Geometrically nonlinear transverse steady-state periodic forced vibration of multi-degree-of-freedom discrete systems with a distributed nonlinearity

A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear free vibrations of two and multi-degree-of-freedom (2-dof) and (N-dof) systems with cubic nonlinearities. This leads to calculation of the nonlinear free modes of vibration and their associated nonlinear frequencies. More recently, this method has also been applied to examine the nonlinear forced transverse steady-state periodic response of 2-dof systems made of two masses and four spiral springs. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of N-dof systems made of N masses and N +2 nonlinear spiral springs. Its dynamic behavior has been described in terms of the mass tensor, the rigidity tensor and the fourth order tensor due to the nonlinearity. In order to solve the nonlinear algebraic system obtained, the corresponding linear free vibration problem was first solved. After determination of the linear eigen vectors and eigen values, a change of basis, from the initial basis – i.e. the displacement basis (DB) – to the modal basis (MB), has been performed using the classical matrix transformation. The new nonlinear algebraic system has then been solved using the so-called first formulation developed in previous works. This has lead to explicit expressions for the nonlinear frequency response function in the neighborhood of each of the N modes. A computer program has been written using Matlab Software. This has allowed the N-dof system nonlinear forced vibrations to be automatically examined, in the neighborhood of each mode, for any distribution of parameters in a systematic and unified way.