Modeling of an Active Boring Bar Tatiana Smirnova, Henrik ˚ akansson

Modeling of an Active Boring Bar Tatiana Smirnova, Henrik ˚ akansson
Modeling of an Active Boring Bar
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
December 2007
Abstract
Vibration problems occurring during internal turning operations in the manufacturing industry urge for adequate passive and/or active control techniques
in order to increase the productivity of machine tools. Usually, passive solutions are based on either boring bars made partly in high Young’s modulus
non-ductile materials such as sintered tungsten carbide or boring bars with
tuned vibration absorbers adjusted to increase the dynamic stiffness in the
frequency range of a certain resonance frequency of the boring bar. By utilizing an active boring bar with an embedded piezoceramic actuator and a
suitable controller, the primary boring bar vibrations originating from the material deformation process may be suppressed with actuator-induced secondary
”anti-” vibrations. In order to design an active boring bar, several issues have
to be addressed, i.e., selecting the characteristics of the actuator, the actuator
size, the position of the actuator in the boring bar, etc. This usually implies
the manufacturing and testing of several prototypes of an active boring bar,
and this is a time-consuming and costly procedure. Therefore, mathematical
models of active boring bars incorporating the piezo-electric effect that enable
the accurate prediction of their dynamic properties and responses are of great
importance. This report addresses the development of a ”3-D” finite element
model of the system ”boring bar-actuator-clamping house”. The spatial dynamic properties of the active boring bar, i.e., its natural frequencies and mode
shapes, as well as the transfer function between actuator voltage and boring
bar acceleration are calculated based on the ”3-D” FE model and compared to
the corresponding experimentally obtained estimates. Two types of approximations of the Coulomb friction force, the arctangent and the bilinear models,
are evaluated concerning modeling contact between the surface of the boring
bar and the clamping house.
Contents
1 Introduction
5
2 Materials and Methods
2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . .
2.1.1 System Overview . . . . . . . . . . . . . . . . . . .
2.1.2 Measurement Equipment and Setup . . . . . . . . .
2.1.3 System Identification . . . . . . . . . . . . . . . . .
2.2 Experimental Modal Analysis . . . . . . . . . . . . . . . .
2.3 Euler-Bernoulli Model of an Active Boring Bar . . . . . . .
2.4 ”3-D” Finite Element Model . . . . . . . . . . . . . . . . .
2.4.1 Model of the system ”boring bar - clamping house”
2.4.2 Model of the Actuator . . . . . . . . . . . . . . . .
2.4.3 Contact Modeling in the Finite Element Analysis .
2.4.4 Coulomb Friction Modeling . . . . . . . . . . . . .
2.4.5 Transient response . . . . . . . . . . . . . . . . . .
2.4.6 Harmonic response . . . . . . . . . . . . . . . . . .
2.5 SDOF Nonlinear Model . . . . . . . . . . . . . . . . . . . .
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3 Results
3.1 Modal Analysis Results . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Natural Frequencies . . . . . . . . . . . . . . . . . . . .
3.1.2 Mode Shapes . . . . . . . . . . . . . . . . . . . . . . .
3.2 System Identification . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Experimentally Estimated Accelerance . . . . . . . . .
3.2.2 Harmonic Response Based on FEM . . . . . . . . . . .
3.2.3 Transient Response Based on FEM . . . . . . . . . . .
3.3 Dynamic Modeling of the Boring Bar with the Coulomb Friction
Force Included . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Transient Response Based on the SDOF Model . . . .
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3.3.2
3.4
Dynamics of the ”3-D” Finite Element Model Enabling
Variable Contact between the Boring Bar and the Clamping House and also with Coulomb Friction Force . . . . . 49
Actuator Receptance . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Summary and Conclusions
56
Chapter 1
Introduction
The internal turning operation is generally considered as one of the most
vibration-prone metal working processes. In such operations, a boring bar
is used to machine deep, precise geometries to required tolerances inside a
pre-drilled hole in a workpiece. A boring bar can usually be characterized as
a slender beam and is generally the weakest link in a machine tool system.
During turning, the material deformation process induces a broad-band excitation of the machine tool, and, as a result, relative dynamic motion between
the boring bar and the workpiece frequently occurs, commonly referred to as
chatter. High levels of boring bar vibration result in poor surface finish, excessive tool wear, tool breakage and severe levels of acoustic noise. Thus, boring
bar vibration has a negative impact on productivity, working environment,
etc. Usually, the high vibration level is excited at the natural frequencies related to low-order bending modes of the boring bar and are dominated by the
bending mode in the cutting speed direction, since it is in this direction that
the cutting force has the largest component [1, 2, 3]. In industry, there are
considerable demands concerning methods improving the stability of internal
turning operations. The motive is mainly to increase the productivity, but the
working environment is also a growing issue.
From at least the beginning of the twentieth century, research in metal
cutting has been devoted to expanding the existing knowledge of cutting dynamics, etc., by means of mathematical modeling and experimental studies
[4, 5, 6, 3, 7, 8, 9]. Most research carried out on the dynamic modeling of cutting dynamics has concerned the prediction of stability limits, i.e., predicting
cutting data for stable cutting [3, 8].
The strategies for controlling boring bar vibrations can be classified into
two directions. The first direction refers to control of cutting data in order
to maintain stable cutting, i.e., to avoid cutting data resulting in chatter or
to continuously vary cutting data in a structured manner to avoid chatter
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Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
[9, 10, 11]. The second direction concerns modifications of the dynamic stiffness
of one or several parts of the chain insert - tool holder - clamping - machine tool,
with the purpose of increasing the system’s resistance to machine tool chatter
[9, 12, 13, 14, 15, 16]. Generally, the boring bar vibration control methods
modifying the dynamic stiffness are divided into two groups: passive and active
control. In passive boring bar vibration control, the dynamic stiffness may be
increased by changing the static stiffness of the bar, e.g., for instance by using a
boring bar produced completely or partly (composite boring bar) of materials
with higher modulus of elasticity such as sintered tungsten carbide [9, 12].
Another passive control strategy is to use passive Tuned Vibration Absorbers
(TVA) to resist machine tool chatter [9, 12]. A TVA consists of a tube which
contains a reactive mass inside a layer of damper oil and is usually built into the
boring bar close to the tool tip [12, 13]. TVA boring bars offer solutions with
a fixed enhancement of the dynamic stiffness, frequently tuned for a narrow
frequency range comprising the fundamental bending modes eigenfrequencies
[9]. On the other hand, active feedback control of turning operations produces
a selective increase of the dynamic stiffness at the actual frequency of the
dominating bending mode [14, 15, 16].
An active control approach was reported by Tewani et al. [14] concerning
active dynamic absorbers in boring bars controlled by a digital state feedback
controller. It was claimed to provide a substantial improvement in the stability
of the cutting process. Browning et al. [15] reported an active clamp for
boring bars controlled by a feedback version of the filtered-x LMS algorithm.
They assert that the method enables one to extend the operable length of
boring bars. Claesson and Håkansson [16] controlled tool vibration by using
the feedback filtered-x LMS algorithm to control tool shank vibration in the
cutting speed direction, without applying the traditional regenerative chatter
theory.
Two important constraints concerning the active control of tool vibration
involve the difficult environment in a lathe and industry demands. It is necessary to protect the actuator and sensors from the metal chips and cutting fluid.
Also, the active control system should be applicable to a general lathe. Pettersson et al. [17] reported an adaptive active feedback control system based on
a tool holder shank with embedded actuators and vibration sensors. This control strategy was later applied to boring bars by Pettersson et al. [18]. Åkesson
et al. [19] reported the successful application of the active adaptive control of
boring bar vibration in industry using an active boring bar with embedded
actuators and vibration sensors.
The active control of boring bar vibration is based on an active boring bar
equipped with embedded actuators and vibration sensors in conjunction with
a feedback controller. The active boring bar typically has an accelerometer
Modeling of an Active Boring Bar
7
attached close to the tool-end, which measures boring bar vibration in the
cutting speed direction. The controller uses the accelerometer signal to produce secondary or ”anti-” vibrations via an actuator embedded inside a groove
milled in the longitudinal direction below the center line of the boring bar. Due
to the piezoelectrical properties of the actuator material, the dynamic control
signal will steer the length expansion of the actuator. The actuator will in
turn apply a bending moment to the boring bar to counteract the primary
vibration excited by the material deformation process [20].
In order to design an active boring bar with embedded actuator, several
issues have to be addressed, i.e., selecting the characteristics of the actuator,
the actuator size, the position of the actuator in the boring bar, etc. Obtaining adequate performance from an active boring bar usually implies the
manufacturing and testing of several prototypes of the active boring bar, this
is a complex, time consuming and costly procedure. Thus, it is likely that the
efficiency of design procedure can be increased by means of dynamic modeling
of active boring bar, for instance, by utilizing ”3-D” finite element modeling. It
is plausible that such a model can be used, e.g., for predicting of the dynamic
properties of an active boring bar, describing the interaction of the boring bar
and actuator, and accounting for nonlinearities introduced into its response by
the contact between the boring bar, clamping house and clamping screws.
A number of research works have been carried out concerning the modeling of a boring bar as a system with a large but finite number of degreesof-freedom, using finite element analysis. Wong [21] used Timoshenko beam
finite elements to model a boring bar. He designed an electromagnetic dynamic absorber and simulated its performance, utilizing modal control and
direct feedback control based on the finite element model of the boring bar.
Wong claims that both controllers succeed in significant damping of boring
bar vibrations at the lowest vibration mode. Nagano [22] used pitched-based
carbon fiber reinforced plastic (CFRP) material to develop a chatter resistant
boring bar with a large overhang. He made an attempt to create a ”3-D” finite
element model in order to predict natural frequencies and improve dynamic
characteristics of the boring bar by modeling embedded steel cores of various
shapes. The cutting performance and stability of the designed boring bars
regarding chatter were investigated experimentally. He claims that utilization
of the CFRP material together with the cross-shaped steel core allows for the
successful stable machining for boring bars with length-to-diameter of more
then seven. Nagano also mentioned the necessity concerning the development
of improved models for the clamping of the boring bar. Later, Sturesson et
al. [23] developed a ”3-D” finite element model of a tool holder shank. They
used normal mode analysis to evaluate the natural frequencies, modal masses
and mode participation factors of the tool holder shank. The modal damping
8
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
was estimated using the free vibration decay method. The spectral densities’
estimates were also utilized to obtain natural frequencies. The results of the
normal mode analysis and spectral densities’ estimates were well-correlated.
Baker et al. [24] used FEM in the stability analysis of a turning operation. He
approximated the cutting force by using the orthogonal cutting force model.
Baker also based his method of stability prediction on the assumption of the
linearity of the tooling structure’s behavior. Two different types of FE models
were considered: firstly, ”1-D” FE models of the tool holder and the workpiece
attached to the rigid base; secondly, ”3-D” FE models of the tool holder and
the workpiece attached to the deformable ”3-D” FE model of machine tool.
The FE models were used to extract structural matrices, with the purpose
of using them in stability analysis. The maximum stable width of cut was
predicted for a set of geometric dimensions of the tool holder and the workpiece (for both types of models) as a function of spindle speed. However, no
experimental results were presented. Later, in [25], Mahdavinejad also tried to
predict maximum width of cut, ensuring stable cutting with the use of a ”3-D”
FE model of a machine tool. Several ”3-D” FE models containing different
machine tool parts were developed. Natural frequencies and the mode shapes
of the tailstock were estimated based on the ”3-D” FE models and compared
with results obtained from experimental modal analysis. The stability lobe
diagrams were obtained for the chuck-center with and without tailstock cases.
In both cases the stability lobe diagrams were calculated based on analytical
considerations and experiments. Results from the modal analysis as well as
the stability lobe diagrams produced with the use of FE models are claimed
to be well-correlated to corresponding experimental estimates.
Thus, it appears as though no work has been carried out on the finite
element modeling of active boring bars with embedded piezoelectric stack actuators.
