A statistical approach to estimate the Lyapunov spectrum in disc brake squeal

The estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis (CEA) in the frequency domain has its fair share of successes and failures. While the CEA is almost standard practice for the automotive industry, time domain methods and the estimation of Lyapunov spectra have not received much attention in brake squeal analyses. One reason is the challenge in estimating the true Lyapunov exponents and their discrimination against spurious ones in experimental data. A novel method based on the application of the Eckmann-Ruelle matrices is proposed here to estimate Lyapunov exponents by using noise in a statistical procedure. It is validated with respect to parameter variations and dimension estimates. By counting the number of non-overlapping confidence intervals for Lyapunov exponent distributions obtained by moving a window of increasing size over bootstrapped same-length estimates of an observation function, a dispersion measure׳s width is calculated and fed into a Bayesian beta-binomial model. Results obtained using this method for benchmark models of white and pink noise as well as the classical Henon map indicate that true Lyapunov exponents can be isolated from spurious ones with high confidence. The method is then applied to accelerometer and microphone data obtained from brake squeal tests. Estimated Lyapunov exponents indicate that the pad׳s out-of-plane vibration behaves quasi-periodically on the brink to chaos while the microphone׳s squeal signal remains periodic.