Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal. Abd Rahim AbuBakar and Huajiang Ouyang*

Transcription

1 Int. J. Vehicle Noise and Vibration, Vol. 2, No. 2, Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal Abd Rahim AbuBakar and Huajiang Ouyang* Department of Engineering, The University of Liverpool, Brownlow St, L69 3GH, UK *Corresponding author Abstract: There are typically two different methodologies that can be used to predict squeal in a disc brake, i.e., complex eigenvalue analysis and dynamic transient analysis. The positive real parts of complex eigenvalues indicate the degree of instability of the disc brake and are thought to associate with squeal occurrence or noise intensity. On the other hand, instability in the disc brake can be identified as an initially divergent vibration response using transient analysis. From the literature it appears that the two approaches were performed separately, and their correlation was not much investigated. In addition, there is more than one way of dealing the frictional contact in a disc brake. This paper explores a proper way of conducting both types of analyses and investigates the correlation between them for a large degree-of-freedom disc brake model. A detailed three-dimensional finite element model of a real disc brake is developed. Three different contact regimes are examined in order to assess the best correlation between the two methodologies. Keywords: finite element method; complex eigenvalue analysis; dynamic transient analysis; disc brake; squeal; correlation. Reference to this paper should be made as follows: AbuBakar, A.R. and Ouyang, H. (2006) Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal, Int. J. Vehicle Noise and Vibration, Vol. 2, No. 2, pp Biographical notes: Abd Rahim AbuBakar is presently a PhD Student at the Department of Engineering, University of Liverpool. He received his BEng in Mechanical Engineering from Universiti Teknologi Malaysia and his Master Degree in Automotive Engineering from the University of Leeds. His research interests are disc brake squeal and vehicle dynamics. Huajiang Ouyang is a Reader at the Department of Engineering, University of Liverpool, UK. He was awarded BEng in Engineering Mechanics in 1982, MEng in Solid Mechanics in 1985 and PhD in Structural Engineering in 1989, in Dalian University of Technology. He has published over a 100 conference and journal papers. His major research interests are structural dynamics, applied mechanics and computational mechanics. Copyright 2006 Inderscience Enterprises Ltd.

2 144 A.R. AbuBakar and H. Ouyang 1 Introduction Squeal noise emanating from car brakes is due to friction-induced vibration or self-excited vibration caused by the rotating disc. Brake squeal occurs at frequency usually above 1 khz (Lang and Smales, 1983) and is described as sound pressure level above 78 db (Eriksson, 2000) or usually at least 20dB above ambient noise level. As a result of squeal noise, carmakers and brakes and friction material suppliers face challenging tasks to improve noise performance and subsequently to reduce high warranty payouts. Akay (2002) stated that the warranty claims because of the Noise, Vibration and Harshness (NVH) issues including brake squeal in North America alone were up to one billion US dollars a year. Brakes squeal has been studied since 1930s by many investigators through experimental, analytical and numerical methods in an attempt to understand, to predict and to prevent squeal occurrence. The readers may refer to Kinkaid et al. (2003) for more details on the above-mentioned three categories of methods. The finite element method has been a popular means to predict squeal in car brakes. This is owing to the fact that the method offers much faster and more cost-efficient solutions than experimental methods, and it can predict squeal noise performance at early stage of design development. It also can capture more realistic representation of disc brakes, including nonlinearities and flexibilities of the disc brake components. There are typically two different analysis methodologies available for predicting car brakes squeal using the finite element method, namely, complex eigenvalue analysis and dynamic transient analysis. Both methodologies have their pros and cons, and these have been discussed by Mahajan et al. (1999) and Ouyang et al. (2005). The positive real parts of the complex eigenvalues indicate the degree of instability of the linear model of a disc brake and are thought to show the likelihood of squeal occurrence or the noise intensity (Liles, 1989). On the other hand, instability in the disc brake can be associated with an initially divergent vibration response using transient analysis. Liles (1989) was the early researcher who incorporated complex eigenvalue analysis with the finite element method whilst Nagy et al. (1994) pioneered dynamic transient analysis with the finite element method. Complex eigenvalue analysis allows all unstable frequencies to be found in one run for one set of operating conditions and hence is very efficient. However, not all unstable frequencies thus obtained can be observed in experiments. Transient analysis is able to predict true unstable frequencies (those found in experiments) in principle if the system model is correct. However it is very time-consuming. Moreover it does not provide any information on unstable modes. It seems that the two methods were performed separately rather than simultaneously to predict instability of the same disc brake. Thus, this raises the question on how well they fare against experimental results and in particular against each other. Dweib and D Souza (1990) and Tworzydlo et al. (1994) performed a linear and a nonlinear stability analysis of a three degrees-of-freedom pin-on-disc model using Routh criterion and eigenvalue analysis, respectively (together with dynamic transient analysis). Contact forces at the pin/disc interface based on experimental data were utilised by Dweib and D Souza (1990). On the other hand, an extended version of the Oden-Martin friction model was used by Tworzydlo et al. (1994) to represent properties of the pin/disc interface. Dweib and D Souza (1990) found that using Routh criterion the steady state sliding motion became unstable if the average value of friction coefficient reached its critical value. The onset of self-excited vibrations was also predicted with the

