Frequency response of a mass grounded by linear and nonlinear springs in series: An exact analysis

Abstract

An exact analytical solution for the frequency response of a system consisting of a mass grounded by linear and nonlinear springs in series was derived. The system is applicable in the study of beams carrying intermediate rigid mass. The exact solution was derived by naturally transforming the integral of the governing nonlinear ODE into a form that can be expressed in terms of elliptic integrals. Hence, the exact frequency–amplitude solution was derived in terms of the complete elliptic integral of the first and third kinds. Periodic solutions were found to exist for all real values of ε and for η > 0 . On the other hand, linear or weakly nonlinear frequency–amplitude responses were found to occur during small-amplitude vibrations ( A ≤ 0.1 ), very large-amplitude vibrations with strong hardening nonlinearity ( A ≥ 10 and ε ≫ 1.0 ), and for all amplitudes when η ≅ 1 . Simulations showed that the system’s periodic response is significantly influenced by the nonlinearity parameter ( ε ), linearity parameter ( η ), and amplitude of vibration ( A ). Besides, the existence of bifurcation points at ε = 0 and different values of η was confirmed. Lastly, an approximate frequency solution obtained using the He’s frequency–amplitude formulation was found to produce errors less than 1.0% for a wide range of input parameters. In conclusion, the present study provides a benchmark solution for verification of other approximate solutions, and the He’s frequency–amplitude formulation can be used to obtain fast and accurate solutions for complex nonlinear systems.