Center manifold and multivariable approximants applied to non-linear stability analysis

TL;DR

This paper combines the center manifold approach, multivariable approximants, and AFT method to analyze non-linear stability and limit cycles in brake vibration models, simplifying complex systems while preserving essential dynamics.

Contribution

It introduces a novel combined methodology for stability analysis of non-linear brake systems without requiring negative friction coefficients.

Findings

Successfully reduced system equations while maintaining dynamics

Predicted limit cycle amplitudes accurately

Validated methods against full system solutions

Abstract

This paper presents a research devoted to the study of instability phenomena in non-linear model with a constant brake friction coefficient. This paper outlines the stability analysis and a procedure to reduce and simplify the non-linear system, in order to obtain limit cycle amplitudes. The center manifold approach, the multivariable approximants theory, and the alternate frequency/time domain (AFT) method are applied. Brake vibrations, and more specifically heavy trucks grabbing are concerned. The modelling introduces sprag-slip mechanism based on dynamic coupling due to buttressing. The non-linearity is expressed as a polynomial with quadratic and cubic terms. This model does not require the use of brake negative coefficient, in order to predict the instability phenomena. Finally, the center manifold approach, the multivariable approximants, and the AFT method are used in order to obtain equations for the limit cycle amplitudes. These methods allow the reduction of the number of equations of the original system in order to obtain a simplified system, without loosing the dynamics of the original system, as well as the contributions of non-linear terms. The goal is the validation of this procedure for a complex non-linear model by comparing results obtained by solving the full system and by using these methods. The brake friction coefficient is used as an unfolding parameter of the fundamental Hopf bifurcation point.