level 1
-112.06
-116.87
-149.03
-155.52
-156.19
Level 2
-110.52
-116.89
-145.72
-154.13
-155.54
Level 3
-109.67
-115.53
-145.56
-153.93
-155.19
Level 4
-110.00
-114.23
-145.77
-154.03
-154.52
Table 10: Angle of mode shapes for the modified boring bar, relative to cutting depth
direction axis.
The results from the linearized setup are presented by Fig. 16 and in Table 11.
In this linear setup, the zy-plane shape is almost identical to those shapes produced
from standard, and modified boring bar measurements, see Figs. 14, 15 and 16. In
the xy-plane the shapes only have a rotation of approximately 10 degrees.
Excitation
Mode 1
Mode 2
Setup 6
Angle of Mode, [Degree]
Level 1 Level 2 Level 3 Level 4
-7.98
-8.46
-8.05
-8.11
-99.72
-100.00
-99.82
-100.07
Table 11: Angle of mode shapes for the linearized boring bar relative to cutting depth
direction axis.
After parameter estimation the Modal Assurance Criterion (MAC) was used to
measure the correlation between the estimated modes shapes. A typical MAC diagram
is presented in Table 12.
Mode [Hz]
526.72
555.67
526.72
1
0.000
555.67
0.000
1
Table 12: The modal assurance criterion matrix coefficients for the two estimated
mode shapes at the resonance frequencies 526.72 Hz and 555.67 Hz. The modes are
estimated using the standard boring bar (clamped with four screws tightened firstly
from the top), the lowest excitation level and the highest tightening torque.
Typical values of the MAC matrices of-diagonal elements are 0.000-0.001, few
values reach 0.007.
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62
Negative Cutting Speed Direction (y+)
0.4
Mode 1
Mode 2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
Feed Direction (z+)
a)
0.25
Negative Cutting Speed Direction (y+)
0.4
0.3
Mode 1
Mode 2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Cutting Depth Direction (x+)
b)
0.6
0.8
Figure 15: The two first mode shapes of the modified boring bar clamped with four
M8 screws, when the clamp screws were tightened firstly from the upper-side, for five
different tightening torques and four different excitation levels. a) in the zy-plane and
b) in the xy-plane.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
Negative Cutting Speed Direction (y+)
0.4
63
Mode 1
Mode 2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
Feed Direction (z+)
a)
0.25
Negative Cutting Speed Direction (y+)
0.4
0.3
Mode 1
Mode 2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Cutting Depth Direction (x+)
b)
1
1.2
Figure 16: The two first mode shapes of the linearized boring bar for four different
excitation levels. a) in the zy-plane and b) in the xy-plane.
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64
3.3
Comparison between the experiments and the Euler-Bernoulli
models
To get an overview of results from the different setups in the experiments and the
Euler-Bernoulli boring bar models relevant results are summarized in this section.
In Fig. 17 a) the magnitude of the boring bar driving point accelerance functions
-clamp screw tightening torque 30 Nm and maximum excitation signal level- for the
six different setups are shown.
Estimates of eigenfrequencies, relative damping and mode shape angle relative to
cutting depth direction for the fundamental boring bar bending modes, max excitation
level and tightening torque 30 Nm, for the six setups, are presented in Table 13.
The multi-span boring bar models are supposed to approximately model the standard boring bar clamped with clamp screws in the clamping housing, thus it is the
experimental and analytical models in cutting speed direction that are most relevant
to compare. The experimental results that are of concern here are those for mode 2
in Table 13.
Setup
1
2
3
4
5
6
Parameters of mode 1
Freq. [Hz] Damp. [%] Angle [◦ ]
525.45
0.93
-21.84
528.58
0.97
-17.74
502.43
1.37
-6.40
482.88
1.51
-70.93
489.11
1.70
-59.47
582.52
2.16
-8.11
Parameters of mode 2
Freq. [Hz] Damp. [%] Angle [◦ ]
555.35
0.90
-110.35
560.83
1.00
-108.26
526.87
1.42
-95.51
512.79
1.05
-154.52
508.22
1.10
-141.06
601.79
0.74
-100.07
Table 13: Eigenfrequencies, relative damping and mode shape angle relative to cutting
depth direction for the six different boring bar setups. Clamp screw tightening torque,
30 Nm and maximum excitation signal level.
For the standard boring bar setups, estimates of eigenfrequencies from the analytical models and from the experiments (clamp screw tightening torque 30 Nm and
maximum excitation signal level) for the fundamental bending mode in the cutting
speed direction are presented together in Table 14.
The integer part of the fundamental eigenfrequencies calculated with the EulerBernoulli models of the standard boring bar setups is identical for both the cutting
speed and the cutting depth direction. By using this information together with the
fundamental eigenfrequencies for mode 1 given in Table 13 information similar to that
given in Table 14 may be extracted for the fundamental bending mode in the cutting
depth direction.
The mode shape estimates from the experimental modal analysis of the six different
setups have components in three dimensions and does not seem to coincide with the
planes defined by the chosen coordinate system (they have components in both the
cutting speed and the cutting depth directions), while the analytical mode shapes from
the analytical models do. However, the mode shapes may still be compared. A number
of typical experimental mode shapes are compared with the Euler-Bernoulli beam
models mode shapes, in the cutting speed direction-feed direction plane, in Fig. 17
b).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
Setup 1
Setup 2
Setup 3
Setup 4
Setup 5
Setup 6
35
Magnitude [dB rel 1(m/s2 )/N]
65
30
25
20
15
440
460
480
500
520
540
560
580
Frequency [Hz]
600
620
640
b)
Negative cutting speed direction (y+)
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
Fixed−Free
FPPF
FPPPF
FSSF
FSSSF
Experimental Data
0.05
0.1
0.15
0.2
Feed direction (z+)
0.25
0.3
b)
Figure 17: a) The magnitude of the boring bar accelerance functions for the six
different setups, from the driving point in cutting speed direction. b) Analytical mode
shapes for five models; one single-span models with fixed-free boundary condition
(black solid line), two multi-span models with pinned boundary condition and two
multi-span models with elastic boundary condition. Also a number of experimental
estimated mode shapes (circles) are overlayed on top of the analytical results.
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66
Analytical model; cutting speed direction
f1 [Hz]
Exp. setup
Mode 2 [Hz]
Fixed-Free
698.33
Setup 6
601.79
Pinned, four screws
Pinned, six screws
Pinned, six screws
527.47
566.92
566.92
Setup 1
Setup 2
Setup 3
555.35
560.83
526.87
Elastic, four M8 screws
Elastic, six M8 screws
Elastic, four M10 screws
Elastic, six M10 screws
Elastic, six M10 screws
519.43
532.09
525.24
541.52
541.52
Setup 1
Setup 2
Setup 3
555.35
560.83
526.87
Table 14: Estimates of the fundamental eigenfrequency in the cutting speed direction
for the experimental setups of the standard boring bar (clamp screw tightening torque
30 Nm and maximum excitation signal level) and the fundamental eigenfrequencies
calculated with the Euler-Bernoulli models of the standard boring bar setups.
4
Summary and Conclusions
Experimental modal analysis has been carried out on two boring bars; one original
standard boring bar and one modified boring bar. The results from the experimental
modal analysis of the two boring bars demonstrate that the different controlled clamping conditions in the experiments yield different dynamic properties of the clamped
boring bars. Hence, the different clamping conditions result in different boundary
conditions along the clamped part of the boring bar. It has also been established that
a boring bar clamped in a standard clamping housing with clamping screws is likely
to have a nonlinear dynamic behavior. The standard clamping housing with clamp
screws is the likely source of the nonlinear behavior. Multi-span Euler-Bernoulli
models of a clamped boring bar incorporating pinned or elastic support boundary
conditions approximating the flexibility of the actual screw clamping of the boring
bar end inside the clamping housing provide significantly higher correlation with experimental modal analysis results compared to a traditional fixed-free Euler-Bernoulli
model.
The fundamental boring bar resonance frequencies decrease with increasing excitation level; see Tables 6 and 7. However, with regard to the behavior of relative
damping as a function of excitation force level; the results from the standard boring
bar indicate that the relative damping for the first mode increases with increasing
excitation force level, while the relative damping for the second mode decreases with
increasing excitation force level; see Table 6. Also, the results from the modified
boring bar give an ambiguous indication of the effects on damping properties; see
Table 7. The clamp screw tightening torque appears to affect the nonlinear behavior
of the boring bar. Variation in the accelerance function estimates which was introduced by the four different excitation force levels seems to be larger for a low screw
tightening torque (10 Nm) than for a high screw tightening torque (30 Nm), see, for
example, Fig. 10 b). By examining, for example, driving point accelerances for the
boring bar with ”linearized” clamping for the four excitation force levels (see Fig. 13)
it seems like the previously observed nonlinear behavior is almost removed. Thus this
supports the conclusion that clamping conditions influence the extent of nonlinearities in boring bar dynamics. The boring bars resonance frequencies increase with
increasing clamp screw torque; see the accelerance magnitude functions in Figs. 9
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
67
, 10, 11 and 12. When changing the number of screws used for clamping, or when
using ”linearized” clamping, changes in dynamic properties of the boring bar (clamping system) are expected. Hence new boundary conditions for the boring bar are
introduced. Also, changing the standard boring bar to the modified boring bar will
alter the dynamic properties of the boring bar - clamping system, i.e. a structural
part of the system is different. The order in which the clamp screws were tightened
(firstly from the upper side or firstly from the underside) had a major influence on
the dynamic properties of the boring bar. This might be observed by comparing the
boring bar driving point accelerances in Fig. 10 with the boring bar driving point
accelerances in Fig. 11. If the clamp screws were tightened firstly from the upper
side, the higher resonance frequency in Fig. 10 a) shows a variation from approximately 552 Hz to 562 Hz, for a change from the lowest to the highest clamp screws
tightening torque. On the other hand, if the clamp screws were tightened firstly from
the underside, the higher resonance frequency in Fig. 11 a) shows a variation from
approximately 495 Hz to 530 Hz, for a change from the lowest to the highest clamp
screws tightening torque. The discrepancies in the results when changing from which
side the boring bars are tightened first, may be due to the difficulty in producing the
exact same clamping conditions when tightening the clamp screws from the bottom
first, compared to tightening the clamp screws from the top first.
Another interesting observation concerns mode shapes and, in particular angles
of the different modes in the cutting depth - cutting speed plane (x-y plane). The
standard boring bar clamped with four M8 screws, tightened firstly from the top has
a first mode with an average mode shape angle or rotation of -20 degrees, relative to
the cutting depth direction (x-direction). The second mode displays an average mode
shape angle (or rotation) of -110 degrees relative to the cutting depth direction, see
Table 9. Changing the clamp screw size or the number of clamp screws affects the
so-called ”mode rotation”, both for the standard boring bar and in the case of the
modified boring bar (see Figs. 14, 15 and 16). In addition, it should be noted that
the boring bar in the linearized setup has rotated fundamental modes; the first mode
has a mode shape angle or rotation of approx. -8 degrees relative to cutting depth
direction (x-direction), and the second mode displays a mode shape angle or rotation
of approx. -100 degrees relative to cutting depth direction (see Table 11).
When comparing the eigenfrequency estimates from the analytical results with
the experimental results (see Table 13), some obvious discrepancy may be observed.
The first and simplest model, the Euler-Bernoulli fixed-free model, overestimates the
first fundamental bending resonance frequency substantially (with approx 100 Hz)
compared to setup 6 and at least with 140 Hz compared to setups (1-3). The EulerBernoulli multi-span models produce eigenfrequency estimates, in the cutting speed
direction, within approx. 6-40 Hz of the experimental eigenfrequency estimates for
setups (1-3). Comparing the mode shapes from the Euler-Bernoulli models with
the experimental mode shape estimates for the clamped standard boring bar, it is
obvious that the fixed-free model shows a too rigid clamping at the base as the
experimental results indicate some deflection adjacent to the clamping housing , see
Fig. 17 b). While the mode shapes provided by the different multi-span models
display a substantially higher agreement with the experimental results. Also, for the
fundamental bending mode in the cutting depth direction, the Euler-Bernoulli multispan models provide higher correlation with the standard boring bar clamped in the
clamping housing with screws compared to Euler-Bernoulli fixed-free model.
The different boring bar clamping conditions result in different boundary con-
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68
ditions for the boring bars in the experiments. Hence, if another boring bar with
different, e.g. length and/or cross-section and/or material as compared to the two
boring bars in this work would be considered. Different boundary conditions produced
by the boring bar clamping on this different boring bar would also result in different
dynamic properties of it. However, identical clamping boundary conditions applied to
the different boring bar and to one of the boring bars in this work would most likely
result in different dynamic properties of the two boring bars. Basically, the dynamic
properties of a clamped boring bar are related e.g. to its physical dimensions, the
material it is made of, the clamping conditions of it.
The following conclusions may be deduced from the results:
• The dynamic properties of a boring bar for different clamping conditions introduced by a clamping housing with clamping screws may differ significantly.
• A clamping housing with clamping screws is likely to provide different clamping conditions/boundary conditions of a boring bar for different clamp screw
torques and if tightened firstly from the upper side or tightened firstly from the
underside.
• A boring bar is likely to exhibit different dynamic properties when clamped or
mounted in the clamping housing with clamping screws by different operators
and from time to time.
• Different dynamic properties of a clamped boring bar are likely to result in
different dynamic properties of the boring bar vibration during machining for
identical cutting data, insert and work material.
• The set of cutting data that enables stable cutting for a given insert and work
material may differ for different clamping conditions.
• A clamping housing with clamp screws is likely to introduce nonlinear dynamic
properties in the boring bar dynamics.
• A multi-span Euler-Bernoulli model compared to a fixed-free Euler-Bernoulli
model is more adequate for the modeling of dynamic properties of a boring bar
clamped in a clamping housing with clamping screws.
5
Future work
In order to further enhance the modeling of boring bars and the actual clamping/boundary conditions imposed by the clamping housing, the finite element method
(FEM) will be considered. Also, the possibility to improve accuracy of boring bar
models by the inclusion of simple nonlinearities will be investigated.
Acknowledgments
The present project is sponsored by the company Acticut International AB in Sweden
which has multiple approved patents and products covering active control technology
for metal cutting.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
69
References
[1] L. Andrén, L. Håkansson, A. Brandt, and I. Claesson. Identification of dynamic
properties of boring bar vibrations in a continuous boring operation. Journal of
Mechanical Systems & Signal Processing, 18(4):869–901, 2004.
[2] L. Andrén, L. Håkansson, A. Brandt, and I. Claesson. Identification of motion of
cutting tool vibration in a continuous boring operation - correlation to structural
properties. Journal of Mechanical Systems & Signal Processing, 18(4):903–927,
2004.
[3] J. Scheuer, L. Håkansson, and I. Claesson. Modal analysis of a boring bar using
different clamping conditions. In In Proceedings of the Eleventh International
Congress on Sound and Vibration, ICSV11, St. Petersburg, Russia, 2004.
[4] R. N. Arnold. Mechanism of tool vibration in cutting of steel. Proc. Inst. Mech.
Eng., 154:261–284, 1946.
[5] S.A. Tobias. Machine-Tool Vibration. Blackie & Son, 1965.
[6] H.E. Merritt. Theory of self-excited machine-tool chatter, contribution to
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[7] R. A. Thompson. The modulation of chatter vibrations. Journal of Engineering
for Industry, pages 673–679, August 1969.
[8] E.W. Parker. Dynamic stability of a cantilever boring bar with machined flats
under regenerative cutting conditions. Journal of Mechanical Engineering Sience,
12(2):104–115, February 1970.
[9] S.M. Pandit, T.L. Subramanian, and S.M. Wu. Modeling machine tool chatter
by time series. Journal of Engineering for Industry, Transactions of the ASME,
97:211–215, February 1975.
[10] S. Kato and E. Marui. On the cause of regenerative chatter due to worpiece
deflection. Journal of Engineering for Industry, pages 179–186, 1974.
[11] G.M. Zhang and S.G. Kapoor. Dynamic modeling and analysis of the boring
machining system. Journal of Engineering for Industry, Transactions of the
ASME, 109:219–226, August 1987.
[12] P.N. Rao, U.R.K. Rao, and J.S. Rao. Towards improwed design of boring bars
part 1: Dynamic cutting force model with continuous system analysis for the
boring bar. Journal of Machine Tools Manufacture, 28(1):33–44, 1988.
[13] F. Kuster. Cutting dynamics and stability of boring bars. Annals of CIRP,
39(1):361–366, 1990.
[14] H. Åkesson, T. Smirnova, L. Håkansson, and I.Claesson. On the development of
a simple and robust active control system for boring bar vibration in industry.
International Journal of Acoustics and Vibration, 12(4), December 2007.
[15] D.J. Inman. Engineering Vibration. Prentice-Hall, second edition, 2001.
70
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[16] W. Weaver, S. P. Timoshenko, and D. H. Young. Vibration Problems in Engineering. John Wilet & Sons., fifth edition edition, 1999.
[17] I.H. Shames and J.M. Pitarresi. Introduction to Solid Mechanics. Upper Saddle
River, new Jersey, Prentice Hall, third edition, 2000.
[18] D.J. Ewins. Modal Testing: Theory and Practice. Research Studies Press, 1984.
[19] J.S. Bendat and A.G. Piersol. Random Data Analysis and Measurement Procedures. John Wiley & Sons, third edition, 2000.
[20] P.D. Welch. The use of fast fourier transform for the estimation of power spectra:
A method based on time averaging over short, modified periodograms. IEEE
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[21] Structural Dynamics Research Laboratory University of Cincinnati. Modal parameter estimation. Course Literature: : http://www.sdrl.uc.edu/academiccourse-info/vibrations-iii-20-263-663, 5 February 2002.
[22] H. Vold, J. Kundrar, G. T. Rocklin, and R. Russell. A multiple-input modal
estimation algorithm for mini-computers. SAE Transactions, Vol 91, No. 1:815–
821, January 1982.
[23] H. Vold and G. T. Rocklin. The numerical implementation of a multi-input modal
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Proceedings, pages 542–548, January 1982.
[24] L. Zhang, H. Kanda, D. L. Brown, , and R. J. Allemang. Polyreference frequency
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Design &Automation, 2nd edition, 1997.
Part III
Estimation and Simulation of the
Nonlinear Dynamic Properties of a
Boring bar
This part is submitted as:
H. Åkesson, T. Smirnova, I. Claesson, T. Lagö and L. Håkansson, ”Estimation and
Simulation of the Nonlinear Dynamic Properties of a Boring bar”, submitted to publication in International Journal of Acoustics and Vibration, August, 2009.
Estimation and Simulation of the Nonlinear
Dynamic Properties of a Boring bar
Henrik Åkesson1,2 , Tatiana Smirnova,
Lars Håkansson and Ingvar Claesson
1
Blekinge Institute of Technology,
Department of Signal Processing,
372 25 Ronneby, Sweden
Thomas T. Lagö
Acticut International AB,
Gjuterivägen 6, 311 32 Falkenberg, Sweden
2
Abstract
In this paper, an initial investigation of the nonlinear dynamic properties
of clamped boring bars is carried out. Two nonlinear single-degree-of-freedom
models with different softening spring nonlinearity are introduced for modeling
the nonlinear dynamic behavior of the fundamental bending mode in the cutting
speed direction of a boring bar. Also, two different methods for the simulation
of nonlinear models are used. The dynamic behavior in terms of frequency response function estimates for the nonlinear models and the experimental modal
analysis of the clamped boring bar is compared. Similar resonance frequency
shift behavior for varying excitation force levels is observed for both the nonlinear models and the actual boring bar.
1
Introduction
In industry where metal cutting operations such as turning, milling, boring and grinding take place, degrading vibrations are a common problem. In internal turning operations, vibration is a pronounced problem, as long and slender boring bars are usually
required to perform the internal machining of workpieces. Tool vibration during internal turning frequently has a degrading influence on surface quality, tool life and
production efficiency. At the same time, such vibrations result in high noise levels.
An extensive number of experimental and analytical studies have been carried out
to study boring bar dynamics. However, most research has usually been carried out on
the dynamic modeling of cutting dynamics and usually concentrates on the prediction
of stability limits [1–5]. In a study concerning the motion of the boring bar, it was
stated that clamped boring bars may display nonlinear dynamic behavior [6]. Later, a
more thorough investigation concerning the clamping conditions of the boring bars [7]
confirmed this assumption regarding nonlinear dynamic behavior.
