Observation and evolution of chaos for a cantilever plate in supersonic flow

Abstract

For a wing-like plate in supersonic flow cantilevered at its root, chaotic motions are studied in this paper. Prior literature has mainly focused on a simply supported plate or the limit cycle oscillations (LCOs) of a cantilever plate. The governing equations are constructed using von Karman plate theory and first-order piston theory. The Rayleigh–Ritz approach is adopted to discretize (and reduce the order of) the partial differential equations of the plate, and the resulting ordinary differential equations (ODEs) are solved numerically by the fourth-order Runge–Kutta (RK4) method. Numerical simulations demonstrate that the evolution of chaos is very complex, and the route to chaos depends on the panel׳s length-to-width ratio a/b. For a/b=1, a period-doubling of periodic motion occurs before transition to chaos. Another route to chaos is via quasi-periodic response directly for a/b=2. The most complicated prechaos and postchaos pattern is for a/b=0.5, which shows the presence of chaos regions with periodicity windows. Additionally, bifurcation diagrams show that certain features of the aeroelastic system such as quasi-periodic motions may be missed with too few Rayleigh–Ritz modes. Time histories, phase plane portraits, Poincaré maps and frequency spectra are used for identifying periodic and chaotic motions.