ABSOLUTE INSTABILITY OR CHAOS? — A TRIBUTE TO DR. DAVID G. CRIGHTON

Abstract

The stabilities of an elastic plate clamped on an infinite, rigid baffle subject to any time dependent force excitation in the presence of mean flow are examined. The mechanisms that can cause plate flexural vibrations to be absolute unstable when the mean flow speed exceeds a critical value are revealed. Results show that the instabilities of an elastic plate are mainly caused by an added stiffness due to acoustic radiation in mean flow, but controlled by the structural nonlinearities. This added stiffness is shown to be negative and increase quadratically with the mean flow speed. Hence, as the mean flow speed approaches a critical value, the added stiffness may null the overall stiffness of the plate, leading to an unstable condition. Note that without the inclusion of the structural nonlinearities, the plate has only one equilibrium position, namely, its undeformed flat position. Under this condition, the amplitude of plate flexural vibration would grow exponentially in time everywhere, known as absolute instability. With the inclusion of structural nonlinearities, the plate may possess multiple equilibrium positions. When the mean flow speed exceeds the critical values, the plate may be unstable and jump from one equilibrium position to another. Since this jumping is random, the plate flexural vibration may seem chaotic.