3 Complex eigenvalue analysis and dynamic transient analysis 145 aforementioned critical value in dynamic transient analysis. Tworzydlo et al. (1994) reported that the onset of limit-cycle vibration confirmed predictions of the complex eigenvalue analysis. However, the authors of these two papers did not provide the values of unstable frequencies, and hence how well the two methods correlated in each paper is not very clear. Hoffmann and Gaul (2003) demonstrated a good correlation when both analysis methods dealt with the same two degrees-of-freedom linear oscillator model. With more realistic representation of a disc brake, von Wagner et al. (2003) used a linear and a nonlinear multi degrees-of-freedom model to predict the instability of the system. They reported that the frequency of the limit-cycle vibration was nearly the same as that predicted in the complex eigenvalue analysis. Mahajan et al. (1999) ran both complex eigenvalue analysis and transient analysis for a large degree-of-freedom disc brake model using the finite element method but comparison was not made between them. In a recent study, Massi and Baillet (2005) showed that the dynamic transient analysis could capture only one of the two unstable frequencies predicted in the complex eigenvalue analysis for a large degree-of-freedom model of a disc brake. They used two different finite element software packages though, namely, ANSYS for the complex eigenvalue analysis and their in-house finite element software called PLAST3 for the dynamic transient analysis. This paper explores a proper way of predicting unstable vibration using complex eigenvalue analysis and transient analysis and attempts to investigate the correlation between the two methods using a single finite element package for a large degree-of-freedom model of a disc brake. Three different contact schemes available in ABAQUS are simulated for the complex eigenvalue analysis and dynamic transient analysis. This is performed to find out which contact scheme can provide a good correlation between the two methods. The information on the correlation or lack of it between the two analysis methods seems very scarce in the literature and is important to brake noise analysts and brake designers. 2 Stability analysis The complex eigenvalue analysis is the preferred method in the brake research community, because it is more mature than the dynamic transient analysis. Furthermore, the complex eigenvalue analysis can provide much faster solutions than the dynamic transient analysis. To perform the complex eigenvalue analysis using ABAQUS, four main steps are required as follows (Kung et al., 2003): nonlinear static analysis for applying brake-line pressure nonlinear static analysis to impose the rotational speed on the disc normal mode analysis to extract natural frequency of undamped system complex eigenvalue analysis that incorporates the effect of friction coupling. In this analysis, the complex eigenvalues are solved using the subspace projection method. The eigenvalue problem can be written in the following form: 2 ( λ M+ λc+ K) y =0 (1)

