Abstract : This contribution proposes a numerical procedure capable of performing nonsmooth modal analysis (mode shapes and corresponding frequencies) of the autonomous wave equation defined on a finite one-dimensional domain with one end subject to a Dirichlet condition and the other end subject to a frictionless time-independent unilateral contact condition. Nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the equation of the dynamics together with the unilateral contact condition. The analysis is achieved using the Wave Finite Element Method, which is a shock-capturing Finite Volume method. As opposed to the traditional Finite Element Method with time-stepping schemes, potentially discontinuous deformation, stress and velocity wave fronts induced by the contact condition are here accurately described, which is critical for seeking periodic orbits. Additionally, the proposed strategy neither introduces numerical dispersion nor artificial dissipation of energy, as required for modal analysis. As a consequence of the mixed time–space discretization, no impact law is required for the well-posedness of the problem in line with the continuous framework. The frequency–energy dependency—shown as backbone curves—provides valuable insight of the system where intricate dynamics such as internal resonance or subharmonic vibration emerges. In contrast to the linear system (without the unilateral contact condition) whose modes are standing harmonic waves, the found autonomous periodic solutions are traveling waves induced by the contact constraints. It is also shown that the periodically-forced vibratory resonances of the system with light structural damping are well predicted by nonsmooth modal analysis. As expected, the open-gap and prestressed configurations at rest show stiffening and softening behaviors respectively.
Are shown below a few of the periodic solutions of interest. The red squares indicate periodicity in time. Dark green stands for large positive displacements while white is for negative displacements. The color scale varies with the animations: the same dark green in distinct animations corresponds to different maximum magnitudes.
First two linear modes of the bar along the grazing trajectory. Unilateral contact conditions are not activated (click right -> view image for optimal quality)
at frequency $\omega_1$
at frequency $\omega_2$
First two nonsmooth modes of the bar. Unilateral contact conditions are activated once per period (click right -> view image for optimal quality)
around frequency $\omega_1$
around frequency $\omega_2$
Purely nonlinear orbits (with no linear counterparts). Unilateral contact conditions are activated once per period but with grazing instants as well (click right -> view image for optimal quality)