CONTROLLED DAMPING OF TWO-FREQUENCY VIBRATIONS

The problem of controlled damping of vibrations in a single-degree-of-freedom mechanical system containing a nonlinearity of the self-oscillatory type is considered. The system is subjected to a low-frequency harmonic excitation. As a measure of intensity of the control action, we use the integral mean-square value of this action. A feedback control is constructed by the method of harmonic linearization in combination with the separation of motions into the fast and slow components in the variational methods. This control is represented in a closed form. The stability of the optimal motions and conditions for their uniqueness are analyzed. Problems of controlled damping of vibrations were previously considered in a number of works. If the system contains prescribed nonlinear terms, the most efficient technique for the control synthesis is the combination of approximate methods with optimal control methods, for example, asymptotic methods [1] or harmonic linearization [2. 3]. In the cited works and in many other publications on the controlled damping of vibrations, problems of reducing the intensity of oscillations caused by external periodic excitations are considered. The control of systems with self-oscillatory nonlinearities and reduction of the intensity of self-oscillations were studied in [4, 5]. There are classes of mechanical systems which are excited by two sorts of vibration sources-an external force and a self-oscillatory component in the system. This is the case, for example, for cutting and other sorts of machining processes. In these cases, the problem of controlled damping of vibrations is also important. In the present paper, the problem is considered for a single-degree-of-freedom mechanical system subject to a low-frequency harmonic excitation. The problem is solved on the basis of the method of harmonic linearization in combination with the separation of motions into the fast (self-oscillating) and slow (forced) components [6] and variational methods.