Motion Equations for the Ball and Beam and the Ball and Arc Systems

Presently, most derivations of the equations of motion for the ball and beam and ball and arc systems model the ball as a point mass. Other derivations apply assumptions related to the ball that lack physical justification. To understand fully the impact the ball has on the equations of motion and controller design, equations of motion are needed that stem from a derivation invoking few assumptions. At that point, simplifying assumptions applied in a consistent manner are possible. This paper derives the equations of motion for each mechanical system using fewer assumptions than what appears in the literature. The development then applies the assumptions needed to convert the derived dynamics to the well-used equations of motion seen in the literature. The presentation shows that some ball and beam models appearing in the literature do not stem from consistent assumptions or correct kinematics. Incorrect kinematics are also present in some of the ball and arc dynamic models. This paper also introduces dimensionless quantities to the dynamic equations of both systems rendering them in dimensionless form. Such formulations allow for comparisons between different systems through the size of the dimensionless quantities. The dimensionless systems are the means used to demonstrate that the equations of motion of each system are the same as the radius of the arc grows without bound. Finally, the paper shows that comparisons between different ball and beam models using these dimensionless numbers are now possible.