Optimal reduction ratio minimizing dissipated energy in a servo system with gear

In order to prevent global warming, it is important to minimize the dissipated energy of machines. This present paper discusses a method for a reduction ratio of gear trains in a mechanical servo system with friction of all types to minimize the dissipated energy. The Coulomb friction of a gear in the servo system is represented by an efficiency of the gear. Even if the system is nonlinear due to Coulomb friction, an analytical optimal velocity function can be obtained by introducing a zero crossing time t_c, that is when an input torque of the gear changes from positive to negative. Since the velocity function is dependent on the system parameters, the moment of inertia and the viscous friction of gear trains are expressed by the reduction ratio n. The dissipated energy due to the optimal velocity function is partially differentiated with respect to the zero crossing time tc and the reduction ratio n. The resultant optimal reduction ratio is compared to a ratio given by a conventional inertia matching method. The results of the simulations indicate that the optimal reduction ratio can greatly reduce the amount of dissipated energy.