A new direct time integration method for the equations of motion in structural dynamics

Abstract

A direct time integration method is presented for the solution of the equations of motion describing the dynamic response of structural linear and nonlinear multi‐degree‐of‐freedom systems. It applies also to large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. The proposed method is based on the concept of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary quasi‐static differential equations under appropriate fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark's, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two.

Stepwise analysis of pantographic beams subjected to impulsive loads

“…As briefly discussed in the introduction, the technical literature is quite packed with contributions aiming to solve this kind of problem, see, e.g., [33–35] and references therein for some recent contributions and Wriggers’ book [36] for a systematic discussion of stepwise integration schemes, as updated up to the first years of this century. Starting from a time discretization, this approach tries to reconstruct the equilibrium path, that is, the pair time t and Lagrangian variables collected in the vectors u and u · , by solving a series of initial value problems.…”

Α mass‐reduction design concept for seismic hazard mitigation

“…The method is simple to implement. It is self‐starting, unconditionally stable, accurate, while it does not exhibit numerical damping …”

BEM for Dynamic Analysis of Plates

“…The parameters of the differential models of either integer or fractional type should satisfy certain restrictions so that non-negative internal work and rate of energy dissipation is predicted. The solution of the resulting integer-order semi-discretized evolution equations is obtained using the timestep integration method presented in Appendix C, while of the fractional order differential equations using the numerical method presented in [22]. Due to the complexity of the viscoelastic models, which include time as an independent variable, exact analytical solutions have been obtained for only a few simplified problems [41,42].…”