Search

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]

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler's paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form. [ABSTRACT FROM AUTHOR]

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]

The article discusses the aspect of the study regarding the paradoxes in the dissipation-induced destabilization or who opened Whitney's umbrella conducted by Oene Bottema. It mentions that the study with damping was compared with the domain of marginal stability and asymptotic stability depending on its characteristic polynomials. It states that he resolved it through the singularity theory on the boundary of two-degrees-of freedom non-conservative systems including the Ziegler's pendulum.