A Study on the Relationship Among Several Friction-Induced Instability Mechanisms Based on Multi-point Contact Nonlinear Dynamical Friction-Induced Vibration Model

Abstract

Brake noise is the hotspot in the field of automotive NVH. Till now, four main frictional squeal mechanisms for the generation of brake noise are highlighted; they are mode coupling instability theory, negative friction-velocity slope instability theory, sprag-slip motion theory, and stick-slip motion theory. However, none of them can explain various frictional squeal phenomena fully. So, it is very important to make clear the relationship among various mechanisms. A nonlinear dynamical friction-induced vibration model with two-point contact and three-degree-of-freedoms is established to explore the mechanism of frictional squeal and the relationship between several main mechanisms. Based on this model, bifurcations of friction coefficient for mode coupling are derived under zero, negative and positive friction-velocity slope conditions. According to the bifurcation friction coefficient for mode coupling and the sign of friction-velocity slope, instability characteristics are summarized in four areas. For each area, real part and imaginary part of instable mode and stick-slip motion are investigated via numerical simulation. At the same time, divergence is calculated and the existence of limit cycle is judged based on Bendixson–Dulac criterion. Based on these results, the relationship among the four mechanisms is more clearly understood.

Appendices

Appendix A

Appendix B

Appendix C

$$ \begin{aligned} \ddot{u} & = \frac{{d^{2} u}}{{d\tau^{2} }},\ddot{v} = \frac{{d^{2} v}}{{d\tau^{2} }},\ddot{w} = \frac{{d^{2} w}}{{d\tau^{2} }},\dot{u} = \frac{du}{d\tau }, \\ \kappa_{11} & { = }1 + \kappa_{1} \cos^{2} \alpha_{1} + \kappa_{3} + \kappa_{4} \cos^{2} \alpha_{2} ,\kappa_{12} = \kappa_{21} = \kappa_{1} \sin \alpha_{1} \cos \alpha_{1} - \kappa_{4} \sin \alpha_{2} \cos \alpha_{2} ,\kappa_{13} = \kappa_{31} = - \kappa_{4} L\,\sin \alpha_{2} \cos \alpha_{2} \\ \kappa_{22} & = \kappa_{1} \sin^{2} \alpha_{1} + \kappa_{2} + \kappa_{5} ,\kappa_{23} = \kappa_{32} = \kappa_{5} L,\kappa_{33} = \kappa_{5} L^{2} \\ f_{i} & = \text{sgn} (V_{\text{reli}} )\mu_{i} ,\mu_{i} = (1 - e^{{ - d\sqrt {\frac{{k_{11} }}{{m_{1} }}} \left| {V_{\text{reli}} } \right|}} )\left\{ {\mu_{k} - (\mu_{k} - \mu_{s} )e^{{ - h\sqrt {\frac{{k_{11} }}{{m_{1} }}} \left| {V_{\text{reli}} } \right|}} } \right\},V_{\text{reli}} = \sqrt {m_{i} /k_{11} } V_{0} + \dot{u} \\ \end{aligned} $$

$u^{*} ,v^{*} ,w^{*}$ solved by the equation:

$$ \left\{ {\begin{array}{*{20}l} {\kappa_{11} u^{*} + \kappa_{12} v^{*} + \kappa_{13} w^{*} = f_{1}^{*} \kappa_{2} v^{*} + f_{2}^{*} \kappa_{5} v^{*} } \hfill \\ {\kappa_{21} u^{*} + \kappa_{22} v^{*} + \kappa_{23} w^{*} = - \frac{{P_{1} + P_{2} }}{{k_{1} }}} \hfill \\ {\kappa_{31} u^{*} + \kappa_{32} v^{*} + \kappa_{33} w^{*} = f_{2}^{*} \kappa_{5} w^{*} L\left( {v^{*} + Lw^{*} } \right)} \hfill \\ \end{array} } \right. $$

Appendix D

When $\beta = 0$, the conjugate eigenvalues of the system are:

$$ \Omega _{1,2}^{\prime \prime 2} = \frac{1}{2}\left[ {(1 + \kappa_{1} + \kappa_{2} ) \pm \sqrt {1 + \kappa_{1}^{2} + \kappa_{2}^{2} + 2\kappa_{1} \cos (2\alpha_{1} - 2\alpha_{2} ) - 2\kappa_{2} \cos (2\alpha_{1} ) - 2\kappa_{1} \kappa_{2} \cos (2\alpha_{2} ) - 2f^{*} \kappa_{2} \kappa_{12} } } \right] $$

Define $\Delta = 1 + \kappa_{1}^{2} + \kappa_{2}^{2} + 2\kappa_{1} \cos (2\alpha_{1} - 2\alpha_{2} ) - 2\kappa_{2} \cos 2\alpha_{1} - 2\kappa_{1} \kappa_{2} \cos 2\alpha_{2} - 2f^{*} \kappa_{2} \kappa_{12}$, therefore:

• (1) When $\Delta > 0,\Omega _{1}$ and $\Omega _{2}$ are conjugate pure imaginary numbers, real parts are zero。

• (2) When $\Delta = 0,\Omega _{1}$ and $\Omega _{2}$ are conjugate pure imaginary numbers, real parts are zero。

• (3) When $\Delta > 0,\Omega _{1}$, and $\Omega _{2}$ are complex conjugate pairs with one real part greater than zero and one less than zero.

Obviously $1 + \kappa_{1}^{2} + \kappa_{2}^{2} + 2\kappa_{1} \cos (2\alpha_{1} - 2\alpha_{2} ) - 2\kappa_{2} \cos 2\alpha_{1} - 2\kappa_{1} \kappa_{2} \cos 2\alpha_{2} > 0$, so we can determine:

When $\kappa_{12} > 0$ and $f^{*} > \frac{{1 + \kappa_{1}^{2} + \kappa_{2}^{2} + 2\kappa_{1} \cos (2\alpha_{1} - 2\alpha_{2} ) - 2\kappa_{2} \cos 2\alpha_{1} - 2\kappa_{1} \kappa_{2} \cos 2\alpha_{2} }}{{2\kappa_{2} \sin 2\alpha_{1} + 2\kappa_{1} \kappa_{2} \sin 2\alpha_{2} }}$, the real part of the eigenvalues of the system is positive.