New York: Consultants Bureau, 1964. — 130 p. — ISBN 978-I-4684-9077-0.One of the most difficult problems in the theory of nonlinear vibrations is the problem of quasi-periodic oscillations. The methods of Poincare or Lyapunov are not applicable to the solution of this problem. The asymptotic method proves to be most effective in this case. Fundamental results in the investigation of quasi-periodlc oscillations by the asymptotic method have been obtained primarily by N. U. Krylov and N. N. Bogolyubov. The present monograph discusses the particular problem, not fully investigated previously of obtaining quasi-periodic solutions of quasi-linear multiple-degree-of-freedom systems when there are several resonance relations between the exciting and natural frequencies. The method for obtaining stationary solutions for this special case is developed in Chapter III and may be considered to be a kind of key to the study of self-induced oscillations of unbalanced rotors. The dynamics of a rotor that has lost stability is described by nonlinear equations and is rather complicated. However, the study of the dynamics is an interesting one in many respects. It can lead, for example, to an estimate of the magnitude of the damping forces required to diminish the amplitudes of the limit cyles to safe values, if not to exclude the loss of stability absolutely. The author did not attempt to discuss the many different factors causing loss of stability of rotors. Only self-induced oscillations produced at supercritical speeds by the action of internal friction are discussed in detail in the present monograph. The theoretical results are compared with experimental data on self-induced oscillations of some modern high-speed spindies observed when the dampers were not fully effective. Films of the path of an oscillating spindle were taken in the experiments with a motion picture camera through a microscope. The author hopes that the methods discussed can be applied to the study of self-induced oscillations of rotors caused by various nonconservative forces, such as external friction or hydrodynamic forces. A slight variation of the method makes it possible to study transient vibrations leading to steady self-induced oscillations; the duration of the process is a matter of interest in failure prevention. In the author's opinion these are promising directions of further studies of self-induced vibrations.
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