Linear and Nonlinear Vibrations of a Column with an Internal Crack

Abstract

In this paper, the results of numerical studies on linear and nonlinear transverse vibrations and instability of a geometrically nonlinear column with an internal crack subjected to Euler’s load are presented. The investigated column is composed of two members. The internal member consists of two rods connected by a pin and strengthened by a rotational spring of stiffness C. The stiffness C of the rotational spring reflects the size of the internal crack. The Hamilton principle was used to formulate the boundary problem. Because of the geometric nonlinearity, the solution to the problem was achieved by means of the perturbation method. The natural vibration frequencies (the linear problem) were computed after obtaining the equations from the first power of the small parameter ε. The orthogonality condition was applied to find and calculate the first correction of vibration frequency. The nonlinear vibration frequency of a column is derived as dependent on vibration amplitude. The results of the numerical calculation are based on vibration frequency, critical loading, and the amplitude–vibration frequency relation.