講演番号
233
文献番号
20175233
Deployment of a robust test-based modal model identification method for trimmed car bodies Bart Peeters 1) Theo Geluk 1) Mahmoud El-Kafafy 2,3) 1) Siemens Industry Software Interleuvenlaan 68, B-3001 Leuven, Belgium (
[email protected]) 2) Vrije Universiteit Brusssel (VUB) Pleinlaan 2, B-1050 Brussel, Belgium 3) Helwan University Cairo, Egypt
ABSTRACT: Whereas the experimental identification of modal models of car bodies in white is a well-established process both on the level of measurement procedures and modal analysis, this is much less the case when dealing with trimmed car bodies. The high amount of damping necessitates the use of many excitation locations, which on its turn makes the identification of modal models challenging. Applications of experimental trimmed body models include static stiffness identification and NVH refinement. This paper discusses the trimmed body application potential of the new iterative MLMM modal parameter estimation method by means of FRF-based static stiffness estimation examples. KEY WORDS: trimmed car bodies, modal model, static stiffness, maximum likelihood
1. INTRODUCTION
leads to sub-optimal results, as for instance evidenced by a
One of main challenges that are now ahead of the modal
degradation of the quality of the fit between the modal model and
analysis community is the estimating of accurate experimentally
the measurements. Hence, a non-reliable modal model and
driven modal models for fully trimmed car bodies. Moving from
consequentially erroneous static stiffness estimates are obtained.
bodies in white to fully trimmed car bodies by including all
In this paper, the potential of a recently developed modal
relevant components contributing to damping, absorption and
parameter estimation method called MLMM for the estimation of
insulation leads to high amounts of damping and high modal
experimentally driven modal models that can be used to estimate
density as well. The high level of damping and the high modal
the structural static stiffness of a trimmed car bodies will be
density necessitate the use of many excitation locations to
presented and discussed. Specific about the MLMM method is
sufficiently excite the modes and to obtain a reliable modal model.
that the well-established concept of maximum likelihood
Two important applications of the experimentally driven modal
estimation is applied to estimate directly a modal model based on
models of trimmed car bodies are static stiffness identification
measured FRFs. Due to the nature of this model, the optimal
and NVH refinement. Static structural stiffness is an important
modal parameters are estimated using an iterative Gauss-Newton
criterion in the automotive structure design as it impacts vehicle
minimization scheme. There are two main benefits of the MLMM
handling, ride comfort, safety and durability. Accurate trimmed
method over the traditional modal estimation methods. The first
car body identification, including global static stiffness
benefit is the capability of the MLMM method to accurately fit an
identification, is therefore of high importance.
FRFs matrix with a very large number of columns (i.e. number of
The more classical modal parameter estimation methods
excitation locations). The second advantage of the MLMM
sometimes fail to achieve a high-quality curve-fit of FRF data
method is that the important constraints like FRFs reciprocity and
measured from fully trimmed car body due to the large amount of
the estimation of real mode shapes can be imposed directly on the
the excitation locations used and the high level of damping.
estimated modal model in the optimization process. The paper
Moreover, the classical modal parameter estimation methods
will be structured along the following lines: section 2 deals with
have rarely the possibility to fully integrate, within the estimated
the FRF-based static stiffness estimation approach; section 3
modal model, some important constraints, which are required for
gives a brief explanation of the MLMM modal parameter
the intended applications, e.g., FRFs reciprocity and real mode
estimation method, section 4 presents the results and discussion
shapes. These constraints are mostly imposed in a subsequent step
for case studies of trimmed car body static stiffness identification,
by altering the estimated modal parameters. Most likely, this
and the conclusions are given in section 5.
