Stability Analysis of a Mass-Sliding Belt System and Experimental Validation as Motivated by the Brake Squeal Problem

Abstract

Purpose

Brake squeal as a dynamic instability phenomenon is a major comfort problem observed in automotive disc brake systems. Thus, it is aimed to investigate the effects of certain operational parameters on squeal initiation.

Methods

The problem is investigated both experimentally and mathematically from the perspective of system stability. Experimentally, a mass-sliding belt experiment is designed and built, with a focus on three key operational parameters. Experiments are conducted at a wide range of these operational parameters. Furthermore, the contact stiffness at the mass and sliding belt contact interface is evaluated via modal tests. Mathematically, a nonlinear mathematical model of the experiment is developed. The model is then linearized through certain assumptions, and the stability of the system is assessed through the linearized mathematical model via complex eigenvalue solution.

Results

Data measured from the experiments are processed in time and frequency domains. Time domain results reveal local dynamic amplifications in time histories of certain operational parameters, which lead to the emergence of super-harmonics in frequency domain. Furthermore, Stribeck type friction characteristic is observed at the mass and sliding belt contact interface. The critical values of dynamic friction coefficient and motor angular speed are obtained for the validation of predictions with experimental data.

Conclusion

A good correlation between the model predictions and experiments is achieved, thus the stability analysis based on linearized model is validated with the experimental data. An extensive understanding about the effects of key operational parameters on system stability is obtained.

Appendices

Appendix A

The change of $\mu$ and ${\mu }_{c}$ with respect to ${F}_{\mathrm{pre}}$ at ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$ is shown in Fig. 25 for ${\theta }_{1}{=\theta }_{2}=27\pi /36 \, \mathrm{rad}$. As in Fig. 19, the system exhibits unstable dynamics in the shaded region. Thus, at the points shown with solid circle in Fig. 25, the system is in stable state. Though, the point given with solid square corresponds to an unstable regime. The experimental time histories and corresponding frequency spectra for these points are shown in Fig. 26. As seen in Fig. 26, the claims related to the stability of the system at the given points are experimentally validated.

Fig. 25

Average dynamic friction coefficient and critical friction coefficient with respect to ${F}_{\mathrm{pre}}$ at ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$ and ${\theta }_{1}{=\theta }_{2}=27\pi /36 \, \mathrm{rad}$. Key: black solid line, $\mu$; black dashed line, ${\mu }_{c}$ black circle, Stable operating point; black square, Unstable operating point; black times, Bifurcation point

Fig. 26

Time and frequency domain representations of experimental data for the stable and unstable operating points at ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$ and${\theta }_{1}{=\theta }_{2}=27\pi /36 \, \mathrm{rad}$. a Acceleration time history, b frequency spectra. Key: blue solid line, Stable operating point at ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$; red solid line, Unstable operating point at ${F}_{\mathrm{pre}}=100 \, \mathrm{N}$ (color figure online)

As seen in Figs. 27 and 28, which are given for the angular configuration of ${\theta }_{1}{=\theta }_{2}=30\pi /36 \, \mathrm{rad}$ and ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$, the model is again validated with the experimental data. Though the system is always in stable state.

Fig. 27

Average dynamic friction coefficient and critical friction coefficient with respect to ${F}_{\mathrm{pre}}$ at ${\omega }_{m}=200 \, \mathrm{rpm}$ and ${\theta }_{1}{=\theta }_{2}=30\pi /36 \, \mathrm{rad}$. Key: black solid line, $\mu$; black dashed line, ${\mu }_{c}$; black circle, Stable operating point; black times, Bifurcation point

Fig. 28

Time and frequency domain representations of experimental data for the stable and unstable operating points at ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$ and ${\theta }_{1}{=\theta }_{2}=30\pi /36 \, \mathrm{rad}$. a Acceleration time history at ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$, b frequency spectra at ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$, c acceleration time history at ${F}_{pre}=150 \, \mathrm{N}$, d frequency spectra at ${F}_{\mathrm{pre}}=150 \, \mathrm{N}$

Appendix B

The change of ${\omega }_{\mathrm{m}}$ and ${\omega }_{\mathrm{mc}}$ with respect to ${F}_{\mathrm{pre}}$ is shown in Fig. 29 for ${\theta }_{1}{=\theta }_{2}=27\pi /36\mathrm{ rad}$. Again, the system exhibits unstable dynamics in the shaded region. Thus, the points shown with a solid circle and a solid square represent stable and unstable states, respectively. As evident from Fig. 30, experimental time histories and frequency spectra that correspond to these points validate the detected state by the linearized model. Furthermore, the critical speeds corresponding to ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$ and ${F}_{\mathrm{pre}}=100 \, \mathrm{N}$ are ${\omega }_{mc}=73 \, \mathrm{rpm}$ and ${\omega }_{\mathrm{mc}}=278 \, \mathrm{rpm}$, respectively.

Fig. 29

Critical motor angular speed with respect to ${F}_{\mathrm{pre}}$ at${\theta }_{1}{=\theta }_{2}=27\pi /36 \, \mathrm{rad}$. Key: black dashed line, ${\omega }_{\mathrm{mc}}$; black cricle, Stable operating point; black square, Unstable operating point; black times, Bifurcation points

Fig. 30

Time and frequency domain representations of experimental data for the stable and unstable operating points at${ \theta }_{1}{=\theta }_{2}=27\pi /36 \, \mathrm{rad}$. a Acceleration time histories; b frequency spectra. Key: blue solid line, Stable operating point at ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$ and ${ \omega }_{\mathrm{m}}=300 \, \mathrm{rpm}$; red solid line, Unstable operating point at ${F}_{\mathrm{pre}}=100 \, \mathrm{N}$ and ${ \omega }_{\mathrm{m}}=100 \, \mathrm{rpm}$ (color figure online)

As seen in Figs. 31 and 32, which are given for the angular configuration of ${\theta }_{1}{=\theta }_{2}=30\pi /36 rad$, the model is again validated with the experimental data. Though the system is always in stable state.

Fig. 31

Critical motor angular speed with respect to ${F}_{\mathrm{pre}}$ at ${\theta }_{1}{=\theta }_{2}=30\pi /36 rad$. Key: black dashed line, ${ \omega }_{mc}$; black circle, Stable operating point; black times, Bifurcation points

Fig. 32

Time and frequency domain representations of experimental data for the stable and unstable operating points at ${\theta }_{1}{=\theta }_{2}=30\pi /36 \, \mathrm{rad}$. a Acceleration time history at ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$ and ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$; b frequency spectra at ${F}_{\mathrm{pre}}=50 \, \mathrm{N}$ and ${\omega }_{\mathrm{m}}=200 \, \mathrm{rpm}$; c acceleration time history at ${F}_{\mathrm{pre}}=150 \, \mathrm{N}$ and ${\omega }_{\mathrm{m}}=100 \, \mathrm{rpm}$; d frequency spectra at ${F}_{\mathrm{pre}}=150 \, \mathrm{N}$ and ${\omega }_{\mathrm{m}}=100 \, \mathrm{rpm}$