Structural transformations of non-conservative systems

The method of structural mappings of gyroscopic systems [1, 2] is developed for systems involving non-conservative positional forces. This technique, considered in the aspect of the legitimate use of the precessional equations of the precessional equations of the applied theory of gyroscopes, enables the difficulties associated with the presence of non-conservative structures in the initial equations to be overcome, and in many cases enables of the Thomson—Tait—Chetayev theorems to be used directly.     

Indefinite damping in mechanical systems and gyroscopic stabilization

This paper deals with gyroscopic stabilization of the unstable system Mẍ + Dẋ + Kx = 0, with positive definite mass and stiffness matrices M and K, respectively, and an indefinite damping matrix D. The main question is for which skew-symmetric matrices G the system Mẍ + (D + G)ẋ + Kx = 0 can become stable? After investigating special cases we find an appropriate solution of the Lyapunov matrix equation for the general case. Examples show the deviation of the stability limit found by the Lyapunov method from the exact value.      

Dissipation induced instabilities in continuous non-conservative systems

In this contribution we analyze the stabilizing and destabilizing effect of small damping for rather general class of continuous non-conservative systems, described by the non-self-adjoint boundary eigenvalue problems. Explicit asymptotic expressions obtained for the stability domain allow us to predict when a given combination of the damping parameters leads to increase or to decrease in the critical non-conservative load. The results obtained explain why different types of internal and external damping so surprisingly influence on the stability of non-conservative systems. As a mechanical example the stability of a viscoelastic rod with small internal and external damping, loaded by tangential follower force, is studied in detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lowering the order of gyroscopic systems

A method for introducing a small parameter into the equations of gyroscopic systems is proposed. It is shown that to solve the problem of the admissibility of simplifications into the equations of gyroscopic systems, one can use results from the investigation of differential equations with a small parameter for a higher derivative.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 381–389, March, 1991.     

Lowering the order of gyroscopic systems

A method for introducing a small parameter into the equations of gyroscopic systems is proposed. It is shown that to solve the problem of the admissibility of simplifications into the equations of gyroscopic systems, one can use results from the investigation of differential equations with a small parameter for a higher derivative.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 381–389, March, 1991.     

Bundles and geometric structures associated with gyroscopic systems

The overview is devoted to topological and geometric structures associated with gyroscopic systems whose action functional S is multivalued. The expediency of constructing and studying them is, in particular, stipulated by the fact that the standard methods of the calculus of variations in the problem with fixed endpoints are not effective for such functionals. One of the methods for overcoming the difficulties arising here is the application of bundles, foliations, connections, and also Riemannian and Lorentz manifolds. In this way, it turns out to be possible to perform the reduction of the two-point problem for S to problems with fixed initial point and movable endpoint for the length functional {ie828-01} of a pseudo-Riemannian manifold foliated over the configurational space of the gyroscopic system considered. As the endpoint manifolds, the leaves of the Riemannian foliation are used, and the correspondence between the extremals of the functionals S and {ie828-02} is stated by using the Ehresmann connection of this bundle. The paper discusses the results on the motions of natural mechanical systems with gyroscopic forces and gyroscopic systems of relativistic type obtained by using the above reduction and also the topological and geometric constructions used in it. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.     

Continuum-wise expansiveness for non-conservative or conservative systems

In this paper, we show that a non-conservative vector field is robustly continuum-wise expansive if and only if it satisfies both Axiom A and the quasi-transversality condition. Moreover, a conservative vector field (divergence-free vector field, Hamiltonian system) is robustly continuum-wise expansive if and only if it is Anosov.     

On the gyroscopic stabilization of conservative systems

We consider conservative systems with gyroscopic forces and prove theorems on stability and instability of equilibrium states for such systems. These theorems can be regarded as a generalization of the Kelvin theorem to nonlinear systems.     

The stability of linear potential gyroscopic systems

The stability of linear potential systems with a degenerate matrix of gyroscopic forces is investigated. Particular attention is devoted to the case of three degrees of freedom. In a development of existing results [Kozlov VV. Gyroscopic stabilization and parametric resonance. Prikl. Mat. Mekh. 2001; 65(5): 739–745], the sufficient conditions for gyroscopic stability are obtained. An algorithm for applying these conditions is proposed using the example of the problem of the motion of two mutually gravitating bodies, each of them being modelled by two equal point masses, connected by weightless inextensible rods.     

Suboptimal feedback control of linear gyroscopic systems

A simplified model of reduced dimensionality is presented for a class of linear gyroscopic systems with quadratic performance indices. This model is based on the concept of weakly coupled subsystems and can be used in the synthesis of suboptimal controllers. Controllers based on this model compare favorably with both optimal and conventional controllers.This research was supported by NSF Grant No. GK-3273 and a grant from the Graduate School of the University of Minnesota.     

Multiple solutions for nonlinear systems with gyroscopic terms

In this paper, we investigate the existence of multiple solutions to some nonlinear systems with gyroscopic terms via variational methods. Some new results are obtained and some results from the literature are improved.     

Structure preserving eigenvalue embedding for undamped gyroscopic systems

This paper concerns the eigenvalue embedding problem of undamped gyroscopic systems. Based on a low-rank correction form, the approach moves the unwanted eigenvalues to desired values and the remaining large number eigenvalues and eigenvectors of the original system do not change. In addition, the symmetric structure of mass and stiffness matrices and the skew-symmetric structure of gyroscopic matrix are all preserved. By utilizing the freedom of the eigenvectors, an expression of parameterized solutions to the eigenvalue embedding problem is derived. Finally, a minimum modification algorithm is proposed to solve the eigenvalue embedding problem. Numerical examples are given to show the application of the proposed method.     

Admissibility of simplified equations in the dynamics of gyroscopic systems

Based on the asymptotic approach of /1/, rigorous mathematical methods are used to single out some known simplified models from the theory of gyroscopic systems and to prove that they may legitimately be employed to solve problems in dynamics (including stability problems). The initial system is of the singularly perturbed type /2/. The use of methods from stability theory /3, 4/ yields conditions under which transition to a simplified (computational) model is permissible. Several papers have been devoted to the solution of such problems for singularly perturbed equations /5/ by methods of Lyapunov theory.     

Asymptotic solutions of Lagrangian systems with gyroscopic forces

We consider Lagrangian systems in the presence of nondegenerate gyroscopic forces. The problem of stability of a degenerate equilibrium pointO and the existence of asymptotic solutions is studied. In particular we show that nondegenerate gyroscopic forces in general have, at least formally, a stabilizing effect whenO is a strict maximum point of the potential energy. It turns out that when we switch on arbitrary small nondegenerate gyroscopic forces, a bifurcation phenomenon arises: the instability properties ofO are transferred to a compact invariant set which collapses atO when the gyroscopic forces are switched off.Work supported by Russian Fund of Basic Research, the Italian Research Council (CNR) and the Italian Ministery of University (MURST)     

On the stability of linear conservative gyroscopic systems

A simple criterion concerning the stability of linear gyroscopic conservative systems is developed. First, a sufficient condition for stability is established. The proof is based on a transformation converting the original autonomous system into a nonautonomous system, and applying Liapunov's direct method to the latter. Then a theorem is given which provides a necessary and sufficient condition for a restricted class of systems. Illustrative examples are provided.