This report addresses the process of developing a ”3-D” finite element
model of the system ”boring bar - actuator - clamping house”. Estimates
of the first two natural frequencies and the corresponding mode shapes have
been produced based on both an initial linear ”3-D” finite element model and
an experimental modal analysis of the active boring bar, and also compared. A
more advanced nonlinear ”3-D” finite element model of the active boring bar,
enabling variable contact between clamping house and boring bar, has also
been considered. As a further extension of this ”3-D” finite element model, a
model that incorporates nonlinear friction force acting between the contacting
surfaces of the boring bar and clamping house has been evaluated. Two different models of the Coulomb friction force, the bilinear and the arctangent,
have been considered. The nonlinear active boring bar FE models were evaluated in comparison with the experimental modal analysis results. Estimates
Modeling of an Active Boring Bar
9
of control path frequency response functions between the voltage applied to
the actuator and the acceleration in the position of the error accelerometer
in the cutting speed direction and in the cutting depth direction, have been
produced. These frequency response functions were estimated both for data
obtained from the numerical simulations using the ”3-D” finite element models
and for experimental data from the lathe. Also, a simple distributed-parameter
Euler-Bernoulli model of the active boring bar has been introduced.
Chapter 2
Materials and Methods
2.1
2.1.1
Experimental Setup
System Overview
The experimental modal analysis and control path estimates were conducted
in a Mazak SUPER QUICK TURN - 250M CNC turning center. The machine
tool has a spindle power of 18.5 kW and a maximal machining diameter of
300 mm, a maximal spindle speed of 4000 (r.p.m.), with 1007 mm between the
centers and a turret capacity of 12 tools (see Fig. 2.1).
Figure 2.1: Mazak 250 SUPER QUICK TURN - 250M CNC turning center.
10
11
Modeling of an Active Boring Bar
Active Boring Bar
The active boring bar was based on a standard boring bar, S40T PDUNR15
F3 WIDAX (see Fig. 2.2). The WIDAX boring bar is made of the material
30CrNiMo8; in the modeling, it is assumed that Young’s elastic modulus E =
205 GP a, density ρ = 7850 kg/m3 and Poisson’s coefficient ν = 0.3.
A
D
z
40
m m
A -A
3 0 0 m m
3 7 m m
x
y
A
3 7 .5 m m
x
Figure 2.2: Top-view and cross-section of a standard boring bar, S40T
PDUNR15 F3 WIDAX.
It has an actuator embedded into a milled space below the center line of
the boring bar and an accelerometer attached close to the insert (see Fig. 2.3).
The coordinate system is defined as follows: x is the cutting depth direction,
y is the negative cutting speed direction and z is the feed direction.
E rro r a c c e le ro m e te r in th e
c u ttin g d e p th d ire c tio n
y
A c tu a to r
z
In se rt
x
E rro r a c c e le ro m e te r in th e
c u ttin g s p e e d d ire c tio n
Figure 2.3: Schematic view of an active boring bar.
Clamping House
As a clamping house, a standard 8437-0 40 mm Mazak holder was used (see
Fig. 2.4). The clamping house is attached to the turret by four screws. The
boring bar can be clamped in the clamping house using either four or six
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
12
screws. In the experiments and simulations presented in the current report,
only four screws were used to clamp the boring bar.
S c r e w p o s itio n s fo r th e c la m p in g o f th e b o r in g b a r
y
x
z
x
S c r e w p o s itio n s fo r a tta c h in g
to th e tu r r e t
a)
y
b)
Figure 2.4: Standard 8437-0 40 mm Mazak holder: a) general view; b) view of
the side for the turret contact.
Piezoelectric Actuator
The piezoelectric stack actuator used in the experiments is made of the piezoelectric material Lead Zirconate Titanate (PZT-5H) and is shown in Fig. 2.5.
The actuator material properties (in the finite element modeling) were chosen
as they are similar to the properties of PZT-5H material, with modifications
made for the strain coefficients d33 and d31 in order to match the specification
for the maximum stroke of the actuator [26, 27]. The actuator specifications
are given in Table 2.1, [26, 27].
Furthermore, the piezoelectric material properties such as the elastic coefficient matrix [cE ], the dielectric matrix [ε] and the piezoelectric matrix [e] for
this material are given in Eq. 1, Eq. 2 and Eq. 3 respectively [26, 27].
Modeling of an Active Boring Bar
Property name
Actuator material
Free expansion ∆La , [m]
Strain coefficient d33 , [m/V]
Max operating voltage (P-P) Vmax , [V]
Density ρ, [kg/m3 ]
Actuator stiffness ka , [N/m]
13
Value
PZT-5H
38 × 10−6
640 × 10−12
150
7500
125 × 106
Table 2.1: Actuator specifications.
⎡
⎢
⎢
⎢
E
[c ] = ⎢
⎢
⎢
⎣
12.72 8.02 8.47
0
0
0
8.02 12.72 8.47
0
0
0
8.47 8.47 11.74 0
0
0
0
0
0
2.30 0
0
0
0
0
0 2.30 0
0
0
0
0
0 2.35
⎤
⎥
⎥
⎥
⎥ × 1010 , [P a]
⎥
⎥
⎦
(2.1)
Figure 2.5: Piezoelectric stack actuator.
⎡
⎢
⎢
⎢
[e] = ⎢
⎢
⎢
⎣
⎤
0
0
−6.62
0
0
−6.62 ⎥
⎥
0
0
23.24 ⎥
⎥ , [C/m2 ]
0
17.03
0 ⎥
⎥
17.03
0
0 ⎦
0
0
0
(2.2)
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
14
⎡
⎤
27.71
0
0
27.71
0 ⎦ × 10−9 , [F/m]
[ε] = ⎣ 0
0
0
30.10
2.1.2
(2.3)
Measurement Equipment and Setup
The following equipment was used to carry out experimental modal analysis:
• 12 PCB 333A32 accelerometers;
• 1 Ling Dynamic Systems shaker v201;
• 1 Gearing & Watson Electronics shaker v4
• 2 Brüel & Kjǽr 8001 impedance heads;
• HP VXI E1432 front-end data acquisition unit;
• PC with IDEAS Master Series version 6.
The boring bar was simultaneously excited in the cutting speed direction
and cutting depth direction by two shakers via impedance heads attached at
the distance l1 = 100 mm from the clamped end of the active boring bar
(see Figure 2.6). The spatial motion of the boring bar was measured by 12
accelerometers and 2 impedance heads glued with the distance of l2 = 25 mm
from each other starting at 25 mm from the free end of the boring bar: 6
accelerometers and one impedance head in the cutting speed direction and 6
accelerometers and the other impedance head in the cutting depth direction.
The following equipment was used to carry out control path identification:
• 2 PCB 333A32 accelerometers;
• 1 KEMO Dual Variable Filter VBF 10M;
• A custom designed amplifier for capacitive loads;
• HP VXI E1432 front-end data acquisition unit;
• PC with IDEAS Master Series version 6.
During the control path identification, the active boring bar, clamped in
the lathe, was excited by means of voltage applied over the actuator. The
response of the boring bar was measured by two accelerometers located at the
error sensor positions, which were at a distance of 25 mm from the free end of
the boring bar in the cutting speed and cutting depth direction respectively.
15
Modeling of an Active Boring Bar
C la m p in g h o u s e
B o r in g b a r
P o s itio n s o f e r r o r
a c c e le r o m e te r s
l2
l2
A c c e le r o m e te r s in th e c u ttin g
d e p th d ir e c tio n
...
l2
l2
l2
l2
l2
l1
P o s itio n s o f
im p e d a n c e h e a d s
y
...
A c c e le r o m e te r s in th e c u ttin g
s p e e d d ir e c tio n
x
z
Figure 2.6: Drawing of the clamped active boring bar with accelerometers and
cement studs for the attachment of impedance heads.
Also system identification was carried out for the transfer paths between
the voltage applied to the actuator and the strain measured at the points P1,
P2, P3 and P4 close to the ”boring bar - actuator” interfaces (see Fig. 2.7).
Four KYOWA strain gages were used for this purpose.
P 3
P 4
S tra in g a g e s
P 2
P 1
x
z
Figure 2.7: Drawing of the active boring bar with strain gages positions.
16
2.1.3
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
System Identification
System identification in general concerns the production of a mathematical
model of the dynamic properties of an unknown system, based on experimentally obtained data. In the case of mechanical systems, system identification
usually implies estimating the frequency response functions between an input
excitation force signal and an output signal which can be displacement, velocity or acceleration. Since it is most convenient to measure the acceleration of
a mechanical system, accelerance frequency response functions are frequently
considered.
Spectral properties The frequency response function can be estimated as
a ratio of the cross-power spectral density between the excitation and the response signals and the power spectral density of the excitation signal. Most of
the non-parametric methods for spectrum estimation are based on the averaging of periodograms [28]. The most general is Welch’s method, which allows
one to average modified periodograms, which are produced based on the windowed sequences of a signal that may be overlapping to a certain extent.
The double-sided power spectral density of a discrete time signal x(n) can
be estimated using Welch’s method as follows [28]:
2
P SD
−j2πnk/N (fk ) =
x
(n)w(n)e
P̂xx
(2.4)
N −1 2
m
M Fs n=0
w (n) m=0 n=0
k
fk =
Fs
N
where N is the block length or periodogram length, k = 0, . . . , N − 1, xm (n) =
x(n + mD), m = 0, 1, ..., M − 1 is the data segment of the signal x(n) with the
length L, M is the number of periodograms, D is the overlapping increment,
i.e., for 50% overlap D=N/2, Fs is the sampling frequency and w(n) is the
window function. In the case of power spectral density estimation, the Hanning
window is commonly used [28].
In practice, the excitation and response signals can be corrupted with noise.
Depending on whether the noise is affecting the measured excitation signal or
response signal, frequency response function estimators appropriate for the
respective case may be utilized. The H1 estimator is usually implemented in
FFT-analyzers [29] and may be used when noise is only assumed to corrupt
the response signal.
1
Ĥ1 (fk ) =
M
−1 N
−1
P SD
(fk )
P̂yx
P SD (f )
P̂xx
k
(2.5)
Modeling of an Active Boring Bar
17
The Ĥ2 estimator is used when noise is assumed to only affect the measured
excitation signal.
Ĥ2 (fk ) =
P SD
(fk )
P̂yy
(2.6)
P SD (f )
P̂yx
k
The quality of the frequency response function estimate can be evaluated
2
, which is the ratio of two estimators
via the coherence function estimate γ̂yx
[30].
2
(fk )
γ̂yx
=
Ĥ1 (fk )
Ĥ2 (fk )
=
P SD
(fk )|2
|P̂yx
P SD (f )P̂ P SD (f )
P̂xx
k
k
yy
(2.7)
The normalized random error of the frequency response function’s magnitude function estimate can be estimated according to [30]:
2
(fk ))1/2
(1 − γ̂xy
εr [|Ĥxy (fk )|] ≈ 2 (f )M
2γ̂xy
k
e
(2.8)
where Me is the equivalent number of averages. The equivalent number of
averages is given by [31]:
Me =
1+2
M
M −1 M −m
m=1
M
(2.9)
(m)
where (m) is given by [31]:
(m) =
N −1
n=0
w(n)w(n + mD)
2
N −1 2
w
(n)
n=0
2
(2.10)
In the case when power and cross-power spectral densities estimates are
produced without the overlapping of data blocks, Me = L/N . This typically
corresponds to the case when burst random noise is used as an excitation
signal.
The normalized random error for the coherence function can be estimated
by [30]:
√
2
2(1 − γ̂xy
(f ))
2
(f )] ≈ (2.11)
εr [γ̂xy
2 (f )M
γ̂xy
e
The normalized random error of the multiple coherence function is given
by [30]:
18
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
√
2
εr [γ̂y:x
(f )]
≈
2
2[1 − γ̂y:x
(f )]
2 (f )(M + 1 − s)
γ̂y:x
e
(2.12)
where s is the number of excitation signals.
