Nonlinear Longitudinal Free 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. Therefore, understanding of longitudinal nonlinear vibration of rod with different boundary conditions and large amplitude is very useful. In this paper, vibration of a rod with different boundary conditions undergoing finite strain, without simplification in strain-displacement relations, 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. The equations with and without damping, are solved with multiple time scales method. In order to verify the accuracy of this method, the results are compared with results of Runge-Kutta numerical method, which have good accuracy. Finally sensitivity analysis for understanding of influence of nonlinear coefficients on rod vibration answer is done.