A combined modal/finite element analysis technique for the dynamic response of a non-linear beam to harmonic excitation

Abstract

In this paper, a method is proposed for modelling large deflection beam response involving multiple vibration modes. Significant savings in computational time can be obtained compared with the direct integration non-linear finite element method. The deflections from a number of static non-linear finite element test cases are transformed into modal co-ordinates using the modes of the underlying linear system. Regression analysis is then used to find the unknown coupled non-linear modal stiffness coefficients. The inclusion of finite element derived modal masses, and an arbitrary damping model completes the governing non-linear equations of motion. The response of the beam to excitation of an arbitrary nature may then be found using time domain numerical integration of the reduced set of equations. The work presented here extends upon the work of previous researchers to include non-linearly coupled multi-modal response. The particular benefits of this approach are that no linearization is imposed, and that almost any commercial finite element package may be employed without modification.

The proposed method is applied to the case of a homogeneous isotropic beam. Fully simply supported and fully clamped boundary conditions are considered. For the free vibration case, results are compared to those of previous researchers. For the case of steady-state harmonic excitation, results are compared with the direct integration non-linear finite element method using ABAQUS. In all cases, excellent agreement is obtained.