The power spectral density estimation parameters used in the production
of the experimental control path frequency response function estimates are
summarized in Table 2.2.
Parameter
Signal type
Excitation frequency range
Sampling frequency, Fs
Number of spectral lines, N
Frequency resolution, ∆f
Number of averages
Window
Overlap
Value
True random
0-1000 Hz
2560
3201
0.3125 Hz
200
Hanning
50 %
Table 2.2: Parameters for the experimental spectral density estimation.
The power spectral density estimation parameters used in the production
of the numerical control path frequency response function estimates are summarized in Table 2.3.
Parameter
Signal type
Excitation frequency range
Sampling frequency, Fs
Number of spectral lines, N
Frequency resolution, ∆f
Number of averages
Window
Overlap
Value
True random
0-1000 Hz
1536 Hz
1536
1 Hz
5
Hanning
50 %
Table 2.3: Parameters for the spectral density estimation concerning the finite
element models.
19
Modeling of an Active Boring Bar
2.2
Experimental Modal Analysis
In the concept of experimental modal analysis, the boring bar is considered
as a multiple degree-of-freedom system. Excitation forces and responses are
measured spatially at discrete positions on the boring bar and collected simultaneously for subsequent parameter estimation or curve fitting. Experimental
modal analysis allows for the identification of a system’s modal parameters:
natural frequencies, mode shapes and relative damping ratios. Experimental
modal analysis serves as a tool for, e.g., the verification and updating of finite
element models [?]. The theory and methods of experimental modal analysis are essentially built on four assumptions: the system is linear, the system
is time-invariant, the system is observable and the system obeys Maxwell’s
reciprocity theorem [32].
Parameter Estimation In the concept of the experimental modal analysis
the boring bar is modeled as a system with Nema degrees-of-freedom. The
boring bar vibration can be described by the equation of motion in matrix
form.
[M]{ẅ(t)} + [C]{ẇ(t)} + [K]{w(t)} = {f (t)}
(2.13)
where Nema is the number-of-degrees of freedom, the matrix [M] is the Nema ×
Nema mass matrix, [C] is the Nema ×Nema damping matrix and [K] is the Nema ×
Nema elastic stiffness matrix. Vector {f (t)} is the space- and time-dependent
load vector. Vector {w(t)} is the space- and time-dependent displacement
vector. Its i-th element contains displacement measured in the point with
coordinates (xi , yi , zi ) and i = 1, ..., Nema at time instant t. The displacement
vector may be written as:
⎧
⎫
⎪
⎪
w(x1 , y1 , z1 , t)
⎪
⎪
⎪
⎪
⎨
⎬
w(x2 , y2 , z2 , t)
{w(t)} =
(2.14)
..
⎪
⎪
.
⎪
⎪
⎪
⎪
⎩ w(x
⎭
Nema , yNema , zNema , t)
The spatial dynamic properties of the boring bar were identified using
the time-domain polyreference least squares complex exponential method [29].
This method is based on the discrete-time version of the impulse response
function matrix:
[h(n)] =
N
ema
r=1
λr Ts n
[A]r e
∗
−λr Ts n
+ [A ]r e
=
2N
ema
r=1
[A]r eλr Ts n
(2.15)
20
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
where [A]r = Qr {ψ}r {ψ}Tr and λr = 2π(−fr ζr + jfr 1 − ζr2 ), {ψ}r is the
Nema × 1 mode shape vector, ζr is the modal damping ratio, fr is the undamped system’s eigenfrequency, Qr is the modal scaling factor and Ts is the
sampling time interval. An estimate of the impulse response function matrix [ĥ(n)] is produced based on the Inverse Fourier Transform of an estimate
of the receptance matrix [Ĥr (fk )]. The estimate of the receptance matrix
[Ĥr (fk )] is assembled based on power spectral density estimates of the measured excitation forces and the cross-power spectral density estimates between
the measured output responses and excitation forces.
Based on the simultaneously estimated impulse responses, between each of
the input forces locations and all of the output response locations, the polyreference least squares complex exponential method utilizes Prony’s method and
the least squares method in the production of global estimates of eigenfrequencies fr , damping ratios ζr and mode shapes {ψ}r [32, 29].
The orthogonality of extracted mode shapes {ψEM A }i and {ψEM A }j , i, j ∈
{1, 2, . . . , Nema }, may be evaluated using the Modal Assurance Criterion [32]:
M ACij =
|{ψEM A }Ti {ψEM A }j |2
({ψEM A }Ti {ψEM A }i )({ψEM A }Tj {ψEM A }j )
(2.16)
The Modal Assurance Criterion may also be used to provide a measure on
the correlation between the experimentally-measured mode shapes {ψEM A }i
and the numerically-calculated mode shapes {ψF EM }j of, e.g., a finite element
model.
M ACij =
|{ψEM A }Ti {ψF EM }j |2
({ψEM A }Ti {ψEM A }i )({ψF EM }Tj {ψF EM }j )
(2.17)
The parameters used in the modal analysis of the clamped active boring
bar can be found in Table 2.4.
2.3
Euler-Bernoulli Model of an Active Boring
Bar
A distributed-parameter Euler-Bernoulli beam model [33, 34] can be used to
describe the dynamic motion of an active boring bar. The Euler-Bernoulli
beam model considers only transverse beam vibrations and ignores shear deformation and rotary inertia [35, 36]. A ”clamped-free” model of the active
boring bar is shown in the Fig. 2.8. The sign convention used for displacement, forces and moment is as follows: forces and displacements in the positive
directions of the coordinate systems axes are positive, and a moment about a
21
Modeling of an Active Boring Bar
Parameter
Signal type
Excitation frequency range
Sampling frequency, Fs
Number of spectral lines, N
Frequency resolution, ∆f
Number of averages
Window
Frequency range for curve fitting
Value
Burst random (90/10)
300-800 Hz
1280
1601
0.3125 Hz
200
Rectangular
400-600 Hz
Table 2.4: Modal analysis parameters.
coordinate system axis that is in the counterclockwise direction if viewed from
the end of the coordinate system axis towards the origin is considered to be
positive. If one assumes that the geometric features of the boring bar tool-end
are negligible to overall boring bar dynamics, i.e., that cross-sectional area A(z)
and cross-sectional moment of inertia around x axis Ix (z) are constants, then
the Euler-Bernoulli partial differential equation of the boring bar transverse
vibrations in the cutting speed direction may be written as [19, 35, 37, 38]:
y
P ie z o c e ra m ic s ta c k a c tu a to r
-m
e
(t)
+ m
a
z
fa2(t)
A c c e le ro m e te r
e
(t)
fa1(t)
z 2(t)
z 1(t)
Figure 2.8: The ”clamped-free” model of the active boring bar.
∂me (z, t)
∂ 2 w(z, t)
∂ 4 w(z, t)
+ EIx
=
ρA
(2.18)
2
4
∂t
∂z
∂z
where ρ is the density of the boring bar material, w(z, t) is the deflection in
y direction, E is the Young’s elastic modulus and me (z, t) is the space- and
time-dependent external moment load per unit length.
The cross sectional area A, the moments of inertia around ”x”-axis Ix and
around ”y”-axis Iy were calculated assuming that the distributed-parameter
22
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
model of the active boring bar has a constant cross-section as shown in Fig.
2.2. The calculated boring bar’s cross-sectional properties are summarized in
Table 2.5.
Property
l
A
I
Cutting speed direction Cutting depth direction
0.2
1.1933 × 10−3
−7
1.1386 × 10
1.1379 × 10−7
Units
m
m2
m4
Table 2.5: Properties used in Euler-Bernoulli model calculations.
To obtain the forced response of the beam, the expansion theorem can be
utilized [36], i.e., the time-domain dynamic response of the boring bar can be
expressed as:
w(z, t) =
∞
ψr (z)yr (t)
(2.19)
r=1
where ψr (z), 1 ≤ r are the normal modes determined by the boundary conditions for the beam and the normalization of the eigenfunctions or mode shapes,
and yr (t) is the modal displacement. Each modal displacement yr (t) can be
determined as a forced response of an undamped mechanical system, which is
usually referred to as Duhamel’s integral [39].
t
1
fload,r (τ ) sin(2πfr (t − τ ))dτ
(2.20)
yr (t) =
mr 2πfr 0
where mr is the modal mass, fr is the undamped system’s eigenfrequency,
fload,r (t) is the generalized load for mode r. In the case of an external moment
load, the generalized load for mode r, fload,r (t) is given by:
l
∂me (z, t)
ψr (z)
dz
(2.21)
fload,r =
∂z
0
where l is the length of the beam, i.e., the length of the boring bar overhang.
Due to the piezoelectric properties of actuator material, the actuator expands under the applied voltage. The expansion of the actuator is, however,
constrained at the ”boring bar - actuator” interfaces. As the actuator exhibits
elastic behavior, some part of its free expansion will be lost and transformed
into loads applied to the boring bar: fa1 (t) and fa2 (t), in parallel with the
”z”-axis. This, in turn will cause displacements z1 (t) and z2 (t) of the boring
bar in ”boring bar - actuator” interfaces. Because of the position of the actuator, i.e., it is placed below the central line of the boring bar with the distance
Modeling of an Active Boring Bar
23
α, the loads applied by the actuator result in the bending and stretching of
the boring bar. The dynamics of the actuator - boring bar interaction can be
described in terms of point receptances at the respective actuator interface of
the active boring bar. Assuming that the Fourier Transform of the responses
Z1 (f ) and Z2 (f ) and of the loads, Fa1 (f ) and Fa2 (f ) (where f is frequency)
exists, the point receptances may be expressed as:
H1 (f ) = −
Z1 (f )
Fa1 (f )
(2.22)
respective
H2 (f ) =
Z2 (f )
Fa2 (f )
(2.23)
Assuming that the actuator operates below its resonance frequency, thus
neglecting the inertial effects of the actuator, the force that the actuator exerts
on the boring bar, Fa (f ) = −Fa1 (f ) = Fa2 (f ), may approximately be related
to the constraint expansion or motion of the actuator, the relative displacement
Z(f ) = Z2 (f ) − Z1 (f ), according to [37]:
Fa (f ) =
Ea Aa
(∆La (f ) − Z(f ))
La
(2.24)
where Fa (f ) is the Fourier Transform of the internal force that the actuator
applies, Aa is the cross-sectional area of the actuator’s piezoelectric material,
Ea is the Young’s elastic modulus of the actuator’s piezoelectric material and
∆La (f ) is the Fourier Transform of the free expansion of an unloaded piezoelectric stack actuator (see Eq. 2.32). If the point receptances at the respective
actuator end are summed to form the receptance HB = H1 (f ) + H2 (f ), the
relative displacement Z(f ) may be expressed as:
Z(f ) = HB (f )Fa (f )
(2.25)
Using a force balance combining Eq. 2.24, Eq. 2.25 and the Fourier Transform
of the expression for the free expansion of an unloaded piezoelectric stack
actuator in Eq. 2.32 yields an expression for the actuator force applied on the
boring bar as a function of actuator voltage V (f ) [37]:
Fa (f ) =
ka ηd33
V (f ) = Hf v (f )V (f )
1 + ka HB (f )
(2.26)
where ka is the actuator equivalent spring constant, η is the number of piezoelectric layers in the actuator, d33 is the piezoelectric strain constant, Hf v (f )
is the electro-mechanic frequency response function between the input voltage
V (f ) and the output actuator force Fa (f ).
24
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
Assume that the natural surface of the boring bar (in the section where
actuator is situated) coincides with the ”z”-axis, and that the ”y”-axis is normal to the natural surface. The distance, then, between the ”boring bar actuator” interface and the natural surface of the boring bar is α, as shown
in Fig. 2.8. Thus, the external moment per unit length applied on the boring
bar by the actuator force Fa (f ) may be approximated as [19]:
me (z, f ) = αFa (f )(δ(z − z1 ) − δ(z − z2 ))
(2.27)
where δ(z) is the Dirac delta function, z1 and z2 are the z−coordinates for
the ”boring bar - actuator” interfaces introduced by the DC-voltage offset
required for the dynamic operation of the actuator and z2 − z1 ≈ La as
max(∆La (t))/La ≤ 0.2% [40]. Expression of the Fourier Transform of Eq.
2.21 with Eq. 2.27 and integration of the resulting equation over the length of
the boring bar yields the following expression for the generalized load of mode
r:
Fload,r (f ) = αFa (f )(ψr (z2 ) − ψr (z1 ))
(2.28)
If z1 = 0, one ”boring bar - actuator” interface is at the clamped end, and
the generalized load of mode r is given by:
Fload,r (f ) = αFa (f )ψr (z2 )
(2.29)
Thus, the frequency-domain dynamic response of the boring bar in the y−
direction may be expressed as [19]:
w(z, f ) =
∞
ψr (z)Hr (f )αHf v (f )V (f )ψr (z2 )
r=1
where Hr (f ) is the frequency response function for the mode r.
(2.30)
Modeling of an Active Boring Bar
2.4
25
”3-D” Finite Element Model
A finite element model of the system ”boring bar - actuator - clamping house”
was built using the commercial finite element software MSC.MARC [41]. The
initial ”3-D” finite element model of the clamped boring bar includes separate models of a boring bar, a clamping house and a model of a piezoelectric
actuator interacting by means of frictionless contact.
The spatial dynamic properties of the ”boring bar - actuator - clamping
house” system were calculated based on the general equation for the dynamic
equilibrium of an undamped system, i.e.,
[M]{ẅ(t)} + [K]{w(t)} = {0},
(2.31)
where [M] is the global Nf em × Nf em mass matrix of the system, [K] is
the global Nf em × Nf em stiffness matrix of the system, w(t) is the spaceand time-dependent Nf em × 1 displacement vector and Nf em is the number
of degrees-of-freedom of the finite element model. The modal analysis was
conducted by using the Lanczos iterative method in the MSC.MARC software
[41, 42]. The dynamic behavior of the system was also examined by means of
transient and harmonic response simulations.
2.4.1
Model of the system ”boring bar - clamping house”
The model of the system ”boring bar - clamping house” consists of two submodels: a sub-model of the boring bar and a sub-model of the clamping house.
The sub-models are connected in terms of variable contact [41]. As a basic
finite element, a tetrahedron was chosen.
The sub-model of the boring bar consists of three parts: the ”body” which
models the part of the boring bar with a constant cross-section, the ”head”
which models the part of the boring bar with a varying cross-section, and the
”tool”, which models the attached insert. In order to simplify the meshing
process and keep down the number of degrees-of-freedom in the system while
still maintaining accuracy within the model, the three boring bar parts were
meshed separately with different element edge lengths (see Table 2.6). These
three parts were connected using ”glue” contact, which implies that the contacting nodes of both parts are tied to each other in such a way that there is
no relative normal or tangential motion between the two parts in these nodes.
The finite element model of the boring bar is shown in Fig. 2.9 a).
For the sake of simplicity, the clamping house and the four clamping screws
were modeled as one body. The sub-model of the clamping house was built
using four-noded tetrahedrons with linear shape functions. The element edge
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
26
Sub-model
Boring bar, ”body”
Boring bar, ”head”
Boring bar, ”tool”
Clamping house
Actuator
Element Edge Length (m)
0.01
0.005
0.005
0.005-0.01
0.01
Polynomial order
2
2
2
1
1
Table 2.6: Finite element size and order of the approximation polynomial used
in the shape functions.
length varies from 0.005 m to 0.01 m, depending on the location of the finite
element in the model, i.e., surfaces of the clamping house that are in contact
with the boring bar have a mesh with higher density than the rest of the
clamping house surfaces. For the clamping house surfaces in contact with the
turret, the following boundary conditions were used: nodal displacements on
the surfaces of the clamping house, which correspond to surfaces of the real
clamping house attached to the turret (see Fig. 2.4 b)) in x−, y− and z−
directions are set to zero. The finite element model of the clamping house is
shown in Fig. 2.9 b).
2.4.2
Model of the Actuator
A ”3-D” finite model of a piezoelectric actuator was developed. Thin layers of
an electrically active ceramic material connected in parallel were modeled as
a stack of eight-noded bricks with piezoelectrical properties, i.e., in addition
to three translational degrees-of-freedom, each node has a fourth degree-offreedom - electric potential. The number of bricks or piezoelectric layers ηa
can be calculated based on the formula for the free expansion of an unloaded
stack actuator [38].
∆La = ηa d33 V (t)
(2.32)
where d33 is the actuator’s strain coefficient, Vmax (t) is the max operating
voltage and ∆La is the free expansion of the actuator. These values are given
in Table 2.1. This yields ηa = 400 bricks or piezoelectric layers in the actuator
stack.
The large number of layers affects the computational complexity of the
finite element model. For this reason, it is desirable to have a model with
a minimal number of degrees-of-freedom ensuring sufficient accuracy. The
number of the peizoelectric layers was reduced with a factor of 10 and the strain
27
Modeling of an Active Boring Bar
y
y
x
x
z
z
a)
b)
Figure 2.9: The 3-D finite element model of a) a boring bar and b) a clamping
house.
constant d33 , which yields the specified free expansion ∆La , was increased with
a factor of 10. Since the reduced number of layers is still large, i.e., 40 layers,
only a principal sketch of the ”3-D” finite element model of the actuator is
shown in Fig. 2.10.
y
x
z
Figure 2.10: Sketch of a ”3-D” finite element model of an actuator.
The electrostatic boundary conditions are shown in Fig. 2.10. The blue
arrows show the nodes with applied negative or zero potential, and the red
arrows correspond to the nodes with applied positive potential. Furthermore,
the large green arrows show the material orientations within the finite elements.
The material orientation in this case determines the direction of current flow
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
28
inside the finite element.
2.4.3
Contact Modeling in the Finite Element Analysis
The adequate modeling of the boring bar dynamic motion requires incorporation of the contact conditions between the boring bar, clamping house and
clamping bolts into the ”3-D” finite element model.
By a numerical contact problem, a complex process of the interaction of two
or more numerical regions’ boundaries may be defined in the computational
domain based on constraints and boundary conditions specified by the physical
nature of the contact between the bodies, e.g., friction or heat transfer [43].
Three objectives of the contact between the boring bar, clamping house
and clamping screws may be stated:
• Detection of the contact between pre-defined contacting bodies;
• Application of constraints to avoid penetration;
• Application of boundary conditions to simulate frictional behavior.
The two basic types of contact bodies are implemented in MSC.MARC,
[41]: deformable and rigid. Thus, there are two types of contact: a contact
between deformable and deformable bodies and a contact between deformable
and rigid bodies.
A ”3-D” deformable body is described by the ”3-D” finite elements, and the
nodes on its external surfaces are defined as potential contact nodes. Their
”3-D” faces form the outer surfaces and are considered as potential contact
segments.
Rigid bodies are usually composed of the analytically defined ”3-D” Non
Uniform Rational Basis (NURBS) surfaces, which are treated as potential
contact segments. Rigid bodies do not deform.
Contact tolerance
Conventionally, the contact between two ”3-D” bodies is implemented in the
following way: one body is determined to be the ”master”, and the other is
determined to be the ”slave” body [44].
The ”master” body is always deformable. The contacting element of the
”master” body is a node. The ”slave” body can be deformable or rigid. The
contacting element of the ”slave” body is a segment which can be assembled
by a patch of the ”3-D” finite element faces in the case of a deformable body
or segments in the case of a rigid body.
Modeling of an Active Boring Bar
29
Deformable-rigid contact
In this case, contact between the deformable ”master” body and the rigid
”slave” body is considered.
In order to simplify the numerical implementation of the contact, contact
tolerance was introduced to determine the distance below which bodies are
considered to be in contact. For instance, node A in Fig. 2.11 is not inside the
contact tolerance interval, thus the ”master” and the ”slave” bodies are not in
contact. The contact tolerance in the MSC.MARC software is defined as 5 %
of the smallest element edge length in the finite element model [41].
" M a s te r"
A
L
T O
L
T O
" S la v e "
Figure 2.11: Contact tolerance.
Contact detection
If node A is within the contact tolerance interval, then the ”master” and
the ”slave” bodies are considered to be in contact (see Fig. 2.12). In the
MSC.MARC software, the contact problem is resolved by using the direct
constraint method [41]. This implies that when contact is detected, the motion
of the contacting bodies is constraint by means of boundary conditions, i.e.,
displacements and nodal forces are recalculated.
When variable contact occurs, the constraints are imposed on degrees-offreedom of node A. The displacement of node A is transformed into the local
coordinate system of the rigid body, which has as its basis the normal {η}
and tangential {τ } vectors. The relative displacement of node A is updated
as follows:
∆{w}η = {v}T {η}
(2.33)
where {v} is the velocity vector of the rigid body. Thus in this case, the
”master” and ”slave” bodies move together with the same speed in the normal
30
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
H
vh
h
H
L
T O
" M a s te r"
A
v
H
t
L
T O
" S la v e "
Figure 2.12: Contact detection.
direction, however the ”master” body can slide on the surface of the ”slave”
body in the tangential direction.
In the case of the glue contact, additional constraint is imposed, such that
no relative tangential motion occurs between the ”master” and the ”slave”
body:
∆{w}τ = {v}T {τ }
(2.34)
Detection of penetration and separation
When the contacting node A moves beyond the contact tolerance interval, it
is considered to be penetrating the ”slave” body. In this case, the iterative
penetration checking procedure is invoked [41].
h
H
L
T O
" M a s te r"
L
T O
A
" S la v e "
Figure 2.13: Penetration detection.
31
Modeling of an Active Boring Bar
Application of Boundary Conditions - Direct Constraint method
In order to conduct contact analysis, different techniques can be utilized, e.g.,
the Lagrange multiplier procedure, the penalty method, hybrid and mixed
methods, and the direct constraint method [41]. The direct constraint method
provides an accurate solution for contact analysis [41]. The principle of the
direct constraint method is enclosed in the application of constraints due to
contact by means of boundary conditions, i.e., normal displacements and nodal
forces.
The problem can be formulated as follows:
[Kaa ] [Kab ]
[Kba ] [Kbb ]
{wa }
{wb }
=
{fa }
{fb }
+
{0}
{fc }
(2.35)
where subscript b is used for the transformed degrees-of-freedom associated
with the nodes in contact, and subscript a is used for the other not transformed degrees-of-freedom; [Kaa ] is a sub-matrix of the global stiffness matrix
which corresponds to the degrees-of-freedom of nodes which are not in contact; [Kbb ] is a sub-matrix of the global stiffness matrix which corresponds to
the degrees-of-freedom associated with nodes in contact; [Kab ] and [Kba ] are
sub-matrices of the global stiffness matrix which corresponds to the degreesof-freedom of nodes in contact respective non-contacting nodes; {wa } is the
space- and time-dependent vector of unknown displacements associated with
non-contacting nodes; {wb } is the space- and time-dependent vector of constrained displacements; {fa } and {fb } are space- and time-dependent vectors of
external forces acting on the constrained and unconstrained degrees-of-freedom
respectively and {fc } is the space and time dependent vector of unknown contact forces. The unknown displacements and contact forces can be found by
means of Gaussian elimination as follows [44]:
[K∗ ]{wb } − {fc } = {f ∗ }
(2.36)
where [K∗ ] and {f ∗ } can be expressed as [44]:
[K∗ ] = [Kbb ] − [Kba ][Kaa ]−1 [Kba ]T
{f ∗ } = {fb } − [Kba ][Kaa ]−1 {fa }
(2.37)
(2.38)
where [K∗ ] is the (Ndof · nc ) × (Ndof · nc ) stiffness matrix, Ndof is the number
of degrees-of-freedom at the node of the finite element (in the case of a 10noded tetrahedron, Ndof = 3), nc is the number of contacting nodes. The
vectors {f ∗ }, {fc }, {wb } have a length of Ndof · nc . Thus, the system of linear
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
32
equations Eq. 2.36 has 2 · Ndof · nc unknown and only Ndof · nc equations. In
order to obtain a unique solution of the system to linear equation Eq. 2.36,
one has to complete this system by using additional conditions, e.g., conditions
of compatibility and equilibrium [44].
