Nonlinear Dynamic Analysis of Conservative Coupled Systems of Mass-Spring via the Analytical Approaches

Abstract

We consider periodic solution for coupled systems of mass-spring. Two practical cases of these systems are explained and introduced. An analytical technique, called the Hamiltonian approach, is applied to calculate approximations to the achieved nonlinear differential oscillation equations. The concept of the Hamiltonian approach is briefly introduced, and its application for nonlinear oscillators is studied. The method introduces an alternative to overcome the difficulty of computing the periodic behavior of the oscillation problems in engineering. The results obtained employing first-order and second-order Hamiltonian approach are compared with those achieved using two other analytical techniques, named the energy balance method and the amplitude frequency formulation, and also to assess the accuracy of solutions, the results were compared with the exact ones. The results indicate that the present analysis is straightforward and provide us a unified and systematic procedure which is simple and more accurate than the other similar methods. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared with the previous approaches such as the perturbation and the classical harmonic balance methods.