During the past decades, the problem of friction-induced vibration and noise has been the subject of a huge amount of works. Various numerical simulations with finite elements models have been largely investigated to predict squeal events. Although a nonlinear analysis is more predictive than Complex Eigenvalues Analysis, one of the main drawbacks of the time analysis is the need of large computational efforts. In view of the complexity of the subject, this approach appears still computationally too expensive to be used in industry for finite element models. In this study, the potential of a new reduced model based on a double modal synthesis (i.e., a classical modal reduction via Craig and Bampton plus a condensation at the frictional interface based on complex modes) for the prediction of self-excited vibrations of brake squeal is discussed. The effectiveness of the proposed modal reduction is tested on a finite element model of a simplified brake system. It will be shown that numerical results of times analysis by applying the proposed reduction correlate well with those of the nonlinear analysis based on a reference model, hence demonstrating the potential of using adapted modal reductions to predict the squeal propensity and to estimate self-excited vibrations and noise.
Brake squeal is the result of friction-generated vibration and noise. It corresponds to a very common problem in industrial application. Thereby friction-induced vibration has become one of the most costly problems and important concerns in automotive engineering. Indeed this instability which results in a harsh noise, especially in the frequency range between 0.1 kHz and 15 kHz, is annoying to passengers’ hearing which leads to consistent customer complaints. Comprehensive overviews on friction-induced vibration and much classic understanding of the mechanism of the disc brake squeal phenomenon can be found in [
Finite element models and numerical simulations have been extensively investigated to detect brake squeal instabilities and to predict amplitudes during squeal events. Nowadays three kinds of analysis can be performed in the design and optimization process of brake systems in order to avoid or at least to reduce the squeal occurrence: eigenvalue analysis to detect frequency instabilities, time analysis, and nonlinear methods [
As previously explained working in the time domain and performing simulations on a full finite element model require prohibitive computational costs. In order to overcome these issues, one solution is to perform model reduction. The use of appropriate reduction bases for the prediction of squeal frequencies and self-excited vibrations during the squeal event and its associated numerical implementation are complex tasks. Recently some efforts have been spent to overcome this problem by considering the potential of using reduction techniques of finite element models. Vermot des Roches [
Recently, Besset and Sinou [
In this section, the finite element model of the brake system is first presented. Then the double modal reduction based on the Craig and Bampton reduction and the interface reduction via complex eigenmodes is briefly introduced.
A simplified model of a brake system (i.e., “pad-on-disc”: a steel pad in sliding contact with an annular disc) is depicted in Figure
Finite element details and material and geometric properties of the simplified brake system.
Disc | Pad | |
---|---|---|
Young modulus | | |
Density | 7 200 | 2 500 |
Number of degrees-of-freedom (dof) | 45 000 | 2 640 |
Height (m) | 0.019 | 0.0128 |
| 0.034 | 0.091 |
| 0.151 | 0.147 |
Pad/disc brake system. (a) Viewing of the 220 contact points (red). (b) Finite element model.
The nonlinear equation of the numerical model can be written in the following form:
In order to reduce the finite element model and more specifically to condense the pad and disc, the Craig and Bampton technique is first used. This reduction technique is one of the most classical processes to reduce the finite element by retaining all the contact nodes at the pad/disc interface and generalized degrees of freedom. The Craig and Bampton technique [
Secondly, a reduction method using complex interface modes is applied in order to condense the pad/disc section (i.e., the remaining physical interface degrees of freedom that are kept unreduced via the Craig and Bampton method). This reduction method is now briefly explained.
In the process, the mass and stiffness matrices (denoted by
By considering the reduced mass and stiffness matrices via the Craig and Bampton method (denoted by
For the reader comprehension, the reduced stiffness matrix
Then, right and left eigenvectors (denoted by
By introducing the matrices
Finally we obtain the reduced brake model:
Algorithm in universal formal pseudocode is provided in [
The numerical simulations and results are decomposed into two main parts: the stability analysis via CEA and the transient nonlinear analysis.
