Nonsmooth modal analysis of a varying cross-sectional area bar in unilateral contact

Abstract

Nonsmooth modes of vibration allow for identification of resonant behaviours and attendant vibratory frequencies in structures prone to unilateral contact conditions on the boundary. The prominent approach for finding nonsmooth modes of vibration entails finding continua of periodic solutions to the system in question. In this paper, nonsmooth modes of a one-dimensional bar of varying cross-sectional area prone to unilateral contact with a rigid obstacle are determined. While numerical and analytical techniques were previously proposed, they were limited to constant cross section bar and could not be applied on the varying area bar for which the classical d’Alembert solution no longer exists. In this article, nonsmooth modal analysis of the varying area bar is performed via a novel treatment of the Signorini conditions within the finite element framework: the nodal boundary method. The nodal boundary method solves the Signorini problem by switching between two sets of shape functions describing either (1) inactive contact motion (motion away from the rigid obstacle) or (2) active contact motion (bar in contact with the rigid obstacle). In the proposed nodal boundary method, the motion of the contacting node does not participate in the resulting governing Ordinary Differential Equation (ODE). Instead, its motion is prescribed by the boundary conditions and is dictated by the motion of internal nodes. The nodal boundary method results in a discontinuous ODE in the internal nodes which can be solved both analytically and via numerical techniques. Solutions obtained by the nodal boundary method exhibit several advantages over existing numerical techniques: no chattering at contact, no penetration of the rigid obstacle, and existence of periodic solutions. Specifically, these periodic solutions are readily detectable via the shooting method with sequential continuation. The nodal boundary method is used successfully for the nonsmooth modal analysis of different models of the varying area bar. Besides, application of the nodal boundary method for nonsmooth modal analysis of the uniform area cantilever bar in Dirichlet or Robin boundary conditions is also demonstrated for comparison with existing literature.