Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping Conditions

Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping Conditions
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö
and Lars Håkansson
March 2007
Abstract
The boring bar is one of the most widely used type of tool holders in metal
cutting operations. The turning process subjects the tool to vibrations, and
cutting in deep workpiece cavities is likely to result in high vibration levels.
The consequences of such vibration levels generally results in: reduced tool
life, poor surface finishing and disturbing sound. Internal turning frequently
requires a long and slender boring bar in order to machine inside a cavity, and
the vibrations generally become highly correlated with one of the fundamental
bending modes of the boring bar. Different methods can be applied to reduce
the vibrations, the implementation of the most efficient and stable methods
require in depth knowledge concerning the dynamic properties of the tooling
system. Furthermore, the interface between the boring bar and the clamping
house has a significant influence on the dynamic properties of the clamped
boring bar. This report focuses on the dynamic properties of a boring bar that
arise under different clamping conditions of the boring bar and are introduced
by a clamping house (commonly used in the manufacturing industry). The
dynamic properties of a boring bar (for different cases of boundary condition
of the boring bar) are presented partly analytically but also experimentally.
Contents
1 Introduction
5
2 Materials and Methods
2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . .
2.1.1 Measurement Equipment and Setup . . . . . . .
2.1.2 Boring Bars . . . . . . . . . . . . . . . . . . . .
2.1.3 Clamping Houses . . . . . . . . . . . . . . . . .
2.1.4 Clamping Conditions . . . . . . . . . . . . . . .
2.2 Experimental Modal Analysis . . . . . . . . . . . . . .
2.2.1 Spectral Properties . . . . . . . . . . . . . . . .
2.2.2 Parameter Estimation . . . . . . . . . . . . . .
2.2.3 Excitation Signal . . . . . . . . . . . . . . . . .
2.3 Analytical Models of the Boring Bars . . . . . . . . . .
2.3.1 Multi-span beam . . . . . . . . . . . . . . . . .
2.3.2 Linearized Model . . . . . . . . . . . . . . . . .
2.3.3 Multi-span Boring bar with Elastic Foundation
2.3.4 Screws - Elastic Foundation . . . . . . . . . . .
2.3.5 Spring Coefficients and Clamping Forces . . . .
2.4 Nonlinear Model . . . . . . . . . . . . . . . . . . . . .
2.4.1 Nonlinear Synthesis . . . . . . . . . . . . . . . .
2.4.2 Ordinary Differential Equation Methods . . . .
2.4.3 Filter Method . . . . . . . . . . . . . . . . . . .
2.4.4 Excitation Signal . . . . . . . . . . . . . . . . .
3 Results
3.1 Experimental Modal Analysis .
3.1.1 Standard Boring Bar . .
3.1.2 Active Boring Bar . . .
3.1.3 Linearized Boring Bar .
3.1.4 Mode shapes . . . . . .
3.1.5 Quality of Measurement
3
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3.2
3.3
3.1.6 Mass-loading . . . . . . . . . . . . . . .
3.1.7 Summary of the Estimated Parameters .
Analytical Models of the Boring Bars . . . . . .
3.2.1 Single-span Model . . . . . . . . . . . .
3.2.2 Multi-span Model . . . . . . . . . . . . .
3.2.3 Multi-span Model on Elastic Foundation
Computer Simulations of Nonlinear Systems . .
3.3.1 Softening Spring Model . . . . . . . . . .
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62
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4 Summary and Conclusions
83
5 Appendix A
88
Chapter 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, whilst also
resulting in severe environmental issues such as high noise levels. By applying,
for example, an active control scheme, these vibrations can be significantly
reduced, with the result of improved workpiece surface finish and increased
tool life [1]. In order to successfully implement such a scheme, the dynamic
properties of system (boring bar - clamping structure) be known, as must the
nature of the disturbing vibrations.
A number of experimental studies have been carried out on mechanisms explaining tool vibration during turning operations [2, 3, 4] and on the dynamic
properties of boring bars [5, 6, 7]. In 1946 the principles of the traditional theory of chatter in simple machine-tool systems were worked out by Arnold [8]
based on experiments carried out on a rigid lathe, using a stiff workpiece but
a flexible tool. In this way he was able to investigate chatter under controlled
conditions. Later in 1965 Tobias [4] presented further investigations of the
chatter phenomena, involving, for example, the chip-thickness variation and
the phase lag of the undulation of the surface. Also, in the same year, Meritt
et al. [2] 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. [9] 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 rel-
5
6
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
ative to the two planes of vibrations. Pandit et al. [10] 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. [11]
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. [5] 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. [6] includes variation of
chip cross-sectional area in their model, whilst Kuster et al. [12] developed a
computer simulation based on a three-dimensional model of regenerative chatter. Walter et al. [13] developed a model of the chuck-workpiece connection
when the workpiece is considered to be weak, using Finite Element Method
(FEM) and experimental studies of a ring shaped weak workpiece; this model
focused on the influence from clamping forces when using jaw chucks. A time
series approach was used by Andrén et al. [7] to investigate boring bar chatter and the results were compared with an analytical Euler-Bernoulli model.
Also, Euler-Bernoulli beam modeling, experimental modal analysis and operating deflection shape analysis were used by Andrén et al. [14] to investigate
the dynamic properties of a clamped boring bar. Results obtained demonstrate
observable differences concerning the fundamental bending modes. They found
that that the bending motion of the first two resonance frequencies is, to a large
extent, in the direction of cutting speed. Scheuer et al. [15] investigated the
dynamic properties of a boring bar, based on experimental modal analysis under different clamping conditions. Two different clamping houses were used:
one clamping from two sides with clamp screws and one circular clamping
sleeve; clamping along the circular surface of the boring bar. Results indicate
that both the eigenfrequencies and the directions of the fundamental bending modes vary for different clamping pressures; in particular, for the circular
clamping sleeve.
The problem of boring bar vibration can be addressed using conventional
methods, such as redesigning the machine tool system, implementing tuned
passive damping or implementing active control [16, 17]
However, the order of stability improvement achieved usually correlates to
the quality and extent of knowledge of the dynamic properties of the tooling
structure -the interface between the cutting tool, or insert, and the machine
tool. Boring bar vibrations are usually directly related to the lower order
bending modes and the dynamic properties of a boring bar installed in a lathe
are directly influenced by the boundary conditions, i.e. the clamping of the
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
7
bar [14]. Following the literature review, it appears that little work has been
done on the clamping properties’ influence on the dynamic properties of a
clamped boring bar. Thus, it is of significance 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. This report focuses on 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 house
of the variety commonly used in industry today. The clamping house 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. To investigate the influence of clamping properties on the dynamic
properties of a clamped boring bar, experimental modal analysis have been
conducted both for a clamped standard boring bar and a clamped active boring
bar under different clamping conditions. Also, analytical Euler-Bernoulli beam
models incorporating clamping flexibility through the use of transverse springs
and rotational springs have been investigated for the modeling of a clamped
boring bar. Finally, some simulations of nonlinear models have also been
studied for observed nonlinear behavior of the clamped boring bar.
Chapter 2
Materials and Methods
Experimental modal analysis has been carried out on different boring bars
for various clamping conditions in order to investigate the changes of the the
clamped boring bar’s dynamic properties.
2.1
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. The
lathe is presented by the photo in Fig. 2.1
y
z
a)
x
b)
Figure 2.1: a) Mazak SUPER QUICK TURN - 250M CNC lathe and b) the
room in the lathe where machining is carried out.
8
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
9
Initially, a right-hand cartesian coordinate system was defined: z in the
feed direction, y in the reversed cutting speed direction and x in the direction
of cutting depth, see Fig. 2.1 b) (upper left corner). Subsequently a sign
convention was defined for use throughout the report. 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, see Fig. 2.2.
y
z
x
y
z
x
Figure 2.2: Right hand definition of the cartesian coordinate system and the
sign convention, where direction shown by the arrows defines the positive direction of displacement, force and moment.
The boring bars were positioned in the operational position, mounted in a
clamping house 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. 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 Brüel & Kjǽr 333A32 accelerometer
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
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
10
•
•
•
•
•
•
•
A custom designed amplifier for capacitive loads.
Active boring bar with embedded piezo ceramic actuator.
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 onto the boring bars with X60 glue (a cold hardener two component
glue). The sensors were evenly distributed along the centerline, on the underside and on the back-side of the boring bar; six accelerometers and one stud
on the respective side (see Figs. 2.3 and 2.4). 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-) see Fig. 2.3. The shakers were positioned relatively close
to the cutting tool.
y
z
a)
x
b)
Figure 2.3: The experimental setup, in a) two shakers suspended from the
ceiling are observable as well as a workpiece and the turret. b) shows a closeup
of the sensors and the shaker configuration on the boring bar.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
B o tto m
V ie w
11
l4
x
l8
l8
l9
z
l5
l6
l1
l2
C e m e n t stu d s
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 2.4: 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 B1 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 = 17, l6 = 100, l7 = 18.5,
l8 = 25 and l9 = 35.
12
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
2.1.2
Boring Bars
Two different boring bars were used in the experimental setup in order to be
able to analyze the changes of the eigenfrequencies and the mode-shapes in
different cases. The first boring bar used in the modal analysis was a standard
”non-modified” boring bar, WIDAX S40T PDUNR15F3 D6G, presented in
Fig. 2.5. The second boring bar used in this experiment was an active borz
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 2.5: 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.”
ing bar, based on the standard WIDAX S40T PDUNR15 boring bar, with an
accelerometer and an embedded piezo-stack actuator, see Fig. 2.6. 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 metal-chips from the material removal process from damaging the accelerometer. The actuator was
embedded into a milled space in the longitudinal direction (z-direction), below
the centerline of the boring bar. By embedding accelerometers and piezo stack
actuators in conventional boring bars, a solution was obtained for the introduction of control force to the boring bar with physical features and properties
that fit the general lathe application. Assuming a constant cross-section along
the boring bar, neglecting the head, the dimensions from Fig. 2.5 b) result in
a cross-sectional area A and a moment of inertia Ix , Iy presented by Table 2.1.
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), see Table 2.2 for material properties.
2.1.3
Clamping Houses
The clamping house is a basic 8437-0 40mm Mazak holder, presented in Fig. 2.7
a) and b), and clamps the tool holder by means of either four or six screws:
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
T o o l
13
y
B o rin g B a r
z
A c c e le ro m e te r
x
A c tu a to r
Figure 2.6: The active boring bar with an accelerometer close to the tool tip
and an embedded piezo-stack actuator in a milled space below the centerline.
Variable
A
Ix
Iy
Value
1.19330295 · 10−3
1.13858895 · 10−7
1.13787080 · 10−7
Unit
m2
m4
m4
Table 2.1: Cross-sectional properties of the boring bar, illustrated in Fig. 2.5
b), where A is the cross-sectional area and Ix , Iy are the moments of inertia
around the denoted axis.
Material composition besides Fe in percent
C
Cr
Ni
Mo
Si
Mn
0.26-0.33 1.80-2.20 1.80-2.20 0.30-0.50 <0.40 <0.60
Material properties
Young’s Modulus
Tensile Strength
Yield Strength
205 GPa
1250 MPa
1040 MPa
S
<0.035
P
<0.035
Density
7850 kg/m3
Table 2.2: Composition and properties of the material 30CrNiMo8.
14
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
two/three from the top and two/three from bottom. The basic holder itself is
mounted onto the turret with four screws.
In addition to the screws, the clamping house also features a guide matching
a track on the turret; this guide positions the clamping house along the z-axis
on the xy plane, whilst the guide pin positions the clamping house on the zaxis, see Fig. 2.7 a). The clamping house has a default thread size of M8 for the
S c re w p o s itio n s fo r a tta c h in g
to th e tu rre t
S c re w p o s itio n s fo r c la m p in g
o f th e b o rin g b a r
y
x
y
G u id e p in
z
z
G u id e
a)
x
b)
Figure 2.7: The clamping house. a) The guide and the guide pin may be
observed on the one word of the clamping house, 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 house to the turrets are shown from the
top side.
screws clamping the boring bar. A second clamping house of the same model
was rethreaded to the thread size M10. Furthermore, a third clamping house,
also of the same type, was used in the construction of a so-called ”linearized”
clamping of the boring bar.
2.1.4
Clamping Conditions
A number of different setups were considered using different boring bars described in section 2.1.2 in conjunction with the different clamping houses. In
the first setup, the reference boring bar was clamped using four M8 class 8.8
screws. The screws were tightened first from the top and then from the underside.
The recommended tightening torque for this class is 26.6Nm, however,
evaluations of the screws revealed that threads remained intact and screws
did not break for a tightening torque of 30Nm.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
15
The second setup involved the same five torques as for the previous setup,
but four screws of size M10 class 8.8, which were, again, tightened first from
the top and then from the underside. As only four clamp screws were used,
the clamping house center screw positions where not used.