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Identification of the static stiffness characteristics on a TB 2. FRF-BASED STATIC STIFFNESS ESTIMATION
structure introduces a couple of complexities. First of all,
Body Static Stiffness is an important parameter when it
identification of a modal model that accurately fits the measured
comes to vehicle design and is identified to enable car body
responses can be difficult due to the high modal density and the
benchmarking, evaluation of different body design variants,
amount of damping that is present. Secondly, a typical body FRF
correlation with CAE and in the end of course in body target
excitation with a limited number of inputs (e.g. 2 – 3) can lead to
definition.
poor stiffness estimation for TB structures. With a too low
A well-established approach for body static stiffness
number of inputs, the global dynamics of the TB structure might
identification is the use of a static stiffness test-bench. The body
not be well excited, resulting in a-symmetric TB mode shapes.
clamping that is required can lead to limitations in result
Using such a TB modal model can result in significant stiffness
repeatability and in addition, difficulties to reproduce the results
overestimations, limiting the use for either benchmarking, CAE
in CAE. Next to that, only a limited number of static loading
correlation or body target setting. To solve this, a significantly
scenarios can be applied to the body structure, limiting the
larger amount of inputs can be used to excite the TB structure,
number of body characteristics that can be identified or compared
proving a more globally distributed excitation of the body
to competitor body structures. For example, identification of a
dynamics. This enables estimation of more symmetric mode
single hard-point (e.g. “Front Strut vertical”) static stiffness will
shapes and consequently, improved static stiffness identification.
not be possible on a static stiffness test bench, while this can be a
However, using a high number of inputs automatically
relevant body design parameter. An
alternative
approach
amplifies the number of FRFs that are used in the modal analysis for
body
static
stiffness
to build the TB modal model. Manual creation of a highly
identification is the so-called “static-from-dynamic method”, in
accurate modal model that not only fits the measured TB FRFs
which free-free measured Frequency Response Functions (FRFs)
precisely but also is reciprocal is highly difficult. Reciprocity of
are used to identify the static stiffness properties of the body
the model is needed, as the model will be used in forced response
structure. Typically, this approach is applied to body-in-white
calculation to do the static stiffness identification. To satisfy these
structures and is, as with a static stiffness test bench, primarily
requirements,
used to quantify the global structural body characteristics of the
identification is needed which is found in the use of the MLMM
body (as the static torsion stiffness). Base principle is that the
modal parameter estimation method (Section 3). MLMM enables
free-free acquired FRFs are used to build a modal model of the
the identification of accurate modal models, also when high
body structure. Static load-cases can be applied to this modal
number of inputs are used (e.g. when large amounts of FRFs are
model while excluding the rigid body properties of the structure,
used to build the modal model), while allowing to identify this
enabling excitation of the body flexibility. Based on the applied
modal model with real and reciprocal modes. An extra advantage
inputs and the calculated body deformations, the static stiffness
is that the modal model identification performed by MLMM
characteristics can be identified. Large advantage of this method
occurs in a far more objective way, significantly reducing the
is that no clamping of the body structure is required, giving the
subjectivity that is typically present in modal model identification.
user more freedom in the applied loading scenarios as well having
This further enhances the robustness and repeatability of the static
better possibilities to reproduce the results in CAE.
stiffness identification procedure.
an
improved
solution
for
modal
model
Given the need for more detailed body target setting to
Using the combined solution of exciting the body structure
enable optimization of the body structure for different attributes,
at multiple locations and application of MLMM method for
this paper discusses a “static-from-dynamic” approach that
modal model identification enables application of the static
enables static stiffness identification on both body-in-white (BIW)
stiffness from dynamic data procedure on both BIW and TB
as well as trimmed body (TB) structures. Next to that, not only
structures. Next to that allowing not only to identify global body
global body static stiffness properties as the torsion stiffness can
stiffness characteristics as torsion, vertical bending, lateral
be identified, also individual suspension-to-body hard-point static
bending, but also quantification of all individual suspension to
stiffness values can be quantified which will be the focus point in
body hard point static stiffness values.
this paper. This allows a more detailed evaluation, benchmarking or CAE correlation of a given body structure on either BIW or TB level.
