Eigenvalue Analysis and Model Reduction in the Treatment of Disc Brake Squeal

By Volker Mehrmann and Christian Schröder

Disc brake squeal is a frequent and annoying phenomenon. It arises from self-excited vibrations caused by friction forces at the pad-rotor interface for an industrial brake model [1] (see Figure 1a and 1b). In order to satisfy customers, the automotive industry has been trying for decades to reduce squeal by changing the design of the brake and the disc. So far, it has found no satisfactory solutions that can be implemented in a systematic way. To improve the situation, several car manufacturers, suppliers, and software companies initiated a joint project, supported by the German Federal Ministry of Economics and Technology, which included two mechanical engineering groups at Technical University (TU) Berlin and TU Hamburg-Harburg, and the numerical analysis group at TU Berlin [1]. The goal of the project was to develop a mathematical model of a brake system with all effects that may cause squeal, to simulate the brake behavior for many different parameters, and to generate a small-scale reduced-order model that can be used for optimization.

The basic finite element (FE) model for analysis and numerical methods is expressed as  the macroscopic equation of motion

\[ M_\Omega\ddot{u}  +D_\Omega\dot{u} +K_\Omega u=f, \quad (1)\]

where \(u\) contains the coordinates (in the FE basis) of the displacements and \(f\) is an external force. \(M_\Omega\), \(D_\Omega\), and \(K_\Omega\in\mathbb{R}^{n,n}\) are large, sparse, parameter–dependent coefficient matrices that collect terms proportional to acceleration, velocity, and displacement, respectively. Here \(M_\Omega\) is a positive semi-definite mass matrix. The nonsymmetric matrix \(D_\Omega\) collects damping and gyroscopic effects, and the nonsymmetric matrix \(K_\Omega\) collects stiffness and circulatory effects. The parameter \(\Omega\) denotes the rotational speed of the brake disc.

Other possible parameters include operating conditions (such as temperature and pad pressure) and material properties (friction coefficient, brake geometry and mass distribution, effects of wear, damping, etc.). The excitation of vibrational modes can be investigated by computing the eigenvalues with positive real parts (associated with the unstable behavior of the model) of the quadratic eigenvalue problem associated with (1). This is still a major computational challenge, particularly if solving for many parameter values.

The goal of the project’s numerical component was to produce from the large-scale system (1) a small-scale system that can be used for optimizing the original system with respect to parameter variations, and in which the system’s complete unstable behavior is present. A particular version of the proper orthogonal decomposition (POD) method was used to generate the reduced-order model. The method computes all eigenvectors associated with right-half-plane eigenvalues for several sample parameter values. This is done by companion linearization and a scaled version of the shift-and-invert Arnoldi method [3] with a tricky shift selection procedure.

The so-called sample matrix is then formed from all these eigenvectors and the equations of motion are projected to the subspace associated with the large singular values of this matrix. This reduces the size of the parametric FE model from one million to about fifty while retaining provably good accuracy for the whole parameter set.

We had access to three FE models: an academic, relatively low-dimensional model, depicted in Figure 1b, which was used to develop the method; and two large-scale models, actually used in industry. Figure 1a shows one of them. While applying the new technique to the industrial models, we noticed that the eigenvalue computation was severely ill-conditioned; actually, the eigenvalue problem was often close to a singular problem. An analysis of the FE models revealed that they contained highly stiff springs used to avoid rigid connections, a modeling trick that has also been observed in a different context [2]. The negative effects of this technique came as a surprise to the industrial partners, and a similar analysis of other car manufacturers’ models demonstrated that this technique is used commonly, although with significant differences in the chosen spring constants.

We devised a sensitivity estimation technique based on pseudo-spectra to automatically detect the presence of these artificial stiff springs and to properly treat the rigid connections associated with the eigenvalue infinity. In combination with effective scaling and shift-and-invert transformations, we implemented and delivered a Python program to the industrial partners. This code will not only be useful in brake squeal studies, but also in other parameter-dependent vibration problems, which have non-proportional damping and other nonsymmetric terms.

To observe the advantage of the new POD method, consider our first (ill-conditioned) industrial model of over 1.2 million degrees of freedom, where the parameter is the scaled rotational speed of the disc. The new method produces better accuracy at lower reduced dimensions when compared to the traditional modal truncation method, as seen in Figure 2. The traditional modal truncation method uses as projection space the eigenvectors associated with the smallest real eigenvalues of a simplified symmetric and definite linear eigenvalue problem obtained by omitting all damping, gyroscopic, and circulatory terms in (1).

Figure 3 shows a striking example observed for the (very ill-conditioned) second industrial model; the eigenvalue approximations of the two methods are often close together but occasionally noticeably distinct, and sometimes even in different half-planes! In this case the traditional modal truncation method would have missed an eigenvalue that leads to squeal. Moreover, the residuals (indicated by the color of the shown marks (o,+)) of the eigenvalues computed by the traditional method \((\approx 10^{-7})\) are much higher than those by POD \((\approx 10^{-11})\).

In conclusion, the new model reduction method, although more costly, produces much smaller reduced models with an accuracy that is far better than the traditional modal truncation method commonly used in industry.

Acknowledgments: The authors are grateful for financial support from the IGF project 16799N of the Research Association GFaI.

References

[1] Gräbner, N., Mehrmann, V., Quraishi, S., Schröder, C., & von Wagner, U. (2015). Numerical methods for parametric model reduction in the simulation of disc brake squeal. Institut für Mathematik, TU Berlin. Manuscript submitted for publication.  

[2] Kannan, R., Hendry, S., Higham, N.J., & Tisseur, F. (2014). Detecting the causes of ill-conditioning in structural finite element models. Comput. Struct., 133, 79–89.

[3] Lehoucq, R.B., Maschhoff, K., Sorensen, D., & Yang, C. (1996). ARPACK Software Package. Rice University.

Volker Mehrmann is full professor at TU Berlin. He is a member of  the German Academy of Science and Engineering (aca-tech), a SIAM Fellow, and past president of GAMM, the International Association for Applied Mathematics and Mechanics. His research interests include numerical mathematics/scientific computing, applied and numerical linear algebra, control theory, and the theory and numerical solution of differential-algebraic equations. He is editor-in-chief of Linear Algebra and its Applications and Chair of the Research Center MATHEON in Berlin. Christian Schröder is a postdoctoral fellow at TU Berlin, where his research interests include numerical linear algebra and model order reduction.