No abstract

The Wave Finite Element Method (WFEM) is implemented to accurately capture travelling waves propagating at a finite speed within a bouncing rod system and induced by unilateral contact collisions with a rigid foundation; friction is not accounted for. As opposed to the traditional Finite Element Method (FEM) within a time-stepping framework, potential discontinuous deformation, stress and velocity wave fronts are accurately described, which is critical for the problem of interest. A one-dimensional benchmark with an analytical solution is investigated. The WFEM is compared to two time-stepping solution methods formulated on a FEM semi-discretization in space: (1) an explicit technique involving Lagrange multipliers and (2) a non-smooth approach implemented in the Siconos package. Attention is paid to the Gibb’s phenomenon generated during and after contact occurrences together with the time evolution of the total energy of the system. It is numerically found that the WFEM outperforms the FEM and Siconos solution methods because it does not induce any spurious oscillations or dispersion and diffusion of the shock wave. Furthermore, energy is not dissipated over time. More importantly, the WFEM does not require any impact law to close the system of governing equations.

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 D 0 and a frictionless unilateral contact condition at x D 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 D 0. The unilateral contact constraint at x D 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.

In the absence of government regulations, the Federation of Canadian Municipalities (FCM) and the Railway Association of Canada (RAC) espouse criteria for the assessment of vibration due to passing trains. Given that many municipalities follow these guidelines when considering the permitting of residential development, the effectiveness of the RAC vibration assessment methods are worth investigating. These include a limit of 0.14 mm/s RMS on amplitudes and a 75 m screening distance for triggering a detailed study of the impacts of railway vibration on developments in proximity to tracks. This paper provides a review of the criteria for their reasonableness at avoiding unwanted and/or undesirable vibration conditions for residential developments. This is accomplished through reference to the generic vibration propagation curves within the latest US Federal Railway/Transit Administrations noise and vibration guidelines and alternative parametric equations for estimating vibration propagation.