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.     

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.     

Stability of nonconservative systems and applications

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

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.     

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)     

Der Einfluß verschiedener Kräftearten auf die Stabilität linearer Systeme

Summary Several general theorems concerning the stability of dynamical systems have been published during the last decades. Starting with a classification of such systems an attempt is made to define the range of validity for these known theorems. Emphasis is laid upon the effect of different forces, as for example velocity-forces (gyroscopic as well as damping or exciting ones) and of positional forces (potential as well as circulatory nonconservative ones). The text of 20 theorems is given.

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Herrn Professor Dr. Hans Ziegler zum sechzigsten Geburtstag gewidmet     

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.     

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.     

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/.     

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)     

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 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.     

On gyroscopic stabilization

The mechanisms of transition between divergence, flutter, and stability for a class of conservative gyroscopic systems with parameters are studied. Two results are obtained which state sufficient conditions for gyroscopic stabilization of conservative systems with an even dimension and a negative definite stiffness matrix. A number of examples are given to demonstrate the feasibility of the results.The work of this author was supported in part by The Danish Technical Research Council through the programme on Computer Aided Engineering Design and by The Danish Natural Science Research Council through the programme on Differential Equations.     

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.     

Gyroscopic Stabilization of 2nd-Order-Systems with Indefinite Damping

We consider the gyroscopic stabilization of the unstable system ẍ + D ẋ + Kx = 0 with positive definite stiffness matrix K. The indefinite damping matrix D is responsible for the instability of the system. The modelling of sliding bearings can lead to negative damping, see [6]. A gyroscopic stabilization of an unstable mechanical system with indefinite damping matrix was investigated in [4] in the case of matrix order n = 2 using the Routh-Hurwitz criterion. The question was raised whether an unstable system can be stabilized by adding a gyroscopic term Gẋ with a suitable skew-symmetric matrix G = −G^T . (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Controllability and observability in the problem of stabilizing steady motions of non-holonomic mechanical systems with cyclic coordinatest

The approach to the solution of stabilization problems for steady motions of holonomic mechanical systems [1, 2] based on linear control theory, combined with the theory of critical cases of stability theory, is used to solve the analogous problems for non-holonomic systems. It is assumed that the control forces may affect both cyclic and positional coordinates, where the number r of independent control inputs may be considerably less than the number n of degrees of freedom of the system, unlike in many other studies (see, e.g., [3–5]), in which as a rule r = n. Several effective new criteria of controllability and observability are formulated, based on reducing the problem to a problem of less dimension. Stability analysis is carried out for the trivial solution of the complete non-linear system, closed by a selected control. This analysis is a necessary step in solving the stabilization problem for steady motion of a non-holonomic system (unlike holonomic systems), since in most cases such a system is not completely controllable.     

Structural Analysis of One Class of Dynamical Systems

We develop the method of structural transformations of dynamical systems (proposed earlier by Koshlyakov) for systems containing nonconservative positional structures. The method under consideration is based on structural transformations that enable one to eliminate nonconservative positional terms from the original system without changing its stability properties.     

Asymptotic properties of the solutions of the equations of motion of gyroscopic systems

The motion of mechanical systems acted upon by gyroscopic and positional forces characterized by a large parameter in the corresponding equations of motion is considered. Periodic solutions of such equations were investigated earlier in [1, 2]. It is proved below that solutions of these equations exist, defined in an interval the length of which is a monotonically increasing unbounded function of the large parameter, and which transfer into the solutions of the corresponding degenerate systems as the large parameter approaches infinity. This function can be specified in more detail if additional assumptions are made regarding the properties of the system and the nature of the forces acting on it.     

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.