InvestIgatIon of the DynamIc ProPertIes of a mIllIng tool holDer

InvestIgatIon of the DynamIc ProPertIes of a mIllIng tool holDer
Investigation of the Dynamic Properties of a Milling Tool
Holder
Investigation of the Dynamic Properties
of a Milling Tool Holder
Henrik Åkesson, Tatiana Smirnova, Lars Håkansson, Thomas Lagö, Ingvar Claesson
Henrik Åkesson, Tatiana Smirnova, Lars
Håkansson, Thomas Lagö, Ingvar Claesson
Blekinge Institute of Technology
Research report No. 2009:07
Copyright © 2009 by individual authors. All rights reserved.
Printed by Printfabriken, Karlskrona 2009.
ISSN 1103-1581
ISRN BTH-RES–07/09–SE
Investigation of the Dynamic
Properties of a Milling Tool Holder
H. Åkesson, T. Smirnova, L. Håkansson,
T. Lagö and I. Claesson
Blekinge Institute of Technology
Research Report No 2009:07
ISSN: 1103-1581, November, 2009.
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
3
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 . . . . . . . . .
4 Summary and Conclusion
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37
4
1
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
1.1.2
5
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
6
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
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
Investigation of the Dynamic Properties of a Milling Tool Holder
7
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.
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
8
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
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.
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].
Investigation of the Dynamic Properties of a Milling Tool Holder
1.3
9
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
10
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
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,
Investigation of the Dynamic Properties of a Milling Tool Holder
11
Figure 6: The DMU 80FD Duoblock milling machine.
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.
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
12
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
13
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
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
14
2
()
()
2
1()
2 ()
S p in d le fra m e
W o rk p ie c e
1
T o o l
3()
()
3
T o o l h o ld e r
()
2
()
3
()
M illin g ta b le
4
()
4 ()
4 ()
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
k
F s is the discrete frequency,
signal x(n), may be produced [38, 40], where fk = N
k = 0, . . . , N − 1, where N is the length of the data segments used to produce the
periodograms and fs is the sampling frequency.
Investigation of the Dynamic Properties of a Milling Tool Holder
15
a)
b)
Figure 10: The experimental modal analysis setup in the Hurco BMC-50 milling
machine.
a)
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.
16
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
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],
−1
.
(4)
Ĥ(fk ) = P̂yx (fk ) P̂xx (fk )
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 =
P̂11 (f )
P̂22 (f )e
j θ̂21 (f )
···
T
j θ̂N 1 (f )
e
.
P̂N N (f )
(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
Investigation of the Dynamic Properties of a Milling Tool Holder
17
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
which
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],
N
Qr {ψ}r {ψ}Tr
Q∗ {ψ}∗r {ψ}H
r
+ r
Ĥ(f ) =
∗
j2πf − λr
j2πf − λr
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
{ψ}
k
l
(7)
M ACkl =
H {ψ}
{ψ}H
{ψ}
{ψ}
k
l
k
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.
18
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
Parameter
Excitation signal
Sampling frequency f s
Block length N
Frequency resolution ∆f
Number of averages L
Burst length
Window
Overlap
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
2.6
19
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.
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.
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
20
2.6.2
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
∂ 2 u(z, t)
∂ 2 u(z, t)
= f (z, t)
(8)
+ 2 EIx (z)
ρA(z)dz
∂t2
∂z
∂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.
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) =0
(9)
u(z, t)|z=0 = 0,
∂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
∂ 3 u(z, t) ∂ 2 u(z, t) = 0,
EI
= 0.
(10)
EI
∂z 2 z=l
∂z 3 z=l
Investigation of the Dynamic Properties of a Milling Tool Holder
2.6.3
21
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 4
Ro − 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
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
22
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.
80
Acceleration [m/s2 ]
Acceleration [m/s2 ]
100
60
40
20
0
−20
−40
−60
−80
−100
0
60
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
Investigation of the Dynamic Properties of a Milling Tool Holder
23
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
PSD [dB rel 1 ((m/s2 )2 /Hz)]
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)]
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
24
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
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)]
PSD [dB rel 1 ((m/s2 )2 /Hz)]
Investigation of the Dynamic Properties of a Milling Tool Holder
25
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.
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.
Node position on the z-axis [cm]
100
Node 2
Node position on the z-axis [cm]
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
26
Node 1
80
60
Node 3
40
20
Node 4
0
−60
−40
−20
0
20
40
60
100
Node 2
80
60
Node 3
40
20
Node 4
0
−40
Node position on the y-axis [cm]
a)
Node 1
−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.
Investigation of the Dynamic Properties of a Milling Tool Holder
0
−10
Phase [degree]
Magnitude [dB] [cm]
0
−20
−30
−40
−50
−1000
−2000
−3000
−4000
−5000
−60
−70
0
27
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.
Node position on the z-axis [cm]
100
Node 2
Node position on the z-axis [cm]
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
28
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
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)
Vibration angle [degree]
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.
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.
Vibration angle [degree]
Investigation of the Dynamic Properties of a Milling Tool Holder
0.1
29
Upper disc
Middle disc
Lower disc
0.05
0
−0.05
−0.1
21
22
23
24
25
Revolutions
26
27
28
29
Vibration angle [degree]
a)
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.
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
30
 
a)
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
3.2
31
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
Accelerance [dB rel 1(m/s2 )/N]
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]
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
32
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.
1
0.9
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
Investigation of the Dynamic Properties of a Milling Tool Holder
33
1
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.
34
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
the two first mode shapes for the respective milling machine setup in the form of
magnitude and angle.
a)
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
Investigation of the Dynamic Properties of a Milling Tool Holder
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
35
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.
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
36
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
M
od
eΨ
[H
z]
1611
1518
1350
1305
1141
986
921
783
k
650
615
z]
de
Mo
Ψ
[H
0
744
755
810
926
993
1064
1128
1510
l
1693
M
od
eΨ
[H
z]
a)
1693
1510
1128
1064
993
926
810
k
755
744
]
de
Mo
Ψ
[Hz
b)
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
37
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
38
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
a)
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.
Investigation of the Dynamic Properties of a Milling Tool Holder
39
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
40
H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson
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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Investigation of the Dynamic Properties of a Milling Tool
Holder
Investigation of the Dynamic Properties
of a Milling Tool Holder
Henrik Åkesson, Tatiana Smirnova, Lars Håkansson, Thomas Lagö, Ingvar Claesson
Henrik Åkesson, Tatiana Smirnova, Lars
Håkansson, Thomas Lagö, Ingvar Claesson
Blekinge Institute of Technology
Research report No. 2009:07
Copyright © 2009 by individual authors. All rights reserved.
Printed by Printfabriken, Karlskrona 2009.
ISSN 1103-1581
ISRN BTH-RES–07/09–SE
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