Forced vibration analysis of nonlinear systems using efficient path-following method

Abstract

In this article, nonlinear forced response of dynamical systems is studied using numerical continuation methods. Several methods are available to calculate nonlinear normal modes. Along with the existing analytical methods, recently, numerical methods, especially the pseudo-arclength continuation method, have attracted many researchers. The pseudo-arclength continuation method is a very powerful method which is capable of handling strongly nonlinear systems. However, as mentioned in recently published article reviews, the computational cost of the method has limited its application. In this research, an updating formula is embedded in the pseudo-arclength continuation algorithm to reduce the computational cost. This modified method is called the efficient path-following method. The assumptions and basis of the efficient path-following method algorithm are same as those presented in other references, but none of them have exploited the updating formula of the efficient path-following method to study the forced response of nonlinear dynamical systems. To investigate the capabilities of the method, forced response of a single-degree-of-freedom Duffing system is computed. It is seen that the efficient path-following method has decreased the computational time significantly up to 70%. The results are in very good conformance with those obtained in other references, which shows the accuracy of this method. To study the ability of the efficient path-following method to handle the multi-degree-of-freedom system, a four-degree-of-freedom nonlinear system is considered, and stable and unstable branches of the solution are computed. It is observed that as the nonlinearity of the system gets stronger, the updating formula becomes more effective.