NORMAL COORDINATE METHOD FOR STUDYING FORCED OSCILLATIONS OF DISSIPATIVE SYSTEMS IN MECHANICS AND ELECTRICAL ENGINEERING

Abstract

The normal coordinate method is used in conservative mechanical systems to reduce two quadratic forms to a sum of squares. In this case, a system of differential equations is split into a system of independent oscillators. A linear dissipative mechanical system with a finite number of degrees of freedom is determined by three quadratic forms: kinetic and potential energy of the system, as well as the Rayleigh dissipation function, which, generally speaking, cannot be reduced to a sum of squares. Conditions are considered under which all three quadratic forms are exactly or approximately reduced to a sum of squares by a single transformation. It is revealed that such systems can be supplemented with normal coordinates in which the system is split into independent second-order systems. This allows one to construct exact or approximate analytical solutions in general form and with an estimated relative error in the case of an approximate solution. The advantages of this approach are shown for problems of theoretical mechanics and electrical engineering, in which analytical solutions are constructed and optimization analysis is carried out. In this case, traditional methods allow only for numerical calculations to be performed for given parameter values.