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Squeal measurement with 3D scanning laser doppler
vibrometer: handling of the time varying system behavior
and analysis improvement using FEM expansion
G. Martin1, E. Balmes1,2, G. Vermot des Roches1, T. Chancelier3
1 SDTools, 44 Rue Vergniaud, 75013 Paris, France
e-mail: martin@sdtools.com
2 ENSAM, PIMM, 151 Bld de l’Hopital, 75013 Paris, France
3 Chassis Brakes International, Drancy, France
Abstract
In the presence of squeal, Operational Deflection Shapes (ODS) are classically performed to analyze
behavior. A simple numeric example is used to show that two real shapes should dominate the response.
This justifies an ad-hoc procedure to extract main shapes from the real brake time measurements. The
presence of two shapes is confirmed despite variations with wheel position and reproducibility tests. To
obtain a high spatial density measurement, 3D Scanning Laser Doppler Vibrometer is interesting but leads
to iterative measurements on a time-varying system. An algorithm to merge sequential measurement and
extract main shapes is detailed. Even with a high-density 3D SLDV measurement, shapes characterizing
the squeal event are still only known on accessible surfaces. Minimum Dynamic Residual Expansion
(MDRE) is thus finally used to estimate motion on a full FE mesh which eases interpretation and
highlights areas where the test and the model contain errors.
1 Introduction
Squeal being an undesired condition, its appearance is never predicted by initial design models, when they
exist, and experiments are needed to understand the exact conditions of occurrence. Two major techniques
are classically used to perform vibration measurements on brake systems: accelerometer measurements
and 3D Scanning Laser Doppler Vibrometer measurement. The specificities of each measurement
techniques are reminded in Table 1.
Table 1: Characteristics of accelerometer and 3D-SLDV measurements
Accelerometer
3D-SLDV
Synchronous measurement
Yes
No
Setup time
Long
Acceptable
Wireframe density
Low
High
Without contact
No
Yes
Rotating parts
No
Yes
Hidden parts
Yes
No
The 3D-SLDV is interesting because the setup time is reasonable, it is without contact, and it can offer a
high number of measured points. However, only visible parts can be measured. Some accelerometers can
be used to add missing information but this still does not include tight areas or junctions. The main
disadvantage of the 3D-SLDV to measure a squeal phenomenon is that measurements are sequential. This
is a problem because as will be highlighted in section 3, brake squeal is a time varying phenomenon.
The first objective of this study was to propose an experimental strategy to extract shapes that characterize
the limit cycle associated with a squeal occurrence. To propose and adequate strategy it is first necessary
to have prior knowledge of the expected properties. Section 2 uses a simple numerical model [1] to
illustrate that the shapes associated with squeal instabilities are expected to be dominated by the
combination of two real modes with contributions that may be quite sensitive to changes.
Knowing that a limit cycle is expected to be composed of a time varying combination of fixed shapes,
Section 3 exploits time-frequency analyses of experimental measurements to demonstrate that indeed two
real shapes dominate the response and have variations that are coherent with the changes due wheel
position. A reproducibility test further demonstrates that, while different experiments can lead to notable
frequency shifts, the shapes underlying the limit cycle are fairly constant. This first analysis is performed
using test on the drum brake test bench shown in figure 1 left.
Figure 1 : Experimental setups. Left drum-brake used for section 3.
Right : disk-brake used for sections 4-5.
Because of the low spatial density, the accelerometer measurements can be difficult to exploit. The second
objective of this paper is thus to present the strategy developed to extract principal shapes from 3D-SLDV
measurements. The algorithm is described in Section 4 and applied on the brake shown in figure 1 right.
Finally, despite a quite high spatial resolution of the laser measurement, many parts are not accessible to
the measurement. As a result, using mode shape expansion techniques to estimate the full FE response
from measurements appears as a necessity to provide understanding of inconsistencies between model and
test and thus pave the way for the proposition of modifications. The Minimum Dynamic Residual
(MDRE) method [2] is applied in Section 5 to illustrate the potential uses. It is shown that expanded
shapes give better understanding of the brake motion and that distribution of the residual energy after
expansion gives insight on modeling errors.
2 Expected shapes in a squeal event: a simple example
To motivate the procedure used to analyze shapes during a squeal event, a simple numeric example is
detailed here. As motivated in [1], the complex mode shapes come from the linearization of the model
around a chosen state (pressure, velocity…). The linearized model provides a system of the form
(1)
where et are displacements, velocities and acceleration, M, C, K are respectively the mass, damping
and stiffness matrices supposed constant. The stiffness matrix can be decomposed in a symmetric part ,
coming from the elastic properties of each components and the linearization of the normal contact loads
and a non-symmetric part linked to the fluctuation of the tangential loads induced by the fluctuation of
the normal loads. It is classical to project the system in the real mode basis , found by solving the
system with and (which is equivalent to consider a friction coefficient ). Projected
in this basis, the complex modes are solution of
(2)
with restitution on the full set of Degrees Of Freedom (DOFs) computed using .