When it comes to nonlinear modeling of the clamping of tools, Yigit et al. [8]
examined and modeled a reconfigurable machine tool structure including weakly non-
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linear joints in consideration of their cubic stiffness. They used a sub-structuring
method called nonlinear receptance coupling and validated the method with experimental data from such a structure. Thus, refereing to the literature review, it seems
like little work has been done on the identification and modeling of the nonlinear dynamic properties of a clamped boring bar. Knowledge regarding nonlinear dynamic
properties of a clamped boring bar may be utilized in the modeling of dynamic properties of boring bars in order to provide increased accuracy in such models. Hence,
it seems to be important to be able to identify the nonlinear dynamic properties of a
clamped boring bar.
Nonlinearities may be caused by several different factors. Common sources of
nonlinearity are, for example; the contact phenomena, in which elements of a system during dynamic motion come into contact with the surrounding environment due
to a large displacement, which, in turn, creates a new set of boundary conditions.
Another example is friction in joints, or sliding surfaces and large forces and/or deformation that cause the properties of the material to behave in a nonlinear manner,
for example plastic deformation [9]. This paper presents an initial investigation of the
nonlinear dynamic properties of boring bars. Two nonlinear single-degree-of-freedom
models with different softening spring nonlinearity have been introduced in order to
model the nonlinear dynamic behavior of the fundamental bending mode in the cutting speed direction of a boring bar. Two different methods for the simulation of
nonlinear models are used, i.e. the Runge-Kutta method, implemented in Matlab [10]
and the digital filter method [11]. The dynamic behavior in terms of frequency response function estimates for the models and the actual clamped boring bar have
been compared.
2
2.1
Materials and Methods
Experimental Setup
The experimental setup and subsequent measurements were carried out in a Mazak
SUPER QUICK TURN - 250M CNC turning center. The CNC lathe has a spindle
power of 18.5 kW and a maximal machining diameter of 300 mm. The distance
between the centers is 1005 mm, the maximum spindle speed is 400 revolutions per
minute (r.p.m.) and the center also has a flexible turret with a tool capacity of 12
tools. Fig. 1 illustrates some of the basic structural parts used during internal turning,
i.e. a workpiece clamped to a chuck and a boring bar clamped ta a clamping housing.
Initially, a right-hand cartesian coordinate system was defined. Subsequently, a
sign convention was defined for use throughout the paper. The coordinate system
and the sign convention are based on the right-hand definition where the directions of
displacements and forces in positive directions of the coordinate axes are considered
positive.
In all setups and during all measurements, the boring bar was positioned in its
operational position, that is, it was mounted to a clamping housing attached to a
turret with screws. The turret could be moved in the cutting depth direction, in the
x-direction, in the feed direction, and in the z-direction, and it could also be made
to rotate about the z-axis for tool change. The turret, etc. was supported by a slide
which in turn, was mounted on the lathe bed. Even though the turret was a movable
component, it was relatively rigid, rendering the dynamic properties of the boring
bars observable.
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 75
T u rre t
S p in d le
Ja w s
W o rk p ie c e
B o rin g b a r
C la m p in g h o u s in g
In se rt
Figure 1: An internal turning setup with a workpiece clamped to a chuck to the left
and a boring bar clamped to a clamping housing to the right.
2.1.1
Boring Bar
The boring bar used in the modal analysis was a standard boring bar, WIDAX S40T
PDUNR15F3 D6G. In Fig. 2, a drawing of the boring bar is shown.
z
3 7 .5 m m
x
3 0 0 m m
y
Ø 4 0 m m
M
C
1
C
C
x
3 7 m m
a)
b)
Figure 2: a) Top-view of the standard boring bar ”WIDAX S40T PDUNR15F3 D6G”,
b) the cross section of the boring bar where CC is the center of the circle and MC is
the mass center of the boring bar.
The standard WIDAX S40T PDUNR15 boring bar is manufactured from the material 30CrNiMo8, (AISI 4330) which is a heat treatable steel alloy (for high strength),
see Table 1 for material properties.
2.1.2
Clamping Condition
The boring bar was positioned in a clamping housing. The clamping housing was
a basic 8437-0 40 mm Mazak holder, presented in Fig. 3 a) and b). The clamping
housing has a circular cavity that the boring bar fits easily into. The clamping is
then carried out by means of screws on the tool side and on the opposite side of the
boring bar by means of either four or six screws: two/three inserted from the top
and two/three inserted from the bottom. The basic holder itself is mounted onto the
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76
Material composition besides Fe, in percent
C
Cr
Ni
Mo
Si
Mn
S
P
0.26-0.33 1.80-2.20 1.80-2.20 0.30-0.50 <0.40 <0.60 <0.035 <0.035
Material properties
Young’s Modulus
Tensile Strength
Yield Strength
Density
205 GPa
1250 MPa
1040 MPa
7850 kg/m3
Table 1: Composition and properties of the material 30CrNiMo8.
turret with four screws. In addition to the screws, the clamping housing also contains
a guide that matches a track on the turret. This guide positions the clamping housing
along the z-axis on the xy plane, whilst the guide pin positions the clamping housing
on the z-axis.
S c re w p o s itio n s fo r a tta c h in g th e
c la m p in g h o u s in g to th e tu rre t
S c re w p o s itio n s fo r th e c la m p in g
o f th e b o rin g b a r
G u id e p in
G u id e
y
z
y
x
a)
z
x
b)
Figure 3: The clamping housing. a) The guide and the guide pin may be observed
on the underside of the clamping housing, whilst the threaded holes for the screws
clamping the boring bar are shown on the right side. b) The screw positions for
attaching the clamping housing to the turrets are shown from the top side.
Six screws of the size M10 were used to clamp the boring bar, three from the top
and three from the bottom. The screws were of the type MC6S norm ”DIN 912, ISO
4762”. The screws were zinc-plated, steel socket, head cap screws with the strength
class 8.8, and a tensile yield strength of Rp02 = 660M P a.
2.1.3
Measurement Equipment and Setup
The following equipment was used in the experimental setup;
•
•
•
•
•
•
12 PCB Piezotronics, Inc. 333A32 accelerometers.
2 Brüel & Kjǽr 8001 impedance head.
1 Brüel & Kjǽr NEXUS 2 channel conditioning amplifier 2692.
OSC audio power amplifier, USA 850.
Ling dynamic systems shaker v201.
Gearing & Watson electronics shaker v4.
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 77
•
•
•
•
•
Hewlett Packard 54601B oscilloscope.
Hewlett Packard 35670A signal analyzer.
Hewlett Packard VXI mainframe E8408A.
Hewlett Packard E1432A 4-16 channel 51.2 kSa/s digitizer.
PC with I-DEAS 10 NX Series.
Twelve accelerometers and two cement studs for the impedance heads were attached
on the boring bars with X60 glue (a cold hardener, two-component glue). The sensors
were evenly distributed along the centerline, on the under-side and on the backside of
the boring bar and six accelerometers and one stud were attached to the respective
sides, see Fig. 4. To excite all the lower order bending modes, two shakers were
attached via stinger rods to the impedance heads, one in the cutting speed direction
(y-) and the other in the cutting depth direction (x-). The shakers were positioned
relatively close to the cutting tool.
B o tto m
V ie w
l4
x
l8
l5
l1
l8
z
l6
l2
C e m e n t stu d s
l9
l3
F r o n t V ie w
l4
B
B
1
B
2
3
y
z
l7
l1
l2
- A c c e le r o m e te r s
l8
l3
l8
l9
Figure 4: Drawings of the boring bar including clamp screws, cement studs and
sensors. The sensors are attached along the underside and the backside of the boring
bar. The threaded holes denoted B1 , B2 and B3 are screw positions indicating where
the boring bar has been clamped from top and bottom. The dimension are in mm,
where l1 = 10.7, l2 = 18, l3 = 101, l4 = 250, l5 = 35, l6 = 100, l7 = 18.5, l8 = 25.
2.2
Experimental Modal Analysis
The primary goal of experimental modal analysis is to identify the dynamic properties
or the modal parameters of the system under examination. In other words, the goal
is to determine the natural frequencies, mode shapes, and damping ratios from experimental vibration measurements. The procedure of modal analysis may be divided
into two parts: the acquisition of data and the parameter estimation or parameter
identification of these data, a process also known as curve fitting [12]. Acquiring good
data and performing accurate parameter identification is an iterative process, based
on various assumptions along the way [12].
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78
2.2.1
Partial Fraction Model
If the frequency response function estimates have well separated modes, the Partial
Fraction Model technique [13] can be used to estimate the modal parameters. The
partial fraction model HL (f ) is described as
HL (f )
=
N
X
r=1
HLr (f )
=
A∗r
Ar
+
j2πf − λr
j2πf − λ∗r
A∗r
Ar
+
j2πf − λr
j2πf − λ∗r
(1)
(2)
Where Ar and λr are the residue and the system pole, respectively, belonging to the
mode r. The last fractional part is close to zero near the natural frequency fr . Thus,
the model can be simplified into;
H(f )
≈
Ĥ(f )
=
A
j2πf − λ
A
j2πf − λ
(3)
(4)
Ĥ(f )(j2πf − λ) = A
j2πf Ĥ(f ) = λĤ(f ) + A.
(5)
(6)
This results in an over determined linear equation system like equation 7, easily
solved by the least-square method using, for example, the the Moore-Penrose pseudoinverse [14]:

where




Ĥ(f0 )
Ĥ(f1 )
..
.
1
1
..
.
Ĥ(fK ) 1




 λ

=

 A

λ = −ζ2πf0 + j2πf0
2.3
Spectral Properties
j2πf0 Ĥ(f0 )
j2πf1 Ĥ(f1 )
..
.
j2πfK Ĥ(fK )
p





1 − ζ 2.
(7)
(8)
Non-parametric spectrum estimation may be utilized to produce non-parametric linear least-squares estimates of dynamic systems [15].
A non-parametric estimate of the power spectral density Pxx (f ), where f is frequency, for a signal x(t) may be estimated using the Welch spectrum estimator [16],
given by:
2
L−1 N −1
k
1 X X
−j2πnk/N w(n)xl (n)e
P̂xx (fk ) =
, fk = Fs
=
LN Fs
N
n 0
(9)
l=0
where k = 0, . . . , N − 1, L is the number of periodograms, N is the length of the data
segments used to produce the periodograms, xl (n) is the sampled signal in segment l
and Fs is the sampling frequency.
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Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 79
Thus, for each input signal x(t) and output signal y(t), a single-input-single-output
system (SISO) system is simultaneously measured and the sampled signals y(n) and
x(n) are recorded. By using, for example, the Welch spectrum estimator [16], the
cross-power spectral density P̂yx (fk ) between the input signal x(n) and the output
signal y(n) and the power spectral density P̂xx (fk ) for the input signal x(n) may be
estimated [6, 15].
A least-squares estimate of a frequency response function between the input signal
x(n) and the output signal y(n) may be determined according to [15]:
Ĥ(fk ) =
P̂yx (fk )
P̂xx (fk )
(10)
and the coherence function as [15]
P̂ (f )P̂xy (fk )
2 (f ) = yx k
γˆyx
.
k
P̂xx (fk )P̂yy (fk )
(11)
The least-square estimate for the SISO system in Eq. 10, can be rewritten for the
multiple-input-multiple-output
(MIMO) system yielding the estimate of the system
h
i
matrix Ĥ(fk ) as [15]
h
i
Ĥ(fk )
=
h
ih
i−1
P̂yx (fk ) P̂xx (fk )
(12)
where [P̂ xx(f k)] is an estimate of the cross spectrum matrix between all the inputs
and [P̂ yx(f k)] is an estimate of the cross spectrum matrix between all the inputs and
outputs.
In the case of multiple inputs, the multiple coherence is of interest as an indication
of the quality of the measurements. The multiple coherence function is defined by the
ratio of that part of the spectrum which can be expressed as a linear function of the
inputs to the total output spectrum (including extraneous noise), and the multiple
coherence function is an extension of the ordinary coherence function from the SISO
case [15].
2.3.1
Excitation Signal
For the experimental modal analysis, burst random was used as the excitation signal.
Based on initial experiments concerning suitable burst length and frequency resolution
(data segment time or data block length time), a burst length of 90% of the data block
length time was selected, see Table 2. Basically, the frequency resolution was tuned
to provide high overall coherence in the analysis bandwidth and the burst length was
tuned to provide high coherence at the resonance frequencies.
2.4
Nonlinear Modeling Methods
The softening spring may be modeled in two different ways, yielding different properties with respect to the displacement. The first of these models yields a force proportional to a nonlinear stiffness coefficient, multiplied by the displacement, squared
with sign. The equation describing a single-degree-of-freedom (SDOF) system with
this type of a softening spring nonlinearity is given by [17]
m
dx(t)
d2 x(t)
+c
+ kx(t) − ks x|x|(t) = f (t)
2
dt
dt
(13)
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80
Parameter
Excitation signal
Sampling frequency f s
Block length N
Frequency resolution ∆f
Number of averages L
Window
Overlap
Frequency range of burst
Burst length
Value
Burst random
10240 Hz
20480
0.5 Hz
200
Rectangular
0%
0-4000 Hz
90%
Table 2: Spectral density estimation parameters.
where m, c and k are the mass, damping and stiffness coefficients of the underlying
linear system, x(t) is the displacement, f(t) is the force, and ks is the nonlinear
stiffness coefficient. The second model yields a force proportional to the nonlinear
stiffness coefficient multiplied by the displacement cubed. Inserted into the equation
of motion describing a SDOF system, this model results in [17]
m
d2 x(t)
dx(t)
+c
+ kx(t) − kc x3 (t) = f (t)
dt2
dt
(14)
where kc is the nonlinear stiffness coefficient.
In order to see if any of the proposed nonlinearities may explain the different
results from the experimental modal analysis, a number of different simulations were
carried out using different parameters.
2.4.1
Nonlinear Synthesis
The are different ways of simulating linear and nonlinear systems: the most common
method to solve ordinary differential equations (ODE) is probably the Runge-Kutta
method implemented in Matlab [18]. Another method is the digital filter method [19].
There are multiple advantages with using ODE solvers: they are rather straightforward to use and they are well known. The disadvantage, however, is that they are
relatively time consuming if large amounts of data are involved. The filter method,
on the other hand, while significantly faster than the ODE solvers [19], are not as
well documented as the ODE solvers with regard to, for example, accuracy and the
ability to handle nonlinear systems [10]. However, for linear systems, the limitations
of the filter method are known [11] and depends on the sampling frequency and the
transformation method used to convert the continuous time parameters to discrete
time parameters [11].
2.4.2
Ordinary Differential Equation Methods
The simulation method used for simulating the nonlinear system is based on an explicit Runge-Kutta of order (4,5), based on the Dormand-Prince pair [18,20], referred
to as ode45 in Matlab. The ode45 method combines a fourth order method and
a fifth order method, both of which are similar to the classic fourth order RungeKutta [20, 21]. The numerical technique solves ordinary differential equations of the
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 81
form [20]
dx(t)
= f (x(t), t),
x(t0 ) = x0 .
(15)
dt
Since the Runge-Kutta method only solves first order differential equations, the
second order differential equations in Eqs. 13 and 14 must be rewritten to coupled
first order differential equations as in Eqs. 18 and 19.
The nonlinear models simulated with the differential equation solvers were based
on the softening spring using the quadratic model in Eq. 16, and the cubed model in
Eq. 17:
gq (x(t))
= ks x|x|(t)
(16)
gc (x(t))
= kc x (t)
(17)
3
where gq (x(t)) and gc (x(t)) replace g(x1 (t)) in Eq. 19 the for respective model. The
model may then be described in state-space formulation as
dx1 (t)
dt
dx2 (t)
m
dt
= x2 (t)
(18)
= −cx2 (t) − kx1 (t) + g(x1 (t)) + f (t)
(19)
where x(t) is the response of the system, and f (t) is the driving force.
2.4.3
Filter Method
The filter method is a time-discrete method for extracting digital filter coefficients
from the analog system using an appropriate transformation method. Thus, the
differential equation is transformed into a difference equation, represented by a digital
filter [19]. The filtering procedure in the discrete time domain is given by
x(n) =
∞
X
k=−∞
h(k)f (n − k)
(20)
where, again, x(n) is the response and f (n) is the input to the system with the impulse
response h(n), but in the discrete time domain.
The transformation may be performed by first dividing the total (multiple degree
of freedom) system into subsystems using the modal superposition theorem and transforming each subsystem’s parameters into filter coefficients. The frequency response
function for a dynamic system may be expressed in terms of modal superposition
as [12]
R
X
Ar
A∗r
H(f ) =
,
(21)
+
j2πf − λr
j2πf − λ∗r
r=1
where R is the number of modes, Ar is the system’s residues belonging to mode r, and
λr is the pole belonging to mode r. The poles and residues may be extracted from a
lumped parameter system, or from a distributed parameter system, or estimated from
experimental modal analysis [22]. Another approach is to directly express the system
as in Eq. 22 and transform the analog filter coefficients into digital filter coefficients.
The transfer function for an analog filter can be expressed as [23]
H(s)
=
d0 + d1 s + . . . + dMa sMa
D(s)
=
C(s)
1 + c1 s + . . . + cKa sKa
(22)
White
Part III
82
where Ma is the order of the polynomial D(s) in the numerator, and Ka is the order
of the polynomial C(s) in the denominator. Transforming the analog filter yields a
digital filter whose z-transform may be expressed as [23]
H(z)
=
B(z)
b0 + b1 z −1 + . . . + bM z −M
=
A(z)
1 + a1 z −1 + . . . + aK z −K
(23)
where M is the order of the polynomial B(z) in the numerator, and K is the order of
the polynomial A(z) in the denominator. In the discrete time domain, the difference
equation describing the filter may be written as [23]
x(n)
=
M
X
m=0
bm f (n − m) −
K
X
k=1
(24)
ak x(n − k).
One of the most common transformation methods is the so-called ”impulse invariant” method, which allows the digital signal to represent the analog signal by an
impulse at sampled intervals, i.e. x(t) → T x(nT ), where T is the sampling period [24]. Other methods include the step invariant, the ramp invariant, the centered
step invariant, the cubic spline invariant and the Lagrange method, each with different properties [11]. The ramp invariant method was used in the simulations and is
defined as:
f (nT + t) =
f (nT + T ) − f (nT )
t + f (nT ),
T
where
0 6 t 6 T.
(25)
This transform method produces zero error at DC, low error at the Nyquist frequency
and low phase distortion [11]. At the same time, the ramp invariant method does produce a large error at a the resonance frequency in comparison to, for example, the
impulse, step and cantered step invariant methods [11]. However, the error introduced in the area of the resonance frequency only becomes large when the resonance
frequency approaches the Nyquist frequency. Therefore, an over-sampling of 20 times
the highest resonance frequency was used in the simulations.
The filter method for simulation of nonlinear systems is carried out by using the
digital filter coefficient for the linear system, and finding the solutions for the nonlinear
difference equation. In the discrete time domain, the nonlinear difference equation
may be written as [19]
x(n)
=
M
X
m=0
bm (f (n − m) − g(x(n − m))) −
K
X
k=1
ak x(n − k).
(26)
Since Eq. 26 contains nonlinear terms, several solutions may exist for x(n) [21]. The
value of x(n) may be found using any of the zero searching algorithms such as the
secant method, bisection method or Newton-Raphson [21] (which was used in this
synthesis).
The models with a nonlinear function g(x(n)) simulated with the filter method
are defined as:
gLin (n) − gs (x(n)) = f (n),
where
gLin (n) − gc (x(n)) = f (n),
where
gs (x(n)) = ks x|x|(n)
(27)
(28)
and
gc (x(n)) = kc x3 (n).
(29)
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 83
The digital filter coefficients were based on the poles and residues estimated from
data acquired during the experiments.
2.4.4
Error Analysis
To be able to draw any relevant conclusion from the simulated data, it is necessary to
know approximately how large the error is. The two previously mentioned methods,
i.e. the ODE and the filter method, introduce different errors. The ODE solver has
both a local error and a global error [18]. The local error over a time step h is of the
order O(hn+1 ). This results in an error less than h5 , where h is the time step and n
the order of the ODE solver. The local error is then propagated with each time step
until it results in the global error. Beside these errors there is of course the numerical
precision which is limited by a 64-bit double-precision representation. Both the local
and the global error can be set in matlab, or, if the step size h is chosen small enough
from the beginning, both the local and the global error will be kept under the default
thresholds [20]. When using the filter method, one more error which is related to the
Laplace to Z-transform conversion must be taken into consideration. To reduce the
effects caused by this error, a suitable transformation method related to the problem
formulation is selected along with a small step size. For the nonlinear simulations
done in this paper, a number of different step sizes were tested including the final
step size of h = 1 · 10−4 . Furthermore, the threshold limit for the Newton-Raphson
zero finder was set to ε = 1 · 10−14 .
