Numerical computation of periodic responses of nonlinear large-scale systems by shooting method

Geometrically nonlinear vibrations of three-dimensional elastic structures, due to harmonic external excitations, are investigated in the frequency domain. The material of the structure is assumed to be linearly elastic. The equation of motion is derived by the conservation of linear momentum in Lagrangian coordinate system and it is discretized into a system of ordinary differential equations by the finite element method. The shooting method is used, to obtain the periodic solutions. A procedure which transforms the initial value problem into a two point boundary value problem, for the periodicity condition, and then it finds the initial conditions which lead to periodic response, is developed and presented, for systems of second order ordinary differential equations. The Elmer software is used for computing the local and global mass and stiffness matrices and the force vector, as well for computing the correction of the initial conditions by the shooting method. Stability of the solutions is studied by the Floquet theory. Sequential continuation method is used to define the prediction for the next point from the frequency response diagram. The main goal of the current work is to investigate and present the potential of the proposed numerical methods for the efficient computation of the frequency response functions of large-scale nonlinear systems, which often result from space discretization of real life engineering applications.