Low-Frequency Free Vibration of Rods with Finite Strain

The governing differential equation for the free vibration of a rod undergoing finite strain is obtained by means of Hamilton’s principle. The equation contains quadratic as well as cubic nonlinear terms. For the low-frequency analysis of rods, the two harmonics solution is considered for the equation. The Galerkin method is employed to convert the partial differential equation to a system of two nonlinear ordinary differential equations. These equations are solved utilizing generalized differential quadrature(GDQ) and continuation methods to obtain the backbone curves and also mode shapes of vibration for rods with two different kinds of boundary conditions.