3
Computer Simulations of Nonlinear Systems
The simulation used a linear component of the models which was based on parameters derived from the experimental modal analysis of the standard boring bar. The
standard boring bar was clamped with six screws that were first tightened from the
bottom. The tightening torque was 10Nm and the excitation level was set at the lowest possible setting. This setup resulted in well separated modes. Thus, the Partial
Fraction Model technique [13] can be used to estimate the modal parameters.
The estimated values are: the resonance frequency f0 = 504.177 Hz, the damping
ζ = 1.714 % and the residue A = −j1.357 · 10−4 . Thus, the linear system may be
expressed as
HL (f ) =
j1.357 · 10−4
−j1.357 · 10−4
+
. (30)
j2πf − (−54.292 + j3.167 · 103 ) j2πf − (−54.292 − j3.167 · 103 )
In Fig. 5, the driving point accelerance functions of the boring bar in the direction of the cutting speed are presented together with the corresponding synthesized
accelerance function.
These parameters are directly applicable to the filter-method when calculating the
filter coefficients. However, when using the ordinary differential equation solvers, the
partial fractions are collected into one polynomial fraction that may be expressed in
terms of the mass, damping and stiffness coefficients m, c and k. Theses parameters
White
Part III
Accelerance [dB rel 1(m/s2 )/N]
84
25
20
15
10
5
Estimated, Level 1
Synthesized, Level 1
0
400
420
440
460
480
500
520
Frequency [Hz]
540
560
580
600
Figure 5: The accelerance function for the driving point of the boring bar in the
cutting speed direction (solid line), and the synthesized SDOF system (dashed line).
were determined using the following relations
A =
f0
=
ζ
=
1
jm4πf0
r
k
1
2π m
c
√
2 mk
(31)
(32)
(33)
which yield a mass of m = 1.632 kg, a damping of c = 126.307 Ns/m and a stiffness
of k = 1.167 · 107 N/m.
3.1
Excitation Signal
True random was selected for the excitation signal. The estimation parameters used
in the nonlinear simulations are presented in Table 3.
Parameter
Excitation signal
Sampling frequency f s
Block length N
Frequency resolution ∆f
Number of averages L
Window
Overlap
Value
True random
10000 Hz
20480
0.4883 Hz
200
Hanning
50%
Table 3: Spectral density estimation parameters and the excitation signal used in the
simulated nonlinear system.
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 85
3.2
Softening Spring Model
The nonlinear softening stiffness coefficients ks and kc in the signed squared and cubic
models were not obtained by direct parameter estimation. The resonance frequency
shifting phenomenon always appears between the accelerance function estimates for
the standard boring bar clamped with screws for different excitation force levels.
Typically, a resonance frequency shift of 5 Hz and, for example, an initial resonance
frequency of 500 Hz render a frequency deviation of 1%, which corresponds to a 10%
deviation in the linear stiffness coefficient. By considering the stiffness deviation,
the stiffness coefficient used in the linear model, the level of excitation force and the
convergence rate in the simulation, the values for the nonlinear stiffness coefficients
were selected as: ks = 5 · 1011 N/m2 and kc = 4 · 1018 N/m3 for the signed squared
and cubic models, respectively. The levels of the excitation force were given the same
ratios as for the experiments with the standard boring bar, and the signal type was
normally distributed random noise, with the peak levels of 100, 200, 300 and 400 mN.
4
Results
The results from the experimental modal analysis and the simulations are presented as
frequency response function estimates in form of accelerance, and the coherence to the
corresponding frequency response function estimate is also presented. Furthermore, a
table with the estimated undamped resonance frequency with corresponding damping
is presented.
4.1
Experimental Results
29
Level1
Level2
Level3
Level4
28
27
26
25
0.9995
0.999
0.9985
24
23
22
0.998
Level1
Level2
Level3
Level4
0.9975
21
20
485
1
2
Coherence γ̂yx
Accelerance [dB rel 1(m/s2 )/N]
The estimated accelerance response function from the experimental modal analysis
is presented in Fig. 6 a) and the corresponding coherence function in Fig. 6 b). The
estimated modal parameters are presented in Table 4.
490
495
500
505
510
515
Frequency [Hz]
a)
520
525
0.997
485
490
495
500
505
510
515
Frequency [Hz]
b)
520
525
Figure 6: a) Driving point frequency response function estimates of the boring bar in
the cutting speed direction for four different excitation levels, and b) the corresponding
multiple coherence.
White
Part III
86
Excitation
Frequency [Hz]
Damping [%]
Level 1
504.18
1.71
Level 2
503.50
1.70
Level 3
503.01
1.72
Level 4
502.57
1.73
Table 4: Resonance frequency and relative damping estimates for the frequency response functions based on the experimental modal analysis.
4.2
Simulation Results
29
Level 1
Level 2
Level 3
Level 4
28.5
28
27.5
27
26.5
26
25.5
25
495
500
505
510
Frequency [Hz]
a)
515
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Fig. 7 a) presents the frequency response function estimates that were produced based
on simulations of the nonlinear model with signed squared stiffness, using the filter
method and the four different excitation levels. Fig. 7 b) presents the corresponding
frequency response function estimates that were produced based on simulations of
the nonlinear model system with cubic stiffness, using the filter method and the four
different excitation levels.
29
Level 1
Level 2
Level 3
Level 4
28.5
28
27.5
27
26.5
26
25.5
25
495
500
505
510
Frequency [Hz]
b)
515
Figure 7: Frequency response function estimates based on simulations of the nonlinear
models using the filter method and four different excitation levels, a) for the presented
model with signed squared stiffness, and; b) for the model with cubic stiffness.
Table 7 gives the estimates of the resonance frequency and the relative damping
for the frequency response functions based on the nonlinear models simulated with the
filter method for the four excitation force levels. The SDOF least-square technique [12]
was used to produce estimates of the resonance frequency and the relative damping.
Excitation
Frequency [Hz]
Damping [%]
Frequency [Hz]
Damping [%]
Squared Model
Level 1 Level 2
503.73
503.42
1.71
1.71
Cubic Model
503.98
503.79
1.71
1.71
Level 3
503.11
1.71
Level 4
502.80
1.71
503.46
1.71
503.01
1.71
Table 5: Resonance frequency and relative damping estimates for the frequency response functions based on the nonlinear models, simulated with the filter method.
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 87
The coherence function estimates are also presented for a narrow frequency range,
including the resonance frequency, and are illustrated for the nonlinear model with
signed squared stiffness in Fig. 8 a) and for the nonlinear model with cubic stiffness
in Fig. 8 b).
1
1
0.9995
0.999
0.999
0.9985
0.9985
0.998
0.9975
0.998
Level1
Level2
Level3
Level4
0.9975
0.997
495
2
Coherence γ̂yx
2
Coherence γ̂yx
0.9995
500
505
510
Frequency [Hz]
a)
Level1
Level2
Level3
Level4
0.997
0.9965
0.996
495
515
500
505
510
Frequency [Hz]
b)
515
Figure 8: Coherence function estimates based on simulations of the nonlinear models,
using the filter method and four different excitation levels, a) for the model with
signed squared stiffness and; b) for the model with cubic stiffness.
29
Level 1
Level 2
Level 3
Level 4
28.5
28
27.5
27
26.5
26
25.5
25
495
500
505
510
Frequency [Hz]
a)
515
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
If the ordinary differential equation solver ode45 in Matlab is used for the four
different excitation levels of the nonlinear model with signed squared stiffness, this
results in the frequency response function estimates shown in Fig. 9 a). Fig. 9 b)
presents the corresponding frequency response function estimates, based on simulations of the nonlinear model system with cubic stiffness, using the ordinary differential
equation solver ode45 in Matlab and the four different excitation levels.
29
Level 1
Level 2
Level 3
Level 4
28.5
28
27.5
27
26.5
26
25.5
25
495
500
505
510
Frequency [Hz]
b)
515
Figure 9: Frequency response function estimates based on the simulation of the nonlinear models, using the ordinary differential equation solver ode45 in Matlab and
four different excitation levels, a) for the model with signed squared stiffness and; b)
for the model with cubic stiffness.
Table 6 presents estimates of the resonance frequency and the relative damping
for the frequency response functions for the four excitation force levels, based on the
White
Part III
88
nonlinear models and simulated with the ordinary differential equation solver ode45 in
Matlab. Also, in this case, the SDOF least-square technique [12] was used to produce
estimates of the resonance frequency and the relative damping.
Squared model
Level 1 Level 2
503.48
502.84
1.70
1.70
Cubic model
503.75
502.71
1.70
1.71
Excitation
Frequency [Hz]
Damping [%]
Frequency [Hz]
Damping [%]
Level 3
502.18
1.70
Level 4
501.54
1.71
500.93
1.75
498.55
1.84
Table 6: Estimates of the resonance frequency and the relative damping for the frequency response functions based on the nonlinear models simulated with the differential equation solver ode45.
1
1
0.999
0.99
0.998
0.98
0.997
0.996
0.995
0.994
Level1
Level2
Level3
Level4
0.993
0.992
0.991
0.99
495
500
505
510
Frequency [Hz]
a)
2
Coherence γ̂yx
2
Coherence γ̂yx
The coherence function estimates are also presented for a narrow frequency range
(including the resonance frequency) and are illustrated for the nonlinear model with
signed squared stiffness in Fig. 10 a) and for the nonlinear model with cubic stiffness
in Fig. 10 b).
515
0.97
0.96
0.95
0.94
Level1
Level2
Level3
Level4
0.93
0.92
0.91
0.9
495
500
505
510
Frequency [Hz]
b)
515
Figure 10: Coherence function estimates based on simulations of the nonlinear models
using the ordinary differential equation solver ode45 in Matlab and four different
excitation levels, a) for the model with signed squared stiffness presented and b) for
the model with cubic stiffness.
4.3
Comparison between Experiment and Simulations
The comparisons between the synthesized accelerance functions based on the parameters estimated from the measured data and the simulated synthesized accelerance
functions based on the simulated data are presented in Fig. 11. The driving point accelerance function estimates based on the experimental data are presented in Fig. 11
a) and the accelerance function estimates based on simulated data using the signed
square nonlinear model and simulated with the filter method are presented in Fig. 11
White
Estimation and Simulation of the Nonlinear Dynamic Properties of a Boring bar 89
29
28.5
28
Level1
Level2
Level3
Level4
27.5
27
26.5
26
25.5
25
495
500
505
Frequency [Hz]
a)
510
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
b). Resonance frequency estimates from the experimental results and all the simulated
models are collected and presented in Table 7.
29
28.5
28
Level 1
Level 2
Level 3
Level 4
27.5
27
26.5
26
25.5
25
495
500
505
Frequency [Hz]
b)
510
Figure 11: Frequency response function estimates a) based on experimental data,
using a standard boring bar clamped with six M10 screws and four different excitation levels, and b) based on simulated data from the signed square nonlinear model,
simulated with the filter method.
Frequency [Hz]
Excitation
Level 1 Level 2
Experimental data
504.18
503.50
Filter method, x|x| model 503.73
503.42
Filter method, x3 model
503.98
503.79
ODE method, x|x| model
503.48
502.84
ODE method, x3 model
503.75
502.71
Level 3
503.01
503.11
503.46
502.18
500.93
Level 4
502.57
502.80
503.01
501.54
498.55
Table 7: Resonance frequency estimates for the frequency response functions based
on the experimental data and the simulated data.
5
Summary and Conclusions
The experimental results obtained during the examination of the dynamic properties
of a clamped boring bar (clamped in the clamping housing with screws), strongly indicate the presence of nonlinearity. Two different nonlinear single-degree-of-freedom
models were simulated in order to investigate if they bear resemblance to the nonlinear dynamic behavior observed for a boring bar clamped in the clamping housing
with screws. In addition, two different simulation methods were used to provide redundancy due to the fact that there are no explicit analytical solutions for the two
different nonlinear single-degree-of-freedom models which can be used as benchmarks.
Both the square with sign stiffness model and the cubic stiffness model show the similar trend in the frequency response function estimates as the experimental results do,
see Tables 7 and 6. The trend is that the resonance frequency decreases with an increasing excitation level; see Fig. 7 and Fig. 9 (produced by the filter method and the
90
White
Part III
ODE solver method, respectively). The coherence function estimates from the data
from the nonlinear systems, simulated with the filter method, display an expected
dip at the resonance frequency, see Fig. 8. By using the filter method to simulate
the comparatively nonlinear systems, the coherence function estimates yield higher
levels in the resonance frequency range of the SDOF systems than those produced by
the ODE solver. It is shown that the simulated results display a similar resonance
frequency shift as the experimental data, as can be seen if comparing Fig. 6 with
Figs. 7 and 9 and also Fig. 11 a) with Fig. 11 b). Thus, including a nonlinearity in
the model of the clamped boring bar is likely to provide a more accurate model of
the actual boring bar’s dynamic properties in comparison to a pure linear model.
Acknowledgments
The present project is sponsored by the company Acticut International AB in Sweden
which has multiple approved patents and products covering active control technology
for metal cutting.
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Part IV
Investigation of the Dynamic
Properties of a Milling Tool Holder
This part is published as:
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson, Investigation of the
Dynamic Properties of a Milling Tool Holder, Research Report No 2009:07, Blekinge
Institute of Technology, ISSN: 1103-1581, November, 2009.
Investigation of the Dynamic Properties of a
Milling Tool Holder
Henrik Åkesson1,2 , Tatiana Smirnova,
Lars Håkansson and Ingvar Claesson
1
Blekinge Institute of Technology,
Department of Signal Processing,
372 25 Ronneby, Sweden
Thomas T. Lagö
Acticut International AB,
Gjuterivägen 6, 311 32 Falkenberg, Sweden
2
Abstract
Vibration problems during metal cutting occur frequently in the manufacturing industry. The vibration level depends on many different parameters such
as the material type, the dimensions of the workpiece, the rigidity of tooling
structure, the cutting data, and the operation mode. In milling, the cutting
process subjects the tool to vibrations, and having a milling tool holder with a
long overhang will most likely result in high vibration levels. As a consequence
of these vibrations, the tool life is reduced, the surface finishing becomes poor,
and disturbing sound appears. In this report, an investigation of the dynamic
properties of a milling tool holder with moderate overhang has been carried out
by means of experimental modal analysis and vibration analysis during the operating mode. Both the angular vibrations of the rotating tool and the vibrations
of the machine tool structure were examined during milling. Also, vibration of
the workpiece and the milling machine was examined during cutting. This report focuses on identifying the source/sources of the dominant milling vibration
components and on determining which of these vibrations that are related to
the structural dynamic properties of the milling tool holder.
96
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Part IV
Contents
1 Introduction
1.1 Literature Review . . . . . . . . . . . . . .
1.1.1 Chatter Theory . . . . . . . . . . . .
1.1.2 Force Models . . . . . . . . . . . . .
1.1.3 Stability . . . . . . . . . . . . . . . .
1.1.4 Vibration Control . . . . . . . . . .
1.1.5 Motivation . . . . . . . . . . . . . .
1.2 Basic Concepts of Metal Cutting in Milling
1.3 Measurement of Forces and Vibrations . . .
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2 Materials and Methods
2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Tool Holder and Tool . . . . . . . . . . . . . . . . . . . .
2.1.2 Cutting Data . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Measurement Equipment and Setup . . . . . . . . . . . .
2.1.4 Spatial Measurements of the Acceleration During Milling
2.1.5 Modal Analysis Setup . . . . . . . . . . . . . . . . . . . .
2.1.6 Excitation Signal for the Experimental Modal Analysis .
2.2 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Operating Deflection Shape Analysis . . . . . . . . . . . . . . . .
2.4 Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . .
2.5 Modal Parameter Estimation . . . . . . . . . . . . . . . . . . . .
2.5.1 Spectral Estimation Parameters . . . . . . . . . . . . . . .
2.6 Distributed Parameter Model of the Milling Tool Holder . . . . .
2.6.1 A Geometrical Model of the Clamped Milling Tool Holder
2.6.2 A Model of Transverse Vibrations . . . . . . . . . . . . .
2.6.3 A Model of Torsional Vibrations . . . . . . . . . . . . . .
2.7 A Finite Element Model of the Milling Tool Holder . . . . . . . .
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3 Results
3.1 Spatial Measurements of Vibration . .
3.1.1 Operating Deflection Shapes .
3.1.2 Angular Vibrations . . . . . . .
3.2 Modal Analysis Results . . . . . . . .
3.2.1 Mode Shapes . . . . . . . . . .
3.3 Distributed Parameter Model Results
3.4 Finite Element Model . . . . . . . . .
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4 Summary and Conclusion
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Investigation of the Dynamic Properties of a Milling Tool Holder
1
97
Introduction
Metal cutting is generally used in the manufacturing industry to machine, e.g., workpieces to desired geometries with certain tolerances. During the machining process,
a number of different machining operations may be involved. There are several different machining operations including turning, milling, drilling, boring, threading,
etc. [1]. Today, there are many advanced machines that have several axes and that
can perform complex milling and turning operations about non-fixed axes [1] by, for
example, rotating or leaning the axis of the spindle. Another example of the type of
advanced operation that modern machines are capable of is the production of an oval
or ellipsoidal cross-section of a workpiece by controlling the tool motion in the radial
direction of the workpiece during turning.
The metal cutting operation may sometimes produce high server vibration levels.
The cause of these vibrations can be attributed to many different factors such as
the cutting parameters, the workpiece material and shape, the tooling structure, the
insert, and the stability of the machine [2]. Thus, there are many different parameters
that influence the stability of the cutting process in milling operations, and there has
been a lot of research done in this area.
1.1
Literature Review
Turning has been one of the most studied metal cutting processes due to the fact that
it is comparatively easy to monitor the forces applied to the tools under controlled
conditions [3]. Turning has also been used to imitate the periodic excitation present
in the milling operation by rotating a non-continuously shaped workpiece that produces an intermittent tooth pass excitation [4]. Many similarities between the cutting
processes in turning operations and in milling operations may be observed, and chatter theory developed for turning operations is also used in milling theory to produce
rough estimates of stability limits [5, 6].
1.1.1
Chatter Theory
Some of the earliest studies on the principles of chatter in simple machine tool systems were produced by Arnold [7] in 1946. In 1965, Tobias [8] presented an extensive
summary of results from a large number of research studies concerning the dynamic
behavior of the lathe application and the chatter theory, and he further developed
the research into the chatter phenomena in consideration of the chip thickness variation and the phase lag of the undulation of the surface. The same year, Merritt [9]
discussed the stability of structures with n-degrees of freedom, assuming the absence
of dynamics in the cutting process. He also proposed a simple stability criterion. Together with Tobias and Merritt, Koenigsberger and Tlusty are also considered to be
the pioneers of chip regeneration formulation for basic chatter theory [10]. Furthermore, Pandit et al. [11] developed a procedure for modeling chatter from time series
by including unknown factors of random disturbances present in the cutting process.
They formulated self-excited random vibrations with white noise as a forcing function. Finally, Kato et al. [12] investigated regenerative chatter vibration caused by the
deflection of the workpiece and introduced a differential equation describing chatter
vibration based on experimental data.
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98
1.1.2
Force Models
There are many force models of various complexity and properties that describe the
cutting process [13–17]. Tlusty [13] presented the relationship between maximum
depth of cut, stiffness of the structure, and a specific cutting coefficient of the cutting
process for turning, where the maximum width of cut is proportional to the static
stiffness and the damping ratio at the cutting tool point of the machine tool. Later,
in 1991, Smith and Tlusty [18] summarized the force models and simulation methods
of the milling process currently used so far. In 1996, Altintas presented a force
model focusing on the helical end mill geometry [14]. In addition to this, Tlusty et
al. presented numerical simulations of the milling dynamics, including saturations
such as the tool jumping out of cut [19]. Also, Engin and Altintas [20] presented
a generalized mathematical model of inserted cutters for the purpose of predicting
cutting forces. The model is capable of considering various insert geometries, angles,
and positions relative to the cutter body.
1.1.3
Stability
In 1981, Tlusty and Ismail [19] studied the basic non-linearity of the cutting processes
by analyzing the vibrations that occur when the tool leaves the workpiece for a part
of the cycle. This was done for both turning and milling and took into consideration
the mode coupling self-excitation. No further conclusions with respect to turning
were made, however, they found that stability boundaries related to the milling were
calculated erroneously by a factor of two to three by the methods known at the time,
thus opening up possibilities for the improvement of stability methods. Furthermore,
an improved method for obtaining stability lobes was presented in 1983 by Tlusty et
al. [21]. In the beginning of the 1990s, the milling operation received more consideration, which resulted in more accurate stability lobes/diagrams based on various
additional properties related to the milling operation [5, 22–27]. A practical example
of how to increase cutting performance by considering stability lobes was presented
by Tlusty et al. [28]. They studied the performance of a long and slender tool in
high speed milling and increased the metal removal rate by choosing the appropriate
tool length with respect to stability lobes that allowed maximum spindle speed in
high speed milling. However, more contributions can be done to this field, since the
variation of machines, tool configuration, inserts, workpiece material, etc., seem to be
virtually unlimited.
