Dynamics of a three-module vibration-driven system with non-symmetric Coulomb’s dry friction

Abstract

In the present paper, a three-module vibration-driven system moving on a rough horizontal plane is modeled to investigate the relation among the system’s steady-state motion, external Coulomb’s dry friction force and internal excitations. Each module of the system represents a vibration-driven system composed of a rigid body and a movable internal mass. Major attention is focused on the primary resonance situation that the excitation frequency is close to the first-order natural frequency of the system. In the case that the external friction is low, the internal excitation is weak and the stick–slip motion is negligible, both methods of averaging and modal superposition are employed to study the steady-state motion of the system. Through a set of algebraic equations, an approximate value of the system’s average steady-state velocity is obtained. Several numerical examples are calculated to verify the validity of the analytical results both qualitatively and quantitatively. It is seen that big quantitative errors will appear if stick–slip motions occur. Then, two mechanisms for the possible stick–slip motions are put forward, which explain the errors on the average steady-state velocity. Numerical simulations verify our analysis on the stick–slip effects and their mechanisms. Finally, to maximize the average steady-state velocity of the system, optimal control problem is studied. It is shown that, in addition to modifying the friction coefficients, the improvement of the system’s efficiency can be provided by changing the initial phase shifts among the three internal excitations.