9 - Equation of Motion

Abstract

This chapter provides an overview of nonlinear equation of motion. The stability analysis is confined to the linear equation of motion. No such general elementary analysis is available for the nonlinear equation of motion. If the physical system is conserving energy, or momentum, then this serves as a computational check on the conservation of the time integration scheme. Explicit schemes are computationally attractive for linear systems of second-order equations because no matrix inversion is required for them. Serious computational differences between explicit conditionally stable and implicit unconditionally stable schemes arise only in nonlinear problems and those leading to a stiff system of linear coupled equations of motion such as typically occur in the elastodynamic problem discretized by finite differences. For the linear single degree of freedom dynamic problem, there exists no fundamental computational distinction between explicit and implicit schemes applied to its solution, and the step size limitation of stability is beyond that dictated by the desire for reasonable accuracy. Serious computational differences between explicit conditionally stable and implicit unconditionally stable schemes arise only in nonlinear, for example, single degree of freedom, problems and those leading to a stiff system of linear coupled equations of motion such as typically occur in the elastodynamic. In nonlinear problems, implicit schemes require the solution of a nonlinear equation at each step while the explicit scheme does not.