Closed form solutions of weakly nonlinear multi-degree of freedom systems with single nonlinear element

In this study, a new method which provides closed form expressions for the frequency domain solutions of multi-degree of freedom systems is introduced. The method is applicable for a multi-degree of freedom system with a single weak nonlinear element. Using this method a multi-degree of freedom system with a single nonlinear element can be put into a form such that solution of the linear part of the system along with the solution of a single nonlinear equation gives the nonlinear system response. The analytic solution of this nonlinear scalar equation is provided for cubic stiffness and Coulomb damping cases. The proposed approach is implemented on case studies of systems with cubic stiffness and dry friction elements. The results show that the method is accurate and completely eliminates the need for an iterative solution; hence, significant time savings can be obtained for large systems. The lack of need for iteration makes the method ideal for large finite element models with a single nonlinear element.