4 146 A.R. AbuBakar and H. Ouyang where M is the mass matrix, C, the damping matrix, which can include friction-induced damping effects as well as material damping contribution, and K is the unsymmetric (because of friction) stiffness matrix. This unsymmetrical stiffness matrix leads to both complex eigenvalues λ and complex eigenvectors y. In the third step stated above, the symmetric eigenvalue problem is first solved, by dropping damping matrix C and the unsymmetric contributions to the stiffness matrix K. Therefore the eigenvalue λ becomes a pure imaginary number λ = iω, and the eigenvalue problem now becomes 2 ( ω M+ K ) z = 0 (2) s where K s is the symmetric stiffness matrix. This symmetric eigenvalue problem then is solved using subspace eigen-solver. The next step is that the original matrices are projected in the subspace of real eigenvectors z and given as follows: M* = [ z, z,..., z ] M[ z, z,..., z ], (3a) T 1 2 n 1 2 T 1 2 n 1 2 n C* = [ z, z,..., z ] C[ z, z,..., z ], (3b) T 1 2 n 1 2 n K* = [ z, z,..., z ] K[ z, z,..., z ]. (3c) Now the reduced eigenvalue problem is expressed in the following form: 2 * n ( λ M* + λc* + K* ) y = 0 (4) which is then solved using the QZ method for a generalised unsymmetrical eigenvalue problem. The eigenvectors of the original system are recovered by the following equation: k [,,..., ] * k y = z z z y (5) 1 2 n where y k is the approximation of the kth eigenvector of the original system. For the dynamic transient analysis, ABAQUS uses central difference rule together with the diagonal lumped mass matrix. The following finite element equation of motion is solved: () t () t () t ex in. Mx = f f (6) At the beginning of the increment, accelerations are computed as follows: () t 1 () t () t ex in x = M ( f f ), (7) where x is the acceleration vector, M, the diagonal lumped mass matrix, f ex, the applied load vector and f in is the internal force vector. The superscript t refers to the time increment. The velocity and the displacement of the body are given in the following equations: ( t+ t) ( t) ( t+ 0.5 t) ( t 0.5 t) t + t ( t) x = x + x (8) 2 ( t+ t) ( t) ( t+ t) ( t+ 0.5 t) x = x + t x (9)

5 Complex eigenvalue analysis and dynamic transient analysis 147 where the superscripts (t 0.5 t) and (t t) refer to mid-increment values. Since the central difference operator is not self-starting because of the mid-increment of velocity, the initial values at time t = 0 for velocity and acceleration need to be defined. In this case, both values are set to zero as the disc is stationary at time t = 0. As opposed to the implicit dynamic integration (ABAQUS/Standard), explicit dynamic integration (ABAQUS/Explicit) does not need a convergent solution before attempting the next time step. Each time step is so small that its stability limit for t is bounded in terms of the highest eigenvalue (ω max ) in the system: 2 t. (10) ωmax This is the reason for efficiency in ABAQUS/Explicit. Because of this explicit solution procedure for dynamic simulations, contact algorithms in explicit and implicit (read Standard in ABAQUS) versions are different. The main differences between the two solvers are listed below. Standard solver uses a pure master slave scheme to enforce contact constraints while Explicit solver, by default, uses a balanced master slave scheme. Therefore alteration in Explicit code is needed by changing a weighted average of contact constraints from WEIGHT = 1.0 (balanced) to WEIGHT = 0.0 (pure). Standard solver and Explicit solver both provide the small sliding contact scheme. However, the small sliding scheme in Standard solver transfers the load to the master nodes according to the current position of the slave nodes. While in Explicit solver, the small sliding scheme always transfers the load through the anchor point. Standard solver, by default, uses a penalty contact algorithm to enforce frictional contact constraints. While Explicit solver, by default, employs a kinematic contact algorithm. The kinematic contact scheme applies sticking constraints in the similar way to the Lagrange multipliers in Standard solver. However, the algorithm is quite different. In order to change this default to the penalty method in Explicit solver, an alteration in Explicit code is needed, which is achieved by adding a keyword, MECHANICAL CONSTRAINTS = PENALTY. Those considerations need to be taken into account so that when performing the complex eigenvalue analysis and transient analysis, both analysis methods use the same contact algorithm and scheme. This is important when comparing results from these two analyses. There are three types of contact formulations available in ABAQUS, namely, infinitesimal, small and finite sliding schemes. By default, in finite sliding contact surfaces may allow for arbitrary separation, sliding and rotation, the slave nodes may come in contact anywhere along the master surface and the load transfers are updated throughout the analysis. Whilst for small sliding the contact formulation assumes that the contact surfaces may undergo arbitrarily large rotations but that a slave node will interact with the same local area of the master surface throughout the analysis. Therefore the slave nodes in contact are not monitored along the entire master surface. The finite and small sliding schemes consider geometric nonlinearities whilst infinitesimal sliding ignores this effect and assumes both the relative motion of the contacted surfaces and the absolute motion of the contacting bodies are small. Therefore, infinitesimal sliding scheme is not suitable for disc brake analysis.

