Errors introduced by finite space and time increments in dynamic response computation

An ilwestigation is made of t he accuracy and stab ili ty of numerica l in teg rat ion method~ when applied to the computation of the dynamic response of st ru ctures to impac t loads. The effect of finit e t im e increments is s tudied both b.v obtaining analytical solu t ions for a single-degree-of-freedom s.vstem and by carry ing ou t nume rical integrat ions for manydegree-of-freedom systems ; t he effect of finite space increments is studied by replacing a continuous beam by a discrete number of elasticall y connected po int masses. It is found t hat: (1) Of t he methods invest iga ted , only Houbolt's is stable when t he time increments are large compared \\' it h the nat ural pe riods of t he sys tem. Errors are in t roduced by H oubolt 's method , in t his case, which result in the damping ou t of t he responses in t he higher modes of vibra tion. All of t he methods give good resul t s when the t im e increment is less t han about 1/30 of t he period in t he highes t frequency mode. (2) The di st ribu ted mass of a beam can be conside red to be co nce ntrated at relatively few mass points for com pu tat ional purposes; us ing a fi ve mass ideali zat ion , t he bending mome nt at t he cente r of a uniform beam is determin ed with good accuracy.