Numerical simulation of the dynamic problem of a generalized system with distributed parameters

Abstract

Introduction. Questions of mathematical modeling of the dynamic problem in the form of a generalized system with one degree of freedom are discussed. Such systems include high-rise tower-type structures. Seismic stability of unique tower-type objects represents a relevant research problem.Aim. To determine the stress-strain state of a studied object under the action of external factors in the form of an earthquake accelerogram.Materials and methods. The methods of structural mechanics, dynamics of structures, and numerical simulation were used. The Lagrange equation was used as a basis for obtaining the motion equation of a generalized system with distributed parameters. The research methodology also included mathematical modeling of the considered systems, their numerical analysis, comparison of the obtained results with literature data.Results. A mathematical model was developed to investigate the stress-strain state of engineering structures under various external, including seismic, effects. The differential equation of the generalized system is solved directly using the method of successive approximations and the Duhamel integral at each time step. The developed algorithm was used to compile a software application in the FORTRAN language followed by obtaining the kinematic and static data of the investigated object. Using the example of a tower-type structure, free vibrations from the action of an instantaneous impulse were investigated. The results from a given earthquake accelerogram are presented.Conclusions. The results obtained on the free vibrations of the object under study agree well with those obtained by numerical simulation. The results obtained by numerical differentiation are effectively identical with those obtained by numerical integration, under the action of various effects. The validity of the results is confirmed by comparing the results obtained by the two methods. The developed software applications can be used for monitoring the state of unique tower-type objects.