6 148 A.R. AbuBakar and H. Ouyang 3 Finite element model Before a complicated model is used for predictions, it should be validated. The FE model of the whole disc brake being studied in this paper has been validated through three stages for complex eigenvalue analysis (AbuBakar et al., 2005). Therefore the material properties of each disc brake component and of the connections between brake components are believed to be valid. However, there are important limitations to the transient analysis functionality in ABAQUS/Explicit. The first limitation is the unavailability of a specific spring element, which is used to connect one component to another for contact interaction. Because of this limitation, the full disc brake model cannot be used. Instead, a reduced disc brake model that consists of two brake pads and a solid disc is used, as shown in Figure 1. The model uses up to 5184 solid elements and approximately about 22,500 degrees of freedom. The second limitation is that only the reduced-integration, eight-node solid element with hourglass control is available in Explicit version whilst full integration eight-nodes solid element is used in Standard version for complex eigenvalue analysis. Different approaches in rotating the disc are inevitable because the Explicit version does not have similar cards as Standard version to activate the boundary condition. MOTION card is used to rotate the disc in the complex eigenvalue analysis while a rigid body that ties to the disc boltholes is used instead in the transient analysis. These differences may slightly affect predicted results later in the analyses that follow. Figure 1 A finite element model of a disc brake The disc is rigidly constrained at its boltholes. For the pads, the leading edge is rigidly constrained in the circumferential direction while the trailing edge is constrained both at the circumferential and at the radial directions. The brake-line pressure is applied directly onto the back plates of the piston and finger pads as illustrated in Figure 2.

7 Complex eigenvalue analysis and dynamic transient analysis 149 Figure 2 Schematic diagram of transient analysis simulation procedure For the transient analysis, the time history of the brake-line pressure and rotational speed are used for describing operating conditions of the disc brake model, as shown in Figure 2. At the first stage, a brake pressure is applied gradually until it reaches t 1, and then it becomes constant. The disc starts to rotate at t 1 and the speed gradually increases up to t 2. Then the rotational speed becomes constant too. For the complex eigenvalue analysis, the simulation procedure is described in the previous section. For this investigation, a constant friction coefficient, µ = which was obtained from squeal experiments, is used while the disc brake is subjected to a brake-line pressure of 0.81 MPa and a rotational speed of 3.22 rad/s. It is important to note that both analysis approaches are performed in the absence of material and friction damping and thermal effect. 4 Simulation results In predicting instability of a disc brake model using the dynamic transient analysis, Hu et al. (1999) used explicit dynamic finite element analysis with penalty method. Chargin et al. (1997) used implicit dynamic finite element analysis with Lagrange multiplier to study aircraft disc brake instability. Baillet et al. (2005) employed explicit dynamic analysis with similar contact constraints to the above to study disc brake squeal. In this paper, three contact regimes, namely, 1) finite sliding with Lagrange multiplier, 2) finite sliding with penalty method and 3) small sliding with Lagrange multiplier, are simulated. Small sliding with penalty method is not available in the Explicit version and thus is not considered here. The first contact regime is considering finite sliding with Lagrange multiplier. Using the complex eigenvalue analysis, three unstable frequencies are predicted at 1.8 khz, 2.8 khz and 8.5 khz, as shown in Figure 3. While for the dynamic transient analysis, displacement in the z-direction at a particular node of the brake pad shows that its

8 150 A.R. AbuBakar and H. Ouyang response decays with time until it levels off at t = 0.05 seconds. This seems to suggest that the disc brake is stable throughout the analysis. Figure 4 illustrates the displacement response. This time history then is converted into the frequency domain using fast Fourier transformation through a Matlab code. The two analyses seem to predict different vibration behaviour of the system. Figure 3 Prediction of unstable frequencies using finite sliding with Lagrange multiplier Figure 4 Time history of displacement at a particular node for finite sliding with Lagrange multiplier The second contact regime is considering finite sliding with penalty method. Under this regime, the complex eigenvalue analysis shows similar results to those of the previous regime where three unstable frequencies are predicted, as depicted in Figure 5. This may suggest that there is no difference between these two contact regimes in the complex eigenvalue analysis of the brake squeal problem. However, the displacement responses over time of these two contact regimes are different in transient analysis even though both show a stable system. This time history of the displacement is shown in Figure 6.