The simplified brake shown in Figure 2 is used as an illustration. The disc (blue) and the backplate
(yellow) are made of steel and the friction (green) is made of an orthotropic softer material, about 5GPa
with an additional Rayleigh damping (see the introduction of [1] for details). The model is clamped at 4
locations at the interface between the backplate and the friction (similar to the one circled in the figure).
Figure 2 : Simple brake model geometry (left) and real mode shapes #8 and #9. (right)
This model has an unstable (negative damping) complex mode at 5050Hz which corresponds to the
interaction between the two real modes #8, with a mostly out-of-plane deformation, and #9 with mostly a
deformation of the pads sliding on the disc.
To confirm that the complex mode shape mainly comes from the interaction between the two real shapes,
the friction coefficient is swept from 0 to 0.2 (not a physical value but still relevant for the interpretation).
The left of Figure 3 shows that the complex modes #8 and #9 are stable at first (as indicated by the red
dots). Then with the increase of the friction coefficient, mode #8 becomes unstable, while the damping of
mode #9 increases.
Figure 3 : Evolution of complex modes with the friction coefficient: roots (left), real mode participation to
the complex mode #8 (middle) and phase between real modes #8 and #9 (right)
From (2), it is possible to look at the evolution of the real mode participation ( of each modal DOF
with the normalization . Figure 3 center thus shows the participation of the real modes to
complex mode #8, which become unstable, evolves from 100% of real mode #8 (blue line) to 70% of real
mode #9 (red line) and 70% of real mode #8. Figure 3 illustrates the significant evolution of the phase
relation between contributions of real modes #8 and #9 to complex mode #8.
To evaluate the pertinence to use only two real modes to interpret the unstable modes, equation (2) is
reduced on the 2 real mode shapes #8 and 9, leading (with the assumption of modal damping) to
(3)
The comparison between the system reduced on the two real modes on the one hand and on all real modes
on the other hand is provided by Figure 4. The accuracy of frequencies is not as good with only two
shapes, but the coupling still occurs and the damping computation is almost unchanged.
Figure 4 : Complex modes frequency (left) and damping (right) for variable friction coefficient. Reduction
on two real modes (dashed lines) and on all real modes (full lines).
The common practice in presence of coupled modes is to try separating the frequencies. Doing so on the
two shape reduced model, by moving away their frequencies with the same percentage (modifying
but not the non-symmetric stiffness part), results in the evolution of the complex frequencies
and damping shown in Figure 5. As mode #9 damping decreases, mode #9 one increases until it is no
longer unstable. The dot lines represent the values of the frequencies and dampings . When the
frequencies are moved away, the influence of the coupling stiffness decreases, and the mode parameters
converge toward the ones without coupling at all.
Figure 5 : Complex mode frequency and damping for varying frequency shift of real modes #8 and #9
3 Time frequency analysis
With the objective of extracting the characteristics of a limit cycle, it is first necessary choose a strategy to
extract shape information. The test case used for illustration is the drum brake of figure 1 left. The
measurement is performed using 38 accelerometers measured simultaneously. The measurement geometry
was built [2] from the numerical mode shapes, using the MSeq algorithm [3], to distinguish the modes in a
frequency band around 900Hz. In the picture, the drum was removed to see the internal instrumentation.
For this brake, a low frequency squeal occurrence (about 900Hz) has been found on vehicle and needed to
be reproduced on bench for further analysis.
To analyze the evolution of the squeal with time, the Gabor transform is used to decompose the signal in
the time-frequency domain. It consists in a Short Time Fourier Transform where at each time step, the
signal is convolved with a Gaussian window whose standard deviation is used to choose the
compromise between time and frequency resolution, fixed by the Heisenberg-Gabor equality
.
The Gabor transform is applied to the measurement on an arbitrary sensor with a very good frequency
resolution which induces a quite poor time resolution . This is acceptable
because the squeal behavior evolution with time is slow. Figure 6 left shows that during the squeal, the
frequency shifts between 906Hz and 913Hz. Three time slices on the right of the figure also highlight the
amplitude evolution of the main resonance. A smaller peak of amplitude, constant in frequency at 900Hz,
is present for each measurement and is due to an harmonic of the power supply of the bench (50Hz). It is
interesting to note that this choice is notably different from earlier practice which used much shorter
buffers in favor of a better time resolution.
