We study the impact of a Hertzian sphere on an infinite Rayleigh beam, which presents both interesting possibilities for numerical approximation as well as new insights in the multiple-contact-event regime. Here, Galerkin projection using an abstract continuous variable replaces the history-dependent convolution in the impact equations by a system of local-in-time ordinary differential equations. The corresponding approximate solutions are validated against direct numerical solution of the convolution-based formulation and show agreement. Unlike the corresponding Euler-Bernoulli beam impact problem, which is governed by a single nondimensional parameter, the Rayleigh beam impact dynamics is governed by two nondimensional parameters. A parametric study reveals three distinct impact regimes: one regime with a single contact followed by permanent separation, a second regime with a single contact that is sustained thereafter, and an intervening regime with initial separation followed by multiple contact events with diminishing amplitudes, leading eventually to sustained contact. Short- and long-time asymptotic analyses are used to characterize the boundaries between the permanent separation and the re-contact regimes in different parameter regimes. The three significant new aspects of this paper are the system itself (Rayleigh beam instead of Euler Bernoulli beam), the model order reduction achieved through Galerkin projection in an abstract domain, and the detailed parametric study leading to a simple qualitative picture of possible behaviours.
Here, we study an undamped Rayleigh beam subjected to a concentrated transverse impulse applied at a point far from the boundaries. For a simply supported finite beam struck at its midpoint, using the modal expansion of the impulse response, a short-time asymptotic series for the response of the struck point is derived. The infinite beam impulse response is obtained as a spectral integral in two ways. The spectral integral is evaluated exactly in terms of modified Bessel functions via Laplace transform calculations. The corresponding short- and long-time asymptotics of the exact integral are consistent with the power series developed earlier and the Euler-Bernoulli beam, respectively. Direct numerical evaluation of the integral with a pre-specified error bound and numerical contour integration is also presented for verification. An ad hoc approximation is given which matches the response well for all times. The relationship between finite and infinite beam responses is examined in some detail. There are three consequences of finite length. First, the coefficients in the power series expansion contain components that decay exponentially with increasing length of the beam. Second, a smooth deviation from the infinite beam response is observed, consistent with the greatest group velocity in the beam, wherein a small precursor is detected which is consistent with broader theory. Third, a slope discontinuity is observed, corresponding to the high-frequency limit of group velocity. The full time series plot from modal expansion satisfactorily shows both the arrival and passage of these high frequency components. This paper opens the door to impact studies that are envisaged in future work.