Mathematical Model of the Mechanics and Dynamics of the Tails in Dinosaurs

Abstract

In this paper, we investigate the effect of tails in the locomotion of dinosaurs. We begin with a survey of physical data for both bipedal and quadrupedal dinosaurs. The ratio of the tail to leg length is compared across many diverse species and a scaling law is developed that relates the tail length, leg length and tail radius. A continuous model for the tail is then developed, and by nondimensionalising a small parameter related to the thinness of the tail simplifies the resulting coupled nonlinear equations. It is shown that the resulting set of equations contain aspects of both beam dynamics and wave propagation. Finally, a discrete version of the tail is derived with the assumption that the sections of the tail are coupled with a stiff joint that allows rotation but does not allow extension. In this model the stiffness of each joint is characterized by an effective spring constant for the th joint and results in a discrete version of the Euler-Bernoulli expression for each of the tail segments. The paper finishes with some preliminary conclusions and directions for future work.