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3. MLMM MODAL PARAMETER ESTIMATION ̂𝑖𝑜 (𝜃, 𝜔𝑘 ) ∈ ℂ the where 𝐻𝑖𝑜 (𝜔𝑘 ) ∈ ℂ the measured FRF, 𝐻
METHOD
modeled FRF, 𝜎𝐻𝑖𝑜 (𝜔𝑘 ) the standard deviation of the measured
3.1. MLMM Theory The iterative MLMM modal parameter estimation method is
FRF for output 𝑜 and input 𝑖 . The maximum likelihood estimates
a multivariable (i.e. MIMO) frequency-domain modal estimation
of 𝜃 (i.e. 𝜓𝑟 , 𝐿𝑟 , 𝜆𝑟 , 𝐿𝑅, and 𝑈𝑅) will be obtained by minimizing
method that uses the modal model as a parametric model to
the cost function ℓ𝑀𝐿𝑀𝑀 (𝜃). This will be done using the Gauss-
represent the measured frequency response functions (FRFs) over
Newton optimization algorithm. To ensure convergence, the
a chosen frequency band. The MLMM method is originally
Gauss-Newton optimization is implemented together with the
introduced in (1) and further improved in terms of the
Levenberg-Marquardt approach, which forces the cost function to
computational time in (2). In (3), the MLMM method is adapted
decrease. To start the optimization algorithm, initial values for all
to take into account some desired and physically motivated
the modal parameters are estimated by the well-known LMS
constraints (e.g., FRFs reciprocity and real mode shapes
Polymax method (6).
estimation) in the optimization process. The MLMM method belongs to the category of the maximum likelihood estimators
3.2. Constrained MLMM: FRFs reciprocity
that is known to be asymptotically consistent and efficient (4)(5).
To identify a reciprocal modal model with the MLMM method
The “MLMM” abbreviation stands for Maximum Likelihood
the residual matrices have to be symmetric and the participation
(estimation of a) Modal Model. The MLMM method optimizes
factors have to be identical (up to a scaling factor) to the mode
the modal model directly instead of optimizing a rational fraction
shape coefficients at the input stations. Therefore, the modal
polynomial model, like in the earlier developed Maximum
model represented by Equation (1) will be altered to be as follows:
Likelihood Estimation methods. The modal model is given as 𝑁𝑚
following considering a displacement over force FRF:
̂ (𝜃, 𝜔𝑘 ) = (∑ 𝐻 𝑟=1
𝑁𝑚
𝜓 𝐿 𝜓 ∗ 𝐿∗ 𝐿𝑅 ̂ (𝜃, 𝜔𝑘 ) = (∑ ( 𝑟 𝑟 + 𝑟 𝑟 ∗ )) + 𝐻 𝑠𝑘 − 𝜆𝑟 𝑠𝑘 − 𝜆𝑟 𝑠𝑘2 𝑟=1
[𝐿𝑅]𝑟𝑒𝑐 + [𝑈𝑅]𝑟𝑒𝑐 𝑠𝑘2
(1)
∗
𝑄𝑟 𝜙𝑟 𝜙𝑟𝐷𝑃 𝑄𝑟∗ 𝜙𝑟∗ 𝜙𝑟𝐷𝑃 + )+ 𝑠𝑘 − 𝜆𝑟 𝑠𝑘 − 𝜆∗𝑟 (4)
+ 𝑈𝑅 with 𝑁𝑚 the number of identified modes, 𝜓𝑟 ∈ ℂ𝑁𝑜×1 the 𝑟 th mode shape, 𝜆𝑟 the 𝑟 th pole, 𝑠𝑘 = 𝑗𝜔𝑘 , ()∗ stands for the complex conjugate of a complex number, 𝐿𝑟 ∈ ℂ1×𝑁𝑖 the 𝑟 th participation factor, 𝐿𝑅 ∈ ℝ𝑁𝑜×𝑁𝑖 and 𝑈𝑅 ∈ ℝ𝑁𝑜 ×𝑁𝑖 the lower and upper residual terms. The lower and upper residual terms are modeling the influence of the out-of-band modes in the considered frequency band. The optimization process tunes the parameters of the modal model to minimize the following quadratic-like cost function so that a best match between the
with 𝑄𝑟 ∈ ℂ the modal scaling factor, 𝜙𝑟 ∈ ℂ𝑁𝑜 ×1 the mode shape vector, 𝜙𝑟𝐷𝑃 ∈ ℂ1×𝑁𝑖 a row vector containing the mode shape elements that correspond to a driving point, [𝑈𝑅]𝑟𝑒𝑐 ∈ ℝ𝑁𝑜×𝑁𝑖 and [𝐿𝑅]𝑟𝑒𝑐 ∈ ℝ𝑁𝑜×𝑁𝑖 the upper and lower residual matrices in which the symmetry property is imposed. The iterative MLMM method will optimize the reciprocal modal model represented by Equation (4) in such a way that the best match between the measurements and the modal model is obtained.