The third setup involved the use of six screws of size M10, with the reference
bar and same torques as previous. The use of six screws involved the use of all
clamping house center screw positions. Setup one and two were then repeated,
using the active boring bar.
In order to accomplish a linearized clamping condition, the standard clamping was modified. 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 was 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
house and glued to it with epoxy to make the clamping rigid, see Fig 2.8.
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
x
W e d g e s
Figure 2.8: The linearized boring bar-clamping house setup.
2.2
Experimental Modal Analysis
The primary goal of experimental modal analysis is to identify the dynamic
properties of the system under examination, the modal parameters; i.e. 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 form these data, also known as curve fitting [18].
16
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
These procedures are often referred to as a discipline of art since the process
of acquiring good data and performing accurate parameter identification is an
iterative process, based on various assumptions along the way [18].
2.2.1
Spectral Properties
Non-parametric spectrum estimation may be utilized to produce non-parametric
linear least-squares estimates of dynamic systems [19].
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 [20], given by:
−1
2
L−1 N
1
k
−j2πnk/N P̂xx (fk ) =
w(n)xl (n)e
(2.1)
, fk = Fs
n= 0
LN Fs
N
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, Fs is the sampling frequency.
Thus, for each input signal x(t) and output signal y(t), a single-inputsingle-output system (SISO system) is simultaneously measured and the sampled signal y(n) and x(n) are recorded. By using, for example, 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 input signal x(n) may be produced [14, 19].
A least-squares estimate of a frequency response function between the input
signal x(n) and the output signal y(n) may be produced according to [19]:
Ĥ(fk ) =
P̂yx (fk )
P̂xx (fk )
(2.2)
and the coherence function as [19]
P̂yx (fk )P̂xy (fk )
2 (f ) =
γˆyx
k
P̂xx (fk )P̂yy (fk )
(2.3)
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 produces as
⎤
⎡
P̂x1 x1 (fk ) P̂x1 x2 (fk ) · · · P̂x1 xQ (fk )
⎥
⎢ P̂
⎢ x2 x1 (fk ) P̂x2 x2 (fk ) · · · P̂x2 xQ (fk ) ⎥
P̂xx (fk ) = ⎢
(2.4)
⎥
..
..
..
...
⎦
⎣
.
.
.
P̂xQ x1 (fk ) P̂xQ x2 (fk ) · · · P̂xQ xQ (fk )
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
17
where the diagonal elements is power spectral densities for the respective input
signal. Also a cross spectrum matrix [P̂yx (fk )] between all the inputs and
outputs may be estimated as
⎤
⎡
P̂y1 x1 (fk ) P̂y1 x2 (fk ) · · · P̂y1 xQ (fk )
⎥
⎢ P̂ (f ) P̂ (f ) · · · P̂
y2 x 2 k
y2 xQ (fk ) ⎥
⎢ y2 x 1 k
P̂yx (fk ) = ⎢
(2.5)
⎥
..
..
..
...
⎦
⎣
.
.
.
P̂yP x1 (fk ) P̂yP x2 (fk ) · · · P̂yP xQ (fk )
The least-square estimate for the (SISO) system in Eq. 2.2, can be rewritten
for the (MIMO system) yielding the estimate of the system matrix Ĥ(fk )
as [19]
−1
Ĥ(fk ) = P̂yx (fk ) P̂xx (fk )
(2.6)
In the case of a multiple inputs, case, the multiple coherence is of interest
as a 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 which is an extension of the ordinary coherence function from the
SISO case. If the inputs are uncorrelated, the multiple coherence γy2p :x (f ) for
the response in point p is given by [19]
γy2p :x (f ) = γy2p x1 (f ) + γy2p x2 (f ) + · · · + γy2p xQ (f )
(2.7)
where Q is the number of inputs and ”:” denotes ”linear dependent on”. However, usually there is are some correlations between the inputs, then the multiple coherence is given by [19]
γy2p :x (f ) = 1 − (1 − γy2p :x1 (f ))(1 − γy2p :x2·1 (f )) · · · (1 − γy2p :xQ·(Q−1)! (f ))
(2.8)
where ”·” denotes ”independent off” and ”!” denotes factorial. The
multiple
coherence may also be expressed using an expanded spectral matrix Pyp xx (f ) ,
who’s determinant is a measure of extraneous on the output and is written as
⎡
⎤
Pyp yp (f ) Pyp x1 (f ) · · · Pyp xQ (f )
⎢ Px y (f ) Px x (f ) · · · Px x (f ) ⎥
1 p
1 1
1 Q
⎢
⎥
Pyp xx (f ) = ⎢
(2.9)
⎥
..
..
..
.
.
⎣
⎦
.
.
.
.
PxQ yp (f ) PxQ x1 (f ) · · · PxQ xQ (f )
Based on the extended spectral matrix the multiple coherence may be expressed
Pyp xx (f ) (2.10)
γy2p :x (f ) = 1 −
Pyp yp (f ) |[Pxx (f )]|
18
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
The normalized random error in frequency response function estimates for
the amplitude function [19] is approximately given by
2
(1 − γ̂y:x
(fk ))
Q·(Q−1)!
(2.11)
εr (|Ĥyx (fk )|) ≈ 2
γ̂y:x
(f
)2(L
+
1
−
Q)
k
e
Q·(Q−1)!
for the phase function it is approximately given by [19]
εr (|Θ̂yx (fk )|) ≈ arcsin εr (|Ĥyx (fk )|)
(2.12)
Finally, an estimate of the normalized random error for the multiple coherence
function is given by [19]
√
2(1 − γ̂y2p :x (fk ))
(2.13)
εr (γ̂y2p :x (fk )) ≈ γ̂y2p :x (fk )(Le + 1 − Q)
where Le is the number of uncorrelated periodograms [19, 20] used in the
average to produce the spectrum estimate.
2.2.2
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 towards 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 an FRF, H(f ), expressed as the receptance between the reference point, input signal q, and the response, output
signal in point p of a structure, may be written as [18],
Hpq (f ) =
N
r=1
A∗pqr
Apqr
+
j2πf − λr j2πf − λ∗r
(2.14)
where r is the mode number, N the number of modes used in the model, Apqr
the residue belonging to mode r between reference point q and response p and
λr is the pole belonging to mode r. The parametric model of the IRF, input
force to output displacement impulse response may be expressed as
hpq (t) =
N
r=1
∗
Apqr eλr t + A∗pqr eλr t
(2.15)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
19
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 MIMOsystems with the purpose of obtaining a global least squares estimates of the
modal parameters. The estimated system matrix [Ĥ(f )] is of size P ×Q, where
P is the number of responses and Q the number of references, and is written
as
⎤
⎡
Ĥ11 (f ) Ĥ12 (f ) · · · Ĥ1Q (f )
⎢ Ĥ (f ) Ĥ (f ) · · · Ĥ (f ) ⎥
22
2Q
⎥
⎢ 21
(2.16)
[Ĥ(f )] = ⎢
⎥
..
..
..
.
.
⎦
⎣
.
.
.
.
ĤP 1 (f ) ĤP 2 (f ) · · · ĤP Q (f )
The procedure of modal parameter estimation starts by determining the
model order of the system under analysis. This can be done with the aid of
a Mode Indicator Function (MIF) and a stability diagram [21]. The function
used was the multivariate MIF and is expressed as [24]
min
||{F(f )}||2 =1
{F(f )}T [H (f )]T [H (f )]{F(f )}
{F(f )}T [H (f )]T [H (f )] + [H (f )]T [H (f )] {F(f )}
= λ(f )
(2.17)
which yields a value 0 ≤ λ(f ) ≤ 1, where [F(f )] is a force vector, [H (f )]
and [H (f )] is the real part and imaginary part, respectively, of the system
matrix [H(f )] and T is the transpose operator. This minimization problem
can be reformulated into an eigenvalue problem as [21]
[H (f )]T [H (f )]{F(f )} = [H (f )]T [H (f )] + [H (f )]T [H (f )] {F(f )}λ(f )
(2.18)
where the smallest eigenvalue λ(f ) corresponds to the minimization problem
in Eq. 2.17. Eq. 2.18 forms an eigenvalue problem of size Q×Q, thus the problem yields the same number of solutions for each frequency as the number of
sources. Plotting all solutions, repeated roots will be detected if the references
excited those modes.
A stability diagram is constructed using estimates of systems poles and
modal participation factors as a function of model order [21]. As the model
order is increased, more and more modal frequencies are estimated but, hopefully, the estimates of the physical modal parameters will stabilize as the correct model order is found. From empirical evaluation of the stability diagram,
the physical ”true” poles seem to asymptotically go to the true values, whereas
computational (nonphysical) poles which arise due to leakage, low signal to
noise ratio (SNR), frequency shift etc, appear more unstructured [21]. Using
the stability diagram with the multivariate MIF overlayed, stable poles which
20
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
appear to have physical correspondence are selected. Along with the poles
and a driving point, real or complex residues are estimated. Mode shapes
were estimated using the frequency polyreference method [25].
As quality assessment of the estimated parameters the FRF’s were synthesizes using the estimated parameters and overlayed with the estimated FRF’s.
Furthermore the Modal Assurance Criterion (MAC)[18] defined by Eq. 2.19.
2
{ψ}
{ψ}H
l
k
M ACkl =
H
{ψ}H
{ψ}
k {ψ}l {ψ}l
k
(2.19)
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.
2.2.3
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 2.3
Coherence function estimates and magnitude functions of frequency response
function estimates were utilized for the selection of burst length and frequency
resolution. 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. In other words, the time of
the dead period of the burst random signal was set so as to be sufficiently long
enough to enable the structural response to decay in order to render influences
from leakage negligible. The data block length was set so as to maintain a
sufficient signal to noise ratio.
Furthermore, four different excitation levels with the proportion {1, 2, 3, 4}
were applied for each of the boundary 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.
2.3
Analytical Models of the Boring Bars
The boring bar has a cross section A(z) and a length of l. Also associated
with the beam is a flexural (bending) stiffness EI(z), where E is Young’s
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
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
21
Value
Burst Random
10240 Hz
20480
0.5 Hz
200
Rectangular
0%
0-4000 Hz
90%
Table 2.3: Spectral density estimation parameters.
elastic modulus for the beam and I(x) is the cross-sectional area moment
of inertia about the ”z axis.” From mechanic theory, the beam sustains a
bending moment M (z, t), which is related to the beam deflection, or bending
deformation u(z, t), by the conservation of momentum. This can be derived as
follows [26]: when the beam deflects, see Fig. 2.9, the longitudinal displacement
y
u (z,t)
s
x
d z r A d x
¶ 2w (t)
¶ t2
z
d q ( z ,t)
d z
d z
Figure 2.9: Deflection model of a beam undergoing pure bending.
θ(z, y, t) will cause a strain (illustrated by the arrow-field in the figure) due
to the rotation of the cross-sectional plane which is parallel with s in Fig. 2.9.
Hooke’s law states that the uniaxial stress σ (or axial force per unit sectional
area) applied to a bar in the z direction is proportional to the strain ε (or
elongation per unit length) within the elastic limit according to
σ
(2.20)
E=
ε
22
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Assuming that the cross-section-plane remains flat after deformation; then the
relation of strain ε and stress σ is given by
∂θ(z, y, t)
∂ 2 u(z, t)
= −y
∂z
∂z 2
2
∂ u(z, t)
σzz (z, y, t) = −yE
∂z 2
εzz (z, y, t) =
(2.21)
(2.22)
The total moment about an axis parallel with the x-axis in the mid-section,
where strain is zero, will cause to stress become
Mx (z, t) =
σzz (x, y, t)ydxdy
(2.23)
y(z) x(z)
using the relation in Eq. 2.22, we get
∂ 2 u(z, t)
Mx (z, t) = −E
∂z 2
y 2 dxdy
(2.24)
y(z) x(z)
where
Ix (z) =
y 2 dxdy
(2.25)
y(z) x(z)
and we get
Mx (z, t) = −EIx (z)
∂ 2 u(z, t)
∂z 2
(2.26)
A model of bending vibration may be derived by examining the force diagram of an infinitesimal element of the beam[27] as indicated in Fig. 2.10.
