An approximate method for one-dimensional structures with strong nonlinear and nonhomogenous boundary conditions

Abstract

There is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non-homogenous boundary conditions. This paper proposes such a method. A distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions. The total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions. The field solution can be obtained in terms of the modal expansion. The modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations, which are solved with the harmonic balance method. Two examples are presented to study the convergence and accuracy of the solution. The first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right. The wave method and the differential quadrature method (DQM) are compared with the proposed method, and are found in good agreement. The second example is a beam with nonlinear torsional boundary absorbers. The method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method (DQEM). The proposed method is also compared with the multi-scale method (MSM) and shows an advantage to deal with strong nonlinearities. The proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions.