new numerical methods for the complex eigenvalue ...

EB2014-SA-007

NEW NUMERICAL METHODS FOR THE COMPLEX EIGENVALUE ANALYSIS OF DISK BRAKE SQUEAL 1

Gräbner, Nils; 2Quraishi, Sarosh *; 2Schröder, Christian ; 2Mehrmann, Volker; 1 von Wagner, Utz 1

Department of Applied Mechanics, Chair of Mechatronics and Machine Dynamics, Technische Universität Berlin, Secr. MS1, Einsteinufer 5, D 10587 Berlin 2 Institut für Mathematik, MA 4-5, TU Berlin, Straße des 17. Juni 136, D 10623 Berlin KEYWORDS - brake squeal; complex eigenvalue analysis (CEA); model reduction; proper orthogonal decomposition (POD); ABSTRACT - The dynamical modeling of a disk brake with respect to squeal and its discretization via the Finite Element Method (FEM), e.g., by using commercial FEM packages, usually results in a large-scale quadratic eigenvalue problem (QEVP) with typically up to a million degrees of freedom. Furthermore, inclusion of squeal relevant physical effects such as gyroscopic and circulatory effects, damping and friction results in a QEVP with parameter dependent, non-symmetric coefficients. To identify the role of different parameters responsible for brake-squeal, a detailed parameter study is necessary, which in-turn requires the solution of many large-scale QEVPs for a variety of choices of the parameter. Thereby complex eigenvalues associated with the audible frequency range should be calculated with high accuracy which is called complex eigenvalue analysis (CEA). The state of the art modal-transformation approaches used in standard FE software converts the QEVP to a space of modal-coordinates. The modal-transformation matrices are typically constructed by solving a symmetric linear eigenvalue problem, which is obtained by dropping the non-symmetric, parameter dependent and damping terms in the QEVP, i.e., by neglecting all the physical effects essential for self-excited vibrations. This simplistic approach empirically works well for the problems where an approximation of the imaginary part of the eigenvalues are required, but for studying the dynamical stability behavior of a brake with respect to squeal, a good approximation of both the real and imaginary parts of the eigenvalues with a positive real part is of crucial interest. In this paper, we present a model-order-reduction approach which takes into account the parameter dependent nature of the damping and stiffness matrices. In our approach, we obtain the model-order-reducing subspace by performing a proper orthogonal decomposition (POD) on the matrix of dominant modes of the non-symmetric QEVP for a variety of parameter choices. Numerical experiments suggest that the new POD based approach is more accurate for the brake squeal problem than state of the art algorithms used in FE programs so far.

INTRODUCTION Brake squeal, usually occurring in the frequency range between 1 to 16 kHz, is a very common phenomenon which occurs when the brakes are suddenly applied in a moving automobile and it is a significant problem in modern brake design [1]. Predicting and eliminating brake squeal is one of the major tasks in development departments in the automotive industry. Physically, brake squeal can be interpreted as noise generated due to self-excited vibration caused by fluctuations in the friction forces at the pad-rotor interface [2]. These vibrations are commonly, although debatably, investigated based on the eigenvalues of linearized models (e.g. [3]). Within the context of the Finite Element Method (FEM), this is popularly known as complex eigenvalue analysis (CEA) [4-7]. According to the Hartman-Grobman theorem [8] or the Lyapunov stability theorem in dynamics, when at least one real part of the eigenvalues is positive, the equilibrium becomes unstable. In fact, this can only explain at best the onset of squeal but not the squeal itself, which is approximately a monofrequent vibration with constant amplitude. This can only be explained by a limit cycle behavior [9] with the problem of possibly coexisting stable silent and squealing limit cycle solutions. Formulating and solving the corresponding nonlinear equations of motion is still a very serious problem. Therefore the examination of linearized FE-models with corresponding CEA will still be an industrial standard tool in the next years. Even though the number of degrees of freedom and computational effort has constantly increased in this, there is still a lack of correspondence between experimental investigations and simulations [10]. This gives the motivation for the examinations in this paper focusing on mathematical procedures while performing the CEA. This article gives an overview of the finite element modelling and analyses the commonly used numerical solution procedures for the high dimensional eigenvalue problem in the context of parametric model order reduction. We discuss model order reduction methods in the context of the simulation and accurate prediction of disk brake squeal. It will be shown that the common procedures include an error in the choice of the subspace. An alternative method called POD (proper orthogonal decomposition) [11] also known as method of snapshots will be shown and compared to the common procedures. A validation of the POD approach is discussed on a simplified brake model. After this an industrial brake model is analyzed by the traditional and the POD approach to compare these methods.

COMMON PROCEDURES TO MODEL BRAKE SQUEAL This section deals with the description and verification of modeling brake squeal by using the finite element method. The pictured method is the standard procedure in PERMAS [12]. The aim of finite element (FE) modeling is to develop a mathematical model which is able to describe the mechanical characteristics detailed enough to derive conclusions regarding the modal stability behavior. These conclusions result from an analysis of the eigenvalues of the system. For eigenvalues with positive real part, a self-excited friction-induced vibration may arise. In particular, the magnitude of the positive real part of an eigenvalue is correlated with the tendency of a brake to squeal [13]. At the beginning of the FE analysis, the initial state of the brake is comparable to that of an unloaded stationary brake. At this point all of the possible contact zones are defined but not in

touch yet. Furthermore, the rotation of the disk is neglected as well. To consider all effects which result from applying the brake pressure and thus from the normal and the friction forces it is essential to go through several refined modeling steps. The general unforced equation of motion has the form ̈

̇

(1)

,

where , and are symmetric mass, damping, and stiffness matrices, and are positive definite and is positive semi-definite and includes the structural damping. In the first step, we perform a linear static analysis (Figure 1). The system under investigation is the disk with the external load from the brake pad. Although the disk is considered stationary, we assign velocity field information to each node to map the friction forces at the contact correctly. The linear static analysis provides the location of contact as well as the normal and friction forces at the contact area (using Coulomb’s model of friction [14, 15]).

Figure 1: (from left to right) unloaded system, contact forces in normal direction due to the brake pressure, friction forces on the brake pad, deformation caused by friction forces.

For the next steps this model will be modified. For this, the state of contact is frozen and the touching points of the contact are constrained in normal direction with multi point constraints (MPCs). The contact forces itself are partially differentiated with respect to the state variables and ̇ to obtain the additional terms and . Here is non-symmetric and describes circulatory effects and the term The resulting system is of the form ̈

(

is symmetric and describes the friction-induced damping.

) ̇

(

)

(2)

,

where the parameter is the rotational speed of the disk. In the second step, the model is loaded with centrifugal forces as if the brake disk would be rotating. With this analysis it is possible to determine the internal stress conditions. Subsequently, the additional terms and are included in the model. The extended equations of motion then have the form ̈

(

) ̇

(

)

,

(3)

where is a skew-symmetric matrix (arising from the modeling of the gyroscopic terms) and is a symmetric matrix describing the geometric stiffness. All of the matrices obtained from FEM (3) are sparse. Using the ansatz ( ) problem (QEVP)

we obtain the parameter dependent quadratic eigenvalue ( ) ( )

( ( )

( ) ( )

( )) ( )

.

(4)

The goal of CEA is to solve the above equation for ( ) and ( ) for various choices of the velocity .

TRADITIONAL APPROACH The standard approach for solving the large-scale ( - dimension) eigenvalue problem (4) is to project it into a smaller dimensional subspace represented as the range of a rectangular (where ) matrix , independent of . The reduced eigenvalue problem then is ̃ ( )̃( )

( ) ( ) ( )) ̃( ) ( ( ) ( ) ̃ ( ) ̃ ( )) ̃( ) , ( ( ) ̃

(5)

̃( ) ( ) ̃( ) ( ) . For this reduced -dimension model all with ̃ the eigenvalues and eigenvectors may be computed with a full dense eigenvalue solution method such as “polyeig” in MATLAB [16]. Typically, this approach works fine whenever contains good approximations of the exact eigenvectors.

In the traditional approach (4) the subspace is determined as the span of the eigenvectors corresponding to the eigenvalues of the smallest magnitude of the simplified EVP (

)

(6)

where is the symmetric and parameter independent part of . The system (6) however, does not include damping and coefficients associated with gyroscopic and circulatory effects. Due to the definiteness of and the associated eigenvalue problem (6) has only purely imaginary eigenvalues and real eigenvectors . The subspace is constructed as a range of the matrix .

(7)

NEW POD BASED APPROACH Similar to the traditional approach the new POD approach is also based on projection into a smaller dimensional subspace . The QEVP (4) includes non-symmetric matrices, damping and velocity dependent terms. Ignoring these terms might lead to inaccurate representation of exact eigenvectors of (4) leading to failure of the traditional approach. To keep the complete information about the model, i.e., non-symmetric matrices, damping and velocity dependence, we build our subspace using eigenvectors obtained from solving (4) for several values of the velocity. In the rest of this section we describe a procedure to compute eigenvectors of (4) for a given velocity and then a POD based approach to build the subspace .

Computing eigenvectors for a given parameter We use a “shift, then first order realization, then invert technique” consisting of three steps: 

Using a shift the QEVP (4) is converted to ( ( )) ( )

( ( )

( )

( ) where, ( ) is non-singular. 

