Motion of a rectangular bar under traction on a horizontal plane

Abstract

This paper forms part of a study into how components move when pushed or pulled. It extends previous work by other authors, who have studied quasi-static pushing, by addressing the dynamics of pulling. In particular, dynamic models are developed for a rectangular bar lying flat on a horizontal planar support and acted upon by a tractive force. Initially, the equations of motion of the bar, a set of six coupled non-linear first order ODEs, are derived by ignoring friction between the bar and the support. The results of numerically integrating the equations to find the bar's trajectory are presented. The integration was implemented using the Bulirsch-Stoer Method. The numerical results show good agreement with closed-form solutions obtained by linearising the equations of motion for small angular displacements. The problem of rough support surfaces is then treated. Expressions for the frictional force and moment exerted on the bar are found in terms of R and α, the polar coordinates of the centre of mass (C.M.) of the bar relative to its instantaneous centre of rotation (C.O.R.). These expressions are used to construct a new set of equations of motion. Two methods of obtaining the trajectory of the bar are described. The first involves direct numerical integration of the equations of motion, as in the case of frictionless supports. The second is based on explicitly searching for R and α using the equations of motion.