modal model and the measurements is obtained: 3.3. Constrained MLMM: Real mode shapes 𝑁𝑖
𝑁𝑜 𝑁𝑓
ℓ𝑀𝐿𝑀𝑀 (𝜃) = ∑ ∑ ∑|𝐸𝑖𝑜 (𝜃, 𝜔𝑘
Imposing that real (normal) mode shapes are estimated is )|2
(2)
𝑖=1 𝑜=1 𝑘=1
test has proportional damping characteristics which is a quite
with 𝐸𝑖𝑜 (𝜃, 𝜔𝑘 ) ∈ ℂ the weighted residual (i.e. the error between ̂𝑖𝑜 (𝜃, 𝜔𝑘 ) represented by the the measured FRF 𝐻𝑖𝑜 (𝜔𝑘 ) and 𝐻 modal model in Equation (1)). This residual is a nonlinear function of the modal model parameters 𝜃, and it is defined as:
𝐸𝑖𝑜 (𝜃, 𝜔𝑘 ) =
̂𝑖𝑜 (𝜃, 𝜔𝑘 ) 𝐻𝑖𝑜 (𝜔𝑘 ) − 𝐻 𝜎𝐻𝑖𝑜 (𝜔𝑘 )
2017 JSAE Annual Congress Proceedings (Spring)
essentially equivalent to the assumption that the structure under hypothetical form of damping. Models with proportional damping assumption form a compromise between the undamped system models from finite element model analysis and the generally viscously damped system models from experimental modal analysis. The hypothesis of proportional damping of a given mode corresponds to a purely imaginary residue matrix
(3)
(7)(8). This corresponds to ℜ(Ψ𝑟 𝐿𝑟 ) = 0 in Equation (1). To
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identify a modal model that incorporates real mode shapes, the constraint ℜ(Ψ𝑟 𝐿𝑟 ) = 0 will be imposed in the optimization process of the MLMM method giving at the end a modal model with purely imaginary residues. In case reciprocity and the real mode shapes constraints are applied simultaneously, which is often needed for advanced engineering based on the experimental modal model, the MLMM method will identify Equation (4) with imposing that ℜ(𝑄𝑟 𝜙𝑟 𝜙𝑟𝐷𝑃 ) = 0 where 𝑄𝑟 , 𝜙𝑟 , and 𝜙𝑟𝐷𝑃 are the same as they are defined in section 3.2. In the optimization process of the MLMM method, it is also taken into account that for each mode the pole remains stable during the iterations (i.e. its real part is
Fig. 1 Decrease of MLMM cost function at each iteration.
negative). Moreover, when applying the reciprocity and real mode shapes constraint simultaneously the frequency mass sensitivity is negative, and the residue at the input point is negative imaginary. More details about the MLMM method (e.g. mathematical implementation, uncertainty derivation …) are presented in (1)(2)(3). 4. APPLICATION TO A TRIMMED CAR BODY A modal test is done in free-free condition on a trimmed car body where the car body is excited at multiple (25+) locations and the resulting acceleration responses are measured in 3 degrees of freedom at a series of output points (40+) distributed over the car body. The aim of this modal test was to obtain an experimentally
Fig. 2 Measured (blue) vs Synthesized FRF (red) – Manual EMA.
driven modal model that will be used for the static stiffness identification. As static stiffness is defined by both the global body characteristics as Torsion, Bending, Shear as well as local body characteristics (as a local bracket stiffness), the modal model has to contain both the global and local dynamics of the body structure. The modal model for the TB has been identified through combined use of Polymax and MLMM, representing the global dynamics up to 100Hz by mode shapes, while the higher frequency (local) flexibility is represented by Upper Residuals. An initial modal model is estimated by applying Polymax to the measured FRFs. Then, the MLMM methods iteratively further optimizes this modal model taking into account that the model verifies the reciprocity and real mode shapes properties. Figure 1 shows an example of the decrease of the MLMM cost function at
Fig. 3 Measured (blue) vs Synthesized FRF (red) – MLMM EMA.
