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.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

The influence of dissipative and constant forces on the form and stability of steady motions of mechanical systems with cyclic coordinates

Mechanical systems with cyclic coordinates subject to dissipative forces with complete dissipation and constant forces applied only to the cyclic variables are considered. Problems of the existence of steady motions in such systems and the conditions for their stability are discussed. It is shown, in particular, that if the Rayleigh function is proportional to the kinetic energy, the stability conditions for the steady motions of the system are the same as or (under certain assumptions) similar to such conditions for steady motions of a corresponding conservative system. The example of a physical pendulum is used to show that such conclusions are generally false: dissipative and constant forces may cause destabilization of stable motions of the system.     

Stabilization of some nonlinear controlled electrohydraulic servosystems

For control systems that model electro-hydraulic servo-actuators, the possibility of stabilization by a linear stationary feedback low is analysed. It is also shown that, when this is possible, equilibria exhibit also asymptotic stability with respect to some relevant state variables.     

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 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.     

Lyapunov functions of the mechanical energy type

When examining the properties of the stability and asymptotic behaviour of a system a Lyapunov function is often used as the total mechanical energy of the system /1–7/. By analogy with the division of the energy into kinetic and potential energy, it is proposed below to construct a Lyapunov function in the form of the sum of two subsidiary scalar functions, such that its derivative on account of the system is estimated using some kind of function of these subsidiary functions. Generalizing the results /8/, we examine the case when the derivative of the Lyapunov function can also take positive values, and the equation of comparison the emerges from the estimate of the Lyapunov function does not permit a separation of variables. V.V. Rumyantsev's theorem /3/ on the asymptotic stability with respect to the velocities of the equilibrium position of a dissipative mechanical system is generalized on the basis of the results obtained.     

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.     

On the approximation of positional control problems

Conditions for the approximation of positional control problems for contrallable systems of sufficiently general form are discussed, covering, in particular, certain classes of distributed-parameter plants. The constructions are based on the results in /1–6/. Analogous questions for programmed control problems were examined, for instance, in /7–14/, and for positional control problems, in /1, 6/. In contrast to the approximation scheme proposed by Krasovskii for general evolution systems /1/, the resolving strategies in the present paper do not depend upon the accuracy ε. In contrast to /6/ a more general case is examined inclusing, in particular, approximation by the Galerkin method and by the difference method of straight lines.     

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.     

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)     

Stabilization of the motions of mechanical systems with variable masses

A holonomic mechanical system with variable masses and cyclic coordinates is considered. Such a system can have generalized steady motions in which the positional coordinates are constant and the cyclic velocities under the action of reactive forces vary according to a given law. Sufficient Routh-Rumyantsev-type conditions for the stability of such motions are determined. The problem of stabilizing a given translational-rotational motion of a symmetric satellite in which its centre of mass moves in a circular orbit and the satellite executes rotational motion about its axis of symmetry is solved.     

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.     

On a Malkin — Massera theorem : PMM vol. 43, no. 6, 1979 pp. 975–979

The results obtained in [1, 2] are complemented by an assertion on asymptotic stability uniform with respect to t₀, x₀, and also are extended to the problems of asymptotic stability with respect to a part of the variables and of optimal stabilization with respect to a part of the variables.     

Remote stationary wave field generated by local perturbing sources in a flow of stratified fluid

A linear formulation is used to study the problem of stationary waves formed in a uniform flow of an inviscid incompressible vertically stratified fluid past a point source or a mass dipole. Formulas are derived representing the characteristics of the wave field in the form of the sum of single integrals. A method is developed for constructing complete asymptotic expansions of the integrals obtained for large distances from the wave generator, including uniform expansions near the leading fronts of the separate modes. Approximate solutions of the problem in question exist (/1–4/ et al.). The behaviour of the characteristics of the wave field near the leading fronts of internal waves was studied in /5, 6/. In the case of a deep liquid the asymptotic form uniform in the neighbourhood of the leading fronts is expressed in terms of Fresnel integrals /5/, and in the case of a liquid of finite depth by Airy functions /6/. Examples of the exact solution of the problem are given in /7/.     

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.     

On the theory of the gyropendulum

Precession equations of motion of the gyropendulum relative to the accompanying Darboux trihedron /1/ and, also, precession equations of the gyropendulum motion relative to the geographic trihedron, considered in /2, 3/, are given a kinematic interpretation. Linear differential equations that define the gyropendulum behavior at finite deflection angles of the rotor axis from the vertical are established for arbitrary motions of its suspension point over the surface of the Earth. These equations have the form of kinematic equations of a solid body spherical motion in terms of Rodrigues-Hamilton parameters, and in the case of stationary base they are in agreement with equations established in /4/. The Liapunov stability ot the gyropendulum equations in both the finite Euler—Krylov angles and in the Rodrigues — Hamilton parameters is proved. Particular cases of integrability in quadratures of the gyropendulum precession equations at finite angles are indicated.     

On the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope

The sufficient conditions for the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope in the case of Bobylev-Steklov integrability are obtained.

It is difficult to expect Lyapunov stability for the unsteady motions of a heavy solid having a fixed point since a dependence of the vibrations frequency on the initial conditions is characteristic for the simplest of them, i.e. periodic motions /1/. Moreover, a rougher property of periodic solutions of the Euler-Poisson equations, orbital stability /2/, is not the subject of special investigations in the dynamics of a solid. The algorithm of the present investigation utilizes the treatment ascribed Zhukovskii /3/ of orbital stability as the Lyapunov stability of motion for a special selection of the variable playing the part of time (see /4/ also) and the Chetayev method /5/ of constructing Lyapunov functions from the first integrals of the equations of perturbed motion. This latter circumstance enables the Chetayev method to be put in one series with the methods used in /1, 4, 6–9/, etc.     

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.     

On the stability of inhomogeneously ageing viscoelastic plates

A method of investigation is proposed and conditions are set up for the stability of viscoelastic inhomogeneously ageing plates of arbitrary shape with a common creep kernel. The form of the stability conditions is found as a function of the surface forces. The stability problem is examined numerically in a finite time interval. The paper touches on the investigations in /1–3/. (See the bibliography of research on the stability of homogeneous viscoelastic systems in /1–5/, for example.)