Assuming the deformation is small enough so that the shear deformation is
much smaller than displacement u(z, t) (i.e., so that the sides of the element
dz do not bend), a summation of forces in the y direction yields
V (z, t) +
∂ 2 u(z, t)
∂V (z, t)
dz − V (z, t) + f (z, t)dz = ρA(z)dz
∂z
∂t2
(2.27)
Here V (z, t) is the shear force at the left end of the element dz, V (z, t) +
∂V (z, t)/∂zdz is the shear force at the right end of the element dz, f (z, t) is
the total external force applied to the element per unit length, and the term
on the right side of the equality sign is the inertial force of the element. The
assumption of small shear deformation used in the force balance is true if
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
u ( z ,t)
y
f ( z ,t)
f ( z ,t)
M
M (z ,t)
z
23
( z ,t) +
d z
¶ M
( z ,t)
d z
¶ z
u ( z ,t)
E , r , I ( z ), A ( z )
u n d e fo rm e d x -a x is
V (z ,t)
r A d z
z
¶ 2u ( z ,t)
¶ t 2
V ( z ,t) +
¶ V ( z ,t)
d z
¶ z
z + d z
Figure 2.10: Simple beam with transverse vibration and a free-body diagram
of a small element of the beam as it is deformed by a distributed force per unit
length, denoted f (z, t).
length divided by the smallest radius of the beam is less then 10 (i.e., for long
slender beams). Next, the moments acting on the element dz about the x axis
through point Q are summed. This yields
∂Mx (z, t)
∂V (z, t)
dz + V (z, t) +
dz dz +
Mx (z, t) − Mx (z, t) +
∂z
∂z
dz
+[f (z, t)dz] = 0 (2.28)
2
Here, the left-hand side of the equation is zero since the rotary inertia of
element dz is assumed to be negligible, simplifying the expression yields
∂Mx (z, t)
∂V (z, t) f (z, t)
V (z, t) −
(2.29)
dz +
+
(dz)2 = 0
∂z
∂z
2
Since dz is assumed to be very small, (dz)2 is assumed to be almost zero, thus
the moment expression becomes
V (z, t) =
∂Mx (z, t)
∂z
(2.30)
This states that the shear force is proportional to the spatial change in the
bending moment. Substitution of this expression for the shear force into
Eq. 2.27 yields
∂ 2 u(z, t)
∂2
M
(z,
t)dz
+
f
(z,
t)dz
=
ρA(z)dz
x
∂z 2
∂t2
(2.31)
Further substitution of Eq. 2.26 into Eq. 2.31 and dividing by dz yields
∂ 2 u(z, t)
∂ 2 u(z, t)
∂2
ρA(z)dz
= f (z, t)dz
(2.32)
+ 2 EI(z)
∂t2
∂z
∂z 2
24
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Eq. 2.32 is often referred to as the Euler-Bernoulli beam equation. The assumptions regarding the beam, used in formulating this model are:
• Uniform along its span, or length, and slender (diameter to length ratio>10).
• Composed of a linear, homogenous, isotropic elastic material, without
axial loads.
• Plane section remains plane.
• The plane of symmetry of the beam is also the plane of vibration, so
rotation and translation are decoupled.
• Rotary inertia and shear deformation can be neglected.
Assuming that the cross-sectional area is constant A(z) = A, the beam equation can be rewritten as
4
∂
u(z,
t)
EI
∂ 2 u(z, t)
+ c2
= 0, c =
(2.33)
2
4
∂t
∂z
ρA
The solution for Eq. 2.33 is subjected to four boundary conditions and two
initial conditions, however, in order to calculate the resonance frequencies and
mode shapes we only need the boundary conditions. A separation-of-variables
solution of the form u(z, t) = u(z)u(t) is assumed, thus the equation of motion
to yields
c2
∂ 2 u(t) 1
∂ 4 u(z) 1
=
−
= (2πf )2
4
2
∂z u(z)
∂t u(t)
The spatial equation results from rearranging Eq. 2.34, which yields
2
∂ 4 u(z)
2πf
−
u(z) = 0
∂z 4
c
(2.34)
(2.35)
By defining
β4 =
(2πf )2
ρA(2πf )2
=
c2
EI
(2.36)
the general solution of Eq. 2.35 can be written as [27]
u(z) = a1 cos βz + b1 sin βz + c1 cosh βz + d1 sinh βz
(2.37)
where a1 , b1 , c1 and d1 are constants of integration determined by the boundary
conditions. The general solution produces infinity solution for β (an infinite
number of resonance frequencies and mode shapes), we denote each solution
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
25
as βr , belonging to mode r. The equation for calculating the natural frequency
is [27]
βr2 EI
(2.38)
fr =
2π ρA
The boundary conditions required in order to solve the spatial equation
from the separation-of-variables solution of Eq. 2.33 are obtained by examining the deflection u(z, t), the slope of the deflection ∂u(z, t)/∂z, the bending
moment EI∂ 2 u(z, t)/∂z 2 and the shear force EI∂ 3 u(z, t)/∂z 3 at all boundaries. From examining these boundary conditions, four equations should be
found and may be written in matrix form as:
[C] {a} = {0}
(2.39)
T
a1 b1 c1 d1
is the vector with unknown constants of
where {a} =
integration and [C] the coefficient matrix determined from the boundary conditions. By equating the determinant of the coefficient matrix to zero, the
characteristic equation and the eigenfrequencies may be found [27]. For each
eigenfrequency fr , three of the four unknown constants of integration can be
found or expressed in terms of the fourth. This is sufficient in order determine
the mode shape {Ψr }. The fourth constant is found using the initial conditions
and determines the participation of each mode in temporal solution. However,
since we only consider the dynamic properties of the system is considered, the
mode shape {Ψr } is normalized.
2.3.1
Multi-span beam
The previous discussion concerned a beam with constant cross-section properties and boundary conditions at each end. In order to apply Euler-Bernoulli
modeling to more complex beam structures with boundary conditions along
the beam at discrete points and/or beam segments with different properties,
the beam may be divided into several sub-beams, also referred to as a multispan beam [28]. Each sub-beam will have the same general solution as the
single span beam in Eq. 2.37, thus the mode shape for each sub-beam my be
expressed as
uj (z) = aj cos β(z − zj ) + bj sin β(z − zj ) +
+cj cosh β(z − zj ) + dj sinh β(z − zj )
(2.40)
26
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
where j is the sub-beam number, J the number of sub-beams and zj the local
coordinate offset. The local coordinate is expressed as
zj =
j−1
lk
(2.41)
k=1
where lk is the length of section 1 ≤ k ≤ J. The equation system will now
consist of 4 times J coupled equations. Thus the coefficient matrix [C] will
have the size 4Jx4J and the vector {a} the size 4Jx1. The eigenfrequencies
and mode shapes are found in the same way as for the single span beam,
i.e. by first finding the solutions to the characteristic equation and then the
corresponding eigenvectors.
2.3.2
Linearized Model
The simplest and most straightforward model of a boring bar is the EulerBernoulli model, which consists of a homogenous single span beam with constant cross-sectional area A(z) = A and constant cross-sectional moment of
inertia I(z) = I. The beam has four boundary conditions, two at each end.
One end is clamped and the other is free, see Fig. 2.11.
z
E , H ,I , A
l
Figure 2.11: Model of a Clamped - 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 l the length of the beam.
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)|z=0 = 0
∂u(z, t) = 0
∂z z=0
(2.42)
(2.43)
The other end is free, thus no bending moment or shear force constrains the
beam at the coordinate z = l when the beam vibrates, this yielding the other
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
27
two boundary condition as
∂ 2 u(z, t) EI
= 0
∂z 2 z=l
∂ 3 u(z, t) EI
= 0
∂z 3 z=l
(2.44)
(2.45)
The general solution for u(z) is then combined with the boundary condition,
which yields
u(z)|z=0
du(z) dz z=0
d2 u(z) EI
dz 2 z=l
d3 u(z) EI
dz 3 z=l
= a1 + c 1 = 0
(2.46)
= b1 + d1 = 0
(2.47)
=
EI (−a1 β 2 cos zβ − b1 β 2 sin βl+
+c1 β 2 cosh βl + d1 β 2 sinh βl) = 0
(2.48)
=
EI (a1 β 3 sin βl − b1 β 3 cos βl+
+c1 β 3 sinh βl + d1 β 3 cosh βl) = 0
(2.49)
and in matrix form
⎡
1
0
1
0
⎢
0
1
0
1
⎢
⎣ −β 2 cos βl −β 2 sin βl β 2 cosh βl β 2 sinh βl
β 3 sin βl −β 3 cos βl β 3 sinh βl β 3 cosh βl
⎤ ⎡
0
a1
⎥ ⎢ b1 ⎥ ⎢ 0
⎥ ⎢
⎥⎢
⎦ ⎣ c1 ⎦ = ⎣ 0
d1
0
⎤⎡
⎤
⎥
⎥ (2.50)
⎦
Setting the determinant of the coefficient matrix equal to zero, i.e. det([C]) =
0, yields the characteristic equation as
2β 5 cos βl cosh βl + β 5 cos2 βl + β 5 sin2 βl + β 5 cosh2 βl − β 5 sinh2 βl = 0(2.51)
since cos2 x + sin2 x = 1 and cosh2 x − sinh2 x = 1 Eq. 2.51 can be simplified
into
cos βl cosh βl + 1 = 0
(2.52)
and βr is determined by finding the roots of this equation.
2.3.3
Multi-span Boring bar with Elastic Foundation
The boring bar was clamped with either two screws on the top and two on
the underside or three screws on the top and three on the underside. In addition, two different bolts were used: M8 and M10. If we consider the clamping
28
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
house 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” [29] as
a boundary condition, instead of the pinned condition. The elastic support
can be seen as two springs in one point, with one spring in the transverse direction; thus, transverse stiffness resistance and one rotational spring exhibits
rotational stiffness resistance. The configurations of the ”elastic support” condition are presented in Fig. 2.12.
k
k
R
R
E , H ,I , A
a )
l1
l
k
l
2
k
T
k
k
R
T
k
R
3
R
b )
E , H ,I , A
l
l1
k
T
l
4
k
T
l
4
k
3
T
Figure 2.12: 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.
Two types of boundary conditions may be categorized from the models presented in Fig. 2.12, where zpos denotes the position of the boundary condition.
One is the ”free” boundary condition, previously expressed as
∂ 2 u(z, t) EI
= 0
(2.53)
∂z 2 z=zpos
∂ 3 u(z, t) = 0
(2.54)
EI
∂z 3 z=zpos
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
29
where there is no bending or shear forces present. The other boundary conditions derive from the ”elastic support” condition and may be expressed as [29]
∂ 2 u(z, t) ∂u(z, t)
EI
= −kR
(2.55)
2
∂z
∂z
z=zpos
∂ 3 u(z, t) = kT u(z, t)
(2.56)
EI
∂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 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”. The pinned boundary
condition can be expressed as
u(z, t)|z=zpos = 0
∂ 2 u(z, t) = 0
EI
∂z 2 (2.57)
(2.58)
z=zpos
The coefficient matrix may now be formulated in order to find the characteristic equations for the different models. The three span model will yield a
12x12 coefficient matrix and the four span model will yield a 16x16 coefficient
matrix. Calculating the determinate of these matrices by hand is fairly timeconsuming and the roots of the characteristic equation are often not possible
to express explicitly [30, 31]. The solutions produced using these models were
found using Matlab and by numerically finding the roots for the characteristic
equations. The boundary conditions for the four models, (three span model
with rotational and transverse springs; three span model without rotational
spring and infinitely stiff transverse spring; four span model with rotational
and transverse springs and four span model without rotational spring and infinitely stiff transverse spring) are presented in appendix A.
2.3.4
Screws - Elastic Foundation
The boring bar was clamped using screws. Since the screws are not rigid
bodies they may, for example, be modeled as flexible bodies using springs. A
model based on a transverse spring and a rotational spring was assumed. Thus,
the two different stiffness coefficients may be calculated using very simplified
models of what is going on whilst clamping the boring bar. Hence, one of
the coefficients corresponds to a transverse spring and the second coefficient
corresponds to a rotational spring. The screws’ end surfaces, which are in
30
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
contact with the boring bar, apply pressure to the bar. This screw pressure
on the boring bar is related to the screws’ tightening torque.
The screws used to clamp the boring bar were of type MC6S norm ”DIN
912, ISO 4762”, presented in Fig. 2.13. The screws are zinc-plated, steel socket,
P / 8
H
b
d
D
b = 6 0 °
P / 2
L
k
S
d
d
2
d
/ 8
5 H
/ 8
H
/ 4
P
1
P / 4
S c re w a x is
a)
b)
Figure 2.13: a) Dimensions of the ISO 4762/DIN 912 socket head cap screw
b) Theoretical metric ISO-thread profile for inside thread and outside thread
without tolerances added, see Table 2.4 for symbol definitions.
head cap screws with the strength class 8.8, with a tensile yield strength of
Rp02 = 660M P a. Two various sizes were used: first M8 and then M10; see
Table 2.4 for dimensions.
Symbol
d
D
s
k
L
b
Symbol
d
d1
d2
P
H
β
M8
8 mm
13 mm
6 mm
8 mm
50 mm
28 mm
M8
8 mm
6.65 mm
7.19 mm
1.25 mm
1.08 mm
60◦
M10
10 mm
16 mm
8 mm
10 mm
50 mm
36 mm
M10
10 mm
8.38 mm
9.03 mm
1.5 mm
1.30 mm
60◦
Description of symbols in Fig. 2.13 a)
Diameter of the screw without head
Diameter of the head
Size or minimal diameter of the hexagon
Hight of the head
Length of the screw without the head
Length of thread part
Description of symbols in Fig. 2.13 b)
Outer diameter of outside thread
√
Inner diameter of outside thread (d1 = 5 8 3 P
)
√
Average diameter of outside thread (d2 = 3 8 3 P )
Pitch
√
Hight of base-triangle (H = 23 P )
Basic profile angle for metric screw threads.
