Geodesics, nonlinear normal modes of conservative vibratory systems and decomposition method

In this paper, the relationship between the nonlinear normal modes (NNMs) of conservative vibratory systems and the geodesics of the corresponding Riemannian manifolds with the Jacobi metric is investigated and a modified Adomian decomposition method for constructing the NNMs is proposed. It is indicated that NNMs in the configuration space are special geodesics of the corresponding Riemannian manifold. These geodesics pass through the origin of the configuration space and with specific directions determined by the total energy of the system. Geodesic equations describing the NNMs become singular at the intersection of the Riemannian manifold and the energy level plane. To solve these singular geodesic equations, the Adomian decomposition method is used with a slight modification. The NNMs of strongly nonlinear vibration systems are constructed via the analytic approximation of the solution of the geodesics. Higher-order approximate NNMs can be constructed by a recursive procedure and the solution series is convergent rapidly. Finally, two examples of nonlinear vibratory systems with two and four degrees of freedom (dof), respectively, are given as illustration. Simulation results verified the effectiveness of this method.