( )

( )) ( )

( )

(8)

( )

( ) and

First order realization of the QEVP (8) in companion form ( ) ( )

where, 

( )

( )

[

( )

],

( )

( ) [

( ) ( ) ( )

(9) ] and ( )

[

( ) ]. ( )

Solve the inverted EVP ( )

( ) ( )

( )

(10)

( )

by “eigs” command in MATLAB [16,17] (implementation of Arnoldi´s method). The eigenvector ( ) can be recovered from ( ). The matrix ( ) does not have to be inverted explicitly. It suffices to solve a linear system with the matrix ( ). By choosing several shifts in positive half plane we find the desired eigenvalues and eigenvectors. POD approach for computing subspace: Proper orthogonal decomposition (POD) [11] is a method to construct a reduced order model via extracting dominant directions from an ensemble of snapshots or measurements for various parameter values. The idea is to solve the QEVP (4) for eigenvectors corresponding to a set of sample velocities and to distill the projection space as the dominant directions in these snapshots. The details are given in Algorithm 1. Algorithm 1: Input: (S1)

( )

( )

FOR ( ) ( ) Compute eigenvectors (associated with the eigenvalues in the positive half plane) of QEVP (4) sampled at velocity .

END ) (S2) Assemble a measurement matrix ̃ ( ( ∑ (S3) Perform the SVD (singular value decomposition) of ̃ Output:

). .

Let us explain this choice of . As mentioned above should contain good approximations of the eigenvectors of (4) corresponding to the eigenvalues of interest for a wide range of the velocities. Consider such an eigenvector ( ) for some unstable eigenvalue, depending on . By interpolation theory ( ) can be well approximated by a linear combination of ( ) (S1) at the sample velocities. So ( ) ( ̃ ). By a second approximation step ( ̃ ) can be approximated by its dominant part, which can be found by the singular value decomposition (SVD) [18]. By computing the SVD ̃

∑

the dominant left singular vectors values provide a good approximation for

,

(11)

corresponding to the -largest singular ̃ ( ).

APPLICATION TO THE BRAKE SQUEAL PROBLEM In this section we validate the POD approach using a simple brake model with degrees of freedom (Figure 2), and we also present some results with a real industrial brake model with degrees of freedom (Figure 3). We compare the results of the new POD based approach with the traditional approach. Validation of the POD approach We compute the eigenvalues and eigenvectors of the QEVP (4) for a fixed value of the parameter, say by formulating it as EVP (10) and using the MATLAB command “eigs” for several shift points in the positive half plane and use these results for the validation or our model reduction approach.

Figure 2: Simple brake model with n ≈ 5000 degrees of freedom.

Figure 3: Industrial brake model with n ≈ 800,000 degrees of freedom

The traditional approach (7) ignores parameter dependence, while the subspace used by POD is constructed for velocities within a range , . Initially we consider sampling points as and almost uniformly subdivide this interval to evaluate convergence when the number of POD samples is increased. The subsequent samples are .

To compare the results of the traditional approach to the POD approach, we compute the relative difference between eigenvalues obtained from the full system (4) and the ones obtained by using the reduced order model (5) ̃ . The relative difference is calculated as (

) (

(12)

)

Next, we compare the accuracy of the eigenvector approximations. To verify the approximation property of the model reduction subspace, we have to make sure that the exact eigenvector is contained in the subspace used for model reduction. We do this by computing the acute angle between the exact eigenvector and the model reduction subspace (

).

(13)

We also compute the angle between the eigenvector for the full system (4) and the eigenvector obtained from the reduced order model (5) (

(

̃ )).

(14)

Finally, we consider the decay of the singular values of the reduced order model with increasing subspace dimension .

Figure 4: Comparison of error in computing eigenvalues and eigenvectors from traditional approach and new POD approach.

The results for samples are plotted by green lines in Figure 4. The red line represents the results from the traditional approach. From these results we can conclude that the new POD based approach is orders of magnitude more accurate when compared with the traditional approach. Moreover the results from the traditional approach do not improve when we increase the dimension of the subspace.

The singular value decay can be used as an indicator for when to stop the subdivision process. When the singular values are lower than a pre-assigned tolerance (e.g. ) we can stop subdividing the parameter space. Results on industrial brake model We build the POD measurement matrix of (4) by solving the QEVP for a few parameter values . We compare the approaches by computing approximate eigenpairs for the test velocity . Since we do not know the exact eigenpairs for the industrial brake model we assess the quality of approximate eigenpairs by measuring the relative residual (

)

(

‖( ‖(

) ‖ )

‖

)

(15)

Figure 5: Eigenvalues of an industrial brake model ( 800,000 dof) computed with POD and Traditional approaches.