Compatibility condition The compatibility condition implies, that when
contact is detected within the contact tolerance, two bodies start to move
together, i.e., Eq. 2.33 is used in the case of variable contact, and Eq. 2.33
and Eq. 2.34 are utilized in the case of glue contact.
Equilibrium condition The contact forces between the contacting node of
the ”master” body {fm } and the contacting segment of the ”slave” body {fsl }
should be equal and opposite, yielding:
{fm } + {fs } = 0
(2.39)
For the glue contact, additional condition is applied:
{fm }T {τ } = {fsl }T {τ } = 0
(2.40)
Deformable-deformable contact
In the case of contact between two deformable bodies, the multi-point constraint is imposed on the contacting node. This means that for each contacting
node, the retained nodes are found from the set of boundary nodes, e.g., for
the 10-noded tetrahedrons, the number of retained nodes is seven - six from
the patch plus the contacting node itself. Retained nodes are used to form a
geometrical surface. After a normal vector {η} to this surface is found, the
analysis proceeds as in the case of the deformable-rigid contact. In the case of
a deformable-deformable contact, there is no ”master”-”slave” relationships;
each contacting body is checked against every other body.
2.4.4
Coulomb Friction Modeling
Generally, the dynamic response of the boring bar during a continuous turning
operation has nonlinear properties [33, 34]. Possible sources of nonlinearities
are, for instance, the intermittent contact between the workpiece and the cutting tool, as well as a nonlinear contact between the boring bar, clamping
screws and the inner surface of the clamping house cavity [33, 34]. Only the
second source of nonlinearities is considered in the present study. However,
the incorporation of a cutting process model is of great importance for further
Modeling of an Active Boring Bar
33
research. When the boring bar is excited by the force originated from the material deformation process, relative motion between the contacting surfaces of
the boring bar, clamping screws and clamping house occurs. Thus, it is likely
that some of the energy introduced into the system by the cutting process
dissipates via friction forces at contacting surfaces between the boring bar and
clamping house [45]. Generally, no lubricating film is used between the boring
bar and clamping house surfaces. Such contact may be considered as a dry
frictional contact. Therefore, the Coulomb model of the friction force may be
used (see Fig. 2.14). The model implies that for relative motion between contact surfaces, the static friction force fst , which is greater than kinetic friction
force fk (due to the difference between the static and kinetic friction coefficients µk < µst ), should be overcome. Thus, the stick-slip friction force can be
expressed as follows [41]:
f=
{v} = 0
|fst | = µst fn ,
fk = −µk fn sign({v}), otherwise
(2.41)
where fn - is a normal force acting on surface or body and {v} is the relative
sliding velocity vector.
f
f
st
f
k
S tic k
v
S lip
Figure 2.14: Coulomb model of dry friction force.
As the result of the dry friction force presence in the system, the relative
motion of contacting surfaces or bodies can be considered as a stick-slip oscillation. Thus, it may affect the dynamic properties of the system ”boring bar
- clamping house”. The friction coefficients µk and µs are usually determined
experimentally. As a kinetic friction coefficient, µk = 0.4 which corresponds
to the typical friction coefficient of unlubricated materials such as chromium
hard steel at low speeds in normal atmospheres against a mild steel counterface
[45].
Two different Coulomb friction force model approximations were incorporated into the ”3-D” finite element model of the system ”boring bar - actuator
- clamping house”: the arctangent model and the bilinear model.
34
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
Arctangent model
This model approximates a discontinuous Coulomb friction force function by
a continuously differentiable function of the relative sliding velocity (see Fig.
2.16) [41].
f
m k fn
J = 0 .0 0 1 v
J
- m k fn
= 0 .0 1
v
v
J = 0 .1 v
Figure 2.15: Arctangent approximation for the Coulomb friction model.
Thus, the friction force is described by [41]:
{v}
2
−1
f = −µk fn tan
sign({v})
π
ϑ
(2.42)
where ϑ - is a value of relative velocity below which sticking occurs. It is
expressed as a percentage of the maximum relative sliding velocity vmax and
usually is taken from the interval ϑ ∈ [0.01vmax , 0.1vmax ] . Small values
of ϑ yield a narrow stick velocity interval, i.e., a closer approximation of the
Coulomb friction model. This also implies that sticking will occur only at very
low velocities, and the slipping mode will be dominant. On the other hand,
large values of ϑ may result in insufficient influence of the friction in the boring
bar model. One way to obtain the initial value of ϑ might be to estimate it
based on transient analysis without incorporated friction force.
Bilinear model
This model is based on relative displacement instead of relative velocity. It
describes a sticking behavior in terms of elastic relative displacements and a
slipping mode in terms of plastic relative displacements [41].
The friction force can be expressed as [41]:
f=
|∆{w}| < δ
− µkδfn ∆{w},
−µk fn sign(∆{w}), otherwise
(2.43)
where δ is the slip threshold or relative sliding displacement below which the
sticking is simulated. The default value of δ is calculated in MSC.MARC as
35
Modeling of an Active Boring Bar
f
m k fn
d
D w
Figure 2.16: Bilinear approximation for the Coulomb friction model.
0.0025 × ¯le , where ¯le is the average edge length of the finite elements defining
the contacting bodies [41]. According to the theory of tribology, the dimension
of a single contacting spot on an engineering surface is in the order of 10−5 m,
which can be assumed as a typical amplitude for the stick-slip oscillation [45].
2.4.5
Transient response
Transient response simulation was conducted with the help of MSC.MARC
software using the direct integration Single-Step Houbolt method [41]. This
algorithm is recommended for nonlinear contact analysis due to its unconditional stability and second-order accuracy. A detailed description of this
method can be found in [46]. The Single-Step Houbolt method uses a constant time step, which is convenient for the calculation of the response signal
synchronized with the excitation signal.
The approximation of the system’s equation of motion Eq. 2.13 by singlestep algorithms in general form can be expressed as [46]:
αm1 [M]{a}n+1 + αc1 [C]{v}n+1 + αk1 [K]{d}n+1 + αm [M]{a}n +
+αc [C]{v}n + αk1 [K]dn = αf 1 {f }n+1 + αf {f }n
(2.44)
{d}n+1 = {d}n + ∆t + β∆t2 {a}n + β1 ∆t2 {a}n+1
(2.45)
{v}n+1 = {v}n + γ∆t{a}n + γ1 ∆t{a}n+1
(2.46)
where {a}n , {v}n and {d}n are the approximations of {ẅ(tn )}, {ẇ(tn )},
{w(tn )} respectively, ∆t is a step size and αm1 , αc1 , αk1 , αm , αc , αk1 , αf 1 , αf ,
β1 , β, γ1 , γ are the algorithmic parameters. The algorithmic parameters may
be tuned by means of conditions of asymptotic annihilation and second-order
accuracy, as well as by certain overshoot behavior imposed on the response
calculated by the algorithm [46].
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
36
According to [46], the algorithm begins with calculation of
{a}0 = [M]−1 ({f0 } − [C]{v0 } − [K]{d0 })
2.4.6
(2.47)
Harmonic response
The harmonic response of the ”3-D” finite element model of the system ”boring
bar - actuator - clamping house” was calculated with the use of MSC.MARC
software. In general, the solution process is based on the Fourier Transform of
the equation motion of the damped system Eq. 2.13.
((j2πf )2 [M] + (j2πf )[C] + [K]){W(f )} = {F(f )}
(2.48)
The excitation force {F} = {f }ej2πf in this case is assumed to have a
constant magnitude at all frequencies. Then, the displacement vector {W}
is found by solving Eq. 2.48 for each frequency from the specified frequency
range. Thus, the receptance frequency response function is found as a ratio of
displacement {W(f )} and force {F(f )} vectors.
{W(f )} = ((j2πf )2 [M] + (j2πf )[C] + [K])−1 {F(f )}
{W(f )}
[Hr (f )] =
= ((j2πf )2 [M] + (j2πf )[C] + [K])−1
{F(f )}
(2.49)
(2.50)
The damping in this case is assumed to be proportional, i.e., [C] = α[M] +
β[K], where the coefficients α and β are chosen such that relative damping ratios at the, for instance, two lowest eigenfrequencies correspond to the relative
damping values estimated by experimental modal analysis. This condition can
be expressed as follows [36]:
ζr =
α
+ βπfr
4πfr
(2.51)
where fr , r = 1, 2, ..., Nema is the rth natural frequency.
One disadvantage of this approach is that the variable contact between the
clamping house and the boring bar is disregarded.
37
Modeling of an Active Boring Bar
2.5
SDOF Nonlinear Model
In order to investigate effects of the arctangent and bilinear approximations of
the Coulomb friction force on the dynamics of a clamped boring bar, the simplest case when the boring bar is described by a SDOF model was considered.
The response of the boring bar during an internal turning operation in the
lathe is usually dominated by the fundamental bending mode in the cutting
speed direction [4, 3]. Therefore, a simple SDOF model with mass m, stiffness
k and damping c corresponding to the modal mass, stiffness and damping of
the respective fundamental bending mode is considered.
A simple SDOF system with nonlinear friction force included, described by
the arctangent model, may be described by the following equation of motion
[41]:
2
mẅ(t) + cẇ(t) + kw(t) + µk mg tan−1
π
ẇ(t)
ϑ
= f (t)
(2.52)
where f (t) is the excitation force.
On the other hand, if the nonlinear friction force is approximated by the
bilinear model, the equation of motion of the SDOF system is as follows:
mẅ(t) + cẇ(t) + kw(t) + µk mgw(t) = f (t), if w(t) > δ
mẅ(t) + cẇ(t) + kw(t) + µkδmg w(t) = f (t), otherwise
(2.53)
(2.54)
The response of the SDOF system under random excitation force was calculated in Matlab with the help of the Runge-Kutta method, the ode45 solver
[47]. The response of the SDOF system was calculated for the range of parameters ϑ and δ, as well as for a set of excitation force levels. The frequency
response functions between the excitation force and the acceleration of the
SDOF system were estimated in order to facilitate the interpretation of results
obtained based on the ”3-D” FE model.
Chapter 3
Results
The results are presented in the following order: firstly, the estimates of the
two fundamental natural frequencies and the corresponding mode shapes based
on experimental modal analysis and the linear ”3-D” FE-model are given. Secondly, the results of the system identification in terms of the frequency response
functions of the so called control path, the accelerances between the actuator
voltage and the acceleration measured by error accelerometers (accelerometers
attached close to the tool tip) both in the cutting speed and cutting depth
directions are presented. Here, experimental modal analysis of the active boring bar, simulation of the harmonic response of the active boring bar finite
element model with linear clamping conditions, and simulation of the transient response of the active boring bar finite element model enabling variable
contact between the boring bar and the clamping house were utilized. Thirdly,
frequency response function estimates for the dynamic response of the control
paths of the ”3-D” FE-model enabling variable contact between the boring
bar and the clamping house with nonlinear Coulomb friction force between
the contacting surfaces of the boring bar, clamping screws and clamping house
are given. Here, results of two different Coulomb friction force models’ influences on the dynamic response of a SDOF system in terms of frequency
response function estimates are reported first. This is done in order to facilitate the interpretation of the frequency response function estimates of the
dynamic response of the control paths of the active boring bar ”3-D” FEmodel with the Coulomb friction force and enabling clamping house variable
contact. The control path accelerance functions are calculated for a range of
values of the governing parameter in both the arctangent and bilinear models.
