Instability and the energy criterion for continuous systems

Abstract

The stability of equilibrium states of elastic structures and other continuous systems under the action of conservative non-gyroscopic forces is often investigated by use of the so-called energy method. According to the energy criterion, instability occurs when the potential energy U, measured from the equilibrium configuration, ceases to be positive definite. The validity of this static approach in particular cases is sometimes shown by exhibiting unstable solutions or by appealing to Rayleigh's Principle to show that a fundamental frequency is imaginary. An equivalent static method is the latent-instability or adjacent-equilibrium-position method. In Section 2 of this paper we consider a general class of conservative, continuous, linear systems and we demonstrate the correctness of the energy criterion in predicting instability for such systems. The method used to show that instability follows when U can be negative is related to the methods of LIAPOUNOFF [l] and CHETAYEV [2] for discrete systems. Instability is also shown to occur when U is not positive definite for nonlinear systems in which the potential energy density is a homogeneous polynomial in the displacements and their spatial derivatives. In Section 3, dissipative forces linear in the velocities are taken into account and it is shown that they do not remove the instability when U can assume negative values.

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