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

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.


Introduction
Modal analysis of nonsmooth mechanical systems, also called nonsmooth modal analysis, has been recently performed on a finite elastic bar of length L subject to a Dirichlet boundary condition at x D 0 and a unilateral contact constraint at x D L [5]. This system satisfies a complete internal resonance condition, i.e. all linear natural frequencies are commensurate with the first one, which has drastic consequences on the nonlinear modal response. To further explore the nonlinear dynamics of this one-dimensional contact problem, a noninternally resonant configuration is instead investigated in the present work. Analytical results were only found for the internally resonant bar and restricted to the first nonlinear mode [1]. Moreover, traditional numerical approaches, in the framework of finite-elements, present numerical issues that hinder the calculation of the modes of vibration of contacting systems [2]. In this work, we propose a semi-analytical technique that employs the exact travelling-wave solution to the one-dimensional wave equation [3, p. 77].

Non-internally resonant finite bar
The system of interest is an unforced, homogeneous elastic bar of length L > 0 and constant cross-sectional area S subject to a conservative unilateral constraint at its right end. Its left end is connected to a rigid support through a spring of stiffness Ä > 0, as depicted in Fig. 1. The displacement, velocity, strain and stress fields are denoted by u.x; t/, v.x; t /, .x; t / and .x; t / respectively. Young's modulus is denoted by E > 0 and > 0 stands for the mass per unit volume. In linear elasticity, the stresses read D E where D @ x u should be infinitesimally small. The unilateral contact force r.t / is related to the stresses by .L; t / D E@ x u.L; t / D r.t /=S . The gap function is defined as g.u.L; t // D g 0 u.L; t / where g 0 is the signed distance between the unrestricted resting configuration and the obstacle. The full formulation reads: Initial conditions where˛D Ä=.ES / and c D p E= . The operators @ . / and @ 2 . / stand for the first and second derivatives of . / with respect to the argument . This formulation possesses a unique solution which conserves the total energy [4]. Non-trivial solutions of the problem satisfying Eqs.
The displacement u.x; t / is then obtained through Eq. (5) provided that function f is defined everywhere on R.
The successive switches in boundary conditions at x D L, reflecting Signorini's condition (3), are incorporated through appropriate functional extensions for the free and contact phases.

Periodic solutions and Nonsmooth Modal Analysis
Nonsmooth modes of vibration (NSMs) are defined as continuous families of periodic solutions satisfying the formulation (1)-(4) together with periodicity conditions in displacement and velocity: 9T > 0 such that u.x; t CT / D u.x; t / and v.x; t CT / D v.x; t /, 8x 2 OE0 I L and 8t > 0. Finding such solutions translates into finding corresponding initial conditions u 0 and v 0 and period T which generate periodic motions. Admissible T -periodic motions involving one contact phase per period, by assumption, are described by a function f that satisfies the following functional equation, arising from u 0 .x/ D u.x; T / and v 0 .
together with functional inequalities mirroring Signorini's conditions at x D L for all t 2 OE0 I T , where t c < T is the contact phase duration. Solving this constrained formulation is a noticeably challenging task. We propose a simultaneous discretization of the space and time domains in order to accurately mimic the propagation of possibly discontinuous velocity waves along the characteristics lines.

Spectrum of nonsmooth vibration
The response of the autonomous elastic bar is depicted in frequency-energy plots (FEPs) where a backbone curve represents a NSM. The backbone curves emerging in the range OE! 1 I 1 are shown in Fig. 2 as sets of  scattered points supposedly belonging to NSMs, where ! k and k for k 2 N are the natural frequencies of the linear system involved in the conditional BC switching: Robin-Neumann and Robin-Dirichlet, respectively. In contrast to NSMs of the internally resonant bar [5] where the energy continuously depends on the frequency, the depicted scattered points indicate more complicated backbone curves. A nonsmooth periodic displacement field is depicted in Fig. 3. This point represents a periodic motion belonging to a backbone curve that emerges in the vicinity of ! 1 . The complicated pattern of the solution is caused by an intricate interplay between various travelling waves embedding the Robin and Signorini boundary conditions. In contrast to linear modes that are purely harmonic functions, the nonsmooth modes of the non-internally resonant system are nonsmooth piecewise-sinusoidal functions.

Conclusions
The annihilation of the full internal resonance condition with the Robin BC generates complicated modal motions that were computed through a semi-analytical approach based on the exact solution of the wave equation.