Error bounds in computational dynamics of rolling half-disk problems: A comparative analysis of integration methods

Modelling and integration of the transient response of freestanding bowl- and potshaped rigid objects under dynamic excitations, like bowls and vases subjected to seismic and environmental actions, pose applied and pure research problems that hold considerable interest in the protection of museum collections and archeological findings. The half disk model with its smooth convex boundary exempt from flat edges at the bottom, though of a simple archetypal nature, deserves consideration for modelling more elaborate bowl-like objects since it allows to circumvent the nontrivial problem of addressing the singularity arising in dynamic balance equations at time instants associated with the impact of flat edges, typically present in vases, while remaining rooted on canonical principles of Lagrangian Mechanics and on ordinary empirical postulates for sliding, rolling and impacting bodies. Accordingly, insights can be gained from the rolling half disk on the response of freestanding objects without resorting to additional heuristic postulates like angular momentum conservation at time instants of impact of flat edges, as proposed by some authors. To properly discern physical dissipation from numerical dissipation in computational analyses, however, a proper investigation of algorithmic damping ratios and computational accuracy becomes mandatory. This contribution investigates the numerical stability and accuracy of the integration schemes employed for the half disk model, bringing a first comparison, based upon spectral radii analysis, among collocation methods, and setting a base for a forthcoming comparison with methods belonging to the Newmark family and with other widely employed approaches like linear and higher order multistep methods.