Fourthly, receptance function estimates between the force produced by the actuator and the displacement of the ”boring bar - actuator” interfaces obtained
based on the simulated transient response of the active boring bar finite element model enabling variable contact between the boring bar and clamping
38
39
Modeling of an Active Boring Bar
house are given. Finally, frequency response function estimates between the
actuator voltage and the strain at four different positions in the axial direction
close to the ”boring bar - actuator” interfaces for both FE model with variable
contact and the actual boring bar are discussed.
3.1
Modal Analysis Results
3.1.1
Natural Frequencies
The two fundamental natural frequencies for the active boring bar estimated
by experimental modal analysis and calculated based on the linear ”3-D” finite
element model are given in Table 3.1.
Model
Experimental Modal Analysis,
(EMA)
”3-D” FEM of the active
boring bar, (FEM)
Mode 1
Frequency, Damping
[Hz]
ratio, [%]
501.638
1.044
496.232
-
Mode 2
Frequency, Damping
[Hz]
ratio, [%]
520.852
1.244
529.046
-
Table 3.1: Two fundamental natural frequencies and corresponding damping
ratios estimates.
3.1.2
Mode Shapes
The mode shapes estimated using experimental modal analysis and calculated
based on the ”3-D” finite element model of the system ”boring bar - actuator
- clamping house” are plotted in Fig. 3.1.
As a quality measure of the mode shapes extracted by experimental modal
analysis, the MAC-matrix was calculated based on Eq. 2.16 [32], yielding:
[M AC]1
M ACEM A1 ,EM A1 M ACEM A1 ,EM A2
=
M ACEM A2 ,EM A1 M ACEM A2 ,EM A2
1.000 0.000
=
0.000 1.000
=
(3.1)
where EM A1 is the mode shape at 501.638 Hz, and EM A2 is the mode shape
at 520.852 Hz.
Mode shape 1 in the cutting depth direction X
1
0.8
0.6
0.4
0.2
EMA
FEM
0
0.1
0.15
0.2
0.25
0.3
Mode shape 1 in the cutting speed direction Y
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
40
0
−0.2
−0.4
−0.6
−0.8
EMA
FEM
−1
0.1
Length of the boring bar, [m]
0.15
1
0.8
0.6
0.4
0.2
EMA
FEM
0.15
0.2
0.25
0.3
b)
0.25
Length of the boring bar, [m]
0.3
Mode shape 2 in the cutting speed direction Y
Mode shape 2 in the cutting depth direction X
a)
0
0.1
0.2
Length of the boring bar, [m]
1
0.8
0.6
0.4
0.2
EMA
FEM
0
0.1
0.15
0.2
0.25
0.3
Length of the boring bar, [m]
c)
d)
Figure 3.1: The first two fundamental mode shapes of the active boring bar:
a) component of mode shape 1 in the cutting depth direction, b) component
of mode shape 1 in the cutting speed direction, c) component of mode shape
2 in the cutting depth direction and d) component of mode shape 2 in the
cutting speed direction (estimated with experimental modal analysis (EMA)
and finite element model (FEM)).
To provide a quantitative measure on the correlation between the mode
shapes from the experimental modal analysis and the mode shapes predicted
by the finite element models, a cross-MAC matrix has been produced based
on Eq. 2.17 [32]. The cross-MAC matrix between the mode shapes calculated
using the ”3-D” finite element model F EM1 at 496.232 Hz and F EM2 at
529.046 Hz and mode shapes estimated using experimental modal analysis
41
Modeling of an Active Boring Bar
EM A1 at 501.638 Hz and EM A2 at 520.852 Hz is given by:
[M AC]2
3.2
M ACF EM1 ,EM A1 M ACF EM2 ,EM A1
=
M ACF EM1 ,EM A2 M ACF EM2 ,EM A2
0.981 0.015
=
0.019 0.983
=
(3.2)
System Identification
This section addresses system identification of two active boring bar transfer
paths or control paths. The two transfer paths are as follows: one between
the actuator voltage and the output signal from the accelerometer measuring
the boring bar vibration in the cutting speed direction close to the insert,
and the other one between the actuator voltage and the output signal from
the accelerometer measuring the boring bar vibration in the cutting depth
direction close to the insert, see Fig. 2.6. These control paths are estimated
for both the actual active boring bar and for the FE models of the active boring
bar.
3.2.1
Experimentally Estimated Accelerance
Two accelerance functions for the active boring bar were estimated: one between the actuator input voltage and the error accelerometer in the cutting
speed direction and one between the actuator input voltage and the error accelerometer in the cutting depth direction. The spectrum estimation parameters and the excitation signal properties are given in Table 2.2. The magnitude
function for the two estimated accelerance functions for the active boring bar
are presented in Fig. 3.2.
3.2.2
Harmonic Response Based on FEM
The harmonic response was simulated using the linear ”3-D” finite element
model of the active boring bar. The results of the harmonic analysis are presented as the magnitude of two accelerance functions between the actuator
voltage and the acceleration (error acceleration) at the two positions corresponding to the error accelerometer positions at the actual boring bar (see
Fig. 3.3). The peak value of the amplitude of the harmonic excitation was
equal to 100 V .
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
42
1
0.9
40
Coherence
2
|Ha(f)| 1 dB rel 1 m/s /V
50
30
20
0.8
0.7
0.6
10
0.5
CSD
CDD
CSD
CDD
0
400
450
500
550
0.4
400
600
450
500
550
600
Frequency, [Hz]
Frequency, [Hz]
a)
b)
Figure 3.2: Magnitude of accelerance function estimates, between the actuator
voltage and the error accelerometer in the cutting speed (CSD) direction and
between the actuator voltage and the error accelerometer in the cutting depth
(CDD) directions, and b) corresponding coherence functions estimates.
80
120
CSD
CDD
|H (f)| 1 dB rel 1 m/s /V
70
2
100
80
60
60
50
40
a
|Ha(f)| 1 dB rel 1 m/s2/V
CSD
CDD
40
20
400
450
500
550
Frequency, [Hz]
a)
600
30
20
400
450
500
550
600
Frequency, [Hz]
b)
Figure 3.3: Magnitude of the accelerance function estimates, between the actuator voltage and the error acceleration in the cutting speed direction (CSD)
and between the actuator voltage and the error acceleration in the cutting
depth direction (CDD), based on the harmonic analysis of the ”3-D” finite
element model of the active boring bar: a) without damping, and b) with
damping in the model.
43
Modeling of an Active Boring Bar
3.2.3
Transient Response Based on FEM
50
1
40
0.95
Coherence
|Ha(f)| 1 dB rel 1 m/s2/V
The transient response of the ”3-D” finite element model, enabling variable
contact between the boring bar and the clamping house, was simulated with
the use of the MSC.MARC software by means of the Single-Step Houbolt
transient operator [41]. A uniformly distributed random excitation signal with
a flat spectrum with an RMS value of 54.628 V was applied to the finite element
model of the actuator. The results of the transient response simulation are
presented as magnitude functions of accelerances between the actuator voltage
and the acceleration at the points corresponding to the error sensor positions
at the actual active boring bar in the cutting speed and cutting depth direction
correspondingly (see Fig. 3.4).
30
20
0.9
0.85
0.8
10
0.75
CSD
CDD
CSD
CDD
0
400
450
500
550
Frequency, [Hz]
a)
600
0.7
400
450
500
550
600
Frequency, [Hz]
b)
Figure 3.4: Magnitude of accelerance function estimates between the actuator
voltage and the acceleration at the error sensor positions in the cutting speed
(CSD), and in the cutting depth (CDD) directions, and b) corresponding coherence functions. Transient analysis of the ”3-D” finite element model of the
active boring bar enabling variable contact between the boring bar and the
clamping house.
An estimated random error for the accelerance function estimate produced
based on the ”3-D” finite element model is shown in Fig. 3.5. The random
error for the accelerance function estimate between the actuator voltage and
the error accelerometer in cutting speed direction at the resonance frequencies
is 0.072 and 0.063. The random error for the accelerance function estimate
between the actuator voltage and the error accelerometer in cutting depth
direction at the resonance frequencies is 0.070 and 0.061 respectively.
The magnitude functions of control path accelerance functions estimated
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
44
0.2
CSD
CDD
Random error
0.15
0.1
0.05
0
400
450
500
550
600
Frequency, [Hz]
Figure 3.5: Random error for the accelerances estimated based on the ”3-D”
finite element model enabling variable contact between the boring bar and the
clamping house.
80
80
70
70
2
|H (f)| 1 dB rel 1 m/s /V
60
50
40
30
20
a
2
|Ha(f)| 1 dB rel 1 m/s /V
experimentally and based on the simulation of the harmonic response of the
linear ”3-D” FE model and the transient response of the ”3-D” FE model
enabling variable contact between the boring bar and the clamping house are
plotted together in the same diagram in Fig. 3.6.
Experiment
FEM Transient
FEM Harmonic
10
0
400
450
500
550
Frequency, [Hz]
a)
60
50
40
30
20
Experiment
FEM Transient
FEM Harmonic
10
600
0
400
450
500
550
600
Frequency, [Hz]
b)
Figure 3.6: Magnitude of accelerance function estimates between the actuator
voltage and the acceleration at the error sensor positions a) in the cutting speed
(CSD), and b) in the cutting depth (CDD) directions obtained experimentally
based on the simulation of the transient response and the harmonic response
of the ”3-D” finite element model of the active boring bar.
Modeling of an Active Boring Bar
3.3
45
Dynamic Modeling of the Boring Bar with
the Coulomb Friction Force Included
In order to improve the control path accelerance functions estimated based on
the ”3-D” finite element model enabling variable contact between the boring
bar and the clamping house, the Coulomb friction force was included in the
model. Two approximations of the Coulomb friction force were utilized: the
arctangent and the bilinear. Dynamic motion of the boring bar excited by the
actuator expansion results in the complex phenomenon of a variable contact
between the boring bar and the clamping house. In order to investigate the
influence of the Coulomb friction force on the dynamic behavior of the boring
bar, accelerance functions were first calculated based on the simplest SDOF
model of the boring bar for a range of values of the governing parameters
and several excitation force levels. Then, the control path accelerances were
calculated based on the ”3-D” finite element model enabling variable contact
between the boring bar and the clamping house for both the arctangent and the
bilinear approximations of the Coulomb friction force. In this case, simulations
of the transient response of the boring bar were carried out for the limited range
of values of the governing parameters (for both the arctangent and the bilinear
model) and a single excitation force level due to the long computational time
required for each simulation.
3.3.1
Transient Response Based on the SDOF Model
In order to simulate the nonlinear SDOF system described by Eq. 2.52 or Eq.
2.53 and Eq. 2.54, the mass m, stiffness k and damping c have to be selected.
For the sake of relevance, these quantities are estimated based on the driving
point accelerance function estimate for the cutting speed direction produced
in the experimental modal analysis of the active boring bar (see Fig. 3.7).
The modal parameters m, k and c can now be estimated based on the
following expressions:
2π 2 fn2
ζn |Ha (fn )|
k
m =
(2πfn )2
√
c = 2ζn mk
k =
(3.3)
(3.4)
(3.5)
where fn is the undamped fundamental natural frequency in the cutting speed
direction, ζn is the corresponding damping ratio, |Ha (fn )| is the value of the
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
46
25
1
20
0.998
0.997
15
Coherence
|Ha(f)| 1 dB rel 1 m/s2/N
0.999
10
5
0.996
0.995
0.994
0.993
0.992
0
0.991
−5
400
450
500
550
Frequency, [Hz]
a)
600
0.99
400
450
500
550
600
Frequency, [Hz]
b)
Figure 3.7: Magnitude of driving point accelerance estimate for the boring bar
in the cutting speed direction (CSD), and b) corresponding coherence function
(produced in experimental modal analysis).
magnitude function of the driving point accelerance function estimate in the
cutting speed direction at the fundamental natural frequency in the cutting
speed direction.
The estimated values of the modal mass, stiffness and damping are as
follows: m = 6.489 [kg], k = 64.467 × 106 [N/m] and c = 427.071 [N s/m].