each iteration. One can see from Figure 1 that the MLMM method successfully minimizes the error between the estimated modal
Next to modal model identification with MLMM, this is also
model and the measured FRFs. Such representation also serves to
done manually. Manually means that different modal models are
verify convergence of MLMM as a function of number of
estimated using the Polymax estimator while applying the desired
iterations.
constraints (FRFs reciprocity and real mode shapes) to the identified model, and then the model that most closely fits the measurements is selected. This manual process is very time consuming and it is highly user-dependent. The comparison
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between the modal model identified by the MLMM and the one identified by the manual process is presented in Figures 2 & 3 by comparing the synthesized FRF to the measured one for a driving point FRF in the vertical direction in the rear body. This comparison allows to demonstrate the impact on the identified modal model quality, as can be seen in the FRF synthesis fit that can be achieved. Figure 2 shows that the manually created modal model, which
Fig. 5 Front Strut and Rear Damper locations.
is basically identified using a linear least squares –based modal parameter estimator, does not fully accurately represents the
A TB modal model is built based on excitation with a limited
measurements. Clear advantage of MLMM usage is demonstrated
number of inputs and a second TB modal model using a large
in Figure 3 where the obtained modal model closely fits the
number of inputs with model fitting done by MLMM. Through
measured FRF. Typical mode shapes identified with MLMM are
vertical load application at the Front Strut and Rear Damper
represented in Figure 4. Using this modal representation of the
location respectively, the static hard-point stiffness for these
dynamic data the static stiffness can be identified by excluding
points is identified for both modal models. To ensure that local
the rigid body behavior from the modal model and calculation of
connection point flexibility does not play a role in this evaluation,
the body deflection under a given applied load.
only the global body dynamics up to 100 Hz is taken into account in this static stiffness comparison. The static stiffness results for the TB modal model based on a limited number of inputs in Figure 6 illustrate the complexity with TB modal models as described in Section 2. Using a limited number of inputs for the modal model, asymmetric TB mode shapes are obtained that result eventually in a-symmetric static stiffness values for suspension to body connection points at the Left and Right side of the body structure. Based on the body structure symmetry, comparable left/right side values are expected at both the Front and the Rear body in vertical direction, which clearly is not the case when using this TB modal model.
Fig. 4 Typical mode shapes identified with MLMM and constituting part of the model used for static stiffness calculations. Fig. 6 Hard-point stiffness: from limited # inputs EMA. The need for multiple inputs for the modal model creation is demonstrated in an example scenario on a trimmed body (Figure
Using multiple inputs for the TB modal model creation and
5 shows a BIW to illustrate hard-point locations) for which the
fitting this data with ML-MM results in improved mode shape
static hard point stiffness is identified at the Front Strut and Rear
symmetry, which is demonstrated by far more symmetric
Damper locations in vertical direction.
left/right static stiffness values of the hard points (Figure 7). In addition, it is clear from Figure 6 and Figure 7 (in which the same y-axis scale has been adopted) that the hard point stiffness values based on a limited number of input based modal model are not
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only a-symmetric, but also much higher than the static stiffness
for example the rear spring or the front lower arm. For both result
values from a multiple input based modal model.
sets in lateral and vertical direction, it can be concluded that for nearly all hard points there is strong symmetry in the static stiffness results, as would be expected on a typical car body structure.