Table 2.4: Dimensions of the ISO 4762/DIN 912 socket head cap screw
The stiffness constants 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 homogenous, having a constant cross-sectional area A, a
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
31
constant cross-sectional moment of inertia I and a length of l. When a screw
is threaded in the clamping house and is clamping the boring bar, a part of the
screw’s tip will not be in contact with the clamping house; thus yielding both
transverse, lateral and bending elasticity. This is due to the fact that the inside
of the clamping house is circular 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. Two different lengths of the screw beam models were selected. The
shortest length represents rigid clamping of the screws within the clamping
house and the longer length represents the case of flexible clamping of the
screw by the clamping house thread.
The screw clamping model is presented in Fig. 2.14, 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 e
T ra n s v e rs e S tiffn e s s
l
E A
Þ
F
k
T
B e n d in g S tiffn e s s
E I
l
Þ
k
R
M
B o rin g b a r
a)
b)
c)
Figure 2.14: a) Sketch illustrating screw clamping of the boring bar, via the
clamping house, b) the transverse stiffness model, and c) the rotational stiffness
model
The spring coefficients from a beam with one end fixed and the other end
subjected to axial (vertical) loading and bending moment are the transverse
spring constant kT and rotational spring constant kR , respectively. These
constants are calculated from beam bending theory [27, 32] as
EA
(2.59)
l
EI
kR =
(2.60)
l
where E is the elasticities modulus, cross-sectional area A, cross-sectional moment of inertia I, and l the length of the beam model. These spring constants
are used in the elastic support models.
kT =
32
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Other important factors which affect clamping conditions include the coupling properties between the screws and the boring bar, and coupling properties between the screws and the clamping house.The force applied by the
clamp screws in the axial direction of the screws on the boring bar surface
may be related to the tightening torque M of the screws, and is identical to
the prestressing force Fp . The tightening torque can be divided into two parts,
one moment MT (arising due to friction in the contact surfaces between the
threads and the geometric relation from the prestressing force) and a second
moment, MC (due to the friction in the contact surface between the screw and
boring bar). The moment due to the threads may be expressed as [33]
MT = Fp rm tan(ϕ + ε)
(2.61)
where rm is the average radius of the screw equal to d22 in Table 2.4, ε is the
angle of friction force component and ϕ is the pitch angle. The moment MC ,
due to the friction between the boring bar and the screw, may be expressed as
MC = Fp µc rc
(2.62)
where µc is friction coefficient between the surfaces, and rc is the average radius
of the contact surface [33], with regard to a contact surface with an inner and
an outer circle. Then the average radius of contact equals
rm
rc =
(2.63)
2
thus, the total moment M equals [33]
M = MT + MC = Fp rm tan(ϕ + ε) + Fp µc rc
(2.64)
Thus, the expression for the force Fp (which the screw is enacts upon the
boring bar when applying the moment M ) may be expressed as
Fp =
M
rm tan(ϕ + ε) + µc rc
(2.65)
The pitch angle ϕ is calculated from the geometry of the screw as
tan ϕ =
P
πdm
(2.66)
where P is the pitch and dm is the average diameter of the screw, equal to
d2 in Table 2.4. The angle of the friction force component ε is related to the
pitch angle ϕ and friction in the thread as
µT
(2.67)
tan ε =
cos β2
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
33
where µT is the friction coefficient between the surfaces of the inner and outer
threads, (i.e. the threads in the clamping house and the threads of the screws)
and β is the profile angle, equal to 60◦ from the ISO-standard presented in
Fig. 2.13.
2.3.5
Spring Coefficients and Clamping Forces
Materials and properties are required in order to model the screws as the
transversal spring and the rotational spring (both previously presented). The
screws’ material is steel (zinc-plated). Whilst the exact type of steel is not
specified, this is in general not relevant. Types of steel vary for different
screw suppliers; usually, more important is the tensile yield strength. However,
according to the supplier of the screws used in this experiment, the elasticity
modulus E is approximately 200 · 109 N/m2 . The dimensions are specified
by the standard and are presented in Table 2.4. The tensile stress area A,
according to the standard ISO 724, is given by
2
π d2 + d1 − H6
(2.68)
A=
4
2
and the moment of inertia I was calculated as
4
H
d
+
d
−
π
2
1
6
I=
4
4
(2.69)
where d1 , d2 and H are given in Table 2.4. The length l is the length of the
”overhang” of the screw; an example was used in which the overhang length
was considered to be 1.5 mm. This length was selected, because the circular
inner diameter of the clamping space of the clamping hose is 40mm and the
height of the boring bar is 37mm, thus resulting in a space from each side of the
boring bar to the circular clamping house boundary of 1.5 mm. Furthermore,
since the boring bar is clamped with screws from both the upper-side and the
underside at the same positions along the z-axis, each axial spring coefficient
includes the longitudinal stiffness from two screws, i.e. two springs in parallel.
The calculated spring coefficients and the spring parameters used in the
spring models are presented in Table 2.5 together with the dimensions and
elasticity modulus.
Estimates of the forces applied by the clamp screws on the boring bar were
calculated using the torque-force relation presented in section 2.3.4, and are
presented in Table 2.6. The screw force was calculated using the dimension
previously presented in Table 2.4, assuming identical friction coefficients for
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
34
A [m2 ]
3.661 · 10−5
5.799 · 10−5
Size
M8
M10
I [m4 ]
1.067 · 10−10
2.676 · 10−10
E [N/m2 ]
l [m]
200 · 109
1.5 · 10−3
kT [N/m]
4.881 · 109
7.732 · 109
kR [Nm]
1.422 · 104
3.568 · 104
Table 2.5: The longitudinal and rotational spring coefficients and the spring
parameters used in the spring models.
the contact surface between the boring bar and the screws, and also the contact
surface between the threads in the clamping house and the screw threads, i.e.
µc = 0.14 and µT = 0.14.
Torque
Force
Screw size
M8 & M10
M8
M10
Tightening torques and clamping forces
10 Nm
15 Nm
20 Nm
25 Nm
30 Nm
9.81 kN 14.71 kN 19.61 kN 24.52 kN 29.42 kN
7.88 kN 11.81 kN 15.75 kN 19.69 kN 23.63 kN
Table 2.6: The clamp screw torque-force relation based on the presented clamp
screw model.
2.4
Nonlinear Model
A linear model may not always be sufficient to explain different results from the
experimental data. Nonlinearities may be caused by several different factors.
A common source of nonlinearity is the contact problem, in which elements
of the system come into contact with the surrounding environment due to
”large” displacement, which, in turn, create a new set of boundary conditions.
Another form of the contact problem is friction in joints, or sliding surfaces.
This problem also involves large forces and/or deformation which may cause
the properties of the material to behave in a nonlinear manner.
All of these nonlinearities are likely to exist to some extent in the boring
bar - clamping house system, perhaps some of them more than others. The
question is if, or to what extent they influence the dynamic behavior of the
boring bar and how they can be determined (if relevant). In these simulations, nonlinearities regarding stiffness was examined, a so-called ”softening
spring” [18, 34, 35]. This nonlinearity was investigated as empirical data has
shown that the fundamental boring bar resonance frequencies display a tendency to move towards lower frequencies as a result of increasing excitation
force level.
The softening spring may be modeled in two different ways, yielding different properties with respect to the displacement. The first of these models
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
35
yields a force proportional to a nonlinear stiffness coefficient, multiplied by
the displacement, squared with sign. The equation describing a SDOF system
with this type of a softening spring nonlinearity is given by [36]
m
dx(t)
d2 x(t)
+c
+ kx(t) − ks x|x|(t) = f (t)
2
dt
dt
(2.70)
where m, c and k are the mass, damping and stiffness coefficients of the underlying linear system, x(t) - the displacement, f(t) - the force, and ks the
nonlinear stiffness coefficient. The second model yields a force proportional
to a nonlinear stiffness coefficient multiplied by the displacement cubed, and
inserted into the equation of motion describing a SDOF system results in [36]
m
dx(t)
d2 x(t)
+c
+ kx(t) − kc x3 (t) = f (t)
2
dt
dt
(2.71)
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: Probably
the most common method to solve ordinary differential equations (ODE) is
the Runge-Kutta method, implemented in Matlab [37]. Another method is
the digital filter method [38].
There are multiple advantages with using ODE solvers: firstly, they are
rather straightforward to use and secondly 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, is significantly
faster than the ODE solvers [38]. But is not as well documented as the ODE
solvers with regarding to, for example, accuracy and the ability to handle nonlinear systems [39]. However, for linear systems, the limitations of the filter
method are known [40] and depends on the sampling frequency and the transformation method used to convert the continuous time parameters to discrete
time parameters [40].
2.4.2
Ordinary Differential Equation Methods
The simulation method used for simulating the nonlinear system is based on
explicit Runge-Kutta of order (4,5) formula, the Dormand-Prince pair[37, 41],
36
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
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 classical
fourth order Runge-Kutta [41, 42]. The numerical technique solves ordinary
differential equations of the form
dx(t)
= f (x(t), t),
dt
x(t0 ) = x0
(2.72)
The Runge-Kutta 4th order method is based on the following expressions
x(t + ∆t) = x(t) + (a1 k1 (x(t), t) + a2 k2 (x(t), t) +
+ a3 k3 (x(t), t) + a4 k4 (x(t), t))∆t
(2.73)
where ∆t is the step size, {a1 , · · · , a4 } and {k1 (x(t), t), · · · , k4 (x(t), t)} are
constants and functions respectively, determined based on the first five terms
of the Taylor series [42]:
1 f (x(t), t)∆t2 +
2!
1
1
+ f (x(t), t)∆t3 + f (x(t), t)∆t4
3!
4!
x(t + ∆t) = x(t) + f (x(t), t)∆t +
(2.74)
Rewriting Eq. 2.74 into Eq. 2.73 yields [42]
1
x(t + ∆t) = x(t) + (k1 + 2k2 + 2k3 + k4 )∆t
6
(2.75)
where
k1 (x(t), t) = f (x(t), t)
∆t
∆t
k2 (x(t), t) = f x(t) +
k1 , t +
2
2
∆t
∆t
k2 , t +
k3 (x(t), t) = f x(t) +
2
2
k4 (x(t), t) = f (x(t) + ∆tk3 , t + ∆t)
(2.76)
(2.77)
(2.78)
(2.79)
The Runge-Kutta method only solves first order differential equations, thus
this requires that the second order differential equations in Eqs. 2.70 and 2.71
are rewritten to coupled first order differential equations as in Eqs. 2.82 and 2.83.
The nonlinear models simulated with the differential equation solvers were
based on the softening spring using the quadratic model in Eq. 2.80, and the
cubed model in Eq. 2.81
gq (x(t)) = ks x|x|(t)
gc (x(t)) = kc x3 (t)
(2.80)
(2.81)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
37
where gq (x(t)) and gc (x(t)) replaces g(x1 (t)) in Eq. 2.83 for respective model.
dx1 (t)
= x2 (t)
dt
dx2 (t)
m
= −cx2 (t) − kx1 (t) + g(x1 (t)) + f (t)
dt
(2.82)
(2.83)
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 [38]. The continuous time filtering may be expressed
in terms of a convolution integral as [43]
∞
h(τ )f (t − τ )dτ
x(t) =
(2.84)
τ =−∞
where x(t) is the response or output signal and f (t) is the input to the system with impulse response h(t). The corresponding filtering procedure in the
discrete time domain is given by
x(n) =
∞
h(k)f (n − k)
(2.85)
k=−∞
where, again, x(n) is the response and f (n) is the input to the system with
the impulse response h(n), but in discrete time domain.
The transformation may be performed by first dividing the total system
(multiple degree of freedom) into subsystems using the modal superposition
theorem and transforming each subsystems parameter into the filter coefficients. The frequency response function for a dynamic system may be expressed in terms of modal superposition as [18]
H(f ) =
R
r=1
Ar
A∗r
+
,
j2πf − λr j2πf − λ∗r
(2.86)
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, a Finite Element Model,
38
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
distributed parameter system, or estimated from experimental modal analysis [27]. Another approach is to directly express the system as in Eq. 2.87
and transform the analog filter coefficients into digital filter coefficients. The
frequency function for an analog filter can be expressed as [43]
D(s)
d0 + d1 s + . . . + dMa sMa
H(s) =
=
C(s)
1 + c1 s + . . . + cKa sKa
(2.87)
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 [43]
b0 + b1 z −1 + . . . + bM z −M
B(z)
=
H(z) =
A(z)
1 + a1 z −1 + . . . + aK z −K
(2.88)
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 [43]
x(n) =
M
m=0
bm f (n − m) −
K
ak x(n − k)
(2.89)
k=1
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 sampling
period [44]. Other methods include the step invariant, ramp invariant, centered
step invariant, cubic spline invariant and Lagrange method; each with different
properties [40]. The ramp invariant method was used in my simulations. This
transform method introduces zero error at DC, low error at Nyquist frequency
and low phase distortion [40]. However, the ramp invariant method introduces
a large error at a resonance frequency than, for example, the impulse, step and
cantered step invariant methods [40]. However, the error introduced in the
area of a resonance frequency only becomes large as 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 performed by using
the digital filter coefficient for the linear system, and finding the solutions for
the nonlinear difference equation. A nonlinear, single degree of freedom system
may be described by differential equation
dx(t)
d2 x(t)
+c
+ kx(t) + g(x(t)) = f (t)
m
2
dt
dt
(2.90)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
39
the corresponding equation in time discrete domain may be written using a t
difference equation as [38]
x(n) =
M
bm (x(n − m) − g(x(n − m))) −
m=0
K
ak f (n − k)
(2.91)
k=1
Since Eq. 2.91 contains nonlinear terms, several solutions may exist for x(n) [42].