These mass, stiffness and damping values were used in the simulations of the
response of the SDOF model with the arctangent and bilinear approximations
of the nonlinear friction force for uniformly distributed broadband random
force with RMS levels of 57.827 N , 289.088 N , 576.647 N , 2888.338 N and
5762.465 N .
The response of the SDOF model with the arctangent approximation of
the Coulomb friction force was calculated for the following parameters of the
relative sliding velocity:
ϑ = [ 0.005ẇmax 0.01ẇmax 0.05ẇmax 0.1ẇmax ]
(3.6)
where ẇmax is the maximum velocity estimated based on the linear model.
The relative damping ratios estimated for each value of parameter ϑ at each
excitation force level are summarized in Table 3.2.
The magnitude function for the accelerance function estimates between
the uniformly distributed random force with an RMS level of 576.647 N and
the acceleration of the SDOF model calculated for the range of values of the
parameter ϑ together with corresponding coherence functions are shown in
Fig. 3.8.
47
Modeling of an Active Boring Bar
Excitation force
RMS level, [N]
57.827
289.088
576.647
2888.338
5762.465
0.005ẇmax
72.447
4.572
3.047
1.994
1.844
Damping ratio, [%]
0.01ẇmax 0.05ẇmax
68.034
29.007
4.572
4.175
3.047
2.847
1.994
1.944
1.844
1.894
0.1ẇmax
17.136
3.746
2.624
1.944
1.894
Table 3.2: Estimates of the relative damping for the frequency response functions based on the SDOF nonlinear model with the arctangent approximation
of the Coulomb friction force.
1
Linear
10%
5%
1%
0.5 %
15
0.98
Coherence
|Ha(f)| 1 dB rel 1 m/s2/N
20
10
0.96
0.94
Linear
10%
5%
1%
0.5 %
5
0.92
0
450
500
550
Frequency, [Hz]
a)
0.9
450
500
550
Frequency, [Hz]
b)
Figure 3.8: a) Magnitude of accelerance functions calculated based on the
response of the SDOF model with the arctangent approximation of nonlinear
force for the different values of the relative velocity ϑ and b) corresponding
coherence functions (excitation force RMS level 576.647 N ).
The response of the SDOF model with the bilinear approximation of the
Coulomb friction force was calculated for the following values of the slip threshold δ: 10−6 [m], 5 · 10−6 [m], 10−5 [m], 5 · 10−5 [m] and 10−4 [m]. Natural
frequency estimates for each of the combinations of δ and the excitation force
levels are given in Table 3.3.
The magnitude function for the accelerance function estimates between
the uniformly distributed random force with an RMS level of 576.647 N and
the acceleration of the SDOF model calculated for the range of values of the
parameter δ together with corresponding coherence functions are shown in Fig.
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
48
Excitation force
RMS level, [N]
57.827
289.088
576.647
2888.338
5762.645
δ = 10−4 ,
[m]
502.5
502.5
502.5
502.5
502.5
Natural frequency, [Hz]
δ = 5 · 10−5 , δ = 10−5 , δ = 5 · 10−6 ,
[m]
[m]
[m]
504
511.5
520
504
508
511.5
504
507
510
502.5
505.5
509.5
502.5
505
509.5
δ = 10−6 ,
[m]
549
537.5
535
532.5
532.5
Table 3.3: Estimates of the natural frequency for the frequency response functions based on the SDOF nonlinear model with the bilinear approximation of
the Coulomb friction force.
3.9.
1
Linear
δ=10−4 m
δ=5⋅10−5 m
δ=10−5 m
δ=5⋅10−6 m
δ=10−6 m
15
0.95
Coherence
2
|Ha(f)| 1 dB rel 1 m/s /N
20
10
Linear
δ=10−4 m
δ=5⋅10−5 m
δ=10−5 m
δ=5⋅10−6 m
δ=10−6 m
0.85
5
0
450
0.9
500
550
Frequency, [Hz]
a)
600
0.8
450
500
550
600
Frequency, [Hz]
b)
Figure 3.9: a) Magnitude of accelerances calculated based on the response
of the SDOF model with bilinear approximation of nonlinear force for the
different values of the slip threshold δ and b) corresponding coherence function
(excitation force RMS level 576.647 N ).
Modeling of an Active Boring Bar
3.3.2
49
Dynamics of the ”3-D” Finite Element Model Enabling Variable Contact between the Boring Bar
and the Clamping House and also with Coulomb
Friction Force
The initial value of the relative sliding velocity ϑ was estimated based on
the transient analysis of the finite element model enabling variable contact
between the boring bar and the clamping house while allowing for no friction.
The maximum velocity in the feed direction of the nodes of the boring bar finite
element model corresponding to the bolts clamping position on the boring bar
was ẇmax = 0.007784 [m/s]. The arctangent friction model was tested for
four different percentage values of the maximum sliding velocity: 10%, 5%,
1% and 0.5 %. The accelerance functions between the actuator voltage and
the accelerations at the point of the active boring bar error sensor positions
were estimated based on the transient analysis of the ”3-D” finite element
model with the arctangent friction model enabling variable contact between
the clamping house and the boring bar for the four different relative sliding
velocities. Magnitude functions for these accelerance functions are plotted
together with the corresponding accelerance function for the finite element
model enabling variable contact between the boring bar and the clamping
house while allowing for no friction, in Figs. 3.10 a) and c). The corresponding
coherence functions are shown in Figs. 3.10 b) and d).
The transient analysis was also carried out on the ”3-D” finite element
model with the bilinear model of the friction force enabling variable contact
between the boring bar and the clamping house. To estimate the slip threshold, the average element length was calculated as ¯le = 6.973 · 10−3 [m], yielding
the slip threshold δ = 1.743 · 10−5 [m]. The bilinear model of the friction force
was tested for four different values of the slip threshold: 1.743·10−5 [m], 5·10−5
[m], 10−5 [m] and 5 · 10−6 [m]. The accelerance functions between the actuator
voltage and the accelerations at the points of the active boring bar error sensor positions were estimated based on the transient analysis of the ”3-D” finite
element model (with the bilinear friction model and enabling variable contact
between the boring bar and the clamping house) for the four different slip
thresholds. Magnitude functions for these accelerance functions are plotted
together with corresponding accelerance function for the finite element model,
enabling variable contact between the boring bar and the clamping house while
allowing for no friction in Figs. 3.11 a) and c). The corresponding coherence
functions are shown in Figs. 3.11 b) and d).
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
50
1
No friction
10%
5%
1%
0.5%
40
30
0.8
Coherence
|Ha(f)| CDD 1 dB rel 1 m/s2/N
50
20
10
0.6
0.4
No friction
10%
5%
1%
0.5%
0
0.2
−10
−20
400
450
500
550
0
400
600
450
a)
600
1
No friction
10%
5%
1%
0.5%
2
|Ha(f)| CSD 1 dB rel 1 m/s /N
550
b)
50
0.8
Coherence
40
30
20
0.6
0.4
No friction
10%
5%
1%
0.5%
0.2
10
0
400
500
Frequency, [Hz]
Frequency, [Hz]
450
500
Frequency, [Hz]
550
600
0
400
450
500
550
600
Frequency, [Hz]
c)
d)
Figure 3.10: Magnitude of accelerance function estimates between the actuator voltage and acceleration in the error sensor positions a) in the cutting
depth direction (CDD) and c) in the cutting speed direction (CSD). Coherence function estimates between the actuator voltage and the acceleration in
the error sensor positions b) in the cutting depth direction (CDD) and d) in the
cutting speed direction (CSD). Based on the transient analysis of the ”3-D”
finite element model of the active boring bar enabling variable contact between
the boring bar and the clamping house and with the arctangent model of the
friction force.
51
Modeling of an Active Boring Bar
1
0.8
40
Coherence
|Ha(f)| CDD 1 dB rel 1 m/s2/N
50
30
20
No friction
δ=1.743⋅10−5 m
δ=5⋅ 10−6 m
δ=1⋅10−5 m
δ=5⋅ 10−5 m
10
0
400
450
500
550
0.6
0.4
No friction
−5
δ=1.743⋅10 m
δ=5⋅ 10−6 m
δ=1⋅10−5 m
δ=5⋅ 10−5 m
0.2
0
400
600
450
a)
600
1
0.8
40
Coherence
|Ha(f)| CSD 1 dB rel 1 m/s2/N
550
b)
50
30
20
No friction
δ=1.743⋅10−5 m
δ=5⋅ 10−6 m
δ=1⋅10−5 m
δ=5⋅ 10−5 m
10
0
400
500
Frequency, [Hz]
Frequency, [Hz]
450
500
Frequency, [Hz]
550
0.6
0.4
No friction
δ=1.743⋅10−5 m
δ=5⋅ 10−6 m
δ=1⋅10−5 m
δ=5⋅ 10−5 m
0.2
600
0
400
450
500
550
600
Frequency, [Hz]
c)
d)
Figure 3.11: Magnitude of accelerance function estimates between the actuator
voltage and the acceleration in the error sensor positions a) in the cutting
depth direction (CDD) and c) in the cutting speed direction (CSD). Coherence
function estimates between the actuator voltage and the acceleration in the
error sensor positions b) in the cutting depth direction (CDD) and d) in the
cutting speed direction (CSD). Based on the transient analysis of the ”3-D”
finite element model of the active boring bar enabling variable contact between
the boring bar and the clamping house and with the bilinear model of the
friction force.
52
3.4
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
Actuator Receptance
−140
1
−150
0.8
−160
Coherence
|Hr(f)| 1 dB rel 1 m/V
The ”boring bar - actuator” interface receptance functions were estimated
based on the transient response of the boring bar under applied random excitation voltage using the ”3-D” finite element model of the boring bar enabling
variable contact between the boring bar and the clamping house. The estimates
of the receptance functions are produced using the calculated displacements
and contact forces in the feed direction collected for the nodes of the boring
bar finite element model corresponding to the actuator-boring bar engagement
(see Fig. 3.12). The random error for the receptance functions estimated based
on the ”3-D” finite element model is shown in Fig. 3.13. The random error for
the receptance function between the actuator voltage and the error receptance
in the feed direction at the second ”boring bar-actuator” interface at the resonance frequencies is 0.138 and 0.089 respectively. The random error of the
receptance function between the actuator voltage and the error receptance at
the first ”boring bar-actuator” interface at the resonance frequencies is 0.047
and 0.081 respectively.
−170
0.6
0.4
−180
0.2
−190
H (f)
2
H (f)
H2(f)
H (f)
1
1
−200
400
450
500
550
Frequency, [Hz]
a)
600
0
400
450
500
550
600
Frequency, [Hz]
b)
Figure 3.12: a) Magnitude function of receptance function estimates for the
”boring bar - actuator” interfaces based on the ”3-D” finite element model of
the active boring bar enabling variable contact between the boring bar and
the clamping house and b) corresponding coherence functions.
It is difficult if not impossible to measure the acceleration and the actuator
force (in the feed direction) of the active boring bar in the interface between
the actuator and the boring bar. For these reasons, the ”boring bar - actuator”
interfaces accelerance functions have not been estimated for the actual boring
bar. The strain at positions close to the actuator interfaces of the actual active
Modeling of an Active Boring Bar
53
1
H (f)
2
H (f)
1
Random error
0.8
0.6
0.4
0.2
0
400
450
500
550
600
Frequency, [Hz]
Figure 3.13: Random error for the receptances estimated based on the ”3-D”
finite element model of the the active boring bar enabling variable contact
between the boring bar and the clamping house.
boring bar may, however, be measured. Thus, frequency response function estimates between the actuator voltage and the strain at four different positions
(see Fig. 2.7) of the active boring bar have been produced based on both the
linear ”3-D” FE model and the model that enables variable contact between
the clamping house and the boring bar. The magnitude functions for these
frequency response functions are shown in Fig. 3.14. Also, frequency response
function estimates between the actuator voltage and the strain at four different
positions (see Fig. 2.7) have been estimated for the actual active boring bar,
and they are illustrated in Fig. 3.15. Furthermore, in Fig. 3.16 the magnitude
of the frequency response function estimates between the actuator voltage and
the boring bar strain for both the ”3-D” FE models and the actual boring bar
are plotted together.
