Statistics of complex eigenvalues in friction-induced vibration

• Friction-induced vibration of simple systems is studied.
• Complex eigenvalue analysis is carried out for stability analysis.
• Uncertainty in significant system parameters is considered and statistics of complex eigenvalues is determined.
• A second-order perturbation method is extended and statistical formulas are derived.

Self-excited vibrations appear in many mechanical systems with sliding contacts. There are several mechanisms whereby friction can cause the self-excited vibration to become unstable. Of these mechanisms, mode coupling is thought to be responsible for generating annoying high-frequency noise and vibration in brakes. Conventionally, in order to identify whether a system is stable or not, complex eigenvalue analysis is performed. However, what has recently received much attention of researchers is the variability and uncertainty of input variables in the stability analysis of self-excited vibrations. For this purpose, a second-order perturbation method is extended and employed in the current study. The moments of the output distribution along with its joint moment generating function are used for quantifying the statistics of the complex eigenvalues. Moreover, the eigen-derivatives required for the perturbation method are presented in a way that they can deal with the asymmetry of the stiffness matrix and non-proportional damping. Since the eigen-derivatives of such systems are complex-valued numbers, it is mathematically more informative and convenient to derive the statistics of the eigenvalues in a complex form, without decomposing them into two real-valued real and imaginary parts. Then, the variance and pseudo-variance of the complex eigenvalues are used for determining the statistics of the real and imaginary parts. The reliability and robustness of the system in terms of stability can also be quantified by the approximated output distribution.