Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stability of non-conservative systems and an estimate of the domain of attraction

The stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces act, is investigated. The condition for asymptotic stability is obtained using the Lyapunov function and an estimate of the domain of attraction is also found in terms of the system being considered. A precessional system is also examined. It is shown that the condition for the asymptotic stability of a system is the condition of acceptability in the sense of the stability of a precessional system. The results obtained are applied to the problem of the stabilization, using external moments, of the steady motion of a balanced gyroscope in gimbals.     

The stability and stabilization of non-linear,non-stationary mechanical systems

Mechanical systems acted upon by extremely non-linear positional forces are considered. The decomposition method is used to determine the sufficient conditions for asymptotic stability of an equilibrium. Problems of stabilizing the equilibrium of non-linear, non-stationary systems with specified potential forces by adding forces of different structure are studied. For systems with a non-stationary, homogeneous, positive-definite potential, the possibility of stabilization by linear dissipative forces, uncharacteristic of linear systems, is established. For systems with an even number of coordinates n ≥ 4, in the presence of dissipative forces with complete dissipation, the possibility of vibrational stabilization by adding circular and gyroscopic forces with coefficients fluctuating about zero is demonstrated.     

Gyroscopic stabilization of nonconservative systems

The problem is considered of the stabilization of a mechanical system having only nonconservative positional forces by adding gyroscopic forces. The gyroscopic stabilization is proved to be always realizable in the case of a degenerate matrix of nonconservative forces and even number of coordinates. If the matrix of nonconservative forces is nonsingular then a possibility of the gyroscopic stabilization is established for all systems whose number of coordinates is divisible by four. For a nonautonomous systemwith nonconservative positional forces and dissipative forces with complete dissipation, some sufficient conditions are obtained for stabilization up to the exponential stability by addition of gyroscopic forces.     

The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

The exponential stability of the unperturbed motion of a non-autonomous mechanical system with a complete set of forces, that is, dissipative, gyroscopic, potential and non-conservative positional forces, is investigated. The problem of stabilizing a non-autonomous system with specified non-conservative forces is considered with and without the use of potential forces. The problem of stabilizing a non-autonomous system with specified potential forces by the action of the forces of another structure is studied. The domain of stabilizability of the relative equilibrium position of a satellite in a circular orbit is found.     

Mechanical systems,equivalent in Lyapunov's sense to systems not containing non-conservative positional forces

Developing results obtained previously (Refs. Koshlyakov VN. Structural transformations of the equations of perturbed motion of a certain class of dynamical systems. Ukr Mat Zh 1997; 49 (4): 535–539; Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. Prikl Mat Mekh 1997; 61 (5): 774–780; Koshlyakov VN, Makarov VL. The theory of gyroscopic systems with non-conservative forces. Prikl Mat Mekh 2001; 65 (4): 698–704; Koshlyakov VN, Makarov VL. The stability of non-conservative systems with degenerate matrices of dissipative forces. Prikl Mat Mekh 2004; 68 (6): 906–913), the general problem of eliminating non-conservative positional structures from the second-order differential equation with constant matrix coefficients, obtained when modelling many mechanical systems, is considered. It is assumed that the matrices of the dissipative and non-conservative positional structures may, in particular, be degenerate. Under fairly general assumptions, theorems containing the necessary and sufficient conditions for a Lyapunov transformation to exist are proved. This converts the initial matrix equation to an equivalent autonomous form (in Lyapunov's sense) with a symmetrical matrix of the positional forces. An illustrative example is considered.     

Stability of nonconservative systems and applications

The paper studies the stability of mechanical systems subjected to dissipative, gyroscopic, and nonconservative positional forces.     

The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems

It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear time-varying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated.     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

Stabilization of the nonautonomous potential systems with the forces of a dissimilar structure

The problem is considered of stabilizing a nonautonomous system given potential forces by adding some dissipative, gyroscopic, and nonconservative positional forces. The stabilizability domain is found for the relative equilibrium of a satellite in the circular orbit.     

