A Semi-Analytical Solution of a Nonlinear Mass-Spring Finite Element Time-Dependent System

Abstract

Second-order nonlinear mass-spring finite element time-dependent systems occurring in science and engineering are considered which generally do not have closed-form solutions and are solved using explicit incremental semi-analytical numerical solution procedures for nonlinear multiple-degree-of-freedom systems. Higher-order equivalent differential equations are derived to enable subsequent values of vectors to be updated using explicit Taylor series expansions. As the time step tends to zero, the values of displacement and velocity are exact in the Taylor series expansions involving as many higher-order derivatives as necessary. The ratio test is done for both the displacement and velocity Taylor series, to automatically adjust the size of time increments to ensure the algorithm's convergency, accuracy and stability. A linear system of two degrees of freedom was initially solved to illustrate how to extend the methods to deal with multiple degrees of freedom systems using matrices and vectors, typically obtained in finite element methods. The incremental semi-analytical solution procedures for nonlinear multiple-degree-of-freedom systems may be used to check results generated by implicit iterative procedures. Further applications of the semi-analytical procedures to time-dependent systems may be extended to time-independent systems that are differentiable in independent variables, such as partial differential equations with many independent variables.