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In this paper we study the combined effects of bearing and rotor asymmetry on the stability of the classical Laval rotor using analytical techniques. This setting intrinsically features equations of motion with periodic coefficients. We obtain closed form approximations for the stability boundaries which give insights in the interaction of different effects which are elsewhere mostly considered in isolated form. Using a classical two degree of freedom model of a rotor we investigate dissipative and follower force type of forcing in combination with internal and external damping in the presence of both rotating and stationary asymmetries. Since in many technical applications the resulting forces are small compared to the elastic restoring forces of the corresponding symmetric system, the former are treated as perturbations. The results are benchmarked with numerical simulations indicating the range of validity of the approximations. Our model is designed to illustrate the most generic effects originating from the coupling of rotor and bearing asymmetries, which cannot be captured in models with constant coefficients. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents new results on instability of general rotating systems with small axial asymmetry and damping. Rotating systems with arbitrary finite degrees of freedom are considered. Instability conditions are derived for nonresonant and resonant rotation speeds characterized by simple and double eigenvalues. As an application, the stability problem of a rotating elastic shaft with small internal and external damping forces and asymmetric imperfections is studied. The numerical example shows validity and accuracy of the derived analytical formulae for the instability regions. [Copyright &y& Elsevier]

Abstract: The classical Chelomei''s problem of stabilization of a statically unstable elastic column by axial harmonic vibration is reconsidered. The excitation frequency is assumed to be arbitrary. Two types of boundary conditions of the column, simply supported and clamped–hinged ends, are considered. Both cases are studied analytically with check by numerical analysis. For undamped columns with the axial force close to the critical stability value, a finite number of triangular stabilization zones appears. This implies that stabilization is attained at medium frequency range (compared to the second eigenfrequency of the unexcited column). The high-frequency stabilization is impossible, as the stabilization regions vanish with the increasing excitation frequency. With addition of internal damping, a continuous stabilization region appears for small excitation amplitudes, which starts at medium frequencies and extends to high-frequency range. The influence of external damping on stabilization regions is shown to be small. [Copyright &y& Elsevier]

A linear vibrational system with multiple degrees of freedom subjected to parametric excitation is considered. It is assumed that the system is statically unstable but close to a critical point, the excitation amplitude and damping are small, and the excitation frequency is arbitrary. A new stabilization condition is derived in terms of integrals depending on eigenfrequencies and modes of the undisturbed conservative system and the symmetric excitation matrix. As a special case, an approximation for high-frequency excitation is deduced from this condition. Influence of damping on stabilization region is shown to be very small. Two examples for systems with one and two degrees of freedom are presented. It is shown that stabilization of statically unstable systems is possible for low, medium and high excitation frequencies. [Copyright &y& Elsevier]

Abstract: A linear oscillatory system with a positive definite mass matrix M, a positive definite stiffness matrix K and a positive semi-definite damping matrix D can be asymptotically stable. In this case the damping D of the system is called pervasive. The condition for the damping to be pervasive was given in a paper by Moran in 1970. Moran''s result has been reinvented in a recent paper from 2004 motivated by a long discussion on “intuitive” examples of marginal and asymptotic stability of systems with incomplete damping. The aim of the present note is to draw attention of the readers to Moran''s result. We reveal the importance of Moran''s criterion for suppression of oscillations by incomplete damping in systems with multiple eigenfrequencies. Finally, we provide extensions to the case of indefinite damping. [Copyright &y& Elsevier]

In this paper we consider the combined effect of rotating and non-rotating asymmetries on a rotor under non-conservative loading, motivated by technical applications in which breaking of symmetries may be used to avoid self-excited vibrations. We consider a rotating disk with two translatory degrees of freedom describing the displacements of the center of the disk with respect to the axis of rotation and two rotational degrees of freedom describing the tilting angles with respect to the axis of rotation. It is shown that breaking of symmetries of the rotating and non-rotating restoring terms has opposite effects for the stability behavior of translatory and rotational motions. Thus, if neither the translatory nor rotational degrees of freedom are dominant, it will be difficult to find robust designs against self-excited vibrations using the approach of breaking symmetries. [ABSTRACT FROM AUTHOR]