Application of a PD controller on two mating gears with elasticity and backlash

The paper deals with the problems related to the mechanical gears, which are commonly used for manipulators and servomechanisms, when elasticity and backlash affect their connection. The dynamic model of two mating gears is determined in three different cases: in the ideal case, in which the gears are rigid and no backlash is present; when the gears are flexible, in absence of backlash; in a realistic case, in which both the elasticity of the gears and the presence of backlash are to be taken into account. The same control algorithm, a PD one, is then applied to the three different cases, under the assumption that only the position and the velocity of the driving gear are available for feedback. The properties of the solutions of the closed-loop systems thus obtained are studied and analytically proven in the paper. Some simulation results complete and confirm the analysis.