Transition of synchronization state during the quasiperiodic route to chaos in coupled friction-induced continuous oscillators

Friction-induced vibration, particularly associated with the squealing problem in disk brake systems, has been a longstanding challenge in the automotive industry. In our research, we employed the synchronization theory to gain insights into the interaction between two coupled cantilever beams attached with tip masses. This proposed model emulates the dynamics of a mountain bike disk brake assembly. The work explores a range of collective behaviors, including synchronized periodic, multi-periodic, quasiperiodic, as well as desynchronized chaotic and quasiperiodic oscillations. Despite numerous studies reported on the synchronization phenomenon in discrete friction-induced oscillatory systems, there appears to be a lack of similar research on continuous systems. This work stands as the first of its kind in exploring the dynamics of synchronization between two coupled continuous systems exhibiting a quasiperiodic route to chaos. A bifurcation study is conducted utilizing the Poincaré points corresponding to the local maxima of oscillation amplitude with respect to the zero-velocity crossing. The results showed the existence of narrow multi-periodic windows during the quasiperiodic route at several locations. Additionally, the existence of multiple quasiperiodic attractors exhibiting different states of synchronization for the same set of parameters is observed. We analyzed the time evolution of the cumulative instantaneous phase difference between the coupled signals and identified distinct states of synchronization possessed by the system. The coupled system undergoes interesting phenomena such as complete phase lock, intermittent phase lock, and phase drifting. Moreover, the transition occurring to the state of synchronization during the quasiperiodic route to chaos is studied employing the phase locking value, the Pearson linear correlation coefficient, and the relative mean frequency. Notably, our findings revealed that while the Pearson correlation can effectively identify both the mode and strength of synchronization, other measures such as phase lock value and relative mean frequency only reflect synchronization strength.