Nonlinear Forced Vibrations of Plates with Oscillating Inclusions

Abstract

These studies focus on the forced vibrations of rectangular plates incorporating oscillating inclusions uniformly distributed in the carrier elastic medium. Plate’s equations of motion consist of the Lamé equation for a carrier material and the equation for oscillators representing the inclusions. The oscillators are acted upon by the nonlinear friction force, described by Coulomb’s modified law, and a cubically nonlinear elastic force. The simply supported forced plate is considered. Using the Galerkin approximation, the problem of plate vibrations is reduced to the study of a system of ODEs. According to numerical and qualitative analyses, the application of harmonic forcing causes the appearance of periodic, quasiperiodic, and chaotic regimes in the system. In particular, the analytical expression is derived for the periodic regimes of the forcing frequency. The scenarios of vibration’s development is identified at the variation of forcing amplitude. The quasiperiodic and chaotic modes are studied by means of Poincaré section technique, and Fourier and Lyapunov spectra analyses. The statistical properties of the sequences of temporal intervals between maxima of system’s solutions are considered in more detail. The sequence extracted from the chaotic trajectory is shown to possess long-term correlations and approximately obeys the Weibull distribution.