Figure 6 : Gabor transform of a squeal measurement (left), time slices (right)
To evaluate in detail the evolution of the behavior, at each instant , the frequency where the amplitude is
maximum is found, and the corresponding shape is extracted. All these shapes are
concatenated and real plus imaginary parts are decomposed with the Singular Value Decomposition
providing sorted real shapes
(4)
with NS the number of sensors and NT the number of time steps.
The family can then be decomposed on these real shapes, leading to
(5)
with
Figure 7 : Time evolution of the main real shapes (left) and relative evolution between the two
main shapes .
Figure 7 left shows the time evolution of and it is clear that the limit cycle shape has only two
significant amplitudes and is thus well represented the by two main real shapes . The relative
evolution between the two main shapes is analyzed on the right and show an amplitude ratio
in the range 0.75-1 and a low phase evolution between -85° and -100°. Thus the limit cycle, despite the
evolution of the complex shape, stays in the same subspace in an analogous way than the numerical result
illustrated in section 2.
It is also important to evaluate reproducibility. For this purpose, a second squeal event has been measured
a day after the first measurement, leading to potential evolution of parameters such as pressure map,
relative component placement or temperature. The time-frequency evolution of this second measurement
and for the same sensor is shown on Figure 8. A quite important frequency shift is also found from 917Hz
to 925Hz and is in average more than 10Hz higher than the first one.
Figure 8 : Gabor transform of the second squeal measurement.
The family containing the two measurement, normalized to have the same weight
(6)
is used and decomposed as in (4) and (5). The evolutions of the amplitudes and , related
to the common main real shapes shown in Figure 9. The predominance of the two main real shapes is a bit
less clear than when the first measurement was decomposed alone (see Figure 7 left), but still there.
Figure 9 : Time evolution of the common main real shape amplitudes for the first measurement
(left) and the second measurement (right).
4 Extraction of principal shapes from laser measurements
Tests limited to accelerometers have a very limited resolution. 3D-SLDV measurements allow a higher
spatial density but lead to sequential measurements. The objective of this section is thus to introduce a
methodology for shape extraction in this context. The test case is the disk brake of figure 1 right. The
measurements have been performed using the Polytec Scanning Laser Doppler Vibrometer PSV500 using
the test geometry visible in Figure 11. This system presents a squeal occurrence at about 4250Hz. To
extract the two main shapes from the sequential measurements, three reference monoaxial accelerometers
have been disposed on the structure as reference: one on the caliper, one on the anchor bracket and one on
the arm.
In order to aggregate the sequential measurements, the hypothesis used is that a subspace of response
shapes of dimension two remains constant during the squeal.
For each point i of the wireframe presented in the introduction, a time measurement is performed (0,1s)
containing three measured signals Xi, Yi, Zi at the scanned point and reference accelerometers
(7)
For each laser position i, the two main real shapes
are extracted using the SVD (4). To
concatenate each measured point, two main shapes at the references are needed. These are
obtained using again an SVD on the restriction at reference accelerometers of all the previously extracted
main shapes
(8)
Finally, for each measured point i, the objective is to find the best linear combination of
, that
minimizes the Euclidian distance with the target shape at references
=
(9)
and the concatenation for each point of
=
(10)
forms the two main shapes.
This procedure has been applied to the 3D-SLDV measurement of the brake disc presented in
introduction. Around 4250Hz, the two main shapes shown in Figure 10 have been extracted.
Figure 10 : Visualization of the two main shapes extracted from all the sequential time measurement with
the proposed method
The first shape shows mainly a deformation of the bracket. The second shape shows another deformation
of the bracket with a deformation of the disc.
5 Expanding the results to full FEM size
Despite the high spatial resolution of the 3D-SLDV measurement, information such as hidden parts or
interfaces between components are missing. Using a FEM to perform expansion is useful for two main
reasons: it evaluates the deformations at all FEM DOFs which helps in analyzing the shapes and
depending on the used algorithm, it can highlight areas where the FEM is doubtful which orients model
updating procedures.
As illustrated in Figure 11, the test wireframe has been constructed in two parts: one in the direct view to
measure the disc, a part of the caliper and a part of the anchor bracket and one through the mirror to
measure the remaining parts of the caliper and the anchor bracket. The experimental wireframe geometry
is overlaid with the FEM and clearly illustrates the missing areas.
Figure 11 : Measurement wireframe on top of the FEM
The two shapes obtained from the exploitation of the limit cycle measurements are used to perform
expansion with a FEM. The MDRE [4] algorithm is used, which consists in finding the shape that
minimizes the weighted sum of two energies: linked to the residual force (not
zero because the model is not exact) and linked to the measurement error (the error between the
measured shape and the observation at sensors of ). The cost function is then written
(11)
with the weighting between the two errors. In more details, to measure the residual force, the static
displacement which it engenders is computed as
(12)
and the deformation energy linked to this residual displacement is the error of model
(13)
To express the measurement error, the expanded shape is observed at sensors using the
observation matrix and compared with the measured shape using the classical Euclidian norm
(14)
From these two errors, the cost function can be put in a matrix form (see [4] and [2]) and thus the
expanded mode shape can be found directly for each value of .
