On bifurcations and chaos of a forced rectangular plate with large deflection loaded by subsonic airflow

Abstract

This paper aims at the bifurcations and chaotic motions of a harmonically driven rectangular plate subjected to a uniform incompressible subsonic airflow. The plate equation of motion is derived by considering the von Karman’s large deflection and Kelvin’s type damping of material. A Galerkin-type solution is applied for the plate stress function and the aerodynamic force. The governing partial differential equation of the system is transformed into ordinary differential equations using the Galerkin method. The divergence instability and the pitchfork-like bifurcation of the plate are explored by theoretical and numerical analysis. The bifurcations of fixed points and periodic motions are thoroughly analyzed. The periodic motions can experience symmetry breaking/restoring bifurcations, period-doubling bifurcations, and saddle–node-like bifurcations, which are vital to the transition between different types of motions. Two typical bifurcation processes feature the bifurcation structure. The first one describes the change between the small and the large periodic orbits; the second one refers to the change between various large periodic orbits. Two criteria are used to predict the chaotic motions, which play a significant role in the transition between the small-orbit and the large-orbit periodic motions. The first one is the classical Holmes–Melnikov’s criterion, and the second one is an approximated criterion that is newly developed from the resonant-response analysis of a reduced system. Results show that the current criterion brings some noticeable improvements compared with Holmes–Melnikov’s criterion.