A linear complementarity problem formulation for periodic solutions to unilateral contact problems

Abstract

Presented is an approach for finding periodic responses of structural systems subject to unilateral contact conditions. No other non-linear terms, e.g. large displacements or strains, hyper-elasticity, plasticity, etc. are considered. The excitation period due to various forcing conditions—from harmonic external or contact forcing due to a moving contact interface—is discretized in time, such that the quantities of interest—displacement, velocity, acceleration as well as contact force—can be approximated through time-domain schemes such as backward difference, Galerkin, and Fourier. The solution is assumed to exist and is defined on a circle with circumference T to directly enforce its periodicity. The strategy for approximating time derivative terms within the discretized period, i.e. velocity and acceleration, is hence circulant in nature. This results in a global circulant algebraic system of equations with inequalities that can be translated into a unique Linear Complementarity Problem (LCP). The LCP is then solved by dedicated and established methods such as Lemke׳s Algorithm. This allows for the computation of approximate periodic solutions exactly satisfying unilateral contact constraints on a discrete time set. The implementation efficiency and accuracy are discussed in comparison to classical time marching techniques for initial value problems combined with a Lagrange Multiplier contact treatment. The LCP algorithm is validated using a simple academic problem. The extension to large-scale systems is made possible through the implementation of a Craig–Bampton type Modal Component Synthesis. The method shows applicability to industrial rotor/casing contact set-ups as shown by studying a compressor blade. A good agreement to time marching simulations is found, suggesting a viable alternative to time marching or Fourier-based harmonic balance simulations.