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
54
1
−90
P1
P2
P3
P4
0.8
Coherence
|H(f)|, 1 dB rel 1/V
−100
−110
−120
0.6
0.4
0.2
−130
−140
400
P1
P2
P3
P4
450
500
550
0
400
600
450
500
550
600
Frequency, [Hz]
Frequency, [Hz]
a)
b)
Figure 3.14: Magnitude of the frequency response function estimates between
the actuator voltage and the boring bar strain, calculated based on the ”3-D”
FE model enabling variable contact between the boring bar and the clamping
house, and b) the corresponding coherence function estimates.
1
−90
P1
P2
P3
P4
0.98
Coherence
|H(f)|, 1 dB rel 1/V
−100
−110
−120
0.96
0.94
0.92
−130
−140
400
P1
P2
P3
P4
450
500
550
Frequency, [Hz]
a)
600
0.9
400
450
500
550
600
Frequency, [Hz]
b)
Figure 3.15: Magnitude of the frequency response estimates between the actuator voltage and the strain measured on the actual active boring bar, and
b) the corresponding coherence function estimates.
Modeling of an Active Boring Bar
55
−90
P1 exp
P2 exp
P3 exp
P4 exp
P1 fem
P2 fem
P3 fem
P4 fem
|H(f)|, 1 dB rel 1/V
−100
−110
−120
−130
−140
400
450
500
550
600
Frequency, [Hz]
Figure 3.16: Magnitude of the frequency response estimates between the actuator voltage and the strain measured on the actual active boring bar and
calculated based on the ”3-D” FE model enabling variable contact between
the boring bar and the clamping house.
Chapter 4
Summary and Conclusions
The ”3-D” finite element model of the system ”boring bar - actuator - clamping
house” resulted in fairly accurate estimates of the boring bar’s fundamental
bending mode eigenfrequencies, 496.2 Hz and 529 Hz (see Table 3.1). Also, it
provided estimates of the fundamental mode shapes that are well-correlated
with the corresponding mode shapes extracted in the experimental modal analysis of the active boring bar (see Fig. 3.1 and the cross-MAC matrix in Eq.
3.1). The discrepancies between the natural frequency and mode shape estimates from the ”3-D” finite element model and the experimental modal analysis may, for instance, be explained by differences between the actual active
boring bar and the finite element model in dimensions and materials properties. Also, the fact that the attachment of the clamping house to the turret is
modeled as infinitely rigid in the finite element model will introduce differences
in the dynamic properties of the finite element model and the active boring
bar.
Of significance is the ability to produce ”3-D” finite element models of
active boring bars which predict the control paths accurately. The following
control paths have been considered: between the actuator input voltage and
the output signal from the accelerometer measuring the boring bar vibration in
the cutting speed direction close to the insert, and between the actuator input
voltage and the output signal from the accelerometer measuring the boring
bar vibration in the cutting depth direction close to the insert.
The ability to produce FE models of active boring bars enabling accurate
modeling of the control paths is important for the development of efficient and
accurate design procedure for active boring bars. The control path frequency
response functions for the actual active boring bar were estimated (see Fig.
3.2 a)). The harmonic analysis of the linear ”3-D” finite element model of the
active boring bar results in control path accelerance estimates with magnitude
levels significantly higher as compared to the control path accelerance estimates
56
Modeling of an Active Boring Bar
57
of the actual active boring bar. This can be observed by comparing the control
path accelerance estimates in Fig. 3.3 a) and Fig. 3.2 a). It is possible
to reduce the magnitude of fundamental resonance frequency peaks in the
control path accelerance function estimates produced based on the linear ”3-D”
finite element model by incorporating proportional damping (see Fig. 3.3 b)).
However, compared to the control path frequency response functions for the
actual active boring bar, the overall magnitudes of the proportionally damped
finite element model control path accelerance functions are significantly higher
(see Fig. 3.2 a) and Fig. 3.3 b)). Thus, the dynamic stiffness of the ”3-D”
finite element model with proportional damping is not sufficient.
The harmonic response simulation of the finite element model is carried out
in MSC.MARC is only enabled for linear systems and uses only the boundary
conditions as defined in the initial phase of the calculations. On the other
hand, the transient response analysis allows for simulation of the active boring
bar’s response based on the ”3-D” finite element model with variable contact between the clamping house and the boring bar. Time variable boring
bar boundary conditions imposed by the clamping house in the finite element
model are enabled. Compared to the linear FE models, the control path accelerance function estimates for the ”3-D” finite element model of the active
boring bar enabling variable contact between the clamping house and the boring bar display significantly improved correlation with the control path accelerance estimates of the actual active boring bar. This might be observed by
comparing Figs. 3.2 a), 3.3 a), b) and 3.4 a) or from the Figure 3.6. It may
also be observed that the ”3-D” finite element model of the active boring bar
allowing variable contact between the clamping house provides an approximation that is stiffer than the actual boring bar. This may be explained by the
fact that boundary conditions used for the attachment of the clamping house
are modeled as infinitely rigid in the FE model. However, in the lathe the
attachment of the clamping house to the turret as well as the attachment of
the turret to the slide, etc., cannot be considered completely rigid and, thus,
flexibility is introduced. By comparing Fig. 3.2 a) and Fig. 3.4 a), it might
be observed that the relative level between each of the resonance peaks for the
two control path accelerances differ between the actual active boring bar and
the FE model enabling variable contact. Thus, the orientation of the bending
”modes” in the cutting depth/cutting speed plane may also differ between the
actual active boring bar and the FE model enabling variable contact.
The possibility to further improve the accuracy of the ”3-D” finite element
model enabling variable contact between the clamping house and the boring
bar by incorporating damping into it has also been addressed. The Coulomb
friction force between the surfaces of the clamping house and the boring bar
was introduced. To facilitate interpretation of the transient response analysis
58
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
results for the ”3-D” finite element model with the Coulomb friction force, simulations of a simple nonlinear SDOF system were initially carried out for two
different Coulomb friction force models; the arctangent model and the bilinear
model. For a SDOF system with the Coulomb friction force approximated with
the arctangent model, it can be observed from damping ratios estimated from
simulations’ results (see Table 3.2) that with an increasing excitation force
level, from 100 N till 10000 N, the influence of the nonlinear friction force on
the system’s damping decreases. This is due to the fact that high excitation
force levels induce vibration with the high velocities and the slip friction force
magnitude is then negligible compared to the forces of the linear part of the
system. At the constant excitation force level, different values of the relative
sliding velocity (ϑ = 0.005ẇmax , 0.01ẇmax , 0.05ẇmax and 0.1ẇmax ) will introduce different levels of damping in the SDOF system (see Table 3.2 and Fig.
3.8 a). With a decreasing value of the relative sliding velocity, greater damping
observed in the frequency response function estimates for the SDOF system
(see Fig. 3.8 a)). Thus, the closer the arctangent friction model approximates
the Coulomb friction force, the influence of the nonlinear friction force on the
response of the system also increases which also is indicated by the coherence
function estimates for the SDOF system in Fig. 3.8 b).
For a SDOF system with the Coulomb friction force approximated with the
bilinear model, it can be observed from the natural frequencies estimated from
simulations’ results (see Table 3.3) that the influence of the nonlinear friction
force on the systems stiffness will decrease with an increasing excitation force
level. Thus, with an increasing force level the vibration displacement will
increase and, as a consequence, the slip friction force magnitude will become
more and more negligible compared to the forces of the linear part of the
system. At the constant excitation force level, different values of the slip
threshold δ from 10−4 m to 10−6 m will result in an increasing resonance
frequency of the structure of approximately 9.5 %, i.e., from 501.5 Hz to 549 Hz
(see Table 3.3 and Fig. 3.9 a)). With a decreasing value of the slip threshold,
greater stiffness is observed in the frequency response function estimates for
the SDOF system (see Fig. 3.9 a)). Thus, with a decreasing slip threshold
the influence of the nonlinear friction force on the response of the system also
increases which also is indicated by the coherence function estimates for the
SDOF system in Fig. 3.9 b).
The dynamic behavior of the ”3-D” finite element model of the system
”boring bar - actuator - clamping house” including the Coulomb friction force
and enabling variable contact is expected to be significantly more complicated
to explain compared to the SDOF system. As time evolves, contact may occur
or it may cease between nodes on the ”3-D” surfaces of the clamping house
and on the boring bar. Thus, the system has time-varying dynamic proper-
Modeling of an Active Boring Bar
59
ties. Moreover, the ”3-D” finite element model constantly changes its state,
meaning that at a certain time instant contact is detected for some nodes, and
friction force influence these nodes, while the other nodes, which were previously in contact, are separated and the friction ceases, etc. The introduction
of the Coulomb friction force approximated with the arctangent model in the
”3-D” finite element model of the active boring bar enabling variable contact
between the clamping house and the boring bar resulted in significantly degraded control path accelerance function estimates for the set of used relative
sliding velocities compared to the case with no friction force in the FE model
(see Figs. 3.10 a) and 3.10 c)). However, replacing the arctangent model
with the bilinear model in the ”3-D” finite element model of the active boring
bar enabling variable contact resulted in improved control path accelerance
function estimates compared to the case with the arctangent friction model
(c.f. Figs. 3.10 and 3.11). These control path accellerance function estimates
display a lower correlation with accelerance function estimates for the actual
active boring bar compared to the the ”3-D” finite element model of the active
boring bar enabling variable contact (compare Figs. 3.2 a), 3.10 a) 3.10 c),
3.11 a) and 3.11 c)). In the bilinear approximation, the friction force is proportional to the relative displacement between contacting bodies within the slip
threshold δ. Vibrations result in relative displacements between the contacting nodes of the boring bar and clamping house. If the relative displacements
are within the slip threshold δ, the friction force introduces an increase in the
stiffness between contacting nodes as compared to no contact. The selection
of slip threshold δ seems to influence the fundamental resonance frequencies
of the system ”boring bar - actuator - clamping house”(see Fig. 3.11 a) and
Fig. 3.11 b)).
The ”boring bar - actuator” interface receptance functions were estimated
based on the transient response of the boring bar under applied random excitation voltage using the ”3-D” finite element model of the boring bar enabling
variable contact between the clamping house and the boring bar. The estimates
of the receptance functions are produced using the calculated displacements
and contact forces in the feed direction collected for the nodes of the actuator
finite element model corresponding to the actuator-boring bar engagement (see
Fig. 3.12). It is difficult if not impossible to measure the acceleration and force
(in the feed direction) of the active boring bar in the interface between the actuator and boring bar. For this reason, the ”boring bar - actuator” interfaces
accelerance functions have not been estimated for the actual boring bar. The
strain at positions close to the actuator interfaces of the actual active boring
bar may, however, be measured. Thus, frequency response function estimates
between the actuator voltage and the strain at four different positions (see Fig.
2.7) of the active boring bar may be produced based on both the ”3-D” FE
60
White
Tatiana Smirnova, Henrik Åkesson and Lars Håkansson
model enabling variable contact between the clamping house and the boring
bar (see Fig. 3.14 a)) and the actual active boring bar (see Fig. 3.15 a)).
By comparing the frequency functions for the actual boring bar and the FE
model, it follows that the FE model lacks damping and is slightly stiffer. Also,
by examining Fig. 3.16 it follows that the orientation of the bending ”modes”
in the cutting depth/cutting speed plane is likely to differ between the actual
active boring bar and the FE model enabling variable contact. One issue that
seems to be of importance to address in future work, is to further improve the
”3-D” FE model of the active boring bar by, e.g., the modeling of the boundary
conditions imposed by the turret on the clamping house.
Acknowledgments
The present project is sponsored by Acticut International AB.
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