9 Complex eigenvalue analysis and dynamic transient analysis 151 At time t = 0.05 seconds the displacement seems to grow only slightly and then diminish. It is thought that this response does not represent instability in the system. Figure 5 Prediction of unstable frequencies using finite sliding with penalty method Figure 6 Time history of displacement at a particular node for finite sliding with penalty method The last contact regime considered in this investigation is small sliding with Lagrange multiplier. From Figure 7, it is seen that there are two unstable frequencies predicted by the complex eigenvalue analysis, i.e., at frequencies of around 1.8 khz and 8.5 khz, which are characterised by a three-nodal-diameters disc mode and an in-plane disc mode, respectively. Those two unstable mode shapes are shown in Figure 8. For transient analysis, the displacement response, as illustrated in Figure 9, is growing with a significant amount during some short periods of time. The time history of the acceleration response of the selected node displays short bursts of limit-cycle oscillation (Figure 10). This finding agrees with previous observations of squeal events by Nack (2000). After converting the displacement response in time domain into the frequency domain, it is found that unstable frequencies are predicted at 1.8 khz and 2.0 khz with high vibration magnitude, as depicted in Figure 11. However, the other unstable frequency predicted by

10 152 A.R. AbuBakar and H. Ouyang the complex eigenvalue analysis is not found in the transient analysis. This is similar to the case of Massi and Baillet (2005) where their transient analysis was able to predict only one of the two unstable frequencies found in the complex eigenvalue. The result seems to suggest that transient analysis has not reached the same degree of sophistication as complex eigenvalue analysis in ABAQUS and thus is unable to reproduce all that can be found by complex eigenvalues analysis. An alternative explanation to this is that complex eigenvalue analysis quite often over-predicts the number of unstable frequencies. Figure 7 Prediction of unstable frequencies using small sliding with Lagrange multiplier Figure 8 Unstable mode shapes of the disc/pads at frequencies of 1.8 khz (left) and 8.5 khz (right) Figure 9 Time history of displacement at a particular node for small sliding with Lagrange multiplier

11 Complex eigenvalue analysis and dynamic transient analysis 153 Figure 10 Time history of acceleration at a particular node for small sliding with Lagrange multiplier Figure 11 Predicted unstable frequencies after converting from time domain for small sliding with Lagrange multiplier From the numerical results of these three contact regimes, it is suggested that small sliding with Lagrange multiplier can produce more reasonable results than the other two contact regimes, where the displacement response contains limit-cycle oscillation. The other contact regimes show convergence of displacement in the transient analysis, which is indicative of a stable system, but an unstable system with three unstable frequencies in the complex eigenvalue analysis. It is also worthwhile to mention that explicit dynamic transient analysis takes more than 24 hours to run before the analysis stops because of divergence when compared with complex eigenvalue analysis, which takes only around 30 minutes to complete one analysis. Incidentally, the same three regimes of finite sliding with penalty method, small sliding with Lagrange multiplier and

12 154 A.R. AbuBakar and H. Ouyang finite sliding with Lagrange multiplier in ABAQUS/Standard (implicit algorithms) are also run for the same problem, but they all diverge prematurely and cannot be used for the purpose of this investigation. One recent review paper on numerical analysis of disc brake squeal found that transient analysis of disc brake squeal is gaining popularity (Ouyang et al., 2005). Because only transient analysis is in theory capable of predicting true squeal behaviour, it is important to further develop this methodology. It is even more worthwhile to develop this parallel methodology after the apparent success of the complex eigenvalue analysis procedure that is specifically suitable for predicting instability of disc brakes and was made available in ABAQUS very recently. The current work is limited to a reduced FE model, which might not be able to lend itself to comparison with experimental results. Further investigations are proposed to get full advantages of a complete FE model to gain a better understanding of both analysis methods in predicting disc brake squeal. This work will be done when relevant software is sufficiently improved. 5 Conclusions This paper explores a proper way of predicting squeal frequencies using complex eigenvalue analysis and dynamic transient analysis, and examines the correlation between the two methodologies for a large degree-of-freedom disc brake model. The FE model and boundary conditions in both analyses must be the same, or as identical as possible (subject to software restrictions) for a fair comparison. Any difference in modelling may bias the finding on the degree of correlation. The complex eigenvalue analysis is performed with an implicit version of ABAQUS whereas the dynamic transient analysis with an explicit version. Three different contact regimes, which are available in both versions of ABAQUS, are explored and simulated. It is found that the finite sliding scheme with Lagrange multiplier or penalty method does not produce a good correlation since the dynamic transient analysis produces a stable system while the complex eigenvalue analysis produces an unstable system with three unstable frequencies. For the small sliding scheme with Lagrange multiplier the dynamic transient analysis can predict very well one of the unstable frequencies, however, misses another one predicted in the complex eigenvalue analysis. Even though transient analysis is capable of predicting squeal events and frequencies of disc brakes in theory, it may not be so in reality because of lack of sophistication in the modelling and algorithms that are available in the Explicit version of ABAQUS. Other software packages that have been widely used in predicting disc brake squeal, such as MSC.Nastran and ANSYS, do not have the capability of ABAQUS complex eigenvalue solver and hence are not thought to warrant such a comparison at present. Acknowledgements The authors would like to thank TRW for providing material data of the disc brake. The first author is grateful for the financial support provided by the Universiti Teknologi Malaysia.