1.1.4
Vibration Control
The common methods used to control vibrations in milling systems utilize the control
mechanisms of the cutting parameters related to the machine. Other methods are
either based on passive vibration absorption or active control that applies a secondary
control force.
Several methods have been developed to find cutting parameters that can be used
to avoid instabilities [29–32]. The most commonly used method is to change some of
the parameters during cutting, i.e., changing the spindle speed or the feed rate of the
workpiece. The goal to be achieved by changing one of the cutting parameters is to
reduce the dynamic feedback into the system and thus to avoid instability. The most
common parameter to change is the spindle speed.
An example of a passive solution was put forth in 2008 by Rashid et al. [33]. They
presented the development and testing of a tuned viscoelastic dampers in a milling
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Investigation of the Dynamic Properties of a Milling Tool Holder
99
operation that were able to attenuate vibrations.
Active solutions have also been proposed, for example, by J.L. Dohner et al. [34]
who developed an active structural control system able to alter the dynamics of the
system. Furthermore, active solution based on embedded piezoelectric actuators in a
palletized workpiece holding system for milling was presented by Rashid et al. [35] in
2006. Another active solution focusing on the spindle was implemented by Denkena
et al. who used a contactless magnetic guide in a milling machine prototype to sense
and actuate harmonic disturbances [36].
1.1.5
Motivation
A major part of the research within metal cutting technology concerns methods for improving the cutting performance and increasing the tool life [3]. Cutting performance
may be defined in terms, of e.g., the material removal rate, the surface roughness,
and the forces the cutting process applies to the different machine parts [3]. Usually,
work concerning methods for improving cutting performance focuses on maximizing
the material removal rate, while keeping the surface finish below a certain roughness
limit [3].
Methods that focus on tool life basically aim at the development of technology able
to increase the time the tools continue to have the desired cutting performance during
machining [3]. Thus, it is important to maintain the required chip formation/chip
breakage and cutting forces, etc., for as long as possible [1].
From the literature review, it can be seen that much work has been done on both
turning and milling theory. Many force models and methods for producing stability
diagrams have been developed. Also, different methods for handling vibration problems have been presented in various articles, see for example section 1.1.4 concerning
the review on vibration control. However, due to the complexity of different cutting
operations, there is still much work to be carried out in order to identify and understand the causes of the problems that arise during machining. It appears that little
experimental work has been done on the identification of the dynamic properties of
milling vibration and spatial dynamic properties of the milling tool holder and the
spindle house. Dynamic modeling of the cutting dynamics is an important research
area for the manufacturing industry. Developments in this area are dependent on,
among other things, knowledge of the dynamic properties of milling vibration during
cutting and the spatial dynamic properties of the milling tool holder and the spindle
house. In order to gain further understanding of the dynamic behavior of the milling
tool holder and the spindle house in the metal cutting process, both analytical and
experimental methods may be utilized. This paper investigates the dynamic properties of the milling vibration and the spatial dynamic behavior of the milling tool
holder and the spindle house during milling and mode shapes and corresponding resonance frequencies for the first two modes of a milling tool holder clamped to milling
machines. For the purpose of the investigation, spectrum analysis, operating deflection shapes analysis (ODS), experimental modal analysis (EMA), FE-modeling, and
distributed-parameter system modeling have been utilized.
1.2
Basic Concepts of Metal Cutting in Milling
A large number of different types of milling cutters, designed for different milling
operations, are available today. Some of the most common types of cutters are end
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mill cutters, ball nosed cutter slot drills, side and face cutters, gear cutters, and
hobbing cutters, see Fig. 1.
a)
b)
c)
d)
Figure 1: a) Side and face cutter b) end cutters, c) slot mill cutter, and d) hobbing
cutter.
Since there are many different types of milling cutters, the understanding of the
cutting parameters and their influence on the machining process is important in order to be able to use them properly [3]. The cutting parameters control the basic
properties of the cutting process where chip formation is one of the crucial parts. As
the milling cutter rotates, the material to be cut is fed into the cutter at a certain
speed denoted as the feed rate, and each tooth of the cutter cuts away small chips
of workpiece material. During the machining of a workpiece, the chip formation process and chip breakage are of vital importance for maintaining an efficient cutting
process. The size and shape of the chip depend on many parameters. The most significant ones are: depth of cut, the feed rate, the cutting speed, the number of teeth,
insert/tooth geometry, and the workpiece material [1]. A simplified drawing of the
material removal process and the cutting setup in milling is presented in Fig. 2.
3 D -v ie w
a
T o p -v ie w
C h ip
In se rt
p
a
D
c
/2
e
v
v
C h ip
C
f
f
In se rt
W o rk p ie c e
z
a
v
D
e
C
c
h 1(a )
/2
v
a
f
Figure 2: A simplified sketch of the material-removal process and the cutting configuration in milling where one tooth forms a chip and removes it from a workpiece. ae
is the width of cut in the radial direction (mm), ap is the cutting depth (mm), Dc is
the diameter of the tool (mm), fz is the feed per tooth (mm/tooth), vC is the cutting
speed (m/min), vf is the feed speed (mm/min), and h1 (α) is the un-deformed chip
thickness (mm) at the angle α (rad).
The cutting speed vC (m/min), is related to the spindle speed n (r.p.m) according
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Investigation of the Dynamic Properties of a Milling Tool Holder
101
to
vc =
nπDc
1000
(1)
where Dc is the diameter of the cutter or tool (mm). The relation between the feed
speed vf (mm/min), and the feed per tooth fz (mm/tooth) is
vf = nzn fz
(2)
where zn is the number of efficient teeth used in the cutter during machining. One of
the parameters which is usually considered in the overall process for optimal efficiency
of the production line is the rate at which material is being removed. The material
removal rate MRR (mm3 /min) depends on the three main cutting parameters: the
feed rate vf , the depth of cut ap (mm), and the width of cut ae (mm). It may be
expressed as
MRR = ae ap vf .
(3)
Another important configuration of the cutting setup involves the entrance and
exit phases of the tool to and from the workpiece. The configurations used are usually
referred to as conventional milling, slot milling and climb milling. These configurations are illustrated in Fig. 3.
v
f
v
f
v
f
Figure 3: Three different cutting configurations. To the left is the climb configuration,
in the center the slot configuration and to the right is the conventional configuration
presented. The thin arrows represent the counterclockwise rotation of the milling tool
seen from the under side of the tool while the thicker strait hollow arrows represent
the feed direction of each cutter.
Conventional milling starts with a thin chip thickness at the entrance phase and
ends with a larger chip thickness at the exit phase. In order for the insert to start to
cut, a sufficient chip thickness must be built up and before the actual cutting starts
workpiece material will slide along the surface [1]. This may result in a deformation
hardening of the surface and also poor surface finish. At the exit phase, the insert will
be exposed to severe tensile stress and the workpiece material might also remain on
the edge of the insert. By contrast, climb milling starts with a large chip thickness and
exits with a thin chip thickness. The insert does not slide or rub the material, which
allows for longer tool life and better surface finish when compared to the conventional
setup. However, climb milling usually expose the machine to larger loads compared
to conventional milling [2].
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102
1.3
Measurement of Forces and Vibrations
The most common method of analyzing the properties and the performance of milling
tools are done by measuring a number of different forces during cutting operations
with the help of dynamometers. The measurement of the forces is carried out by either
using a table-mounted dynamometer or using a spindle-mounted dynamometer. The
table-mounted dynamometer is mounted on the table of the milling machine and any
component to be milled can be fixed over the dynamometer. Forces in x, y and
z directions may be measured and the coordinate systems of the measured signals
stay fixed relative to the milling table. There are also tables/fixtures that measure
the ”feed force”, the ”deflection force” and the moment applied to the table. The
spindle-mounted dynamometer, which is mounted between the spindle and milling
tool, usually measures the cutting forces in the x, y and z directions and moment
applied to the spindle, but in this case the x-y coordinate system is rotating relative to
the table. In other words, the x-y coordinates rotate with the milling tool. Examples
of these types of dynamometers are presented in Fig. 4.
a)
b)
c)
Figure 4: Three types of dynamometers; in a) a table-mounted dynamometer measuring forces in the x-y-z directions is presented, b) shows a table-mounted dynamometer
measuring torque and c) is a spindle-mounted dynamometer.
When measuring the vibrations of a milling tool holder or a milling tool, laser
vibrometers are usually used [37]. This method requires a line of sight and may
limit the conditions for the machining. For example, the use of cooling liquids may
not be possible. Also, the chips removed during the cutting process might interfere
with the measurement. Other types of vibration sensors that may be utilized for the
measurement of milling tool vibration are the strain gauge and the piezo film. Such
sensors usually require amplifiers mounted on the tool holder and wireless communication such as telemetric equipment to transfer the sensor signals to data acquisition
systems. For non-rotating parts, accelerometers are commonly used for vibration
measurements.
2
2.1
Materials and Methods
Experimental Setup
The first milling machine used in the experiments was a Hurco BMC-50 vertical CNC
machining centre. The spindle was of the ATC type, which means that the spindle
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Investigation of the Dynamic Properties of a Milling Tool Holder
103
speed can be varied between 10-3000 rpm in steps of 20 rpm, and the maximum torque
was 428 Nm, see Fig. 1. The second machine used in the experiments was a DMU
80FD Duoblock which is a 5-axis milling machine, see Fig. 6. In addition to boring
and milling operations, this machine can also carry out turning operations in a single
machine setup. This is possible because it has a rotary table which can rotate with
up to 800 rpm. It has a maximal a torque of 2050 Nm and a holding torque of 3000
Nm. The spindle has a maximal rotation speed of 8000 rpm and a maximal torque of
727 Nm.
Figure 5: The Hurco BMC-50 milling machine.
2.1.1
Tool Holder and Tool
The milling tool holder is the interface between the spindle and the tool which holds
all the inserts. The milling tool holder used in the experiments was of the type
E3471 5525 22160 which has an overhang of 140mm and a diameter of 48mm, see
Fig. 7. Mounted on the tool holder was the tool R220.69-0050-12-7A presented to
the left in Fig. 7. This tool has a cutting diameter of Dc = 50mm and a seventeeth,zn = 7 (inserts) configuration. The insert used in the tool configuration was
XOMX120408TR-M12 T250M.
The material type of the milling tool holder is SS-2511 (EN-16NiCrS4) and the
material composition and properties are presented in Table 1.
2.1.2
Cutting Data
Three cutting parameters were considered in the experiments: cutting depth, spindle
speed and table feed rate. While two of the cutting parameters were kept constant, the
third was changed in five small steps. This was done for each parameter. In Table 2,
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Figure 6: The DMU 80FD Duoblock milling machine.
1 4 0 m m
In s e rts
4 8 m m
4 0 m m
T o o l
T o o lh o ld e r
5 0 m m
Figure 7: The tool holder E3471 5525 22160 is illustrated with the tool R220.690050-12-7A mounted. The tool is configured with seven inserts of the type
XOMX120408TR-M12 T250M.
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Investigation of the Dynamic Properties of a Milling Tool Holder
105
Material composition besides Fe, [%]
C
Si
Mn
P
S
Cr
Ni
0.13-0.18 0.15-0.40 0.7-1.1 0.035 0.050 0.60-1.00 0.80-1.20
Material properties
Young’s Modulus Poisson’s ratio Mass density Tensile strength
210 GPa
0.3
7850 kg/m3
207 MPa
Table 1: Composition and properties of the material EN-16NiCrS4.
the cutting data used in the experiments are given. In the table, it is also observable
that the width of the cut ap (how much of the workpiece is removed in the y-direction
per tool pass) varied slightly, see Fig. 8. These variations were, however, inevitable
due to the settings of the cutting data used in the experiments. The influence of these
small changes is likely to be insignificant in respect to the degree of forces expected
from the overall setup.
Setup
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Spindle speed n
[rev/min]
1401
1401
1401
1401
1401
1401
1465
1528
1592
1656
1401
1401
1401
1401
1401
Table feed vf
[mm/min]
1401
1401
1401
1401
1401
1401
1401
1401
1401
1401
1401
1501
1601
1701
1801
Cutting depth ap
[mm]
1
2
3
4
5
2
2
2
2
2
2
2
2
2
2
Width of cut ae
[mm]
26.0
25.6
25.2
24.4
24.0
26.0
25.8
25.4
25.2
25.0
24.6
24.2
23.8
23.4
23.0
Table 2: The cutting data used during the milling measurements.
2.1.3
•
•
•
•
•
•
•
•
Measurement Equipment and Setup
12 PCB Piezotronics, Inc. 333A32 accelerometers.
2 Brüel & Kjǽr 8001 impedance head.
1 Brüel & Kjǽr NEXUS 2 channel conditioning amplifier 2692.
OSC audio power amplifier, USA 850.
Ling dynamic systems shaker v201.
Gearing & Watson electronics shaker v4.
Hewlett Packard VXI mainframe E8408A.
Hewlett Packard E1432A 4-16 channel 51.2 kSa/s digitizer.
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n
a
a
e
v
v
C
p
f
ie c e
p
k
r
W o
Figure 8: The side milling configuration of the cutting setup during the milling measurements.
•
•
•
•
2.1.4
PC with I-DEAS 10 NX Series.
Custom designed slit disk for measuring angular frequency.
Rotec 5.3.
Autodesk Inventor.
Spatial Measurements of the Acceleration During Milling
To examine the spatial dynamic behavior of milling machine components during the
milling process, the acceleration at a number of different spatial locations on the
structure was measured simultaneously. The accelerometers had to be positioned on
non-rotating parts. Thus six accelerometers were positioned on the spindle frame and
three accelerometers on the workpiece, see Fig. 8 (for the setup on the Hurco milling
machine). The sensor setup on the DMU 80FD Duoblock milling machine was almost
the same. The only difference was that instead of using nine accelerometers for the
four positions defined in Fig. 9, twelve accelerometers were used. Thus, all directions
were measured in the four nodes in the DMU 80FD Duoblock milling machine setup.
Furthermore, the angular velocity along the tool holder was also measured at three
positions, see Fig. 9. Two disks with 500 uniformly distributed gaps on the tool holder
and a reflector tape on the spindle, together with optical scanning, was used to handle
the measurements of angular velocities. All data from the milling measurements were
collected with a sampling frequency of fs = 51200 Hz, using a VXI Mainframe, Matlab
and VibraTools SuiteTM.
The workpiece material was carbon steel SS1312 (EN 10 025) and the different
workpieces used in the experiments had approximately the dimensions of 70x60x530
mm (y, z, x), see Fig. 9. The workpiece was clamped to the milling table which moved
in the x-direction, resulting in a continuous cutting process along the workpiece, see
Fig. 8.
2.1.5
Modal Analysis Setup
The next step was to examine the dynamic properties of the milling tool holder
mounted in the spindle. This was done using two shakers that excited the tool holder
close to the tool in two orthogonal directions, see Fig. 10 a). Each shaker excited
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Investigation of the Dynamic Properties of a Milling Tool Holder
+ Z
- X
2
(t)
(t)
2
+ Z 1(t)
W o rk p ie c e
q
Z 1
T o o l
- Y 3(t)
(t)
3
T o o l h o ld e r
- Y 2 (t)
S p in d le fra m e
- X
107
(t)
M illin g ta b le
q
q
Z 2
Z 3
(t)
(t)
+ X
4
(t)
+ Y 4 (t)
- Z 4 (t)
A c c e le ro m e te rs o n w o rk p ie c e
Figure 9: The sensor configuration during the milling measurements conducted on the
Hurco BMC-50 milling machine. Six accelerometers where positioned on the spindle
frame and three on the workpiece. Three sensors for the velocity measurements of
the spindle and the tool holder were positioned according to the illustration.
the tool holder via a stinger rod connected to an impedance head, thus measuring
the driving point in the respective direction, see Fig. 10 b). At the same time,
the acceleration at 11 other locations along the tool holder and spindle frame was
measured, see Fig. 10 where the modal analysis setup on the Hurco BMC-50 milling
machine is presented.
The modal analysis setup in the DMU 80FD Duoblock milling machine was almost identical to the setup in the Hurco BMC-50 milling machine. The differences
between the sensor setups concerns the number of sensors and positions are presented
in Fig. 11.
2.1.6
Excitation Signal for the Experimental Modal Analysis
All the measurements that were performed as a basis for the experimental modal
analysis were done using the excitation signal burst random, 80% noise and 20%
silent.
2.2
Spectral Properties
Non-parametric spectrum estimation may be utilized to produce non-parametric linear least-squares estimates of dynamic systems [38]. By using the Welch spectrum
estimator [39], the cross-power spectral density P̂yx (fk ) between the input signal
x(n) and the output signal y(n), and the power spectral density P̂xx (fk ) for the input
signal x(n), may be produced [38, 40], where fk = Nk F s is the discrete frequency,
k = 0, . . . , N − 1, where N is the length of the data segments used to produce the
periodograms and fs is the sampling frequency.
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108
+ Z
+ X
+ Y
Im p e d a n c e h e a d
A c c e le ro m e te r
a)
b)
Figure 10: The experimental modal analysis setup in the Hurco BMC-50 milling
machine.
1
2
3
4
5
6
1
2
3
4
T o o l
T o o l h o ld e r
S p in d le h o u s e
- Z
a)
5
T o o l
T o o l h o ld e r
S p in d le h o u s e
- Z
b)
Figure 11: Cross-section view in the y-z plane of the spindle house and the tool
holder illustrating the positions of the sensors measuring acceleration and force in the
y direction for the EMA setup in the two milling machines, the sensor configurations
are identical for the x-z plane. The black squares represents the accelerometers and
the black rectangular represents the impedance heads. In a) the EMA setup in the
Hurco milling machine is presented where the positions of the sensor one to six are
placed along the z-axis according to {260, 180, 140, 100, 60, 20} mm from the tool tip
and in b) the EMA setup for the DMU 80 milling machine is shown and the positions
of the sensor one to five are placed along the z-axis according to {560, 370, 100, 60,
20} mm from the tool tip.
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Investigation of the Dynamic Properties of a Milling Tool Holder
109
In the case of a multiple-input-multiple-output (MIMO) system with P number
of responses and Q number of references, an estimate of the cross-spectrum matrix
[P̂xx (fk )] between all the inputs is produced, where the diagonal elements are power
spectral densities (PSDs) for the respective input signal and the of-diagonal consists
of cross-spectral densities. Also, a cross-spectrum matrix [P̂yx (fk )] between all the
inputs and outputs may be estimated in the same way.
The least-square estimate for a MIMO system may be written as [38],
h
i h
ih
i−1
Ĥ(fk ) = P̂yx (fk ) P̂xx (fk )
.
(4)
In the case of multiple inputs, the multiple coherence is of interest as a measure of
the quality of the MIMO system’s estimates [38].
2.3
Operating Deflection Shape Analysis
The spatial motion of a machine or a structure during real operating conditions may
be investigated using operating deflection shapes analysis (ODS). By simultaneous
measurements of N responses at discrete points on a structure the forced spatial
motion of the machine or structure, either at a moment in time, or at a specific
frequency may be estimated [40]. Thus, by considering the phase and amplitude of
the response signals from e.g. N accelerometers distributed on an operating structure,
it is possible to produce estimates of operating deflection shapes for the operating
structure. The amplitude is measured by either power spectrum or power spectral
density estimates depending whether the signal is tonal or random [38, 41]. And
the phase between each spatial position is estimated from cross-power spectra or
cross-power spectral densities [38, 41]. An estimate of a frequency domain operating
deflection shape may be constructed as follows [40]:
{ODS(f )}RM S =
nq
P̂11 (f )
q
P̂22 (f )ej θ̂21 (f )
···
q
j θ̂N 1 (f )
P̂N N (f )e
oT
.
(5)
Where P̂nn (f ) are e.g. estimated power spectra and ej θ̂n1 (f ) are phase functions
of cross-power spectra P̂n1 (f ), n ∈ {2, · · · , N }.
2.4
Experimental Modal Analysis
The primary goal of experimental modal analysis is to identify the dynamic properties
of the system under examination or the modal parameters. In other words, the
purpose is to determine the natural frequencies, mode shapes and damping ratios
from experimental vibration measurements. The procedure of modal analysis may be
divided into two parts: the acquisition of data followed by the parameter estimation
or parameter identification that can be determined with these data, a process also
known as curve fitting [42]. Acquiring good data and performing accurate parameter
identification is an iterative process, based on various assumptions along the way [42].
