Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
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IssuesArchive of Issues2022-5pp.1035-1043

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A.G. Petrov, "Existence of Normal Coordinates for Forced Oscillations of Linear Dissipative Systems," Mech. Solids. 57 (5), 1035-1043 (2022)
Year 2022 Volume 57 Number 5 Pages 1035-1043
DOI 10.3103/S0025654422050223
Title Existence of Normal Coordinates for Forced Oscillations of Linear Dissipative Systems
Author(s) A.G. Petrov (Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 119526 Russia,
Abstract A linear dissipative mechanical system with a finite number of degrees of freedom is defined by three quadratic forms: the kinetic and potential energy of the system, as well as the dissipative Rayleigh function. It is known that one can always introduce normal coordinates in which the kinetic and potential energies are reduced to the sum of squares with some coefficients. The third quadratic form, in general, does not reduce to a sum of squares. This study discusses the conditions under which all three quadratic forms can be reduced to a sum of squares by a single transformation. For such systems, one can introduce normal coordinates, in which the system is split into independent systems of the second order, and their analysis is greatly simplified. Examples of the analysis of forced oscillations of linear dissipative systems for two and three degrees of freedom are given.
Keywords quadratic forms, canonical form, small vibrations, friction forces
Received 18 August 2021Revised 25 October 2021Accepted 15 November 2021
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