Optimization of oscillations of mechanical systems with friction

A generalized method for finding optimal control of the amplitude of one-dimensional oscillations in the vicinity of the equilibrium position for a scleronomous multidimensional mechanical system with friction is proposed. The oscillatory degree of freedom of the system does not lend itself to direct control. Its movement is influenced by other, directly controlled degrees of freedom. They are being chosen as the control functions. The number of directly controlled coordinates can include both positional and cyclic coordinates. The method does not use conjugate variables in the sense of the Pontryagin’s maximum principle and does not increase the dimension of the original system of differential equations of motion. The effectiveness of the proposed method is demonstrated by the example of a specific oscillatory mechanical model with dry and viscous friction.