Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint

Abstract

The present contribution describes a numerical technique devoted to the nonsmooth modal analysis (natural frequencies and mode shapes) of a non-internally resonant elastic bar of length L subject to a Robin condition at x = 0 and a frictionless unilateral contact condition at x = L. When contact is ignored, the system of interest exhibits non-commensurate linear natural frequencies, which is a critical feature in this study. The nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the boundary conditions. In order to find a few of the above families, the unknown displacement is first expressed using the well-known d’Alembert’s solution incorporating the Robin boundary condition at x = 0. The unilateral contact constraint at x = L is reduced to a conditional switch between Neumann (open gap) and Dirichlet (closed gap) boundary conditions. Finally, T-periodicity is enforced. It is also assumed that only one contact switch occurs every period. The above system of equations is numerically solved for through a simultaneous discretization of the space and time domains, which yields a set of equations and inequations in terms of discrete displacements and velocities. The proposed approach is non-dispersive, non-dissipative and accurately captures the propagation of waves with discontinuous fronts, which is essential for the computation of periodic motions in this study. Results indicate that in contrast to its linear counterpart (bar without contact constraints) where modal motions are sinusoidal functions “uncoupled” in space and time, the system of interest features nonsmooth periodic displacements that are intricate piecewise sinusoidal functions in space and time. Moreover, the corresponding frequency-energy “nonlinear” spectrum shows backbone curves of the hardening type. It is also shown that nonsmooth modal analysis is capable of efficiently predicting vibratory resonances when the system is periodically forced. The pre-stressed and initially grazing bar configurations are also briefly discussed.