A linearization technique for the dynamic response of nonlinear continua

Abstract

The efforts of this dissertation are directed toward the development of a technique for understanding the dynamic response of structural elements governed by nonlinear partial differential equations. This technique is based on the concepts of the equivalent linearization method which relies on obtaining an optimal linear set of equations to model the original nonlinear set. In this method, the linearization is performed at the continuum level. At this level, the equivalent linear stiffness and damping parameters are physically realizable and are defined in such a way that the method can be easily be incorporated into finite element computer codes. Three different approaches to the method are taken with each approach based on the minimization of a distinct difference between the nonlinear system and its linear replacement, Existence and uniqueness properties of the minimizat4on solutions are established. The method is specialized for the treatment of steady-state solutions to harmonic excitation and of stationary response to random excitation. Procedures for solving the equivalent linearization are also discussed. The method is applied to three specific examples: one dimensional, hysteretic shear beams, thin plates governed by nonlinear equations of motion and the same nonlinear thin plates but with cutouts. Solutions via the equivalent linearization method using the stress difference minimization compare well with Galerkin's method and numerical integration. The last example is easily handled by the continuum equivalent linearization technique, whereas other methods prove to be inadequate.