A Time Variational Method for the Approximate Solution of Nonlinear Systems Undergoing Multiple-Frequency Excitations

The main objective of this paper is to use the time variational method (TVM) for the nonlinear response analysis of mechanical systems subjected to multiple-frequency excitations. The system response, which is composed of fractional multiples of frequencies, is expressed in terms of a fundamental frequency that is the greatest common divisor of the approximated frequency components. Unlike the multiharmonic balance method (MHBM), the formulation of the proposed method is very simple in analyzing the systems with more than two excitation frequencies. In addition, the proposed method avoids the alternate transformation between frequency and time domains during the calculation of the nonlinear force and the Jacobian matrix. In this work, the performance of the proposed method is compared with that of numerical integration and the MHBM using three nonlinear mechanical models undergoing multiple-frequency excitations. It is observed that the proposed method produces approximate results during the quasi-periodic response analysis since the formulation includes an approximation of the incommensurate frequencies to commensurate ones. However, the approximation error is very small and the method reduces a significant amount of computational efforts compared to the other methods. In addition, the TVM is a recommended option when the number of state variables involved in the nonlinear function is high as it calculates the nonlinear force vector and the Jacobian matrix directly from the displacement vector. Moreover, the proposed method is far much faster than numerical integration in capturing the steady-state, quasi-periodic responses of the nonlinear mechanical systems.