A Gaussian sinc-collocation approach for a whipping cantilever with a follower shear force at its tip

A spatial discretization scheme, based on a set of Gaussian sinc functions, is proposed for the temporal projection of a set of partial di erential equations (PDEs), describing the non-linear dynamics of an elastic–plastic hardening–softening (EPHS) cantilever, subjected to a follower shear force at its tip. The dynamics so described correspond to planar whipping of a pipe conveying uid, ruptured near a rightangled bend. The constitutive EPHS moment curvature relationship used here follows the earlier work of Reid et al. (Proceedings of the Royal Society of London, Series A 1998; 454:997–1029). Compared to the more classical Lagrangian polynomial-based collocation functions, the Gaussian sinc functions have better localization properties. Moreover, for a relatively large number of collocation points, use of such functions does not lead to numerical over ow or under ow problems, often associated with the use of higher order polynomial collocation functions. The spatial discretization leads to a set of non-linear ordinary di erential equations (ODEs) in time, which are in turn integrated via a fourth order adaptive Runge–Kutta scheme. Some numerical results for a cantilever whipping in a plane are presented to further illustrate the present approach. The method is a step forward towards the development of a mesh-free non-linear beam element, suitable for dynamic analyses of pipe networks and pipe-on-pipe impact problems