2.5
Modal Parameter Estimation
There are several different methods for the identification of the modal parameters [42,
43]. There are two basic curve fitting methods. One consists of curve fitting in the
frequency domain using measured frequency response function (FRF) data and a
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110
parametric model of the FRF. The other method employs curve fitting toward the
measured impulse response function (IRF) data using a parametrical model of the
IRF [42]. Many methods use both domains, depending on
h which
i parameter that
estimated [42]. A parametric model of the FRF matrix, Ĥ(f ) , expressed as the
receptance between the reference points, or the input signals, and the responses or
the output signals, may be written as [42],
h
N
i X
Qr {ψ}r {ψ}Tr
Q∗ {ψ}∗r {ψ}H
r
+ r
Ĥ(f ) =
∗
j2πf
−
λ
j2πf
−
λ
r
r
r=1
(6)
where r is the mode number, N is the number of modes used in the model, Qr is the
scaling factor of mode r, {ψ}r is the mode shape vector of mode r, and λr is the pole
belonging to mode r.
Because two sources (references) were used during data acquisition, a method
capable of handling multi-references is required. One such method is the polyreference
least square complex exponential method developed by Vold [44, 45]. This method
is defined for identification of MIMO-systems with the purpose of obtaining a global
least-square estimate of the modal parameters. While this method was used in this
work, the mode shapes were estimated using the frequency domain polyreference
method [46]. The modal scaling method used was unity modal mass [43].
To assess the quality of the estimated parameters, the FRF’s were synthesized using the estimated parameters and overlayed with the estimated FRF’s. Furthermore,
the Modal Assurance Criterion (MAC) [42] defined by
2
{ψ}H
k {ψ}l
(7)
M ACkl =
H
{ψ}H
k {ψ}k {ψ}l {ψ}l
was used as a measure of correlation between the mode shape {ψ}k belonging to mode
k, and the mode shape {ψ}l belonging to mode l, where H is the Hermitian transpose
operator.
2.5.1
Spectral Estimation Parameters
The estimation parameters used for the spectral density estimates, frequency response
functions and operating deflection shapes are presented in Table 3, Table 4 and Table 5
respectively.
Parameter
Excitation signal
Sampling frequency f s
Block length N
Frequency resolution ∆f
Number of averages L
Burst length
Window
Overlap
Value
Cutting process
51200 Hz
40960
1.25 Hz
20
Hanning
50%
Table 3: Spectral density estimation parameters used in the production of the milling
tool holder spectra during continuous machining.
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Investigation of the Dynamic Properties of a Milling Tool Holder
Parameter
Excitation signal
Sampling frequency f s
Block length N
Frequency resolution ∆f
Number of averages L
Burst length
Window
Overlap
111
Value
Burst random
51200 Hz
40960
1.25 Hz
200
80%
Rectangular
0%
Table 4: Spectral density estimation parameters used in the production of the frequency response functions for the modal analysis.
Parameter
Excitation signal
Sampling frequency f s
Block length N
Frequency resolution ∆f
Number of averages L
Burst length
Window
Overlap
Value
Cutting process
51200 Hz
40960
1.25 Hz
6
Hanning
50%
Table 5: Spectral density estimation parameters used in the production of the operating deflection shapes.
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2.6
Distributed Parameter Model of the Milling Tool Holder
The milling tool holder may be considered to be a beam with the cross section A(z)
and the length l. The Euler-Bernoulli beam theory may be utilized to approximately
model a milling tool holder’s lower order bending modes [40,47]. The Euler-Bernoulli
beam theory is generally considered for slender beams that have a diameter to length
ratio exceeding 10 as this ratio allows the effects of shear deformation and rotary
inertia to be ignored [48]. As a result, this theory tends to slightly overestimate the
eigenfrequencies. This problem increases when dealing with the eigenfrequencies of
higher modes [48].
2.6.1
A Geometrical Model of the Clamped Milling Tool Holder
The milling tool holder has a complex structure and a cone interface is used for the
particular clamping mechanism that attaches the tool holder to the spindle. Furthermore, the tool holder consists of a cylindrical shaft with a lip towards the spindle and
in the center of the tool holder is cooling channel. In the model, spindle and the tool
holder are assumed to be clamped rigidly. The geometry of the tool holder has also
been simplified into a pipe in the analytical model. The cross-section of the milling
tool holder and the corresponding analytical model are illustrated in Fig. 12 together
with the assumed clamping.
R ig id c la m p in g o f
c o n e su rfa c e
l
2 R
2 R
o
l
2 R
i
o
2 R
i
Figure 12: a) The cross-section of the milling tool holder and in b) the simplified
analytical model, where l=140.90 mm is the length of the overhang, Ro = 24.00 mm is
the radius of the tool holder and Ri = 10.25 mm is the radius of the coolant channel.
The cross-sectional properties of the simplified model are presented in Table 6.
Variable
A
I
J
Value
1.4795 · 10−3
2.5191 · 10−7
4.0838 · 10−7
Unit
m2
m4
m4
Table 6: The cross-sectional properties of the milling tool, where A is the area, I is
the moment of inertia and J is the polar moment of inertia.
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Investigation of the Dynamic Properties of a Milling Tool Holder
2.6.2
113
A Model of Transverse Vibrations
The Euler-Bernoulli differential equation describing the transversal motion of the
milling tool holder in the y-direction may be written as [48]
∂ 2 u(z, t)
∂ 2 u(z, t)
∂2
ρA(z)dz
EI
(z)
= f (z, t)
(8)
+
x
∂t2
∂z 2
∂z 2
where A(z) is the milling tool holder’s cross-sectional area, E is Young’s elastic modulus for the tool holder, I(z) is the cross-sectional area moment of inertia about the
”x axis”, ρ is the density, t is the time, u(z, t)is the deflection in the y-direction and
f (z, t) is the external force per unit length. It is assumed that both the cross-sectional
area A(z) and the flexural stiffness EI(z) are constant along the milling tool holder.
Eq.8 is often referred to as the Euler-Bernoulli beam equation. The model assumes
that the following assumptions regarding the beam and its plane are true:
• The beam is uniform along its span-, or length-, and slender (diameter to length
ratio¿10).
• The beam is composed of a linear, homogenous, isotropic elastic material without axial loads.
• The plane section remains plane.
• The plane of symmetry of the beam is also the plane of vibration so that rotation
and translation are decoupled.
• Rotary inertia and shear deformation can be neglected.
To model the milling tool holder, a Fixed-Free Euler-Bernoulli beam model was
applied. The beam has four boundary conditions, two at each end. One end is
clamped and the other is free, see Fig. 13.
z
r ,E ,G ,A ,I ,J
l
Figure 13: Model of a Fixed - Free beam, where ρ is the density, E is the elasticity
modulus (Young’s coefficient), G is the shear modulus, A is the cross-sectional area,
I is the moment of inertia, J is the polar moment of inertia and the length of the
beam l = 140.9 mm.
The clamped side of the beam will be fixated. Thus the displacement and the slope
of the displacement in this point z = 0 will equal zero and the two first boundary
conditions become
∂u(z, t) u(z, t)|z=0 = 0,
=0
(9)
∂z z=0
The other end is free, so that no bending moment or shear force constrains the
beam at the coordinate z = l when the beam vibrates. This yields two other boundary
conditions that can be described as
∂ 2 u(z, t) ∂ 3 u(z, t) EI
= 0,
EI
= 0.
(10)
∂z 2 z=l
∂z 3 z=l
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2.6.3
A Model of Torsional Vibrations
In the same way as for the transverse vibration model, a vibration model for the
torsional vibrations may be derived by considering the equation of motion for an infinitesimal element of the beam [48]. The differential equation describing the torsional
motion for the milling tool holder around and along the z axis may be written as [48]
2
G ∂ θ(z, t)
∂ 2 θ(z, t)
−
= τ (z, t)
(11)
2
∂t
ρ
∂z 2
where θ(z, t) is the angular deflection, G is the shear modulus, ρ is the density and
τ (z, t) is the externally applied torque load per unit length. The clamped boundary
condition is applied to where the milling tool holder is connected to the spindle and
yields zero deflection. At the other end there is no torque in the case of the free
vibration model. Thus, the boundary conditions for a milling tool holder with a
coolant channel modeled as a hollowed shaft becomes
θz (z, t)|z=0 = 0
π
Ro4 − Ri4 Gθz (z, t)|z=l = 0
2
(12)
(13)
where Ro is the outer radius of the milling tool holder and Ri is the radius of the
coolant channel. The relation between the shear modulus G and the elasticity modulus
E is given by [49]
G=
E
2(1 + ν)
(14)
where ν is Poisson’s ratio.
2.7
A Finite Element Model of the Milling Tool Holder
The milling tool holder was modeled in a CAD program and a finite element analysis
was conducted to estimate the natural frequencies and mode shapes of the tool holder.
The finite element mesh of the milling tool holder, consisted of 73470 nodes and 42728
elements, is presented in Fig. 14 where the white lines represent the borders of the
elements connected at the nodes.
In the FE analysis the clamping surface on the back of the holder, i.e. behind the
lip to the left in Fig. 14, was constrained to be fixed for all degrees of freedoms.
3
Results
This experimental investigation resulted in a large amount of vibration data that
was collected from both experimental setups from both milling machines. However,
the results presented in this report only constitute a small part of the investigation,
but they represent the essence of the results. The results from the experimental examination are presented in terms of measured acceleration signals as a function of
time and power spectral densities of the acceleration signals. Operating deflection
shapes were estimated for one of the milling machines and are presented. Also, results from experimental modal analysis of the tool holder mounted in the milling
machine are given. Finally, resonance frequencies and mode shapes calculated based
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Investigation of the Dynamic Properties of a Milling Tool Holder
115
Figure 14: The finite element mesh of the milling tool holder consisting of 73470
nodes and 42728 elements.
on distributed-parameter system models of the milling tool holder were generated as
well as the corresponding results from the finite element analysis of the milling tool
holder.
3.1
Spatial Measurements of Vibration
In order to get an overview of the measured acceleration signals during machining,
the acceleration of the workpiece moving in the feed direction (+x4 ) on the DMU
80FD Duoblock milling machine during machining is presented in the time domain in
Fig. 15.
60
80
Acceleration [m/s2 ]
Acceleration [m/s2 ]
100
60
40
20
0
−20
−40
−60
−80
−100
0
40
20
0
−20
−40
−60
2
4
6
8
10
Time t [sec]
a)
12
14
16
1.8
1.82
1.84
1.86
Time t [sec]
b)
1.88
1.9
Figure 15: a) Accelereation of the workpiece in the feed direction (+x4 ) during a
milling operation performed in the DMU 80FD Duoblock milling machine. The radial depth ae was 23 mm, the axial depth ap was 2 mm, the feed speed vf was
1401 mm/min and the spindle speed n was 1401 rev/min b) and the corresponding
acceleration record zoomed in.
Results in terms power spectral density estimates of workpiece vibration in the
feed direction (+x4 ) are presented in Fig. 16 a) and b) for five different axial depths
(ap = 1, 2, 3, 4, 5 mm) in the DMU 80FD Duoblock milling machine. The radial
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40
Depth=1mm
Depth=5mm
30
20
10
0
−10
−20
−30
−40
−50
−60
400
600
800
1000
1200
1400
Frequency [Hz]
a)
1600
1800
PSD [dB rel 1 ((m/s2 )2 /Hz)]
PSD [dB rel 1 ((m/s2 )2 /Hz)]
depth was ae = 26.0, 25.6, 25.2, 24.2, 24.0 mm, the feed speed vf was mm/min and the
spindle speed n was 1401 r.p.m. The periodic components found in the power spectral
density estimates in Fig. 16 a) are related to the spindle speed n. Furthermore, a
broadband response to an underlying structure may be observed in Fig. 16 a). Power
spectral density estimates workpiece vibration, zoomed in frequency to the interval
of the dominating resonance peak at approx. 770 Hz for, for the five different axial
depths are presented in Fig. 16 b).
40
30
20
10
0
−10
−20
Depth=1mm
Depth=2mm
Depth=3mm
Depth=4mm
Depth=5mm
−30
−40
−50
−60
700
750
800
850
Frequency [Hz]
b)
900
Figure 16: Power spectral densities of workpiece vibration in the feed direction (+x4 )
during milling for different axial depths ap in the DMU 80FD Duoblock milling
machine. In a) (for the gray solid line ap = 1 mm and for the black solid line
ap = 5 mm) and the radial depth ae was 26 mm and 24 mm respectively, the
feed speed vf was mm/min and the spindle speed n was 1401 r.p.m. b) Zoomed
in frequency to the interval of the dominating resonance peak for the five different axial depths (ap = 1, 2, 3, 4, 5 mm) and the corresponding radial depth was
ae = 26.0, 25.6, 25.2, 24.2, 24.0 mm.
Results in terms power spectral density estimates of workpiece vibration in the
feed direction (+x4 ) when changing the feed speed vf are presented in Fig. 17 for five
different feed speeds (vf = 1401, 1501, 1601, 1701, 1801 mm/min) in the DMU 80FD
Duoblock milling machine. The radial depth was ae = 24.6, 24.2, 23.8, 23.4, 23.0 mm,
the axial depth ap was 2 mm and the spindle speed n was 1401 r.p.m. In Fig. 17 a) no
particular changes can be observed, but when zooming in on the peaks as illustrated
in Fig. 17 b) a small difference in magnitude is observable.
Results in terms power spectral density estimates of workpiece vibration in the
feed direction (+x4 ) when changing the spindle speed n are presented in Fig. 18 a),
for five different spindle speeds (n = 1401, 1465, 1528, 1592, 1656 r.p.m.) in the DMU
80FD Duoblock milling machine. The radial depth was ae = 26.0, 25.8, 25.4, 25.2, 25.0
mm, the axial depth ap was 2 mm and the feed speed vf was 1401 mm/min. In Fig. 18
a), it is observable how the frequency of the harmonics changes with the change of
spindle speed. In Fig. 18 b), typical power spectral density estimates of workpiece
vibration in the feed direction (+x4 ) during machining conducted in the Hurco BMC50 milling machine is presented. The radial depth ae was 26 mm, the axial depth
ap was 1 mm, the feed speed vf was 1401 mm/min and the spindle speed n was
1401 r.p.m. Also, when carrying out the machining in the Hurco BMC-50 milling
machine both a large number of narrow-banded peaks and a broadband response of
10
5
0
−5
−10
−15
−20
1401 mm/min
1501 mm/min
1601 mm/min
1701 mm/min
1801 mm/min
−25
−30
−35
−40
750
775
800
825
Frequency [Hz]
a)
850
PSD [dB rel 1 ((m/s2 )2 /Hz)]
PSD [dB rel 1 ((m/s2 )2 /Hz)]
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Investigation of the Dynamic Properties of a Milling Tool Holder
117
4
2
0
−2
−4
1401 mm/min
1501 mm/min
1601 mm/min
1701 mm/min
1801 mm/min
−6
−8
814
815
816
817
818
Frequency [Hz]
819
820
b)
Figure 17: a) Power spectral densities of workpiece vibration in the feed direction
(+x4 ) during milling for five different feed speeds (vf = 1401, 1501, 1601, 1701, 1801
mm/min) in the DMU 80FD Duoblock milling machine. The radial depth was ae =
24.6, 24.2, 23.8, 23.4, 23.0 mm, the axial depth ap was 2 mm and the spindle speed n
was 1401 r.p.m. b) Corresponding spectra zoomed in at one of the harmonics.
an underlying structure may be observed in the spectra, see Fig. 18 b).
3.1.1
Operating Deflection Shapes
To obtain information on how the spindle frame vibrates relative to the workpiece,
spatial measurements of the acceleration of these structural parts were carried out.
Accelerometer positions and measurement directions on the spindle frame and the
workpiece are illustrated in Fig. 19 a). To facilitate illustration of the operating
deflection shapes, the spindle frame and the workpiece are simplified into a skeleton
structure where the measurement positions are illustrated by black circles, defined as
nodes, as also shown in Fig. 19 a). In Fig. 19 b) the simplified skeleton structure of
the spindle frame and the workpiece is shown in the y-z plane and in Fig. 19 c) it is
shown in the x-z plane.
Observe that the fourth node in the skeleton structure is fixed on the workpiece
and thus moving away from the other nodes, along the x-axis, as the tool is cutting
the workpiece. The spatial motion of this structure has a complex behavior and
changes with time. However, an operating deflection shape at one of the dominant
peeks in the spectral density previously presented (see Fig. 16), i.e. at 780 Hz, was
estimated during a short time sequence and is presented in Fig. 20 a) and b). The
deformation shape is presented with arrows in the figure. Observe that the size of the
arrows does not represent the absolute magnitudes of the four positions deflection;
their magnitudes are displayed in an enlarged scale to make them observable.
In order to show the complex spatial behavior of the measurement positions on
the spindle frame and the workpiece, a trajectory for node two is presented during
a time sequence of 15.625 ms, see Fig. 21 a) and b). The trajectory was produced
by filtering the acceleration signals with a band-pass filter having a center frequency
at 780 Hz. The frequency response function for the band-pass filter is presented in
Fig. 22.
By combining the trajectory plots for each of the four measurement positions
PSD [dB rel 1 ((m/s2 )2 /Hz)]
10
5
0
−5
−10
−15
−20
1400 rev/min
1464 rev/min
1528 rev/min
1591 rev/min
1655 rev/min
−25
−30
−35
−40
750
770
790
810
Frequency [Hz]
a)
830
850
PSD [dB rel 1 ((m/s2 )2 /Hz)]
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10
5
0
−5
−10
−15
−20
−25
−30
−35
−40
400
600
800
1000
1200
1400
Frequency [Hz]
b)
1600
1800
Figure 18: a) Power spectral densities of workpiece vibration in the feed direction
(+x4 ) during milling for five different spindle speeds (n = 1401, 1465, 1528, 1592, 1656
r.p.m.) in the DMU 80FD Duoblock milling machine. The radial depth was ae =
26.0, 25.8, 25.4, 25.2, 25.0 mm, the axial depth ap was 2 mm and the feed speed vf was
1401 mm/min. b) Power spectral density of workpiece vibration in the feed direction
(+x4 ) performed in the Hurco BMC-50 milling machine. The radial depth ae was
26 mm, the axial depth ap was 1 mm, the feed speed vf was 1401 mm/min and the
spindle speed n was 1401 r.p.m.
+ Z
- Y
+ X
a)
b)
c)
Figure 19: a) presents a 3d-view of the spindle frame, tool holder and the milling table
with the workpiece. The measurement positions shown as black circles connected by
straight black lines forming a skeleton structure, b) present the skeleton structure in
the y-z plane and c) presents the skeleton structure in the x-z plane.
100
Node 2
Node position on the z-axis [cm]
Node position on the z-axis [cm]
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Investigation of the Dynamic Properties of a Milling Tool Holder
Node 1
80
60
Node 3
40
20
Node 4
0
−60
−40
−20
0
20
40
60
100
Node 2
Node 1
80
60
Node 3
40
20
Node 4
0
−40
Node position on the y-axis [cm]
a)
119
−20
0
20
40
60
80
Node position on the x-axis [cm]
b)
z-axis
z-axis
Figure 20: Operating deflection shape for the spindle frame and the milling table
with the workpiece at the frequency 780 Hz, estimated during machining. The radial
depth ae was 24.4 mm, the axial depth ap was 4 mm, the feed speed vf was 1401
mm/min and the spindle speed n was 1401 r.p.m. In a) the shape is presented in the
y-z plane and in b) the shape is presented in the x-z plane.
y-axis
a)
x-axis
y-axis
x-axis
b)
Figure 21: Trajectory plot of the measured acceleration signals in node two at 780 Hz
during machining. The radial depth ae was 24.4 mm, the axial depth ap was 4 mm,
the feed speed vf was 1401 mm/min and the spindle speed n was 1401 rev/min. a)
and b) viewed from two different perspectives perspective.
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0
−10
Phase [degree]
Magnitude [dB] [cm]
0
−20
−30
−40
−50
−2000
−3000
−4000
−5000
−60
−70
0
−1000
400
800
1200
Frequency [Hz]
a)
1600
2000
−6000
0
400
800
1200
Frequency [Hz]
b)
1600
2000
Figure 22: The frequency response function for the filter used in the production of
the time domain ODS, a) is the magnitude and b) is the phase.
on the spindle frame and workpiece and plotting them together with the skeleton
structure its spatial motion for a short time interval may be illustrated as in Fig. 23.
Observe that the ellipses are displayed in a enlarged scale to make them observable.
3.1.2
Angular Vibrations
The angular vibrations of the milling tool holder were measured at three different
positions, two on the tool holder and one at the spindle close to the clamping of the
tool holder. In Fig. 24, the angular vibrations of the three positions versus the number
of revolutions of the tool holder are shown in the same diagram. In this figure, the
angular vibrations during approximately the first 35 revolutions are measured prior
to engagement of the tool in the workpiece. Note how, during the first revolutions,
when no machining is carried out, the angular vibration of the tool holder and the
spindle are still observable.
