Method of Normal Coordinates for Studying Forced Oscillations of Dissipative Systems in Mechanics and Electrical Engineering
A. G. Petrov1, V. A. Rumyantseva2
1Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia
2Bauman Moscow State Technical University, Moscow, Russia
Keywords: Lagrange method, quadratic forms, normal coordinates, dissipative systems, electric circuit
Abstract
The normal coordinates 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 dissipative function, which, generally speaking, cannot be reduced to a sum of squares. Conditions are considered under which all three quadratic forms are reduced to a sum of squares by a single transformation exactly or approximately. It is shown that, for such systems, normal coordinates can be introduced 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 numerical calculations to be performed for given parameter values.
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