In Figure 5, the found eigenvalues in and near the positive half plane up to a frequency of 3000 Hz are depicted. The color indicates the size of the residual. The following observations can be made  The residuals from the new POD approach ( ) are considerably smaller than those from the traditional approach ( ).  The traditional and the POD approach disagree in the number of unstable eigenvalues.  The POD space is lower dimensional ( ) compared to the space obtained from the traditional approach ( ), rendering the subsequent small dimensional computations more efficient.

CONCLUSION In this paper, the common procedure to simulate brake squeal by using the complex eigenvalue analysis in the FEM is discussed. In this procedure a high dimensional parameter dependent quadratic eigenvalue problem has to be solved. It is shown that the traditional approach to solve the problem may be erroneous and therefore we presented an alternative method based on the proper orthogonal decomposition (POD). It is a model-order-reduction approach which takes into account the parameter dependent nature of the damping and stiffness matrices. The validation with a simple brake model suggests that the new POD based approach is more accurate than state-of-the-art algorithms used in FE programs so far. Application to an industrial brake model show that the traditional approach may fail to detect eigenvalues in the positive half plane which correspond to unstable vibrational modes. We have considered single-parameter studies with angular velocity as a parameter in our brake model, but our POD subspace construction technique is general and permits variation in other parameters, like material stiffness, brake pad pressure, friction coefficient, mass distribution, wear-condition and others as well. In the future, we plan to include the effects of varying mass distribution to find out permissible parameter values that would result in a silent brake using our POD approach. Furthermore, we plan to include nonlinearities in the model and discuss several methods to solve this problem by extending the POD method.

ACKNOLEDGEMENT: The IGF project 16799N of the Research Association GFaI has been supported by the Federal Ministry of Economics and Technology on the basis of a decision by the German Bundestag.

We would like to thank our partners within the AiF project, the companies Intes GmbH (Hr. Dr. M. Klein, Hr. Dr. N. Wagner) and Audi AG (Hr. S. Kruse).

REFERENCES [1]

J. Fieldhouse, “A proposal to predict the noise frequency of a disc brake based on friction pair interface geometry”, SAE Paper 1999-01-3403, 17th Annual brake colloquium and engineering display, Miami Beach, FL, USA, 1999.

[2]

A. Akay, “Acoustics of friction”, Acoustical Society of America Journal, Vol. 111, pp. 1525–1548, 2002.

[3]

G. Spelsberg-Korspeter, P. Hagedorn, “Complex eigenvalue analysis and brake squeal: Traps, shortcomings and their removal”, SAE Int. J. Passeng. Cars - Mech. Syst., Vol. 5, Issue 4, pp. 1211–1216, 2012.

[4]

A. AbuBakar, H. Ouyang, “Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal”, International Journal of Vehicle Noise and Vibration, Vol. 2, Number 2, pp. 143-155, 2006.

[5]

P. Liu, “Analysis of disc brake squeal using the complex eigenvalue method”, Applied Acoustics, Vol. 68, Issue 6, pp. 603–615, 2007.

[6]

A. Papinniemi, “Brake squeal: a literature review”, Applied Acoustics, Vol. 63, Issue 4, pp. 391–400, 2002.

[7]

N.M. Kinkaid, “Automotive disc brake squeal”, Journal of Sound and Vibration, Vol. 267, Issue 1, Pages 105–166, 2003.

[8]

P. Hartman, “A lemma in the theory of structural stability of differential equations”, Proceedings of the American Mathematical Society, Vol. 11 , pp. 610–620, 1960.

[9]

D. Hochlenert, U. von Wagner, “How do nonlinearities influence brake squeal?”, Proceedings of 29th SAE brake colloquium, pp. 179-186, 2011.

[10]

U. von Wagner,” Methods for Rapid Development of Silent Brakes – Actual Research and Future Prospects”, 32nd International μ Symposium, VDI Fortschritt Berichte 773, pp. 268 – 299, 2013.

[11]

G. Kerschen, J. Golinval, A. Vakakis, L. Bergman, “The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview”, Nonlinear Dynamics, Vol. 41, pp. 141–170, 2005.

[12]

INTES, “PERMAS Product Description”, Version 14, Stuttgart, 2012.

[13]

G. Liles, “Analysis of disc brake squeal using finite element methods”, SAE Paper 891150, 1989.

[14]

D. Dowson, “History of Tribology”, Professional Engineering Publishing, Bury St Edmunds, Suffolk, 1998.

[15]

E. Rabinovicz, “Friction and Wear of Materials”, John Wiley and Son, New York, 1965.

[16]

MATLAB version R2013a, Natick, Massachusetts, 2013.

[17]

R. B. Lehoucq, D. C. Sorensen, C. Yang, “ARPACK Users Guide: Solution of LargeScale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods”, SIAM, 1998.

[18]

G. Golub, C. Reinsch, "Singular value decomposition and least squares solutions", Numerische Mathematik, Vol. 14, Issue 5, pp. 403–420. 1970.