All three sensors show a good agreement on the angular vibrations when there
is no cutting, see Fig. 25 a). However, during machining a discrepancy between the
angular vibrations measured by the sensor on the spindle and the angular vibrations
measured by the sensors on the tool holder is observable, see Fig. 25 b).
By plotting a waterfall diagram of the order spectra of the angular tool holder
vibrations closest to the tool, it is obvious that the main angular vibration is directly
related to the first order, see Fig. 26. The radial depth ae was 24.0 mm, the axial depth
ap was 5 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401
rev/min. To facility observability of the peaks of the higher orders in the order spectra
the first order was excluded from them and they were again plotted in a waterfall
diagram as illustrated in Fig. 26 b). Also, in this figure an underlying broadband
dynamic angular response of the tool holder may be observed. The seventh, 14:th
and 21:th order of the spindle speed are slight higher than the direct neighboring
orders, see Fig. 26 b). These orders are also the first, second and third order of the
tooth-passing frequency.
100
Node 2
Node position on the z-axis [cm]
Node position on the z-axis [cm]
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Investigation of the Dynamic Properties of a Milling Tool Holder
Node 1
80
60
Node 3
40
20
Node 4
0
−60
−40
−20
0
20
40
60
Node position on the y-axis [cm]
a)
100
Node 2
121
Node 1
80
60
Node 3
40
20
Node 4
0
−40
−20
0
20
40
60
80
Node position on the x-axis [cm]
b)
Figure 23: Spatial motion of the spindle frame and the workpiece for a short time
interval, based on band pass filtered acceleration signal measured during cutting.
The radial depth ae was 24.4 mm, the axial depth ap was 4 mm, the feed speed vf
was 1401 mm/min and the spindle speed n was 1401 r.p.m. The ellipses represent
the motion of the measured nodes for the frequency 780 Hz. The circle on each
ellipse represents a synchronization point for all the nodes (measurement positions)
at a certain time instant and is followed by a solid ellipse line which indicates the
direction of the motion. In a) the motion is presented in the y-z plane and in b) the
motion is presented in the x-z plane.
Vibration angle [degree]
0.4
0.3
0.2
Upper disc
Middle disc
Lower disc
0.1
0
−0.1
−0.2
−0.3
−0.4
15
20
25
30
35
40
Revolutions
45
50
55
60
Figure 24: Angular vibrations of tool holder and the spindle, measured at two positions on the tool holder and at one position on the spindle. The first 35 revolutions
are measured prior to engagement of the tool in the workpiece directly followed by
the engagement phase. After approx. 40 revolutions the material removal process is
carried out according to the selected cutting data. The radial depth ae was 24.2 mm,
the axial depth ap was 2 mm, the feed speed vf was 1501 mm/min and the spindle
speed n was 1401 rev/min.
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Part IV
122
Vibration angle [degree]
0.1
Upper disc
Middle disc
Lower disc
0.05
0
−0.05
−0.1
21
22
23
24
25
Revolutions
26
27
28
29
a)
Vibration angle [degree]
0.6
Upper disc
Middle disc
Lower disc
0.4
0.2
0
−0.2
−0.4
401.5
402
402.5
403
403.5
Revolutions
404
404.5
405
405.5
b)
Figure 25: The angular vibrations of the milling tool holder and the spindle during
machining when the radial depth ae was 24.2 mm, the axial depth ap was 2 mm, the
feed speed vf was 1501 mm/min and the spindle speed n was 1401 rev/min. a) Shows
the angular vibrations prior to cutting and b) shows the angular vibrations during
cutting.
A n g u la r d is p la c e m e n t [d e g re e ]
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Investigation of the Dynamic Properties of a Milling Tool Holder
123
1 4 0 6 .6 0
0 .3 0
0 .2 0
1 4 0 6 .5 5
0 .1 0
1 4 0 6 .5 0
0
0
P M
[R
d
ee
sp
e
l
in d
S p
1 4 0 6 .4 5
5
1 0
1 5
1 4 0 6 .4 0
2 0
O rd e rs
2 5
1 4 0 6 .3 5
]
2 5 ( g1 0
- 2
1 4 0 6 .6 0
)
1 4 0 6 .5 5
2 0
1 5
1 4 0 6 .5 0
1 0
1 4 0 6 .4 5
5
0
S p in d
le s p e
e d [R
P M ]
A n g u la r d is p la c e m e n t [d e g re e ]
a)
0
1 4 0 6 .4 0
5
1 0
1 5
O rd e rs
2 0
2 5
3 0
1 4 0 6 .3 5
b)
Figure 26: Waterfall plot of the order spectra of the angular vibration during 30 s of
machining. The radial depth ae was 24.0 mm, the axial depth ap was 5 mm, the feed
speed vf was 1401 mm/min and the spindle speed n was 1401 rev/min. a) presents
the order spectra with the first order included in the plot and b) presents the order
spectra when the first order has been removed.
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Part IV
124
3.2
Modal Analysis Results
20
10
0
−10
−20
−5X:+1X
−5X:+3X
−5X:+4X
−5X:+5X
−5X:+6X
−30
−40
400
600
800
1000
1200
1400
Frequency [Hz]
a)
1600
1800
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Results in terms of accelerance function estimates and coherence function estimates
from the experimental modal analysis (EMA) carried out on two milling machines
are presented first in this section next to each other. In other words, the accelerance
function estimates from the Hurco BMC-50 milling machine are presented in Fig. 27
a), Fig. 28 a) and Fig. 29 a) and the accelerance function estimates from the DMU
80FD Duoblock milling machine are presented in Fig. 27 b), Fig. 28 b) and Fig. 29 b).
The two EMA setups differed slightly between the two machines, see section 2.1.5.
A significant peak is noticeable around 650 Hz in the accelerance functions produced
from the EMA carried out in Hurco BMC-50 milling machine and a peak around 750
Hz is noticeable in the accelerance functions produced from the EMA carried out in
the DMU 80FD Duoblock milling machine.
20
10
0
−10
−20
−5X:+1X
−5X:+2X
−5X:+4X
−5X:+5X
−5X:+6X
−30
−40
400
600
800
1000
1200
1400
Frequency [Hz]
b)
1600
1800
Figure 27: Accelerance magnitude function estimates between the force input in xdirection and the acceleration responses in x-direction based on the experimental
modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the
DMU 80FD Duoblock milling machine.
Typical multiple coherence function estimates obtained during the experimental
modal analysis are illustrated in Fig. 30. The coherence function estimates presented
in Fig. 30 shows values above 0.9 for most frequencies between 350 Hz and 1800 Hz.
This indicates that the level of forces and accelerations was fairly good in the region
of interest, that is, between 350 Hz up to 1400 Hz. Some dips may be observed around
800 Hz in estimates done from both EMA setups and one larger around 1500 Hz in the
coherence function estimate from the EMA carried out in the DMU 80FD Duoblock
milling machine.
3.2.1
Mode Shapes
Based on the accelerance functions estimated from the modal analysis setups carried
out in the Hurco BMC-50 milling machine and in the DMU 80FD Duoblock milling
machine, a number of resonance frequencies were estimated. The estimated resonance
frequencies and their relative damping are presented in Table 7.
Furthermore, for each resonance frequency a corresponding mode shape of the
spindle house - tool holder system was estimated. A figure (Fig. 31) defining the
positions of the sensors together with two tables ( Table 8 and Table 9) presents
20
10
0
−10
−5Y:+1Y
−5Y:+3Y
−5Y:+4Y
−5Y:+5Y
−5Y:+6Y
−20
−30
−40
400
600
800
1000
1200
1400
Frequency [Hz]
a)
1600
1800
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
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Investigation of the Dynamic Properties of a Milling Tool Holder
125
20
10
0
−10
+5Y:−1Y
+5Y:−2Y
+5Y:−4Y
+5Y:−5Y
+5Y:−6Y
−20
−30
−40
400
600
800
1000
1200
1400
Frequency [Hz]
b)
1600
1800
20
10
0
−10
−20
−5Y:+1X
−5Y:+3X
−5Y:+4X
−5Y:+5X
−5Y:+6X
−30
−40
400
600
800
1000
1200
1400
Frequency [Hz]
a)
1600
1800
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Figure 28: Accelerance magnitude function estimates between the force input in ydirection and the acceleration responses in y-direction based on the experimental
modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the
DMU 80FD Duoblock milling machine.
20
10
0
−10
−20
+5Y:+1X
+5Y:+2X
+5Y:+4X
+5Y:+5X
+5Y:+6X
−30
−40
400
600
800
1000
1200
1400
Frequency [Hz]
b)
1600
1800
Figure 29: Accelerance magnitude function estimates between the force input in ydirection and the acceleration responses in x-direction based on the experimental
modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the
DMU 80FD Duoblock milling machine.
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Part IV
126
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
*:+6Y
*:+6X
0.1
0
400
600
800
1000
1200
1400
Frequency [Hz]
a)
1600
1800
Multiple Coherence γ 2
Multiple Coherence γ 2
1
0.9
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
*:−5Y
*:+5X
0.1
0
400
600
800
1000
1200
1400
Frequency [Hz]
b)
1600
1800
Figure 30: Typical coherence functions between the force inputs (the asterisk denotes
all inputs) and the acceleration responses +6Y,-5Y (dashed line) and +6X,+5X (solid
line) from the experimental modal analysis measurements, a) in the Hurco BMC-50
milling machine and b) in the DMU 80FD Duoblock milling machine.
Mode
1
2
3
4
5
6
7
8
9
10
Hurco milling machine
Freq. [Hz] Damping [%]
615.4
3.91
650.4
2.58
783.0
0.60
920.6
1.49
985.5
1.23
1141.1
1.75
1305.3
1.96
1349.9
1.68
1518.0
4.11
1611.3
0.38
DMU 80FD milling machine
Mode Freq. [Hz] Damping [%]
1
744.1
2.68
2
755.4
2.61
3
809.6
2.38
4
925.8
1.24
5
992.8
2.03
6
1064.0
1.34
7
1128.2
1.50
8
1509.5
1.17
9
1692.8
1.04
Table 7: Estimated resonance frequencies and their relative damping coefficients from
the modal analysis setups carried out in the Hurco BMC-50 milling machine and in
the DMU 80FD Duoblock milling machine.
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Investigation of the Dynamic Properties of a Milling Tool Holder
127
the two first mode shapes for the respective milling machine setup in the form of
magnitude and angle.
T o o l
T o o l h o ld e r
S p in d le h o u s e
- Z
T o o l
T o o l h o ld e r
S p in d le h o u s e
a)
- Z
b)
Figure 31: Cross-section view of the spindle house and the tool holder illustrating the
mode shape for the first bending mode. In a) for the Hurco milling machine and in
b) for the DMU 80 milling machine.
Hurco milling machine
mode at 615.4 Hz
Position Norm. mag Angle [◦ ]
1
0.05
65.4
2
3
0.36
50.4
4
0.58
47.9
5
0.75
44.7
6
1.00
45.3
DMU 80FD milling machine
mode at 744.1 Hz
Position Norm. mag Angle [◦ ]
1
0.02
47.5
2
0.01
48.3
3
0.28
54.2
4
0.69
56.2
5
1.00
57.1
-
Table 8: Mode shape table presenting magnitude values for each measurement position
and the angle relative to the x-axis, for the first mode estimated for respective milling
machine setup.
In order to be able to evaluate the quality of the estimated modal parameters,
a synthesized accelerance function are produced and overlaid on top of the corresponding estimated accelerance function. In Fig. 32 and Fig. 33, the driving point
accelerance functions and the transfer accelerance functions between x and y direction
in the driving point position, are presented together with their synthesized functions.
The synthesized functions show good agreement with the estimated accelerance functions. In order to check the quality of the estimated mode shapes, a MAC matrix was
produced. These matrixes are presented in Fig. 30, and as can be seen the orthogonality between the two first modes are excellent, while a high correlation exists for
higher order modes.
3.3
Distributed Parameter Model Results
The results from the distributed parameter models of a clamped milling toolholder,
in terms of natural frequency estimates, for both bending and torsional modes, are
presented in Table 10.
The mode shapes, based on the distributed parameter models of a clamped milling
tool holder, for the first and second bending mode as well as for the first torsional
mode is presented in Fig. 35. Note that the first and second bending mode shapes are
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Part IV
128
Hurco milling machine
mode at 650.4 Hz
Position Norm. mag Angle [◦ ]
1
0.03
-24.2
2
3
0.35
-39.3
4
0.55
-42.1
5
0.72
-45.1
6
1.00
-44.5
DMU 80FD milling machine
mode at 755.4 Hz
Position Norm. mag Angle [◦ ]
1
0.01
-42.2
2
0.01
-41.8
3
0.22
-35.6
4
0.68
-33.9
5
1.00
-32.8
-
15
−5X:+5X, Estimated
−5X:+5X, Synthesized
10
5
0
−5
−10
−15
−20
400
600
800
1000
1200
1400
Frequency [Hz]
1600
1800
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Table 9: Mode shape table presenting the magnitude values for each measurement
position and angle relative to the x-axis, for the second mode estimated for respective
milling machine setup.
15
−5Y:+5X, Estimated
−5Y:+5X, Synthesized
10
5
0
−5
−10
−15
−20
400
600
800
1000
1200
1400
Frequency [Hz]
1600
1800
20
10
0
−10
−20
−30
−40
−50
−4X :+4X, Estimated
−4X :+4X, Synthesized
400
600
800
1000
1200
1400
Frequency [Hz]
1600
1800
Accelerance [dB rel 1(m/s2 )/N]
Accelerance [dB rel 1(m/s2 )/N]
Figure 32: The magnitude of the synthesized and the measured accelerance functions
for the milling tool holder when clamped in the Hurco BMC-50 milling machine, a)
between the force signal and the acceleration signal from location 5X and b) between
the force signal at location 5Y and the acceleration signal from location 5X.
20
10
0
−10
−20
−30
−40
−50
−4X:−4Y, Estimated
−4X:−4Y, Synthesized
400
600
800
1000
1200
1400
Frequency [Hz]
1600
1800
Figure 33: The magnitude of the synthesized and the measured accelerance functions
for the milling tool holder when clamped in the DMU 80FD Duoblock milling machine, a) between the force signal and the acceleration signal from location 4X and b)
between the force signal at location 4Y and the acceleration signal from location 4X.
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Investigation of the Dynamic Properties of a Milling Tool Holder
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
615
650
783
921
986
1141
1305
1350
1518
l
1611
0
744
755
810
926
993
1064
1128
1510
l
1693
M
od
eΨ
[H
z]
1611
1518
1350
1305
1141
986
921
783
k
650
615
Ψ
de
Mo
a)
]
[Hz
M
od
eΨ
[H
z]
129
1693
1510
1128
1064
993
926
810
k
755
744
b)
Ψ
de
Mo
]
[Hz
Figure 34: a) MAC matrix presenting the correlation between the estimated mode
shapes. In a) the matrix based on the mode shapes estimated from the Hurco BMC-50
milling machine setup, while in b) the matrix is based on the mode shapes estimated
from the DMU 80FD Duoblock milling machine setup.
Mode
1
2
3
Frequency [Hz]
1902.34
11921.74
6321.53
Type of mode
First bending
Second bending
First torsional
Table 10: Natural frequency estimates based on the distributed parameter model of
a clamped milling tool holder.
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Part IV
130
in the transverse direction, while the first torsional mode shape represents a rotation
deformation of the structure around and along its own centerline.
Normalized mode shape
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
First bending mode
Second bending mode
First torsional mode
0.02
0.04
0.06
0.08
0.1
0.12
Distance from clamping [m]
0.14
Figure 35: The first and second bending modes together with the first torsional mode.
Calculated based on the distributed-parameter system models of a clamped milling
tool holder.
3.4
Finite Element Model
The first six natural frequencies estimated based on the milling tool holder FE-model
are presented in Table 11. The corresponding mode shapes are illustrated in Fig. 36to
Fig. 38.
Mode
1
2
3
6
7
8
Frequency [Hz]
1437.7
1440.4
5417.6
6728.7
6744.2
7789.2
Type of mode
First bending
First bending
First torsional
Second bending
Second bending
First longitudinal
Table 11: Natural frequency estimates based on FE analysis of a clamped milling tool
holder.
4
Summary and Conclusion
A number of different machining measurements have been conducted which show good
agreement with the expectations grounded in cutting theory. A pronounced periodicity is present in acceleration measurements carried out during milling operations.
This periodicity is directly related to the spindle speed n and also to the harmonics
observable in Fig. 16 a) and b). Furthermore, when increasing the cutting depth ap
the vibration level increases as can be seen in Fig. 16 b). This also agrees with the
theory [1], showing that increasing cutting depth results in increasing cutting forces,
and thus also in greater excitation levels. When changing the feed speed (feed rate),
no significant changes are observed although a small increase in vibration level with
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Investigation of the Dynamic Properties of a Milling Tool Holder
a)
131
b)
Figure 36: a) The milling toolholder mode shape belonging to the natural frequency
at 1437.7 Hz, and b) the mode shape belonging to the natural frequency at 1440.4
Hz, estimated with a FE model. The solid part represents the shape of the mode and
the transparent part drawn in thin lines is the un-deformed part.
a)
b)
Figure 37: a) The milling toolholder mode shape belonging to the natural frequency
at 5417.6 Hz, and b) the mode shape belonging to the natural frequency at 6728.7
Hz, estimated with a FE model. The solid part represents the shape of the mode and
the transparent part drawn in thin lines is the un-deformed part.
a)
b)
Figure 38: a) The milling toolholder mode shape belonging to the natural frequency
at 6744.2 Hz, and b) the mode shape belonging to the natural frequency at 7789.2
Hz, estimated with a FE model. The solid part represents the shape of the mode and
the transparent part drawn in thin lines is the un-deformed part.
132
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Part IV
increasing speed may be noticed, see Fig. 17. The dependence of the harmonics on
the spindle speed is observable in Fig. 18 where the spindle speed was changed. In
all the acceleration records, an underlying dynamic response may be observed. This
is confirmed from the results obtained from the modal analysis carried out in both
milling machines; compare the acceleration spectra in Fig. 18 b) with the accelerance
function in Fig. 27 a) where a peak near 620 Hz is present in both figures. The results
from the modal analysis, presented in Table 8 and Table 9, suggest that this is the
first bending mode of the tool holder when the tool holder is clamped in the Hurco
BMC-50 milling machine. This can also be seen when comparing the acceleration
spectra in Fig. 16 with the accelerance function in Fig. 28 b), where a peak near
790 Hz is present in both figures. The results from the modal analysis, presented in
Table 8 and Table 9, from the setup in the DMU 80FD Duoblock milling machine
suggest this is the first bending mode of the tool holder. The difference in frequency
of the first bending modes of the tool holder in the different machines indicates a more
rigid clamping in the DMU 80FD Duoblock milling machine. Also the fact that the
natural resonance frequencies estimated for both EMA setups in the same frequency
range, presented in Table 7, are in general higher in the case of the DMU 80 Duoblock
milling machine compare to the case of the Hurco BMC-50 milling machine, supports
this conclusion.
The operating deflection shape analysis of the spindle frame - workpiece of the
milling machine provided information concerning their spatial motion during machining. The deflection shape of the spindle frame-workpiece, can also be connected to
the first bending mode, and the complex behavior of the shape may be explained by
the fact that the milling tool holder is rotating while it is assumed to have a motion
that is in itself related to a first bending mode. This assumption is however, not
confirmed since no accelerometer measured the bending vibration of the milling tool
holder during cutting.
To move on, angular vibration measurements of the milling tool holder showed the
significance of the various orders. For example, the first order of the spindle speed
had a major impact on the angular vibration level which suggests a significant unbalance of spindle - tool holder system. An unbalance will generally introduce transverse
vibration directly related to the first order of the spindle speed. However, due to the
sensor setup, the discs mounted on the milling tool holder, will because of the unbalance, be displaced from the rotation center and thus influence the angular velocity
measured by the sensors. The first order of the spindle speed was already present in
the angular vibration even before cutting took place, see the angular vibration before
cutting in Fig. 25 a). It may also be noticed that the largest vibrations for all measurements in all nodes occur at a frequency, where one of the orders of the spindle
speed coincides with a fundamental bending mode of the milling tool holder. The
main purpose with using rotation sensors was to discover any angular motion relating
to torsional modes of the milling tool holder. One important conclusion from the
measurement was that no such dynamic behavior could be observed or demonstrated.