Fig. 7 Hard-point stiffness: from multiple inputs EMA. This highlights further the TB modal model complexity discussed in Section 2, having a limited number of inputs for the EMA resulting in a-symmetric mode shapes and stiffness over
Fig. 9 Hard-point static stiffness in vertical direction.
estimation. Using multiple inputs for the modal model and fitting the data with MLMM fixes this, resulting in symmetric and realistic stiffness estimates. Through application of the hard point static stiffness identification approach to all connection points of the suspension
These results give insight in the static stiffness distribution over the body structure and are used for different applications as identification of body weak-points, benchmarking, evaluation of body structural modifications or correlation with CAE results.
with the car body structure, insight is acquired on the static
With this, the application possibilities for the “static-from-
stiffness distribution over the body. Hard point lateral static
dynamic method” are significantly increased, as both BIW and
stiffness results for both Front and Rear body structure are shown
TB structures can now be analyzed accurately for static stiffness
in Figure 8 and vertical static stiffness results in Figure 9. At the
performances. Identification of the individual hard point static
body front, we are considering the front Strut to body and the front
stiffness values
suspension (lower arm, stabilizer bar) to front subframe
characteristics also gives a much more detailed insight in the body
connection points. At the body rear, we are considering the rear
static characteristics, for example when benchmarking against a
spring to body and the rear suspension to rear subframe (lower
competitor body structure.
next to the global body static stiffness
arm, toe link, upper arm, and stabilizer bar) connection points. For confidentiality reasons actual values cannot be shown for the static stiffness numbers.
5. CONCLUSION This paper discussed the applicability of a recently developed modal parameter estimation method called MLMM to trimmed car bodies’ static stiffness identification. To obtain reliable values for the static stiffness of trimmed car body, which is challenging in terms of damping level and the modal density, the car body has to be excited at many locations. This results in FRFs matrix with huge number of columns. It was found that the traditional linear least squares-based modal parameter estimators fail to deliver a modal model that verifies some important properties (e.g., reciprocity and real mode shapes) and accurately fits an FRFs matrix with a very large number of columns. The presented
Fig. 8 Hard-point static stiffness in lateral direction.
results showed that the introduced MLMM modal parameter estimation method was capable to deliver a high quality
Results in both lateral and vertical direction are shown for hard
reciprocal modal model that incorporates real mode shapes even
point pairs, i.e. both the left and the right side body connection of
when a huge number of excitation locations are used for the modal
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testing. The results showed symmetric distribution of the estimated static stiffness over the car body as would be expected on a typical car body structure, but was in the past not always achieved with prior-art methods. The new MLMM modal parameter estimation method also has application potential for trimmed body NVH refinement and CAE correlation. This is currently being investigated and will lead to future publications. REFERENCES (1) M. El-Kafafy, T. De Troyer, B. Peeters, P. Guillaume, Fast Maximum-Likelihood Identification of Modal Parameters with Uncertainty Intervals: a Modal Model-Based Formulation, Mechanical System and Signal Processing,Vol. 37, pp. 422-439 (2013). (2) M. El-Kafafy, G. Accardo, B. Peeters, K. Janssens, T. De Troyer, P. Guillaume, A Fast Maximum Likelihood-Based Estimation of a Modal Model, In Proceedings of IMAC 33, a Conference and Exposition on Structural Dynamics, Orlando (FL), USA, 2015. (3) M. El-Kafafy, B. Peeters, P. Guillaume, T. De Troyer, Constrained maximum likelihood modal parameter identification applied to structural dynamics, Mechanical Systems and Signal Processing,72–73, pp. 567-589 (2016). (4) R. Pintelon, J. Schoukens, System Identification: A Frequency Domain Approach, Wiley IEEE Press. 2012. (5) P. Guillaume, P. Verboven, S. Vanlanduit, Frequency-domain maximum likelihood identification of modal parameters with confidence intervals, Proceedings of the 23rd International Seminar on Modal Analysis.. Leuven, Belgium (1998). (6) B. Peeters, H. Van der Auweraer, P. Guillaume, J. Leuridan, The PolyMAX frequency-domain method: a new standard for modal parameter estimation? Shock and Vibration 11:395-409 (2004). (7) E. Balmes, Frequency domain identification of structural dynamics using the pole/residue parametrization, Proceedings of the 14th International Modal Analysis Conference. Dearborn, MI, USA (1996). (8) W. Heylen, S. Lammens, P. Sas, Modal Analysis Theory and Testing, Heverlee: Katholieke Universiteit Leuven, Department Werktuigkunde. ( 2016).
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