On the influence of gyroscopic forces on the stability of steady-state motion : PMM vol. 39, n 6, 1975, pp. 963–973

The question of the influence of gyroscopic forces on the stability of steady-state motion of a holonomic mechanical system when the forces depend upon the velocities of only the position coordinates was answered by the Kelvin-Chetaev theorems [1] on the influence of gyroscopic and dissipative forces on the stability of equilibrium. However, if the gyroscopic forces depend as well on the velocities of the ignorable coordinates, then their influence on the stability of steady-state motions can, as the two problems in [2] show, prove to be entirely different from the influence of gyroscopic forces depending only on the velocities of the position coordinates. In this paper we investigate the influence of gyroscopic forces depending linearly on the velocities of the generalized coordinates, including the ignorable ones, on the stability of the steady-state motion of a holonomic conservative system. We prove that when the gyroscopic forces applied with respect to the ignorable coordinates are given as total time derivatives of certain functions of the position coordinates, the gyroscopic forces can both stabilize as well as destabilize the steady-state motion. Under certain conditions, this influence is also preserved for the action of dissipative forces depending on the velocities of only the position coordinates. In the case of action of dissipative forces depending also on the velocities of the ignorable coordinates, we have indicated the stability and instability conditions of the steady-state motion. Examples are considered. In conclusion, we discuss the conditions under which the application of gyroscopic forces to the system is equivalent to adding terms depending linearly on the generalized velocities to the Lagrange function.     

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.     

非保守线性陀螺系统的稳定性*

本文研究具有势力,陀螺力、循环力和瑞利阻尼的非保守线性力学系统的稳定性。借助于瑞利商证明了三个稳定性定理,这些定理给出的稳定性判据不依赖于瑞利商,因此方便实用。     

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.     

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 the steady-state motions of systems with quasicyclic coordinates

The stability of the steady-state motions of a system with quasicyclic coordinates under the action of potential and dissipative forces and also forces which depend on the quasicyclic velocities is investigated. The results are applied to the problem of the stability of the steadystate plane-parallel motions of a rotor on a shaft which is set up in elasticated bearings with a non-linear reaction /1/.

The stability of the stationary motions and relative equilibria of systems with a single cyclic (quasicyclic) coordinate has previously been investigated /2/ from a common point of view. The question of the stability of the stationary motions of systems with quasicyclic coordinates under the action of constant and dissipative forces has been considered in /3/. The results obtained in /2/ have been generalized /4/ to systems with several cyclic (quasicyclic) coordinates and, additionally, a third regime of uniform motions, which includes the regime considered in /3/, has also been investigated.     

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.     

Stabilizing and destabilizing effect of small velocity‐dependent forceson non‐conservative systems: new results

A theory of the destabilization paradox in general non‐conservative systems with small dissipative and gyroscopic forces is presented. The problem is investigated by the approach based on the sensitivity analysis of multiple eigenvalues. An explicit asymptotic expression for the critical load as a function of the dissipation and gyroscopic parameters allowing to calculate a jump in the critical load is derived. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dissipation-induced subcritical flutter in the acoustics of friction

We consider a gyroscopic system under the action of small dissipative and non–conservative positional forces, which has its origin in the models of rotating elastic bodies of revolution in frictional contact such as the singing wine glass or the squealing disc/drum brakes. The spectrum of the unperturbed gyroscopic system forms a spectral mesh in the plane ‘frequency versus gyroscopic parameter’ with double semi–simple purely imaginary eigenvalues at the nodes. In the subcritical range of the gyroscopic parameter the eigenvalues involved into the crossings have the same Krein signature and thus their splitting due to changes in the stiffness matrix, which break the rotational symmetry of the body, cannot produce complex eigenvalues and, therefore, flutter. We establish that perturbation of the gyroscopic system by the dissipative forces with the indefinite matrix can lead to the subcritical flutter instability even if the rotational symmetry is destroyed. With the use of the perturbation theory of multiple eigenvalues we explicitly find the linear approximation to the domain of the subcritical flutter, which turns out to have a conical shape. The orientation of the cone in the three dimensional space of the parameters, corresponding to gyroscopic, damping, and potential forces, is determined by the sign of an explicit expression involving the entries of both the damping and potential matrices. With the use of a time–dependent coordinate transformation we demonstrate that the conical zones of flutter for the original autonomous system coincide with the zones of the subcritical parametric resonance of the rotationally symmetric flexible body with the load moving in the circumferential direction. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attitude stabilization of a rigid body in conditions of decreasing dissipation

The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the most complicated one in the problem of stability analysis of mechanical systems with a nonstationary parameter at the vector of dissipative forces. The lemma of the estimate from below for the norm of the restoring torque in the neighborhood of the stabilized motion of a rigid body and two theorems on asymptotic stability of the stabilized motion of a body are proven. It is shown that the sufficient conditions of asymptotic stability found in the theorems are close to the necessary ones. The results of numerical simulation illustrating the conclusions obtained in this study are presented.