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 [42] (which
was used in this synthesis).
The models with a nonlinear function g(x(n), simulated with the filter
method were based on both the quadratic model in Eq. 2.80, and the cubed
model in Eq. 2.81, both representing the softening spring.
gLin (n) − gs (x(n) = f (n),
gLin (n) − gc (x(n) = f (n),
where
where
gs (x(n)) = ks x|x|(n)
gc (x(n)) = kc x3 (n)
(2.92)
(2.93)
The digital filter coefficients were based on the poles and residues estimated
from experimental measured data.
2.4.4
Excitation Signal
True random was selected for the excitation signal so that the resolution and
number of averages may be alter after the simulation results were produced.
The estimation parameters used in the nonlinear simulations are presented in
Table 2.7.
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
True Random
10000 Hz
20480
0.5 Hz
800
Hanning
50%
-
Table 2.7: Spectral density estimation parameters and excitation signal used
in the simulated nonlinear system.
Chapter 3
Results
Firstly, this section presents results from the experimental modal analysis of
boring bars, under different clamping conditions and excitation force levels.
The results presented constitute a small part of an extensive investigation of
the dynamic properties of boring bars for various configurations and setups;
however they represent the essence of the experimental results. Secondly, this
section present results based on analytical Euler-Bernoulli models of boring
bars with different spans, and simple models of boring bar clamping. Finally,
results from simulations of the nonlinear models are given.
3.1
Experimental Modal Analysis
Shaker excitation was used for the experimental modal analysis of the boring bars. The utilized spectrum estimation parameters and excitation signals’
properties are given in Table 2.3. 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 fundamental bending 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. Fig. 3.1 illustrates
typical frequency response function estimates based on the same measurement
locations for input force and output response at the boring bar. These frequency response function estimates are produced using different clamp screw
tightening torques and/or a different excitation level.
Based on the frequency function estimates, the modal parameters are then
estimated using the poly-reference technique. A frequency range covering the
40
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
41
Accelerance [dB rel 1(m/s2 )/N]
35
30
25
20
15
10
5
0
−5
−10
−15
300
350
400
450
500
550
600
650
700
750
800
540
550
560
570
580
590
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
35
30
25
20
15
10
490
500
510
520
530
Frequency [Hz]
b)
Figure 3.1: a) The accelerance of the boring bar driving point response in the
direction of cutting speed (y-), using the standard boring bar, four screws of
size M8 and tightened first from the top using five different tightening torques
and four different excitation levels. b) The corresponding estimates zoomed in
around the first resonance frequencies.
42
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
significant part near the resonance frequencies was selected, i.e. ±100 to ±200
Hz around the resonance peaks. A Multivariate Indicator Function (MIF) was
produced from all the driving point data and then overlayed on top of a stability diagram. The stability diagram consist of poles calculated using different
model orders up to a given order (see Fig. 3.2 for a typical stability diagram
with the corresponding indicator functions). When stable poles corresponding
4 5
N u m b e r o f P o le s
4 0
3 0
S ta b le
2 0
V e c to r
D a m p in g
F re q u e n c y
N e w P o le
1 0
M IF 1
M IF 2
2
3 9 8
5 0 0
6 0 0
7 0 4
F re q u e n c y [H z ]
Figure 3.2: Typical stability diagram for experiments conducted on the boring
bar. In order to identify the location and number of poles, two indicator
functions are overlayed on the stability diagram.
to the modes of interest have been selected, residues are estimated using the
driving points. Finally, the mode shapes are estimated using all the measured
FRF:s. In order to check whether the estimated parameters are functional
or not, they are used to synthesize a number of FRF:s, among those are the
driving points. Furthermore, orthogonality of the mass scaled mode shapes are
checked using the MAC matrix. If any of the checks indicate on nonfunctional
estimates, the maximum model order used to create the stability diagram is
changed and new stable poles are selected. Also erroneous or strange FRF:s
may be disregarded when the different models are calculated. These steps are
performed until acceptable results are achieved, or until (what seems to be
when) the best possible results given the data are achieved.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
43
Results from six various setups, described by Table 3.1 are presented.
Setup
Number
1
2
3
4
5
6
Boring bar
Standard
Standard
Standard
Active
Active
Linearized
Configuration
Number of Screws Screw Size
four
M8
six
M10
six
M10
four
M8
six
M10
-
Tighten first from
top
top
bottom
top
top
-
Table 3.1: The configuration of the different setups from which experimental
modal analysis results are presented.
3.1.1
Standard Boring Bar
When clamping the standard boring bar so that the bottom side of the boring bar is clamped against the clamping house (i.e. the screws are tightened
from the topside first and subsequently from the bottom-side) the fundamental boring bar resonance frequencies increases with increasing tightening, see
Fig. 3.3. In this setup, screws of size M8 were used; the spectrum estimation
parameters and excitation signal is presented in Table 2.3. 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 3.1. As can be seen in Fig. 3.4 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 3.2 and Table 3.3.
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 house
with four M8 screws.
When clamping the standard boring bar so that the bottom side of the
boring bar is clamped against the clamping house, the fundamental boring
bar resonance frequencies increases with increasing tightening, see Figs. 3.5
a) and b). As can be seen in Fig. 3.6, the fundamental boring bar resonance
frequencies decreases slightly with increasing excitation level.
44
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
10Nm
15Nm
20Nm
25Nm
30Nm
26
24
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
24
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
b)
Figure 3.3: The accelerance 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, using five different tightening torques. a) the driving point
in cutting speed direction (y-) and b) the driving point in negative cutting
depth direction (x-).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
45
Accelerance [dB rel 2(m/s2 )/N]
32
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
24
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
a)
Accelerance [dB rel 2(m/s2 )/N]
32
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
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
b)
Figure 3.4: The accelerance of the boring bar response using the standard
boring bar, four screws of size M8 and when clamp screws where tightened
firstly from the upper-side of the boring bars, using two different tightening
torques and four different excitation levels. a) the driving point in cutting
speed direction (y-) and b) the driving point in negative cutting depth direction
(x-).
46
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Resonance Frequency, Mode 1 [Hz]
Torque
10Nm
15Nm
20Nm
25Nm
30Nm
Level 1
509.52
518.13
523.84
526.64
526.72
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
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
Table 3.2: Estimates of the fundamental boring bar resonance frequencies
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. 3.3 a) and b) respectively. The grey rows of mode 1 and mode
2 correspond to the boring bar frequency response functions Fig. 3.4 a) and b)
produced for the four different excitation levels.
Damping of Mode 1 [%]
Torque
10Nm
15Nm
20Nm
25Nm
30Nm
Level 1
0.99
0.97
0.88
0.87
0.86
Level 2
1.04
1.00
0.91
0.88
0.88
Level 3
1.08
1.03
0.94
0.90
0.90
Damping of Mode 2 [%]
Level 4
1.14
1.08
0.96
0.92
0.93
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 3.3: The relative damping estimates for the fundamental boring bar
bending modes 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. 3.3 a) and b) respectively. The grey rows of mode
1 and mode 2 correspond to the boring bar frequency response functions in
Fig. 3.4 a) and b), produced for the four different excitation levels.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
47
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
10Nm
15Nm
20Nm
25Nm
30Nm
26
24
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
24
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
b)
Figure 3.5: The accelerance 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, using five different tightening
torques. a) the driving point in cutting speed direction (y-) and b) the driving
point in negative cutting depth direction (x-).
48
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Accelerance [dB rel 1(m/s2 )/N]
32
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
24
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
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
22
20
18
16
14
490
500
510
520
530
540
550
560
570
580
Frequency [Hz]
b)
Figure 3.6: The accelerance 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 upper-side of the boring bar, using two different
tightening torques and four different excitation levels. a) the driving point in
cutting speed direction (y-) and b) the driving point in negative cutting depth
direction (x-).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
49
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 Figs. 3.7 and 3.8 with Figs. 3.5 and 3.6.
3.1.2
Active Boring Bar
The active 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 active boring bar as were performed with the standard boring bar. From the results presented in Figs 3.9
and 3.10 it is clear that the dynamic properties of the active boring bar have
changed significantly, mostly with regard to cutting speed direction (compare
with the results from the standard boring bar Figs. 3.3 and 3.4). 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 3.4 and Table 3.5.
Resonance Frequency, Mode 1 [Hz]
Torque
10Nm
15Nm
20Nm
25Nm
30Nm
Level 1
449.83
466.62
478.76
482.29
484.39
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
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
Table 3.4: Estimates of the fundamental boring bar resonance frequencies
based on all measurements, using the setup in which the active 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. 3.9 a) and b) respectively. The grey rows of mode 1 and
mode 2 correspond to the boring bar frequency response functions in Figs. 3.10
a) and b), produced for the four different excitation levels.
When the active boring bar is clamped with six, size M10 screws, results
obtained resemble those derived from clamping the same bar with size M8
screws, see Figs. 3.9 and 3.11. The largest differences may be observed in the
frequency function estimates in cutting speed direction, see Fig. 3.11 a).
50
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Accelerance [dB rel 1(m/s2 )/N]
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
24
22
20
18
16
14
420
440
460
480
500
520
540
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
24
22
20
18
16
14
420
440
460
480
500
520
540
Frequency [Hz]
b)
Figure 3.7: The accelerance of the boring bar response using the standard
boring bar, six screws of size M10 and when clamp screws were tightened
firstly from the underside of the boring bar, using five different tightening
torques. a) the driving point in cutting speed direction (y-) and b) the driving
point in negative cutting depth direction (x-).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
51
Accelerance [dB rel 1(m/s2 )/N]
32
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
22
20
18
16
14
420
440
460
480
500
520
540
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
26
24
22
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
16
14
420
440
460
480
500
520
540
Frequency [Hz]
b)
Figure 3.8: The accelerance of the boring bar response using the standard
boring bar, six screws of size M10 and when clamp screws were tightened firstly
from the underside of the boring bar, using two different tightening torques and
four different excitation levels. a) the driving point in cutting speed direction
(y-) and b) the driving point in negative cutting depth direction (x-).
52
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
26
24
22
20
10Nm
15Nm
20Nm
25Nm
30Nm
18
16
14
440
450
460
470
480
490
500
510
520
530
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
24
22
20
18
16
14
440
450
460
470
480
490
500
510
520
530
Frequency [Hz]
b)
Figure 3.9: The accelerance of the boring bar response using the active boring
bar, four screws of size M8 and when clamp screws were tightened firstly from
the upper-side of the boring bar, using five different tightening torques. a)
the driving point in cutting speed direction (y-) and b) the driving point in
negative cutting depth direction (x-).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
53
Accelerance [dB rel 1(m/s2 )/N]
32
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
22
20
18
16
14
440
450
460
470
480
490
500
510
520
530
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
26
24
22
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
16
14
440
450
460
470
480
490
500
510
520
530
Frequency [Hz]
b)
Figure 3.10: The accelerance of the boring bar response using the active boring
bar, four screws of size M8 and when clamp screws were tightened firstly from
the upper-side of the boring bar, using two different tightening torques and
four different excitation levels. a) the driving point in cutting speed direction
(y-) and b) the driving point in negative cutting depth direction (x-).
54
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
10Nm
15Nm
20Nm
25Nm
30Nm
26
24
22
20
18
16
14
440
450
460
470
480
490
500
510
520
530
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
10Nm
15Nm
20Nm
25Nm
30Nm
30
28
26
24
22
20
18
16
14
440
450
460
470
480
490
500
510
520
530
Frequency [Hz]
b)
Figure 3.11: The accelerance of the boring bar response using the active boring
bar, six screws of size M10 and when clamp screws were tightened firstly from
the upper-side, using five different tightening torques. a) the driving point in
cutting speed direction (y-) and b) the driving point in negative cutting depth
direction (x-).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
55
Accelerance [dB rel 1(m/s2 )/N]
32
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
22
20
18
16
14
440
450
460
470
480
490
500
510
520
530
510
520
530
Frequency [Hz]
a)
Accelerance [dB rel 1(m/s2 )/N]
32
30
28
26
24
22
20
18
16
14
440
Level 1, 10Nm
Level 2, 10Nm
Level 3, 10Nm
Level 4, 10Nm
Level 1, 30Nm
Level 2, 30Nm
Level 3, 30Nm
Level 4, 30Nm
450
460
470
480
490
500
Frequency [Hz]
b)
Figure 3.12: The accelerance of the boring bar response using the active boring
bar, six screws of size M10 and when clamp screws were tightened firstly
from the upper-side, using two different tightening torques and four different
excitation levels. a) the driving point in cutting speed direction (y-) and b)
the driving point in negative cutting depth direction (x-).
