Application of Weight Functions in Nonlinear Analysis of Structural Dynamics Problems

This paper presents a weighted residual method with several weight functions for solving differential equation of motion in nonlinear structural dynamics problems. Order of variation of acceleration is assumed to be quadratic in each time step in which polynomial of displacement would contain five unknown coefficients. Five equations are required for determination of these coefficients in each time step. These equations are obtained from initial conditions, satisfying equation of motions at both ends, and weighted residual integration. In this study, four procedures are considered for weight function to be used in the weighted residual integration as; unit weight function, Petrov-Galerkin's weight function, least square weight function, and collocation weight function. Due to higher order of acceleration in the proposed method, the results indicate better and more accurate responses. Among the tested functions, the unit weighted function method demonstrated to be non-dissipative and its numerical dispersion showed to be clearly less than the common Newmark's linear acceleration method. Also critical time step duration in stability investigation for weighted function procedure showed to be larger than the critical time step duration obtained by other methods used in the nonlinear structural dynamics problems.