The analytical models together with the finite element model provided rough estimates of where in frequency the natural frequencies of the various modes may be
expected to be found and how the mode shapes will look like. This is important when
setting up the measurement and selecting sensor configuration. Both the analytical
model and the finite element model were configured in order to overestimate the natural frequencies. In the real setups, the milling tool holder was configured with a tool
that made the structure longer and at the same time adding mass to the end of the
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Investigation of the Dynamic Properties of a Milling Tool Holder
133
structure. The clamping of the milling tool holder was furthermore assumed to be
infinitely rigid for the model, which is not the case in reality.
One of the most interesting results, can be found in the accelerance function
in Fig. 29, where significant peaks near 800, 1000 and 1600 Hz were found in the
HurcoBMC-50 milling machine setup, while peaks near 925 was found in the DMU
80FD Duoblock milling machine setup. These peaks were not found in the accelerance
functions in Fig. 27 and Fig. 28. The presence of the peaks in the accelerance function
in Fig. 29 suggests that the structure is sensitive to forces applied in the orthogonal
direction with respect to the direction of the response. If this result is a property
of a milling tool holder clamped in a spindle and supported by bearings, or if it is
something else, needs further investigations. However, it should be noted that the
similar results (concerning the orthogonal sensitivity) were obtained for two different
milling machines.
Acknowledgments
The present project is sponsored by the company Acticut International AB in Sweden
which has multiple approved patents covering active control technology.
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Mark Regelbrugge, Chi-Man Kwan, Roger Xu, Bill Winterbauer, and Keith
Bridger. Mitigation of chatter instabilities in milling by active structural control.
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Part V
Preliminary Investigation of Active
Control of a Milling Tool Holder
This part is based on the publication:
H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson and T. Lagö Preliminary investigation of active control of a milling tool holder, ACTIVE 2009 - The 2009 International Symposium on Active Control of Sound and Vibration, Ottawa, Canada,
20-22 August, 2009.
Preliminary Investigation of Active Control
of a Milling Tool Holder
Henrik Åkesson1,2 , Tatiana Smirnova,
Lars Håkansson and Ingvar Claesson
1
Blekinge Institute of Technology,
Department of Signal Processing,
372 25 Ronneby, Sweden
Thomas T. Lagö
Acticut International AB,
Gjuterivägen 6, 311 32 Falkenberg, Sweden
2
Abstract
In milling, vibration is a common problem. During cutting, the cutting
process excites the tool and the tool holder continuously. Having a milling tool
holder with a long overhang will easily result in high vibration levels. The
vibration level depends on many different parameters such as material type,
dimensions of the workpiece and tool holder, cutting data, etc. High vibration
levels result in reduced tool life, poor surface finish and disturbing sound. There
are different methods to reduce the vibration levels, e.g. trial and error by
an operator, introducing a passive tuned damper or applying active vibration
control. This paper presents an active vibration control solution for a milling
tool holder and preliminary results from the active control. The major challenge
of transferring electrical power while maintaining signal quality to and from
a rotating object is discussed and the proposed solution to this challenge is
presented.
1
Introduction
Vibration problems during metal cutting occur frequently in the manufacturing industry. The vibration level depends on many different parameters such as material
type, the dimensions of the workpiece, the rigidity of the tooling structure and the
cutting data. The milling process is typically characterized by the usage of a tool
with one or several teeth that rotates about a fixed axis [1], (there are several types of
milling machines today that have several axes and that can perform complex milling
operations about a non-fixed axis). The cyclic variation of forces that arise due to
the rotating tool in the machining process might excite any of the natural modes of
the milling machine structure. One part of the machine tool structure is the milling
tool holder, and using a long overhang for this part will most likely result in high
vibration levels.
140
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Part V
Several methods have been developed to find cutting parameters that may reduce
instabilities during machining [2, 3]. One alternative method is to change some of the
parameters during the milling operation [2], i.e. changing the spindle speed or the
feed rate of the workpiece. The goal with changing one of the cutting parameters is
to reduce the feedback and thereby avoid instability. Active solutions have also been
proposed, for example by J.L. Dohner et al. [4], where an active structural control
system was developed to alter the system’s dynamics.
In order to determine which method should be used and how, the dynamic properties of the system have to be thoroughly analyzed. One study concerning the dynamic
properties of one type of milling tool holder and the dynamic properties of two milling
machines have been reported in [5]. This paper claims that the vibrations display a
high correlation with the two fundamental bending modes of the milling tool holder.
This paper discusses the implementation and properties of an active milling tool
holder with vibration sensors and an embedded piezoceramic actuator. It furthermore
presents a solution for transferring the vibration sensor signal and the electrical actuator power signal between a rotating active milling tool holder and a non-rotating
control unit placed, for example, outside the milling machine. Results from the measurement of the power and sensor signals are presented for a number of different
rotation speeds. Finally, results from initial experiments concerning the active control of a milling tool holder vibrations are also presented.
2
Materials and Methods
The milling machine used in the experiments was a Hurco BMC-50 vertical CNC
machining centre. The spindle was of the ATC type, i.e. the speed can be varied
between 10-3000 RPM in steps of 20 RPM and the maximum torque is 428 Nm, see
Fig. 1.
Figure 1: The Hurco BMC-50 milling machine.
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Preliminary Investigation of Active Control of a Milling Tool Holder
2.1
141
Active Tool Holder, Tool and Inserts
The milling tool holder is the interface between the spindle and the tool which holds
all the inserts. The milling tool holder used in the experiments was a modified tool
holder based on a standard tool holder of the type E3471 5525 16160 which has an
overhang of 160 mm and a diameter of 38 mm, see Fig. 2 a).
An actuator was embedded a small distance from the center line in order to be
able to produce a bending moment. The milled space for the actuator can be observed
in Fig. 2 a). On the milling tool holder, a tool of the type R220.69-0040-12-5A was
mounted and is illustrated in Fig. 2 b). This tool has a cutting diameter of Dc =
40 mm and a five teeth, zn = 5 (inserts) configuration. The insert used in the tool
configuration was XOMX120408TR-M12 T250M.
M illin g to o l
A c tiv e m illin g to o l h o ld e r
M ille d s p a c e fo r a c tu a to r
a)
3 8 m
m
4 0 m
m
3 8 m
m
1 6 0 m
m
4 0 m
m
b)
Figure 2: a) The modified tool holder and b) the tool with the five teeth mounted.
2.2
Slip Ring Device
In order to provide electrical transfer paths for power and sensor signals to a rotating
milling tool holder, the slip-ring method was used. A prototype able to transfer 15
ampere, 200 volt at a rotating speed of 8000 RPM (for a tool holder with a diameter
of max 50 mm), or a velocity of 60 m/s, at the contact surfaces, was designed. Two
poles are required for transferring power to one actuator and two poles are required
to connect to one sensor, i.e. if wires are used. An alternative is to transfer power
to two actuators and a wireless sensor signal transmitter. The sliding contacts are
spring mounted and have two contact surfaces each towards the slip rings so that in
case of large vibrations the risk of losing electrical contact is reduced. The slip ring
device prototype enables the use of four sliding contacts per slip ring, thus providing
eight contact surfaces if needed. From each slip ring, a conducting wire is accessible
from the bottom side of the slip ring device for the connection of actuators and/or
sensors. A sketch of the prototype is presented in Fig. 3 a) and a photo is presented
in Fig. 3 b).
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Part V
142
R o ta tin g p a rt
F ix e d p a rt
S lip rin g s
S lid in g c o n ta c t
S lid in g c o n ta c t
a)
b)
Figure 3: a) A sketch of the slip ring device with four poles and four double sliding
contacts. b) A photo of the first slip ring device prototype.
2.3
Experimental Setup
Four different experiments were conducted.
Firstly, the quality of the signals over the slip ring device was tested at the maximum rotation speed of the spindle, i.e. n =3000 RPM. This was done by connecting
the four conductors in the rotating part in pairs - resulting in two input poles and two
output poles - and using broad band noise, 0-50000 Hz, as the input signal measuring
the input and the output signals simultaneously.
Secondly, the actuator and the sensors were connected to the slip ring device.
The actuator was excited in the frequency range of 0-5kHz and a channel estimation/control path - the transfer function between the voltage over the actuator and
the response of the vibration sensor - was performed using either the piezo-film or
the accelerometer for a number of different RPM, from 100 RPM up to 1000 RPM in
steps if 100.
Furthermore, in order to determine the dynamic properties of the clamped milling
tool holder by means of experimental modal analysis, an impulse hammer was used.
The milling tool holder was excited close to the milling tool tip with an impulse
hammer at opposite sides to the accelerometer attached to the milling tool holder,
see Fig. 4.
The fourth experiment involved cutting with and without active control. As the
control algorithm, the filtered-x algorithm was used, and the controller was set to a
frequency range of 0 to 4000 Hz. A block diagram of the main structure of all of the
setups is presented in Fig. 4.
2.4
Cutting Parameters
In order to test active control of the milling tool holder vibrations, we cut a workpiece
consisting of the material SS1312. The workpiece had approximately the dimensions
70x50x500 mm (y, z, x), see Fig. 5. The workpiece was clamped to the milling table,
and the milling table moved in the x-direction, resulting in a continuous cutting
process along the workpiece, see Fig. 5. The cutting data used in the active control of
the milling vibration were: feed speed vf was set to 1000 mm/min, the spindle speed
n = 700 RPM, cutting a radial depth ae = 10 mm and an axial depth ap equal to
2mm. During the cutting process, the active control algorithm was turned on and off.
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Preliminary Investigation of Active Control of a Milling Tool Holder
S u rfa c e w h ic h is
c la m p e d
143
T o o l h o ld e r
H P 3 5 6 7 0 A
S ig n a l a n a ly z e r
C h 1
N I U S B -9 2 3 3
C h 1 , C h 2
A m p . 1
A m p . 2
K e m o F ilte r
S lip rin g d e v ic e
P ie z o film
K e m o F ilte r
A c tu a to r
D S P
T o o l
A c c e le ro m e te r
Figure 4: The main structure of the experimental setups. Note that only one sensor
is connected to the slip ring device at a time. The slip ring device is represented as a
block in the figure in order to present the actuator and the piezo-film configurations,
otherwise these components are covered by the slip ring device.
n
a
a
e
v
v
C
p
f
ie c e
p
k
r
W o
Figure 5: The side milling configuration of the cutting setup during the milling measurements.
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Part V
144
3
Results
Results from the preliminary examination of the slip ring path and the control path
involving the slip ring device are presented in terms of magnitude of frequency response estimates together with the quality measure in terms of coherence [6]. Then,
the results from the active control setup are presented in terms of power spectral
density estimates [6].
3.1
Setup One, Slip Ring Test
0
1
−0.1
0.98
2
Coherence γ̂yx
Magnitude [dB rel 1(V/V]
The first experimental setup shows a good transfer of the signal with a signal loss of
less than 0.05 dB over a wide frequency range and a coherence close to one, see Fig. 6.
−0.2
0.96
−0.3
0.94
−0.4
0.92
−0.5
0.9
−0.6
−0.7
0
5
10
15
Frequency [kHz]
a)
20
25
0.88
0
5
10
15
Frequency [kHz]
b)
20
25
Figure 6: a) The frequency response function estimate between the input noise and
the output noise passed through the slip ring device at 3000 RPM, with the conductors
connected in pairs. And b) the corresponding coherence function estimate.
3.2
Setup Two, Control Path Estimation
The second experimental setup involves both the actuator and the sensors. First,
a control path estimate is conducted with one sensor at a time when the spindle is
not rotating, see Fig. 7. It is observable that the accelerometer sensor has better
signal quality compared to the piezo film in the frequency range of 200 Hz to 1000
Hz, see the coherence in Fig. 7 b). It should be noted that the accelerometer has
a known sensitivity of 100mV/g while the piezo film’s sensitivity is unknown. Also,
the voltage/force relation for the actuator is specific for the particular configuration
of the milling tool holder and has not been calibrated. Thus, the transfer function
estimates are presented as sensor voltage over actuator voltage.
Since the accelerometer provided the best performance, it was used for the control
path estimates for the various spindle speeds. Results of the control path estimates
from 100 RPM up to 1000 RPM in steps of 100 RPM are presented in Fig. 8. In Fig. 8
a) only small changes are observable in the control path frequency response function
estimates for different RPM. However, in the corresponding coherence functions in
Fig. 8 b) it is observable that the quality of the coherence is reduced in the frequency
range of 50 to 800 Hz for increasing RPM.
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Preliminary Investigation of Active Control of a Milling Tool Holder
1
10
0.9
0
0.8
2
Coherence γ̂yx
Magnitude [dB rel 1(V/V]
145
−10
0.7
0.6
−20
0.5
−30
0.4
0.3
−40
0.2
−50
Piezo−film & Actuator
Accelerometer & Actuator
−60
0
0.5
1
1.5
2
2.5
3
3.5
Frequency [kHz]
4
4.5
Piezo−film & Actuator
Accelerometer & Actuator
0.1
0
5
0
0.5
1
1.5
2
2.5
3
3.5
Frequency [kHz]
a)
4
4.5
5
b)
2
Coherence γ̂yx
Magnitude [dB rel 1(V/V]
Figure 7: a) The frequency response function estimate between the actuator and the
vibration sensor, when the spindle is not rotating. b) The corresponding coherence
function estimate.
0
1
0.8
−10
0.6
−20
0.4
−30
0.2
−40
−50
−60
0
1
2
3
Frequency [kHz]
a)
4
50
1000
800
600
400
200
RPM
0
1000
800
600
400
200
RPM
0
1
2
3
4
5
Frequency [kHz]
0
b)
Figure 8: a) Control path estimates between the actuator and the accelerometer,
when the spindle rotates from 100 RPM up to 1000 RPM in steps of 100 RPM. b)
The corresponding coherence function.
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146
3.3
Setup Three, Impact Test
1
50
Piezo−film
Accelerometer
40
0.9
0.8
2
Coherence γ̂yx
Magnitude [dB rel 1 V/N]
In the third setup, an experimental modal analysis was conducted using an impulse
hammer. Both sensors were used at the same time, since the spindle did not rotate.
The frequency response function estimate is presented as voltage over force from the
sensor signals. From the magnitude function in Fig. 9 a), the difference between strain
and acceleration with respect to frequency is obvious. It should also be observed that
the piezo-film is a capacitive sensor. With the help of the coherence function in Fig. 9
b), the signal quality of the sensors may be compared. The quality of the signal from
the accelerometer seems to be better when compared to the signal from the piezofilm. The piezo-film signal quality is degraded by a 50 Hz disturbance and related
harmonics, and this results in periodic dips in the coherence function.
0.7
30
0.6
0.5
20
0.4
10
0.3
0.2
0
Piezo−film
Accelerometer
0.1
−10
0
500
1000
1500
Frequency [Hz]
a)
2000
2500
0
0
500
1000
1500
Frequency [Hz]
b)
2000
2500
Figure 9: a) The frequency response function estimate between the impulse hammer
and the vibration sensors. b) The corresponding coherence function estimate.
3.4
Setup Four, Active Control
The results from the cutting experiment are presented as power spectral densities
of the acceleration signal from the milling tool holder, using two different frequency
resolutions. Also, a wide frequency range is presented together with zoomed-in power
spectral density estimates. Firstly, a resolution of δf = 12500/4096 ≈ 3 Hz was used
giving a fairly high resolution of the power spectral density estimates. As can be
seen in Fig. 10, the power spectral density estimates have several harmonics with a
distance of 700/60 ≈ 11.7 Hz which is related to the spindle speed. In the power
spectral density estimates in Fig. 10 a), an underlying dynamic system is observable.
In Fig. 10 b), the corresponding power spectral density estimates for the frequency
range of 590 Hz to 640 Hz are shown.
Secondly, a spectral resolution of 12500/256 ≈ 49 Hz was used in the power spectral density estimates of the milling tool holder acceleration and these spectra are
presented in Fig. 11.
PSD [dB rel 1 ((m/s2 )2 /Hz)]
PSD [dB rel 1 ((m/s2 )2 /Hz)]
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Preliminary Investigation of Active Control of a Milling Tool Holder
30
Without Control
With Control
25
20
15
10
5
0
−5
400
600
800
1000
Frequency [Hz]
1200
1400
147
40
Without Control
With Control
35
30
25
20
15
10
5
590 595 600 605 610 615 620 625 630 635 640
Frequency [Hz]
a)
b)
30
Without Control
With Control
25
20
15
10
5
0
−5
0
1000
2000
3000
Frequency [Hz]
a)
4000
PSD [dB rel 1 ((m/s2 )2 /Hz)]
PSD [dB rel 1 ((m/s2 )2 /Hz)]
Figure 10: a) Spectral densities estimates of the acceleration signal from the milling
tool holder without active control (solid line) and with active control (dashed line).
b) The corresponding power spectral density estimates zoomed in.
21
Without Control
With Control
20
19
18
17
16
15
14
13
12
500
600
700
800
900
1000
Frequency [Hz]
1100
1200
b)
Figure 11: a) Spectral densities estimates of the acceleration signal from the milling
tool holder without active control (solid line) and with active control (dashed line).
b) The corresponding spectra zoomed in.
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4
Summary and Conclusions
Preliminary results from experiments using an active milling tool holder have been
described in this paper. A method to transfer electrical signals and electrical power
to a rotating tool holder has been presented. The slip ring device seems to be a
successful way of transferring signals and power without the loss of any significant
signal quality. This is shown in Fig. 6, where the conductors have been connected in
pairs at the rotating side, while rotating with the maximum rotation speed of this
milling machine. Results concerning the control path estimates showed a slight loss of
signal quality. This may, however, be explained by the fact that the sensors pick up
other vibrations than those originating from the actuator. Thus, uncorrelated noise
affects the control path estimate when the spindle is rotating. In addition to this, the
impact test confirms the quality of the frequency response function estimates during
the measurements performed during rotation. During the experiments, the bending
of the milling tool holder, the acceleration close to the tool tips and the excitement
of the tool holder with the actuator were successfully sensed by the setup. When
it comes to the active control feasibility study, some attenuation of the milling tool
holder vibration has been introduced. However, this may well be rectified when more
work has been carried out to find the optimal performance of the suggested setup.
Acknowledgments
The present project is sponsored by the company Acticut International AB in Sweden
which has multiple approved patents covering active control technology.
References
[1] Milton C. Shaw. Metal Cutting Principles. Oxford University Press, second edition, 2005.
[2] Y. Altintas and Philip K. Chan. In-process detection and suppression of chatter
in milling. International Journal of Machine Tools and Manufacture, 32(3):329 –
347, 1992.
[3] T. S. Delio. Chatter recognition and control system. US Patent No. 5,170,358,
MLI Manufacturing Laboratories, Inc., Gainesville, FL, October 1992.
[4] Jeffrey L. Dohner, James P. Lauffer, Terry D. Hinnerichs, Natarajan Shankar,
Mark Regelbrugge, Chi-Man Kwan, Roger Xu, Bill Winterbauer, and Keith
Bridger. Mitigation of chatter instabilities in milling by active structural control. Journal of Sound and Vibration, 269(1-2):197 – 211, 2004.
[5] H. Åkesson, T. Smirnova, L. Håkansson, I. Claesson, and T. Lagö. Investigation of
the dynamic properties of a milling structure; using a tool holder with moderate
overhang. In Proceedings of the 27th International Modal Analysis Conference,
Orlando, 9-12 February 2009.
[6] J.S. Bendat and A.G. Piersol. Random Data Analysis And Measurement Procedures. John Wiley & Sons, third edition, 2000.
Part VI
Noise Source Identification and
Active Control in a Water Turbine
Application
This part is based on the publication:
H. Åkesson, A. Sigfridsson, T. Lagö, I. Andersson and L. Håkansson, Noise Source
Identification and Active Control in a Water Turbine Application, In Proceedings of
The Sixth International Conference on Condition Monitoring and Machinery Failure
Prevention Technologies, Dublin, Ireland, 23-25 June, 2009.
Noise Source Identification and Active
Control in a Water Turbine Application
Henrik Åkesson1,2 , Andreas Sigfridsson and Thomas T. Lagö
1
Acticut International AB
Gjuterivägen 6, 311 32 Falkenberg, Sweden
Ingvar Andersson
Turab AB Förrådsgatan 2, 571 39 Nässjö, Sweden
Lars Håkansson
Blekinge Institute of Technology,
Department of Signal Processing,
372 25 Ronneby, Sweden
2
November 25, 2009
Abstract
When optimizing a water turbine application for high efficiency, it is not
uncommon that disturbing vibrations occur. This may happen even when the
generator is not used at maximum power. The challenge is to find the root
cause of the high vibration levels. When the vibrations have started, more or
less everything vibrates. This paper presents an approach that has been utilized
in a real-life situation using multiple tools and a methodology to break down
the global vibrations into cause and effect. The paper can act as an example of
how modern vibration analysis and methodology can be utilized when vibration
levels are too high and the cause of these vibrations is uncertain. This paper
presents a vast range of data and shows multiple measurements that have been
performed. The paper also shows how the results from these measurements
can be used to locate the vibration source. In addition to this, a solution that
may resolve this problem is presented together with the attenuation result. The
successful attenuation of the structural vibration using the proposed solution
confirms the conclusions with the aid of the method presented in the paper.