13 References Complex eigenvalue analysis and dynamic transient analysis 155 AbuBakar, A.R., Ouyang, H., Li, L. and Siegel, J.E. (2005) Brake Pad Surface Topography Part II: Squeal Generation and Prevention, SAE Paper Akay, A. (2002) Acoustics of friction, Journal of the Acoustic Society of America, Vol. 111, No. 4, pp Baillet, L., Linck, V., D Errico, S., Laulagnet, B. and Berthier, Y. (2005) Finite element simulation of dynamic instabilities in frictional sliding contact, Journal of Tribology, Vol. 127, No. 3, pp Chargin, M.L., Dunne, L.W. and Herting, D.N. (1997) Nonlinear dynamics of brake squeal, Finite Elements in Analysis and Design, Vol. 28, pp Dweib, A.H. and D souza, A.F. (1990) Self-excited vibrations induced by dry friction Part II: stability and limit cycle analysis, Journal of Sound and Vibration, Vol. 137, No. 2, pp Eriksson, M. (2000) Friction and Contact Phenomenon of Disc Brakes Related to Squeal, PhD Thesis, Faculty of Science and Technology, Uppsala University, Sweden. Hoffmann, N. and Gaul, L. (2003) Effects of damping on mode-coupling instability in friction induced oscillations, ZAMM Z. Angew. Math. Mech., Vol. 83, No. 8, pp Hu, Y., Mahajan, S. and Zhang, K. (1999) Brake Squeal DOE Using Nonlinear Transient Analysis, SAE Paper Kinkaid, N.M., O Reilly, O.M. and Papadopolous, P. (2003) Review of automotive disc brake squeal, Journal of Sound and Vibration, Vol. 267, pp Kung, S., Steizer, G., Belsky, V. and Bajer, A. (2003) Brake Squeal Analysis Incorporating Contact Conditions and Other Nonlinear Effects, SAE Paper, Lang, A.M. and Smales, H. (1983) An approach to the solution of disc brake vibration problems, Proceedings of IMechE, C37/83, pp Liles, G.D. (1989) Analysis of Disc Brake Squeal Using Finite Element Methods, SAE Paper Mahajan, S.K., Hu, Y.K. and Zhang, K. (1999) Vehicle Disc Brake Squeal Simulations and Experience, SAE Paper Massi, F. and Baillet, L. (2005) Numerical analysis of squeal instability, International Conference on Emerging Technologies of Noise and Vibration Analysis and Control, November, pp Nack, W.V. (2000) Brake squeal analysis by finite elements, International Journal of Vehicle Design, Vol. 23, Nos. 3 4, pp Nagy, L.I., Cheng, J. and Hu, Y. (1994) A New Method Development to Predict Brake Squeal Occurrence, SAE Paper Ouyang, H., Nack, W., Yuan, Y. and Chen, F. (2005) Numerical analysis of automotive disc brake squeal: a review, International Journal of Vehicle Noise and Vibrations, Vol. 1, Nos. 3 4, pp Tworzydlo, T.T., Becker, E.B. and Oden, J.T. (1994) Numerical modelling of friction-induced vibrations and dynamic instabilities, Applied Mechanics Review, Vol. 47, No. 7, pp von Wagner, U., Jearsiripongkul, T., Vomstein, T., Chakraborty, G. and Hagedorn, P. (2003) Brake Squeal: Modeling and Experiments, VDI-Report 1749, pp