Nonlinear longitudinal forced vibration of a rod undergoing finite strain

Rods are one of significant engineering’s structures and vibration analysis of a rod because of extended application of it in engineering is very important. Due to large amplitude or to excitations at frequencies close to their resonance frequencies, these structural elements can experience large amplitude, hence nonlinear vibration. Therefore, understanding of longitudinal nonlinear vibration of rods with different boundary conditions and large amplitude is very useful to reach an appropriate designing. In this paper, vibration of a rod with different boundary conditions undergoing finite strain, without simplification in strain–displacement equation, is investigated. For obtaining governing equation, Green–Lagrange strain, structural damping and Hamilton principle are used and then Galerkin method is employed to convert nonlinear partial differential equation to nonlinear ordinary differential equation. In spite of many papers that only use of cubic term for nonlinearity, the governing equation has quadratic and cubic terms. At the first, free vibration equations are solved with multiple time scales method and for verifying the accuracy of this method, the results are compared with results of Runge–Kutta numerical method, which have good accuracy. Then, forced vibration is investigated in the cases of primary resonance and hard excitation including sub-harmonic and super-harmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including nonlinear coefficients, damping coefficient and external excitation are discussed