| Abstract: | A large body of recent literature has been devoted to the topic of motions of mechanical systems forced by oscillatory inputs (see, for example, References 7, 12, 15, 17 and 29). A common feature in most of this work (with Reference 7 being the exception) is that the results apply principally to kinematic problems, and the equations of motion do not involve drift terms. The main object of study in this paper is the class of bilinear control systems with oscillatory (periodic) inputs. We shall show that included in this class are models of mechanical system dynamics which are obtained through a process of reduction and linearization about operating points. In particular, we study the stability of such systems under high-frequency, periodic forcing. The averaged potential is defined for linearizations of the systems on the (reduced) configuration space, and it is shown that stable motions of the forced system are associated with minimum values of this quantity. We use both classical averaging theory as well as a novel geometric argument to provide parallel but independent assessments of the use of the averaged potential in carrying out a stability analysis. A salient feature of the geometric approach is that we are able to justify the use of an energy-like quantity to determine Lyapunov stability in conservative mechanical systems. This makes contact with a growing body of literature on the use of energy methods for stability and control design (e.g. References 7, 8, 18, 24, 26 and 27). We briefly describe the connection with previous work on the averaged potential. |