56
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Relative Damping of Mode 1 [%]
Torque
10Nm
15Nm
20Nm
25Nm
30Nm
Level 1
1.40
1.22
1.17
1.16
1.36
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
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 3.5: The relative damping estimates for the fundamental boring bar
bending modes based on all measurements, using the setup in which the active
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. 3.9 a) and b) respectively. The grey
rows of mode 1 and mode 2 correspond to the boring bar frequency response
functions in Figs. 3.10 a) and b), level produced for the four different excitation
levels.
3.1.3
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. 3.13 and Table 3.6,
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. 3.13.
Mode 1
Frequency [Hz]
Damping [%]
Mode 2
Frequency [Hz]
Damping [%]
Modal Parameters
Level 2 Level 3
584.15 583.13
2.04
2.16
Modal Parameters
Level 1 Level 2 Level 3
602.25 602.07 601.92
0.76
0.75
0.75
Level 1
583.82
2.12
Level 4
582.52
2.16
Level 4
601.79
0.74
Table 3.6: The resonance frequency and the relative damping of the linearized
boring bar, estimated with poly-reference technique.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
57
Accelerance [dB rel 1(m/s2 )/N]
36
Level 1
Level 2
Level 3
Level 4
34
32
30
28
26
24
22
20
560
570
580
590
600
Frequency [Hz]
a)
610
620
630
Accelerance [dB rel 1(m/s2 )/N]
36
Level 1
Level 2
Level 3
Level 4
34
32
30
28
26
24
22
20
560
570
580
590
600
Frequency [Hz]
b)
610
620
630
Figure 3.13: The accelerance of the boring bar response using the linearized
setup and with four different excitation levels. a) the driving point in cutting speed direction (Y-) and b) the driving point in negative cutting depth
direction (X-).
58
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
3.1.4
Mode shapes
A boring bar mode shape shows the spatial deformation pattern of the bar
for that particular mode and thus for each degree of freedom measured on the
boring bar, in both amplitude and spatial phase. This section presents all the
mode shapes estimated from the three different setups: the standard boring
bar, the active 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. 3.14, a) and b) respectively. The angle of rotation around
z-axis (relative the cutting depth direction for each measurement) is presented
in Table 3.7. The mode shapes in xy-plane illustrated in Fig. 3.14 b) and the
corresponding values in Table 3.7 show an average rotation of approximately
20 degrees.
Angle of Mode 1, [Degree]
Torque
10Nm
15Nm
20Nm
25Nm
30Nm
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 3.7: Angle of mode shapes for the standard boring bar, relative to cutting
depth direction axis.
Measurements derived from the active boring bar differ somewhat from
those obtained with the standard boring bar. Mode shapes are presented in
Fig. 3.15 and the values of the angle of rotation in Table 3.8. The shapes are
almost identical in the yz-plane, but in the xy-plane the shapes rotate around
the z-axis. Table 3.8, demonstrates a trend of clockwise rotation with increasing torque, as well as counterclockwise rotation with increasing excitation level;
this applies to both modes. The 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.
The results from the linearized setup are presented by Fig. 3.16 and in
Table 3.9.
In this linear setup, the zy-plane shape is almost identical to those shapes
produced from standard, and active boring bar measurements, see Figs. 3.14,
3.15 and 3.16. In the xy-plane the shapes only have a rotation of approximately
10 degrees.
Negative Cutting Speed Direction (y+)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
0.4
59
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
0.3
Negative Cutting Speed Direction (y+)
Feed Direction (z+)
a)
0.4
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
0.8
1
1.2
Cutting Depth Direction (x+)
b)
Figure 3.14: 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.
Negative Cutting Speed Direction (y+)
60
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
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
0.3
Negative Cutting Speed Direction (y+)
Feed Direction (z+)
a)
0.4
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
0.6
0.8
Cutting Depth Direction (x+)
b)
Figure 3.15: The two first mode shapes of the active boring bar clamped with
four M8 screws, when the clamp screws were tightened firstly from the upperside, for five different tightening torques and four different excitation levels. a)
in the zy-plane and b) in the xy-plane.
Negative Cutting Speed Direction (y+)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
0.4
61
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
0.3
Negative Cutting Speed Direction (y+)
Feed Direction (z+)
a)
0.4
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
1
1.2
Cutting Depth Direction (x+)
b)
Figure 3.16: 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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Angle of Mode 1, [Degree]
Torque
10Nm
15Nm
20Nm
25Nm
30Nm
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
Level 4
-18.50
-27.58
-55.76
-67.69
-70.50
Angle of Mode 2, [Degree]
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 3.8: Angle of mode shapes for the active boring bar, relative to cutting
depth direction axis.
Angle of Mode, [Degree]
Excitation Level 1 Level 2 Level 3
Mode 1
-7.98
-8.46
-8.05
Mode 2
-99.72 -100.00 -99.82
Level 4
-8.11
-100.07
Table 3.9: Angle of mode shapes for the linearized boring bar relative to cutting
depth direction axis.
3.1.5
Quality of Measurement
Since the frequency response function estimates are based on the linear H1
estimation method, a measure of the linear relation between input and output signals may be represented by the coherence function, or (as this case has
several sources), the multiple coherence function. Typical multiple coherence
function estimates obtained during the experiment are illustrated in Fig. 3.17;
observe that the level in Fig. 3.17 b) starts from 0.99. 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
Fig 3.18. Typical values of the of-diagonal elements are 0.000-0.001, few values
reach 0.007.
3.1.6
Mass-loading
As previously mentioned, all the sensors, cables and shakers etc. affect the
structure and thus also the relationship between the frequency response function estimate the ”true” frequency response function. It is nice to have an
estimate close to the true frequency response function, the purpose was, however, to examine the influence of different clamping condition on the the boring bar’s dynamic system. In order to acquire information concerning the
sensors’ influence on the resonance frequency and the damping of the boring
bar, a measurement using an impulse hammer and two accelerometers was
conducted. Both impedance heads and all the accelerometers were removed,
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
63
1
2
Multiple Coherence γ̂yx
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
300
350
400
450
500
550
600
650
700
750
800
640
660
680
700
Frequency [Hz]
a)
1
2
Multiple Coherence γ̂yx
0.999
0.998
0.997
0.996
0.995
0.994
0.993
0.992
0.991
0.99
500
520
540
560
580
600
620
Frequency [Hz]
b)
Figure 3.17: The multiple coherence corresponding to typical frequency response function estimates, where solid lines represent cutting speed direction
(y-) and dotted lines represent negative cutting depth direction (x-).
64
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Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
M A C v a lu e
1
0 .8
0 .6
0 .4
0 .2
0
M
.7 2
5 2 6
o d
e [
H
5 5
5 .6
7
z]
.6 7
5 5 5
52
6 .7
2
]
[H z
e
d
M o
Figure 3.18: The modal assurance criterion matrix coefficients for the two
estimated mode shapes at the resonance frequencies 526.72 Hz and 555.67 Hz,
where the of-diagonal values are equal to 0.000. 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.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
65
with the exception of the accelerometers at the driving point. Thus, only two
accelerometers were glued to boring bar providing the mass-loading when hitting the boring bar, (compared with 14 accelerometers and two impedance
head connected to the shakers) see Fig. 3.19. Once again, the periodic disturbances are present but insignificant. Compare Fig. 3.19 with Fig. 3.13, that
is, the estimates from the hammer excitation with shaker excitation, both for
the linearized structure.
The estimates show a resonance frequency shift of approximately 20 Hz
for the driving point in both directions. Mass-loading lowers the resonance
frequency from approximately 604 Hz to 583 Hz in the cutting depth direction,
and from approximately 620 Hz to 602 Hz in the cutting speed direction whilst
still using two acceleromters.
3.1.7
Summary of the Estimated Parameters
A number of phenomenon may be observed in each individual experimental
setup. The estimated parameters for max excitation level and max tightening
torque, 30Nm, are presented in Table 3.10 in order that difference may be
observed.
Setup
1
2
3
4
5
6
Parameters of Mode 1
Freq. [Hz]
Damp. [%]
Angle [◦ ]
525.45
528.58
502.43
482.88
489.11
582.52
0.93
0.97
1.37
1.51
1.70
2.16
-21.84
-17.74
-6.40
-70.93
-59.47
-8.11
Parameters of Mode 2
Freq. [Hz]
Damp. [%]
Angle [◦ ]
555.35
560.83
526.87
512.79
508.22
601.79
0.90
1.00
1.42
1.05
1.10
0.74
-110.35
-108.26
-95.51
-154.52
-141.06
-100.07
Table 3.10: Eigenfrequencies, relative damping and mode shape angle relative
to cutting depth direction for the six different boring bar setups. Clamp screw
tightening torque, 30Nm and maximum excitation signal level.
3.2
Analytical Models of the Boring Bars
This section presents results from a number of different Euler-Bernoulli models
of the boring bar, including different simple models of boring bar clamping.
The first model assumed rigid clamping of the boring bar by the clamping house. 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 house. Thus, at each pinned boundary condition, the boring bar
66
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Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Accelerance [dB rel 1(m/s2 )/N]
35
Direction −y6
Direction −x6
30
25
20
15
10
540
560
580
600
620
640
660
Frequency [Hz]
a)
1
0.99
2
Coherence γ̂yx
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
540
Direction −y6
Direction −x6
560
580
600
620
640
660
Frequency [Hz]
b)
Figure 3.19: a) The accelerance of the boring bar response using the linearized
setup, two accelerometers and an impulse hammer. The solid line represents
the driving point in cutting speed direction (y-), the dashed line is the driving
point in negative cutting depth direction (x-). b) The corresponding coherence
functions; observe that the plot shows the coherence from 0.91 to 1.
Accelerance [dB rel 1(m/s2 )/N]
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
35
30
67
Setup 1
Setup 2
Setup 3
Setup 4
Setup 5
Setup 6
25
20
15
440
460
480
500
520
540
560
580
600
620
640
Accelerance [dB rel 1(m/s2 )/N]
Frequency [Hz]
a)
Setup 1
Setup 2
Setup 3
Setup 4
Setup 5
Setup 6
35
30
25
20
15
440
460
480
500
520
540
560
580
600
620
640
Frequency [Hz]
b)
Figure 3.20: The accelerance of the boring bar response for the six different
setups, a) the driving point in cutting speed direction and b) the driving point
in cutting depth direction.
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model will be rigidly clamped in the cutting speed direction without any rotational constraints about the clamping position. All the Euler-Bernoulli models
of the boring bar assume a homogenous constant cross-section, i.e. E(z) = E,
ρ(z) = ρ A(z) = A, Ix (z) = Ix and Iy (z) = Iy .
3.2.1
Single-span Model
The simplest model is the single span model with rigid clamping at one end
and no clamping (free) at other. The boring bar is assumed to be entirely
contained by the clamping house, which has a length of 100mm, and clamping
ends where the clamping house ends. Thus the model is a fixed-free beam with
the length of 200mm. The first three resonance frequencies in the cutting speed
direction and in the cutting depth direction are presented in Table 3.11, and
the three first mode shapes in Fig. 3.21. Depending on the direction of the
Direction of mode
Cutting speed direction (y-)
Cutting depth direction (x-)
f1 [Hz]
698.33
698.11
f2 [Hz]
4376.36
4374.98
f3 [Hz]
12253.94
12250.08
Table 3.11: The first three resonance frequencies of the Euler-Bernoulli fixedfree model with a length of l = 200mm.
1
Normalized mode shape
0.8
Mode 1
Mode 2
Mode 3
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
Feed direction (z+) [m]
Figure 3.21: The first three mode shapes in cutting speed direction and cutting
depth direction of the Euler-Bernoulli fixed-free model.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
69
transverse motion (cutting speed or cutting depth) assumed to be modeled by
the Euler-Bernoulli beam, the shape of the boring bar cross-section will result
in a slightly different moment of inertia, see Table 2.1. Thus, the resonance
frequencies in respective direction will also differ.