The stability analysis that is the first step in order to predict the squeal propensity of a numerical brake model is only briefly presented in this study. This is explained by the fact that a comprehensive study was conducted previously by the authors on this subject in [
In the second part of this section, numerical simulations of the transient nonlinear self-excited vibrations will be presented. More specifically, efficiency of the reduction technique based on complex modes will be illustrated. All the results will be analyzed and compared to a reference calculation based on the finite element model without any interface condensation.
The stability analysis of the numerical brake model can be performed by using the CEA on the reduced finite element model (see (
The stability of a brake system is determined by the sign of real part of eigenvalues: an equilibrium is asymptotically stable if all eigenvalues have negative real parts; it is unstable if at least one eigenvalue has positive real part. In this last case instability is generated from the equilibrium point and self-excited vibration occurs.
Figures
Stability analysis based on the reduced model (red marks) and the reference model (black lines). (a) Real parts. (b) Imaginary parts.
Based on the previous CEA results (see Figure
Firstly the time series (transient dynamics and self-excited vibrations), then the spectral content by using short time Fourier transform (to visualize faster time-varying signals), and finally the limit cycles and the Fast Fourier Transform (to represent the nonlinear stationary signal in the frequency domain) are studied in detail. The displacements selected to illustrate the results correspond to one degree-of-freedom of the disc in the direction normal to the disc surface (i.e., the direction normal to the frictional interface), one degree-of-freedom of the disc in the direction tangential to the disc surface.
For each case, results from the reduction process based on complex interfaces modes are compared with those from the reference model. Convergence versus the number of interface modes has been conducted (results not presented in the following in order not to overload the paper content). In the following, it has been chosen to show results with 15 interface modes.
Figures
Transient simulations and spectrograms for one degree-of-freedom of the disc at the normal component: (a, b)
Transient simulations and spectrograms for one degree-of-freedom of the disc at the tangential component: (a, b)
As the friction coefficient is increased to
Frequency of the unstable mode via CEA and time analysis.
CEA (reference and reduced model) | Time analysis (reference) | Time analysis (reduced model) | |
---|---|---|---|
| 402 Hz | 394 Hz | 394 Hz |
| 400 Hz | 367 Hz | 370 Hz |
Limit cycles and Fast Fourier Transform on the stationary self-excited vibrations for one degree-of-freedom of the disc at the normal component: (a, b)
Limit cycles and Fast Fourier Transform on the stationary self-excited vibrations for one degree-of-freedom of the disc at the tangential component: (a, b)
Increasing the friction coefficient further to
Finally, all the results via the reduction technique based on complex modes (for the time series, the spectral content, and the limit cycles with the Fast Fourier Transform) are in perfect agreement with the reference performed via the finite element model without any interface condensation. This illustrates the efficiency and robustness of the proposed reduction method. The computation time and the number of degrees of freedom of the numerical model are extensively reduced by using the reduction process based on complex interfaces modes. This demonstrates the potential of the proposed method (see Table
Computational costs of the proposed reduced method and the reference model.
Reference | Reduced model | |
---|---|---|
Number of dof at the pad/disc interface | 1320 | 15 |
Time | ≈200 hours | ≤1 hour |
Extension of the double modal synthesis approach [
The proposed methodology has the potential to drastically improve the computational costs. It is not dependent on the particular choices of modeling (geometry of the brake system, friction model, or local nonlinearities in the system) and it appears easily implementable on a finite element model.
In addition, numerical results highlight that a nonlinear analysis is required to predict accuracy of the squeal events. The applicability of the proposed modal reduction strategy to a full industrial brake system will be the next step in a near future. Then appropriate comparisons and correlations with experiments could be investigated.
The authors declare that they have no competing interests.
J.-J. Sinou acknowledges the support of the Institut Universitaire de France.