1
Introduction
In Sweden, power plants are commonly situated along water courses. Today, there
are around 1900 hydroelectric power stations of various sizes, 700 producing a power
of 1.5 MW or more and 1200 small-scale power plants producing a power less than
1.5 MW. Together they generate approximately half of the Swedish electrical power
capacity. Hydroelectric power is a renewable energy resource and is thus considered
to be a so-called a green source of energy. It is therefore likely to be beneficial to the
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environment to use this sort of energy resource. By building dams, water is trapped
and can be controlled to flow through tunnels in the dam, turning turbines and driving
generators, see Fig. 1. However, when optimizing a water turbine application for high
efficiency, it is not uncommon that disturbing vibrations occur. Occasionally, the
vibrations cause damage to the power plant. Vibrations may be caused by various
blade configurations even when the generator is not used at maximum power. These
harmful vibrations have been experienced at several power plants and a solution to
this problem has been pursued.
P o w e r h o u se
R e s e rv o ir
P o w e r lin e s
G e n e ra to r
In ta k e
P e n s to c k
T u rb in e
R iv e r
Figure 1: Sketch of a hydroelectric power station.
In order to find a suitable solution to the problem, a first step is to identify
and understand the cause of the vibrations. The power plant station presented in
this paper is principally configured according to Fig. 1 and the real power house is
presented in Fig. 2.
2
Materials and Methods
The first step was to examine the vibration problem experienced in the hydroelectric
power plant. In order to get an appropriate and detailed description of the actual
problem, vibrations were measured and recorded during operation of the power plant
and then used to perform operating deflection shape (ODS) analysis [1, 2]. The operating deflection shapes of a structure provide information on the structure’s spatial
deformation, either for a certain time instance or for a certain frequency, depending
on the analysis method. In order to examine whether the problem identified from the
ODS analysis is related to the dynamic properties of the structure, an experimental
modal analysis was performed. Also, coarse Finite Element Models (FEM) were used
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Noise Source Identification and Active Control in a Water Turbine Application 153
—
Figure 2: The transformation station together with long-distance power lines (left
side in the photo), and power house where the turbine and the generator are located
(right side in the photo).
to optimize sensor locations and get an indication of what motions to expect. Finally,
after identifying the vibration problem, active control was implemented in order to
attenuate the vibrations and also to confirm whether the previous analysis was correct
or not.
The structures of interest are both the rotating turbine and its bearings along
with the supporting frame. The main bearings are inside the blue frame illustrated
in the model in Fig. 3 b).
2.1
Operating deflection shapes
To examine the behavior of the bearing frame during operation mode, the acceleration
at a number of different spatial locations of the structure (was measured simultaneously. By considering the phase and amplitude of the response signals from the
accelerometers on the operating structure, it is possible to produce an estimate of the
ODS. The amplitude is estimated by either auto-power spectra or auto-power spectral density estimates depending on whether the signal is tonal or random [3, 4]. The
phase between each spatial position is then estimated from the cross-power spectrum.
Thus, the expression of the ODS in terms of power spectra is given by [2]:
{ODS(f )}RM S =
n q
P̂11 (f )
q
P̂22 (f )ej θ̂21 (f )
···
q
P̂N N (f )ej θ̂N 1 (f )
oT
(1)
where P̂nn (f ) is the estimated power spectrum and ej θ̂n1 (f ) is the phase function
of the estimated cross-spectrum P̂n1 (f ) for n ∈ {2, . . . , N }.
For the ODS measurements, eleven different sensor locations were selected, six on
the top of the generator and five on the bottom side of generator frame, see Fig. 4
and Fig. 5.
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a)
b)
Figure 3: In a) the penstock is observable as the large dark tube going around the
red beams which support the concrete structure. In the center, the generator shaft
is located. Above this floor, the generator windings are located while below it the
turbine blades are turned by the water flow. In b) an overview of the whole generator
shaft (grey) and the main bearing frame (blue) is presented. The bottom side of the
bearing frame is reachable from the room presented in a).
a)
b)
Figure 4: a) The top cover of turbine one. Below the floor plates, the generator windings are rotating. b) Six accelerometers were glued to two different rigid structures
close to the shaft, measuring in the x- (left), y- (into picture) and z- (up) directions.
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The acceleration occurring at the measurement positions was measured simultaneously during various operating conditions. In connection with this, the following
conditions were examined:
1. Three power conditions: the turbine generating 2.6 MW, 1.5 MW and 3.1 MW.
2. Turbine standing still, (second turbine running in the background).
3. Turbine stopping, (shaft de accelerating).
4. Turbine starting, (shaft accelerating).
Figure 5: Five accelerometers measuring vibrations in the z-direction were glued on
the bottom side of the bearing frame.
2.2
Modal Analysis
The primary goal of experimental modal analysis is to identify the dynamic properties
by obtaining the modal parameters of the system under examination; i.e. to determine
the natural frequencies, mode shapes and damping ratios from experimental vibration
measurements [1]. The frequency transfer function Ĥ(fk ) between the input signal
x(n) and the output signal y(n) for a dynamic system may be estimated according
to [4]:
Ĥ(fk ) =
P SD
(fk )
P̂yx
P SD (f )
P̂xx
k
, fk =
k
Fs
N
(2)
P SD
where 0 6 k 6 N/2.56, F s is the sampling frequency, P̂yx
(fk ) is the crosspower spectral density between the input signal x(n) and the output signal y(n), and
P SD
(fk ) is the power spectral density for the input signal x(n).
P̂xx
In the previous ODS measurements, a number of accelerometers were used. However, in the experimental modal analysis, an excitation force has to be produced in
order to be able to estimate the frequency response functions. This is normally done
using an impulse hammer or a shaker. In this case, when the structure under investigation is large and relatively rigid, a shaker is preferable in order to be able to
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inject sufficient vibration energy into the structure. The largest shaker available at
the time was used for experimental modal analysis and was mounted to one of the
beams of the bearing frame (position 3 in Fig. 5). The force applied to the structure
was measured simultaneously with the accelerometers previously described so that
the frequency response estimates could be produced.
a)
b)
Figure 6: a) An overview of the shaft, the bearing frame (covered with plates) and a
shaker attached to one of the rigid beams. b) A close-up illustrating how the shaker
was attached to the bearing frame.
2.3
Actuator Setup
The actuators used in the implementation of active control were inertia mass actuators
and are presented in Fig. 7 a). They can give a maximum force of 5200 N and have
been tuned to work in the range of 18 to 26 Hz, see Fig. 7 b). This results in a
resonance frequency of 22 Hz, a vibrating mass of 38.5 kg and a spring stiffness of
736 N/mm.
Two different positions were tested in order to evaluate the performance. Three
actuators were first mounted on the top of the generator structure, see Fig. 8 a), and
then secondly two actuators were positioned on the bottom side of the bearing frame,
see Fig. 8 b) and c). As a reference signal to the controller, an accelerometer placed
beside one of the actuators was used. During the active control measurements, two
different operating conditions were used, first when the generator produced 3.2 MW
and secondly when the generator produced 3.4 MW.
2.4
The Active Control Algorithm
As a digital controller, a single channel feedback filtered-x LMS algorithm implemented in DSP was used. The algorithm is suitable in this application and gives a
robust solution [5]. The block diagram of the feedback filtered-x LMS algorithm is
shown in the Fig. 9 and is described by equations (3-7).
The feedback filtered-x LMS algorithm with a leakage coefficient is defined by the
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Noise Source Identification and Active Control in a Water Turbine Application 157
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6000
Force − springs
Force − motor
Force − actuator
5000
Force [N]
4000
3000
2000
1000
0
10
20
30
a)
40
50
Frequency [Hz]
60
70
80
b)
Figure 7: a) A picture of one of the inertial actuators mounted on the bearing frame.
b) The force response of the actuators.
following equations:
y(n) = wT (n)x(n)
e(n) = d(n) + yC (n)
w(n + 1)
(3)
(4)
= γw(n) − µxĈ (n)e(n)
(5)
where µ is the adaptation step size and
xĈ (n) =
"I−1
X
i=0
ĉi x(n − i), . . . ,
I−1
X
i=0
#T
ĉi x(n − i − M + 1)
(6)
is the filtered reference signal vector, which usually is produced by filtering the
reference signal x(n) with an FIR-filter estimate ĉ(i), i ∈ 0, 1, . . . , I − 1 of the forward
path. w(n) is the adaptive FIR filter coefficient vector, y(n) is the output signal from
the adaptive FIR filter, e(n) is the error signal, measured by the accelerometer, yC (n)
is the secondary vibration, Ĉ is the estimate of the forward path, d(n) is the primary
vibration, x(n) = [x(n), . . . , x(n − M + 1)] is the reference signal vector and x(n) is
related to the delayed error signal as
x(n) = e(n − 1).
(7)
This algorithm uses an estimate of the forward path (D/A converter, amplifier,
actuator and structural path) to produce an adequate direct gradient estimate that
enables a minimization of the mean square error in the control application. The error
signal is produced by the sum of the primary vibration signal and the secondary
vibration induced by the actuator and transferred through the forward path. The
estimate of the forward path was made with the use of the LMS algorithm prior to
the implementation of active control.
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a)
b)
c)
Figure 8: a) The actuators mounted from the top of the generator frame. b) The
actuators mounted from the bottom side of the bearing frame where c) is the same
configuration as in b) illustrating how the actuators are supposed to act on the bearing
frame.
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Noise Source Identification and Active Control in a Water Turbine Application 159
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F IR filte r
w (n )
x (n )
E s tim a te o f th e
fo rw a rd p a th Cˆ
x
Cˆ
(n )
A d a p tiv e
a lg o rith m
y (n )
F o rw a rd p a th
C
y C (n )
d (n )
å
e (n )
D e la y
Figure 9: Block diagram of the feedback filtered-x LMS algorithm.
3
Results
The results from the ODS measurements are presented as linear power spectra since
the responses contain significant tonal components. The operating deflection shape is
presented for the ”first mode” as well as for the finite element model analysis. For the
experimental modal analysis result, a frequency response estimate is presented. For
the active control setup, some representative results of vibration with and without
control in form of power spectra are presented.
3.1
Vibration Spectrum and Operating Deflection Shape
Initially, vibration levels for three different power operating conditions of the power
plant are presented, see Fig. 10. Two peaks that are more significant than the others
may be observed here, one at 21 Hz and a second peak at approximately 800 Hz.
Since the spectra has the unit acceleration and it is the level of displacement which
is of importance, the first peak is larger and thus likely to be much more harmful in
comparison to the second peak. This is the case because the acceleration is the second
derivative of displacement and thus the magnitude of the displacement will decline
with a factor of (j2πf )2 with the frequency when compared with the acceleration.
The frequency of 21 Hz is the same as the blade wheel frequency.
The power spectra from the accelerometers placed on the bottom side of the
bearing frame generated when the turbine is running and when it has been stopped
is presented in Fig. 11. It is clear that the vibrations have almost disappeared and
the residual vibrations originate from the second turbine in the power plant station.
It may also be observed that the peak at 800 Hz is almost unnoticed by the sensors
located on the bottom side of the bearing frame.
The operating deflection shape is presented in Fig. 12 and illustrates clearly the
larger motion towards the center as well as the fact that both sides operate in phase.
3.2
Finite Element Model
The first bending mode from the finite element model of the bearing frame shows the
same behavior as measured during the operating modes, i.e. the largest motion is in
the center of the frame, see Fig. 13.
Lin. Spectrum [m/s2 rms]
Z1+,2.6MW
Z2+,2.6MW
Z1+,1.5MW
Z2+,1.5MW
Z1+,3.1MW
Z2+,3.1MW
0.6
0.5
0.4
0.3
0.2
0.1
0
200
400
600
800
Frequency [Hz]
1000
Lin. Spectrum [m/s2 rms]
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0.6
0.5
0.4
0.3
0.2
0.1
0
15
1200
Z1+,2.6MW
Z2+,2.6MW
Z1+,1.5MW
Z2+,1.5MW
Z1+,3.1MW
Z2+,3.1MW
16
17
18
19
20
21
22
Frequency [Hz]
a)
23
24
25
b)
Z6−, 2.6MW
Z6−, Turbine still
0.25
0.2
0.15
0.1
0.05
0
200
400
600
800
Frequency [Hz]
a)
1000
1200
Lin. Spectrum [m/s2 rms]
Lin. Spectrum [m/s2 rms]
Figure 10: a) Power spectrum from two positions on top of the generator for three
different power operating modes of the power plant. b) Zoomed-in power spectrum.
0.25
Z6−, 2.6MW
Z6−, Turbine still
0.2
0.15
0.1
0.05
0
15
16
17
18
19
20
21
22
Frequency [Hz]
23
24
25
b)
Figure 11: a) Power spectrum from an accelerometer glued to the bottom side of the
bearing frame when the turbine is running and when the turbine is off. b) Zoomed-in
power spectrum.
Figure 12: The operating deflection shape at 21 Hz. The length of the arrows represents the magnitude of the displacement measured by the sensors positioned at the
locations indicated by the red circles.
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Noise Source Identification and Active Control in a Water Turbine Application 161
—
a)
b)
Figure 13: A finite element model, a) the un-deformed body and in b) the deformed
body together with the ”wired” un-deformed body.
3.3
Modal Analysis
Accelerance [dB rel 1(m/s2 )/N]
The frequency response function estimated from the signals obtained during the experimental modal analysis contained a lot of noise. This was due to the relatively
rigid structure as compared to the force capacity of the shaker. However, it is possible
to get an indication of the dynamics of the structure. Two peaks at 21 and 23 Hz are
observable, see Fig. 14.
−40
−50
−60
−70
−80
−90
−100
−110
−120
10
15
20
25
30
35
40
45
Frequency [Hz]
50
55
60
Figure 14: The magnitude of the accelerance function estimate between the position
where the shaker is attached (position 3) and the accelerometer close to the shaft
(position 6), see Fig. 6.
3.4
Active Control
An overview of the vibration spectra is presented in Fig. 15 to show the levels of
vibration with and without attenuation. The vibrations are from an accelerometer
mounted on the top of the generator denoted Z7+. It should be observed that the
active control system does not handle the vibration peak at 800 Hz, but attenuates
the intended vibration peak at 21 Hz by an order of six.
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1.2
Lin. Spectrum [m/s2 rms]
Lin. Spectrum [m/s2 rms]
Z7+,3.4MW, Control Off
Z7+,3.4MW, Control On
1
0.8
0.6
0.4
0.2
0
Z7+,3.4MW, Control Off
Z7+,3.4MW, Control On
1
0.8
0.6
0.4
0.2
0
780
100 200 300 400 500 600 700 800 900 1000
Frequency [Hz]
a)
Lin. Spectrum [m/s2 rms]
0
1.2
785
790
795
Frequency [Hz]
800
805
b)
0.9
0.8
Z7+,3.4MW, Control Off
Z7+,3.4MW, Control On
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
15
16
17
18
19
20
21
22
Frequency [Hz]
23
24
25
c)
Figure 15: a) The spectra of vibration signals from the top of the generator with and
without control. b) zoomed-in spectra that show the peak at 800 Hz. c) zoomed-in
spectra that show the vibration at 21 Hz during attenuation.
In Fig. 16, the vibration levels are presented with and without control for two different actuator configurations. The measured signal is from an accelerometer mounted
on top of the generator denoted Z9+.
4
Summary and Conclusions
In this paper, an effective method that is able to identify vibration sources has been
presented. An experiment was conducted where large vibrations were measured during various operating conditions, showing a stable tonal component at 21 Hz. The
dynamics of the bearing structure indicate a dynamic weakness at the same frequency
the shaft is excited with. Also, a solution of how to attenuate the vibrations has been
proposed and a first implementation has demonstrated the successful attenuation of
these vibrations. However, the performance of the active system may be enhanced
further. For example, the number of actuators and the size of each actuator, the
number of sensors used in the control algorithm as well other parameters in the digital controller, are still interesting to evaluate in order see what is possible to achieve
while maintaining costs at a minimum.
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Noise Source Identification and Active Control in a Water Turbine Application 163
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Lin. Spectrum [m/s2 rms]
1
0.9
0.8
Z9+,3.2MW, Control Off, Actuator Up
Z9+,3.2MW, Control On, Actuator Up
Z9+,3.2MW, Control Off, Actuator Down
Z9+,3.2MW, Control On, Actuator Down
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
15
16
17
18
19
20
21
22
23
24
25
Frequency [Hz]
Figure 16: Spectrum of an accelerometer signal from an accelerometer glued to the top
side of the generator (denoted Z9+), for four different configurations. The actuators
have been positioned on the top side of the generator and to the bottom side of the
bearing frame, while active control has been active and turned off.
Acknowledgments
The present project is sponsored by the company Acticut International AB in Sweden
which has multiple approved patents covering active control technology. The project
has been performed in collaboration with Turab AB in Sweden.
References
[1] D.J. Inman. Engineering Vibration. Prentice-Hall, second edition, 2001.
[2] L. Andrén, L. Håkansson, A. Brandt, and I. Claesson. Identification of motion of
cutting tool vibration in a continuous boring operation - correlation to structural
properties. Journal of Mechanical Systems & Signal Processing, 18(4):903–927,
2004.
[3] F. Harris. On the use of windows for harmonic analysis with the discret fourier
transform. In Proc. of the IEEE, volume 66, 1978.
[4] J.S. Bendat and A.G. Piersol. Random Data Analysis And Measurement Procedures. John Wiley & Sons, third edition, 2000.
[5] S.M. Kuo and D.R. Morgan. Active Noise Control Systems. Telecommunications
and Signal Processing. Wiley Interscience, 1996.
Furthermore, vibration in milling has also been
studied in relation to milling tool holders with a
long overhang.A basic investigation concerning the
spatial dynamic properties of the tool holders of
milling machines, both when not cutting and during
cutting, has been carried out. Also, active control
of milling tool holder vibration has been investigated and a first prototype of an active milling tool
holder was implemented and tested.The challenge
of transferring electrical power while maintaining
good signal quality to and from a rotating object is
addressed and a solution to this is proposed.
Finally, vibration is also a problem for the hydroelectric power industry. In Sweden, hydroelectric power plants stand for approximately half of
Sweden’s electrical power production and are
also considered to be a so-called green source of
energy. When renovating water turbines in smallscale hydroelectric power plants and modifying
them to optimize efficiency, it is not uncommon
that disturbing vibrations occur in the power
plant. These vibrations have a negative influence
on the production capacity and will wear various
components quickly. Occasionally, these vibrations
may cause severe damage to the power plant. To
identify this vibration problem, experimental modal analysis and operating deflection shape analysis
were utilized. To reduce the vibration problem, active control using inertial mass actuators was investigated. Preliminary results indicate a significant
attenuation of the vibrations.
Henrik Åkesson
ISSN 1653-2090
ISBN 978-91-7295-172-3
2009:05
2009:05
Analysis of Structural Dynamic Properties
Vibration in metal cutting is a common problem in
the manufacturing industry, especially when long
and slender tool holders or boring bars are involved in the manufacturing process. Vibration has a
detrimental effect on machining. In particular the
surface finish is likely to suffer, but tool life is also
most likely to be reduced. Tool vibration also results in loud noise that may disturb the working
environment.
The first part of this thesis describes the development of a robust and manually adjustable
analog controller capable of actively controlling
boring bar vibrations related to internal turning.
This controller is compared with an adaptive
digital feedback filtered-x LMS controller and it
displays similar performance with a vibration attenuation of up to 50 dB.
A thorough experimental investigation of the
influence of the clamping properties on the dynamic properties of clamped boring bars is also
carried out in second part of the thesis. In relation to this, it is demonstrated that the number
of clamping screws, the clamping screw diameter
size, the screw tightening torque and the order
the screws are tightened, have a significant influence on a clamped boring bar’s eigenfrequencies
as well as on its mode shape orientation in the
cutting speed - cutting depth plane. Also, an initial
investigation of nonlinear dynamic properties of
clamped boring bars was carried out.
and Active Vibration Control Concerning Machine Tools and a Turbine Application
ABSTRACT
Analysis of Structural Dynamic
Properties and Active Vibration
Control Concerning Machine
Tools and a Turbine Application
Henrik Åkesson
Blekinge Institute of Technology
Doctoral Dissertation Series No. 2009:05
School of Engineering
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