3.2.2
Multi-span Model
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 house, and one corresponding to the boring bar clamped
with six screws in the clamping house. The eigenfrequencies and mode shapes
(eigenfunctions) for the two models were calculated in Matlab by finding the
roots to the characteristic equation produced using the boundary conditions
presented in Appendix A, and using Eq. 2.39, etc. The results for the two
different models are presented in Table 3.12. When the fixed clamping model
Using four screws
Direction of mode
f1 [Hz]
Cutting speed direction (y-) 527.47
Cutting depth direction (x-) 527.30
Using six screws
Cutting speed direction (y-) 566.92
Cutting depth direction (x-) 566.74
f2 [Hz]
3390.18
3389.11
f3 [Hz]
9539.40
9536.39
3575.59
3574.46
10058.76
10055.59
Table 3.12: The first three resonance frequencies in cutting speed direction
and cutting depth direction for the Euler-Bernoulli models, with the following
boundary conditions; free-pinned-pinned-free and free-pinned-pinned- pinnedfree.
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. 3.22.
3.2.3
Multi-span Model on Elastic Foundation
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. These two models 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
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1
Normalized mode shape
0.8
Mode 1
Mode 2
Mode 3
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.25
0.3
Feed direction (z+) [m]
a)
1
Normalized mode shape
0.8
Mode 1
Mode 2
Mode 3
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
Feed direction (z+) [m]
b)
Figure 3.22: The first three mode shapes, for a) the free-pinned-pinned-free
model of the boring bar and b) the free-pinned-pinned-pinned-free model of
the boring bar.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
71
Table 2.5. The length of the clamp screw overhang was selected to 1.5mm.
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 3.13, and mode shapes are shown in
Fig. 3.23.
Using four M8 screws
Direction of mode
f1 [Hz] f2 [Hz]
Cutting speed direction (y-) 519.43 3303.79
Cutting depth direction (x-) 519.27 3302.81
Using six M8 screws
Cutting speed direction (y-) 532.09 3335.17
Cutting depth direction (x-) 531.94 3334.20
Using four M10 screws
Cutting speed direction (y-) 525.24 3346.84
Cutting depth direction (x-) 525.08 3345.83
Using six M10 screws
Cutting speed direction (y-) 541.52 3398.74
Cutting depth direction (x-) 541.36 3397.74
f3 [Hz]
9257.16
9254.48
9278.08
9275.46
9404.65
9401.83
9484.36
9481.62
Table 3.13: The first three resonance frequencies in the cutting speed direction
and in the cutting depth direction, for the two multi-span boring bar models,
with flexible clamping boundary conditions.
3.3
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, clamped with four screws, tightened firstly from the top. The tightening torque was 30Nm and the excitation level was lowest. For the purpose of
simplification, only the mode in cutting speed direction, estimated from the
driving point, was used, thus the linear part HL (f ) of the model only consists
of one degree of freedom, which in terms of receptance may proximately be
written as
HL (f ) =
A∗
A
+
j2πf − λ j2πf − λ∗
(3.1)
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1
Normalized mode shape
0.8
Mode 1
Mode 2
Mode 3
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.25
0.3
Feed direction (z+) [m]
a)
1
Normalized mode shape
0.8
Mode 1
Mode 2
Mode 3
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.05
0.1
0.15
0.2
Feed direction (z+) [m]
b)
Figure 3.23: The first three mode shapes for the Euler-Bernoulli boring bar
model, with boundary conditions a) free-spring-spring-free (four clamp screws)
and b) free-spring-spring-spring-free (six clamp screws).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
73
where
λ = −ζ2πf0 + j2πf0
1 − ζ2
(3.2)
The values are: resonance frequency f0 = 555.675 Hz, damping ζ = 0.966 %
and the residue A = −j1.001 · 10−4 . Thus the linear system may be expressed
as
HL (f ) =
−j1.001 · 10−4
j1.001 · 10−4
+
(3.3)
j2πf − (−33.727 + j3491.246) j2πf − (−33.727 − j3491.246)
Fig. 3.24 displays a diagram if the synthesized SDOF system accelerance function, corresponding to the receptance in 3.1. This figure also presents an
estimate of the driving point accelerance of the boring bar in the direction
of cutting speed, and a synthesized two-degrees-of-freedom system accelerance
function response for two fundamental boring bar modes. These parameters
Accelerance [dB rel 1(m/s2 )/N]
35
30
Measured
Synthesized 1DOF
Synthesized 2DOF
25
20
15
10
5
500
510
520
530
540
550
560
570
580
590
600
Frequency [Hz]
Figure 3.24: The accelerance function for the driving point of the boring bar
in the cutting speed direction, the synthesized sdof system and the synthesized
two-degrees-of-freedom system.
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, which may be expressed in terms of the mass, damping and stiffness coefficients m, c and k.
74
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Theses parameters were determined using the following relations
1
jm4πf0
k
1
f0 =
2π m
c
ζ = √
2 mk
A =
(3.4)
(3.5)
(3.6)
which yields a mass of m = 1.431 kg, a damping of c = 96.503 Ns/m and a
stiffness of k = 17.440 · 106 N/m.
3.3.1
Softening Spring Model
The nonlinear softening stiffness coefficients ks and kc in the signed squared and
cubic model were not obtained by direct parameter estimation. The resonance
frequency shifting phenomena always appears between 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 renders 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 = 8 · 1012
N/m2 and kc = 4 · 1019 N/m3 for the signed squared and cubic model, 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 peak levels 100, 200, 300, 400 mN.
Fig. 3.25 a) presents the frequency response function estimates that were
produced based on simulations of the nonlinear model with a signed squared
stiffness, using the filter method and the four different excitation levels. Fig. 3.25
b) presents the corresponding frequency response function estimates produced
based on simulations of the nonlinear model system with a cubic stiffness,
using the filter method and the four different excitation levels. Table 3.14
gives 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 [18] was used to produce estimates of resonance frequency and relative damping. Fig. 3.26 a) presents the coherence functions for simulations
of the nonlinear model with a signed squared stiffness, using the filter method
and the four different excitation signals. Fig. 3.26 b) shows the coherence
Accelerance [dB rel 1(m/s2 )/N]
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
32
30
75
Level 1
Level 2
Level 3
Level 4
28
26
24
22
20
530
535
540
545
550
555
560
565
570
560
565
570
Accelerance [dB rel 1(m/s2 )/N]
Frequency [Hz]
a)
32
30
Level 1
Level 2
Level 3
Level 4
28
26
24
22
20
530
535
540
545
550
555
Frequency [Hz]
b)
Figure 3.25: 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.
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Excitation
Frequency [Hz]
Damping [%]
Frequency [Hz]
Damping [%]
Squared Model
Level 1 Level 2
552.06 548.47
1.00
1.06
Cubic Model
555.03 553.31
0.98
1.01
Level 3
544.87
1.17
Level 4
541.30
1.31
550.52
1.14
546.98
1.42
Table 3.14: Resonance frequency and relative damping estimates for the frequency response functions based on the nonlinear models, simulated with the
filter method.
functions estimates for the simulations of the nonlinear model with a cubic
stiffness, using the filter method and the four different excitation levels. The
coherence function estimates are also presented for a narrow frequency range
including the resonance frequency and are illustrated for the nonlinear model
with a signed squared stiffness in Fig. 3.27 a) and for the nonlinear model with
a cubic stiffness in Fig. 3.27 b).
If the ordinary differential equation solver ode45 in Matlab is used for simulations of the nonlinear model with a signed squared stiffness for the four
different excitation levels, it results in the frequency response function estimates shown in Fig. 3.28 a). Fig. 3.28 b). presents corresponding frequency
response function estimates, based on simulations of the nonlinear model system with a cubic stiffness, using the ordinary differential equation solver ode45
in Matlab and the four different excitation levels. Table 3.15 presents estimates
of the resonance frequency and the relative damping for the frequency response
functions based on the nonlinear models, simulated with the ordinary differential equation solver ode45 in Matlab, for the four excitation force levels.
Also, in this case, the SDOF least square technique [18] was used to produce
estimates of resonance frequency and relative damping. Fig. 3.29 a) gives the
coherence functions for the simulations of the nonlinear model with a signed
squared stiffness, using ode45 in Matlab and the four different excitation signals. Fig. 3.29 b) shows the coherence functions’ estimates for simulations
of the nonlinear model with a cubic stiffness, using the ordinary differential
equation solver ode45 and the four different excitation levels. The coherence
function estimates are also presented for a narrow frequency range (including
the resonance frequency) and are illustrated for the nonlinear model with a
signed squared stiffness in Fig. 3.30 a) and for the nonlinear model with a
cubic stiffness in Fig. 3.30 b).
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
77
1
0.9
0.8
2
Coherence γ̂yx
0.7
0.6
0.5
0.4
0.3
0.2
Level 1
Level 2
Level 3
Level 4
0.1
0
0
500
1000
1500
2000
2500
3000
Frequency [Hz]
a)
1
0.9
0.8
2
Coherence γ̂yx
0.7
0.6
0.5
0.4
0.3
0.2
Level 1
Level 2
Level 3
Level 4
0.1
0
0
500
1000
1500
2000
2500
3000
Frequency [Hz]
b)
Figure 3.26: 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.
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1
0.9
0.8
2
Coherence γ̂yx
0.7
0.6
0.5
0.4
0.3
0.2
Level 1
Level 2
Level 3
Level 4
0.1
0
530
535
540
545
550
555
560
565
570
Frequency [Hz]
a)
1
0.9
0.8
2
Coherence γ̂yx
0.7
0.6
0.5
0.4
0.3
0.2
Level 1
Level 2
Level 3
Level 4
0.1
0
530
535
540
545
550
555
560
565
570
Frequency [Hz]
b)
Figure 3.27: 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.
Accelerance [dB rel 1(m/s2 )/N]
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
32
30
79
Level 1
Level 2
Level 3
Level 4
28
26
24
22
20
530
535
540
545
550
555
560
565
570
560
565
570
Accelerance [dB rel 1(m/s2 )/N]
Frequency [Hz]
a)
32
30
Level 1
Level 2
Level 3
Level 4
28
26
24
22
20
530
535
540
545
550
555
Frequency [Hz]
b)
Figure 3.28: Frequency response function estimates based on 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.
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1
0.9
0.8
2
Coherence γ̂yx
0.7
0.6
0.5
0.4
0.3
0.2
Level 1
Level 2
Level 3
Level 4
0.1
0
0
500
1000
1500
2000
2500
3000
Frequency [Hz]
a)
1
0.9
0.8
2
Coherence γ̂yx
0.7
0.6
0.5
0.4
0.3
0.2
Level 1
Level 2
Level 3
Level 4
0.1
0
0
500
1000
1500
2000
2500
3000
Frequency [Hz]
b)
Figure 3.29: Coherence function estimates based on 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.
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
81
1
0.99
2
Coherence γ̂yx
0.98
0.97
0.96
0.95
0.94
0.93
0.92
Level 1
Level 2
Level 3
Level 4
0.91
0.9
530
535
540
545
550
555
560
565
570
Frequency [Hz]
a)
1
0.99
2
Coherence γ̂yx
0.98
0.97
0.96
0.95
0.94
0.93
0.92
Level 1
Level 2
Level 3
Level 4
0.91
0.9
530
535
540
545
550
555
560
565
570
Frequency [Hz]
b)
Figure 3.30: 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.
82
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
Excitation
Frequency [Hz]
Dampamping [%]
Frequency [Hz]
Damping [%]
Squared Model
Level 1 Level 2
553.73 551.97
0.98
1.00
Cubic Model
555.35 554.64
0.98
0.98
Level 3
550.18
1.02
Level 4
548.36
1.06
553.44
1.00
551.82
1.06
Table 3.15: Estimates of resonance frequency and relative damping for the
frequency response functions based on the nonlinear models simulated with
the differential equation solver ode45.
Chapter 4
Summary and Conclusions
The results from the experimental modal analysis of boring bars demonstrate
that a boring bar clamped in a standard clamping house with clamping screws
has a nonlinear dynamic behavior. Also, the results indicate that the standard
clamping house with clamp screws is the likely source of the nonlinear behavior.
The experimental modal analysis results from the boring bar clamped in a
”linearized” standard clamping house with steel wedges and epoxy glue indicated a significant reduction in non-linear dynamic behavior. Thus, a boring
bar clamped in a standard clamping house with clamping screws has significant, nonlinear dynamic properties. Different excitation force levels will not
yield identical frequency response function estimates for the same transfer path
in the boring bar (see any of the figures presenting boring bar FRF:s for different excitation force levels, i.e. Figs. 3.4, 3.6, 3.8, 3.10 and 3.12). Based on
a large number of measurements, a trend may be observed; the fundamental
boring bar resonance frequencies decrease with increasing excitation level; see
Table 3.2 and 3.4 which summarize estimated resonance frequencies. 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 damping for the
first mode increases with increasing excitation force level, while damping for
the second mode decreases with increasing excitation force level; see Fig. 3.3.
Also, the results from the active boring bar give an ambiguous indication of
the effects on damping properties; see Fig. 3.5.
The clamp screw tightening torque appears to affect the nonlinear behavior
of the boring bar. Variation in the FRF:s which was introduced by the four
different excitation force levels seems to be larger for a low tightening torque
(10Nm) than for a high tightening torque (30Nm), see, for example, Fig. 3.4.
