Chaotic motion of a composite laminated plate with geometric nonlinearity in subsonic flow

Abstract

The bifurcation and chaotic motion of a two-dimensional (2D) composite laminated plate with geometric nonlinearity subjected to incompressible subsonic flow and transverse harmonic excitation is investigated. Based on von Karman's large deformation theory and incompressible subsonic aerodynamic model, the equation of motion of the composite laminated plate is established using the Hamilton's principle. The variable separation method is adopted to transform the equation of motion of the laminated plate into nonlinear ordinary differential equations (ODE). For the first-order expansion of the transverse displacement, the critical divergence velocity corresponding to the pitchfork bifurcation of the laminated plate is obtained by analyzing the stiffness term in the nonlinear ODE and the Melnikov's method is adopted to predict the chaotic motion of the plate after the bifurcation. The effects of the flow velocity and the amplitude and angular frequency of the external excitation on the chaotic motion of the plate are analyzed. Numerical simulations of the transverse displacement–time history, phase portrait, Poincaré map and bifurcation diagrams of the transverse displacement are used to verify the validity of the analytical results. For higher-order expansion of the transverse displacement, the critical divergence velocity is obtained by analyzing the stiffness matrix in the ODEs. The displacement–time histories and phase portraits of the transverse displacement obtained from higher-order expansions are compared with those obtained from the first-order expansion. The effects of the ply angles of the laminated plate on the critical divergence velocity are also discussed for both the first-order expansion and higher-order expansion of the transverse displacement. It can be seen from the results that the critical divergence velocity of the laminated plate decreases with the increasing ply angle. The parameters of the flow velocity and the amplitude and angular frequency of the external excitation for generating the chaotic motion of the plate obtained by the numerical simulations are within the range predicted by the Melnikov's method. Comparing with the results obtained by the higher-order expansions of the displacement, the first-order expansion can qualitatively reflect the dynamic characteristics of the composite laminated plate in subsonic flow.