Energy Conserving Equations of Motion for Gear Systems

A system of two meshing gears exhibits a stiffness that varies with the number of teeth in instantaneous contact and the location of the corresponding contact points. A classical Newtonian statement of the equations of motion leads to a solution that contradicts the fundamental principle of mechanics that the change in total energy in the system is equal to the work done by the external forces, unless the deformation of the teeth is taken into account in defining the direction of the instantaneous tooth interaction force. This paradox is avoided by using a Lagrange’s equations to derive the equations of motion, thus ensuring conservation of energy. This introduces nonlinear terms that are absent in the classical equations of motion. In particular, the step change in stiffness associated with the introduction of an additional tooth to contact implies a step change in strain energy and hence a corresponding step change in kinetic energy and rotational speed. The effect of these additional terms is examined by dynamic simulation, using a system of two involute spur gears as an example. It is shown that the two systems of equations give similar predictions at high rotational speeds, but they differ considerably at lower speeds. The results have implications for gear design, particularly for low speed gear sets.