Also, experimental modal analysis results involving the so-called ”linearized”
boring bar clamping, support the conclusion that clamping conditions influence
the extent of nonlinearities in boring bar dynamics. By examining, for example,
83
84
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
driving point accelerances in the boring bar with ”linearized” clamping for the
four excitation force levels (see Fig. 3.13), it can be seen that only insignificant
differences are present. Thus, nonlinear behavior on the part of the boring
bar seems to be almost removed (within the level of normal experimental
uncertainty).
Another example of dynamic behavior on the part of the clamped boring
bar is exhibited in the change in fundamental boring bar resonance frequencies
with changing clamp screw tightening torques. Boring bar dynamics display
increasing resonance frequency with increasing clamp torque; see Figs. 3.3, 3.5,
3.7, 3.9 and 3.11.
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 and obvious. Hence, new boundary conditions for the
boring bar are introduced. Also, changing the standard boring bar to the active 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 (first from the upper-side or first 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. 3.5 with the boring bar driving point accelerances in Fig. 3.7. If
the clamp screws were tightened firstly from the upper side, the higher resonance frequency in Fig. 3.5 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. 3.7 a) shows a variation from
approximately 495 Hz to 530 Hz, for a change from the lowest to the highest
clamp screws tightening torque. These results may have arisen 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). Assuming that the boring bar is rigidly clamped, and that the boring
bar has a homogenous cross-section in the x-y plane, this would result in one set
of mode shapes in the cutting speed direction (y-direction) and one set of mode
shapes in the cutting depth direction (x-direction). However, this is not the
case in the results presented -the transverse sensitivity of the accelerometers is
not sufficient to explain the obtained deviations of mode shape angles in the xy plane. Transverse sensitivity would only explain mode shape angles of up to
2-3 degrees. There are two possible explanations for this phenomenon. Firstly,
the assumption of a constant cross-section may not be true. The major part
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
85
of the boring bar in the length direction has a constant cross-section; however,
this is not the case for the section of the boring bar head to which the tool is
attached; see Table 2.1 and Fig. 2.5. Secondly, the clamping conditions may
effect the mode shape rotation 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 3.7. It is also possible to notice a trend in the first mode for clockwise
rotation with increasing torque; such a trend is not significant for the second
mode, see Fig. 3.14. 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 active boring bar (see Figs. 3.14, 3.15 and 3.16).
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 3.9). In comparison to
the ”linearized” clamping case, the use of six size M10 clamp screws (tightened
firstly from the top) resulted in a similar rotation of the fundamental modes,
constituting a difference of approx. 2 degrees; see Table 3.10.
The Euler-Bernoulli boring bar models provide rough approximations of
the low-order resonance frequencies and the corresponding spatial shapes of
the modes. Also, (in the x-y plane) the Euler-Bernoulli models will provide
one set of mode shapes in the cutting speed direction (y-direction) and one
set of mode shapes in the cutting depth direction (x-direction). The first and
simplest, fixed-free model overestimates the lower fundamental resonance frequencies by approx. 170 Hz, and the upper fundamental resonance frequency
by approx. 140 Hz compared to the most rigidly clamped boring bar using six M10 clamp screws. Compared with the linearized boring bar setup
the fixed-free Euler-Bernoulli model overestimates the lower fundamental resonance frequencies by approx. 115 Hz, and the upper fundamental resonance
frequency by approx. 100 Hz. Furthermore, the Euler-Bernoulli model yields a
0.2 Hz difference in frequency between the two fundamental resonance frequencies, while, the experimental results from, for example, the linearized boring
bar setup displays a 20-30 Hz difference in fundamental resonance frequencies.
It may be assumed that the linearized setup will display a difference in fundamental resonance frequencies which adheres to the Euler-Bernoulli model.
However, experimental results indicate a difference of approximately 20 Hz (see
Fig 3.13 and Table 3.6). It is obvious the fixed-free model will overestimate
the fundamental resonance frequencies since it assumes rigid clamping which
86
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
is not the case in reality. In addition, the fixed-free model does not consider
the influence of shear deformation and rotary inertia in the beam, meaning
that resonance frequencies will be overestimated.
The multi-span models are assumed to be more realistic, yet simplified,
models of boring bar clamping conditions. The results from the multi-span
model (using the pinned boundary screw positions) illustrate that displacement
of the boring bar is likely to occur between the screws, inside the clamping
house. Since this configuration allows motion over a longer span than the
simple fixed-free model, it also produces lower resonance frequencies. For
this reason, the Euler Bernoulli model is more appropriate for the pinned
boundary condition than for the fixed-free, since the length to diameter ratio
has increased, even though this ratio is still below the recommended value of
10.
The effects of different screw dimensions and properties on the boring bar
may be investigated by using elastic foundations. The results in Table 3.13
show that the fundamental resonance frequencies for the Multi-span model
increase with an increasing number of clamp screws. An M8 screw features a
10 Hz increase, whilst an M10 screw features a 15 Hz increase. This increase
in fundamental resonance frequency (for the Multi-span model) also occurs as
clamp screw size is increased, so that (using four screws) yields an approximate
5 Hz increase, whilst changing from M8 to M10 (using six screws) yields approx.
a 10 Hz increase. This result is confirmed by experimental results which yield a
frequency change of approx. 5-15 Hz, depending on the clamp screw tightening
torque (see Figs 3.3 and 3.5). Mode shapes from the all the Euler-Bernoulli
models are fairly similar. However, inside the clamping house, the mode shapes
differ significantly between the fixed-free model and the multi-span models. In
the case of the multi-span models, the mode shapes have a spatial deflection
inside the clamping house, while, for the fixed-free model, the mode shapes
have no deflection inside the clamping house.
Experimental results strongly indicate that the boring bar (clamped in the
clamping house with screws) possesses nonlinear dynamic properties. Two
different nonlinear single-degree-of-freedom models were simulated in order to
investigate if they bear resemblance the dynamic behavior of the boring bar
clamped in the clamping house 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-degreeof-freedom models which can be used as benchmark. Both the square with
sign stiffness model and the cubic stiffness model show a similar trend in frequency response function estimates as the experimental results (see Tables 3.14
and 3.15). The trend is decreasing resonance frequency with increasing excitation level; see Figs. 3.25 and Figs. 3.28 (produced by the filter method and the
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
87
ODE solver method, respectively). The coherence function estimates for the
input and output signals of the nonlinear SDOF systems simulated with the
filter method display an expected dip at the resonance frequency that increases
with increasing excitation level, see Fig. 3.26. By using the ODE solver method
to simulate the comparatively nonlinear systems, the coherence function estimates assume comparatively slightly higher levels in the resonance frequency
range of the SDOF systems than the filter method.
Chapter 5
Appendix A
The three span model without rotational springs and infinitely stiff transverse
springs will have boundary conditions; Free-Pinned-Pinned-Free; yielding the
equations as
d2 u1 (z) =0
EI dz2 z=0
3
u1 (z) =0
EI d dz
3
z=0
u1 (z)|
z=l1
du1 (z)
du2 (z) +
dz
dz
z=l1
2
2
u1 (z)
u2 (z) EI d dz
+ EI d dz
2
2
z=l1
d3 u1 (z)
d3 u2 (z) EI dz3 + EI dz3 =0
u2 (z)|
z=l1 +l2
du2 (z)
du3 (z) +
dz
dz
z=l1 +l2
2
2
u2 (z)
u3 (z) EI d dz
+ EI d dz
2
2
z=l1 +l2
d3 u2 (z)
d3 u3 (z) EI dz3 + EI dz3 z=l1 +l2
d2 u3 (z) EI dz2 z=l1 +l2 +l3
d3 u3 (z) EI dz3 =0
z=l1
z=l1 +l2 +l3
88
=0
=0
=0
=0
=0
=0
=0
=0
(5.1)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
89
The three span model with transverse springs and rotational springs will have
boundary conditions; Free-Elastic-Elastic-Free; yielding the equations as
2 u (z) 1
EI d dz
=0
2
z=0
3 u (z) 1
EI d dz
=0
3
z=0
2
2
u1 (z)
u2 (z) 1 (z)
EI d dz
+ kR dudz
+ EI d dz
=0
2
2
z=l1
3 u (z)
3 u (z) 1
2
EI d dz
− kT u1 (z) + EI d dz
=0
3
3
z=l1
2 u (z)
2 u (z) 1
2
+ EI d dz
=0
EI d dz
2
2
z=l1
3
3
u1 (z)
u2 (z) EI d dz
+ EI d dz
=0
3
3
z=l1
(5.2)
2
2
u2 (z)
u3 (z) 2 (z)
EI d dz
+ kR dudz
+ EI d dz
=0
2
2
z=l1 +l2
d3 u2 (z)
d3 u3 (z) =0
EI dz3 − kT u2 (z) + EI dz3 z=l1 +l2
2 u (z)
2 u (z) 2
3
EI d dz
+ EI d dz
=0
2
2
z=l1 +l2
3 u (z)
3 u (z) 2
3
+ EI d dz
=0
EI d dz
3
3
z=l1 +l2
2
u3 (z) =0
EI d dz
2
z=l1 +l2 +l3
3 u (z) 3
EI d dz
=0
3
z=l1 +l2 +l3
90
White
Henrik Åkesson, Tatiana Smirnova, Thomas Lagö and Lars Håkansson
The four span model without rotational springs and infinitely stiff transverse springs will have boundary conditions; Free-Pinned-Pinned-Pinned-Free;
yielding the equations as
d2 u1 (z) =0
EI dz2 z=0
3
u1 (z) EI d dz
=0
3
z=0
u1 (z)|
z=l1
du1 (z)
du2 (z) +
dz
dz
z=l1
2
2
u1 (z)
u2 (z) EI d dz
+ EI d dz
2
2
z=l1
d3 u1 (z)
d3 u2 (z) EI dz3 + EI dz3 =0
u2 (z)|
z=l1 +l2
du3 (z) + dz z=l1 +l2
2 u (z) 3
EI d dz
2
z=l1 +l2
d3 u3 (z) EI dz3 z=l1 +l2
=0
z=l1
du2 (z)
dz
2
u2 (z)
EI d dz
+
2
3
u2 (z)
EI d dz
+
3
u3 (z)|
z=l1 +l2 +l3
du4 (z) + dz z=l1 +l2 +l3
2
u3 (z)
d2 u4 (z) +
EI
EI d dz
2
dz 2
z=l1 +l2 +l3
d3 u3 (z)
d3 u4 (z) EI dz3 + EI dz3 z=l1 +l2 +l3
d2 u4 (z) EI dz2 z=l1 +l2 +l3 +l4
d3 u4 (z) EI dz3 z=l1 +l2 +l3 +l4
du3 (z)
dz
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
(5.3)
Analysis of Dynamic Properties of Boring Bars
Concerning Different Clamping Conditions
91
The four span model with transverse springs and rotational springs will
have boundary conditions; Free-Elastic-Elastic-Elastic-Free; yielding the equations as
2 u (z) 1
=0
EI d dz
2
z=0
3 u (z) 1
EI d dz
=0
3
z=0
2
2
u1 (z)
u2 (z) 1 (z)
+ kR dudz
+ EI d dz
=0
EI d dz
2
2
z=l1
3 u (z)
3 u (z) 1
2
EI d dz
− kT u1 (z) + EI d dz
=0
3
3
z=l1
2 u (z)
2 u (z) 1
2
EI d dz
+ EI d dz
=0
2
2
z=l1
3
3
u1 (z)
u2 (z) EI d dz
+ EI d dz
=0
3
3
z=l1
2
2
u2 (z)
u3 (z) 2 (z)
EI d dz
+ kR dudz
+ EI d dz
=0
2
2
z=l1 +l2
3 u (z)
3 u (z) 2
3
− kT u2 (z) + EI d dz
=0
EI d dz
3
3
z=l1 +l2
(5.4)
2 u (z)
2 u (z) 2
3
EI d dz
+ EI d dz
=0
2
2
z=l1 +l2
d3 u2 (z)
d3 u3 (z) EI dz3 + EI dz3 =0
z=l1 +l2
2
2
u3 (z)
u4 (z) 3 (z)
EI d dz
+ kR dudz
+ EI d dz
=0
2
2
z=l1 +l2 +l3
3 u (z)
3 u (z) 3
4
EI d dz
− kT u3 (z) + EI d dz
=0
3
3
z=l1 +l2 +l3
2 u (z)
2 u (z) 3
4
+ EI d dz
=0
EI d dz
2
2
z=l1 +l2 +l3
3 u (z)
3 u (z) 3
4
EI d dz
+ EI d dz
=0
3
3
z=l1 +l2 +l3
2
u4 (z) EI d dz
=0
2
z=l1 +l2 +l3 +l4
3
u4 (z) =0
EI d dz
3
z=l1 +l2 +l3 +l4
Acknowledgments
The present project is sponsored by the Foundation for Knowledge and Competence Development and the company Acticut International AB.
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