Nevertheless, full resolution of MDRE is generally not accessible [5], so that the reduction proposed in [2]
is a major contribution. The FEM is reduced on a basis combining the response to unit loads at sensors, for
the result to be at least as good as the static expansion, and the free modes of the structure (considering the
friction coefficient equals to zero), for the expansion to be exact if the model matches perfectly the
measurements. This family of shapes is orthonormalized with respect to mass and stiffness matrices,
leading to the basis
(15)
with the free mode shapes and the part of the response to unit loads at sensors that is
orthogonal to the free modes (this part will be called later enrichment).
Furthermore, this model reduction can be useful if the expansion is used in combination with an updating
procedure, to speed up the parametrized studies. It also permits a quick evaluation of the MDRE result
with several values of the parameter which allows more or less measurement error.
Rather than absolute measurement and model errors, relative errors are introduced to ease
interpretation of the role of parameter . The relative measurement error is defined by
(16)
and the relative model error by
(17)
Because of the chosen reduction basis, the DOFs can be decomposed in those linked to the free modes
and those linked to the enrichment . In this basis, the stiffness matrix is diagonal with first the
mode pulsations
and then the pseudo-pulsations
. From this property, the model error can be
decomposed in the part linked to the free modes and the part linked to the enrichment
(18)
Several expansions with an increasing from 1 to 1e10 have been performed on the disc brake test case.
The evolution of the relative measurement and model errors is shown in Figure 12.
Figure 12 : Evolution of relative model and measurement error with . Left: first main shape, right:
second.
For a very low value of , the relative model error is really low but the expanded shape does not
match the measurements. When increasing the value of , the measurement error decreases with at first an
incresing model error linked to the enrichment (up to ). Then, the measurement error continues
to decrease but with a strong increase of the model error linked to the free modes: to represent the
measurement well, the free modes are used but do not satisfy well the mechanical equilibrium of the
model. The free mode subspace seems thus relevant but does not have the good frequency distribution.
Choosing an intermediate (5e+05), the expanded mode shapes are show in Figure 13.The first one is
dominated by the deformation of the right column and the right side of the caliper. The second shape is
mostly a deformation of the whole caliper and again the right column. The analysis of the squeal behavior
with the expanded shapes is simpler than with only the deformations at sensors. It can help providing
modification propositions to impact the phenomenon.
Figure 13 : Expansion result for the two extracted experimental shapes form the limit cycle measurements.
The expansion can also be used to analyze the model quality. Indeed, the energy distribution coming from
the residual forces shows areas of the model where the mechanical equilibrium is poor. This residual
energy can be split in two: the residual energy related to the free mode shapes, and the residual energy
related to the enrichment shapes.
Figure 14 shows that the residual energy related to the enrichment shapes is mostly a concentration of
energy near the sensor locations
Figure 14 : Model error on the enrichment shapes
Figure 15 shows residual energy distribution on the free mode shapes. This energy is more global with
nevertheless several concentration areas: in the sliding contact between the pad and the disc (A), at each
contact between the pad and the caliper (B) and between the columns and the caliper (C). Using these
residual energy maps helps defining a relevant parametrization of the model to perform model updating if
needed.
Figure 15 : Model error on the free mode shapes
6 Conclusion
This study showed that the interpretation of the squeal behavior as an interaction between two real shapes
makes sense, not only numerically but also experimentally. The reproducibility of the phenomenon,
despite frequency shifts, is good when the subspace is compared instead of the complex shapes directly:
the limit cycle lies in a two-dimensional subspace with a slow variation of relative amplitude and phase.
The time/frequency analysis strategy retained is very simple and more advanced techniques such as the
Hilbert transform could be considered [6]. Tracking of modes during the cycle and before the squeal
instability would be a useful complement to the test performed here. It would correspond to force
appropriation and could be coupled to auto-resonance techniques [7].
The two-dimensional subspace is sufficiently stable to allow a piecewise reconstruction combining
reference sensors and 3D-SLDV measurements. This leads to field measurements of squeal shapes, but
still only measures accessible surfaces and thus fail to give indications about the behavior of junctions.
Finally, the expansion of the two shapes extracted from the limit cycle measurements provides a fully
detailed estimation of the test shapes. The proposed reduction methodology enables practical studies on
industrial models and the results can be interpreted in terms of shapes of interest and possibly as a basis to
localize errors to be used in model updating analysis. Combining expansion and updating is a clear
perspective of this work.
References
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[5] N. Nifa, “Solveurs performants pour l’optimisation sous contraintes en identification de
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