Analytical Solution to Non-Linear Spring Mass System Free Vibration through Stepwise Linearization

Abstract

Classical method of solution of free vibration of non-linear spring mass systems has been iterative in nature as no methodology has been developed so far to obtained close form solutions. Proper time step needs to be defined upfront for obtaining realistic convergent solutions. This involves thorough understanding of the system dynamics is essential and is very challenging to the engineers at times to obtain the solution. Many a time the solution diverge and time step needs to be modified accordingly. The methodology presented in this paper linearize the non-linear spring and get piecewise linear solutions. The piecewise solutions are related to each other from continuity conditions at the piece wise linear interfaces. Free vibrations of un-damped and damped systems are presented in this paper. In order to ensure continuous motion in tensile and compressive domains, under damped spring mass system alone is considered for analysis. The methodology developed is non-iterative and hence can be developed into a software code easily to obtain the solutions without worrying about convergence issues. Validation of the solution shows that the step wise linear solutions produce sufficient accuracy even at small number of linear steps.