On stability of stationary motion of a nonconservative nonholonomic rheonomic system

One considers, in this paper, the motion of a mechanical system in a nonstationary field of potential and positional forces, subject to the action of rheonomic holonomic and nonholonomic linear homogeneous constraints. Assuming that differential equations of motion of the system considered satisfy the conditions for the existence of Painlevé's integral of energy, formulated in [Painlevé, P., 1897. Leçons sur l'intégration des équations de la Mécanique, Paris] and [Appell, P., 1911. Traité de mécanique rationnelle, T. II, Dynamique des systémes – Mécanique analitique, Gauthier-Villars, Paris] and generalized in [Čović, V., Vesković, M., 2004. On stability of motion of a rheonomic system in the field of potential and positional forces, BAMM-1720/2004, No-2233, 93–100] and [Čović, V., Vesković, M., 2005. Hagedorn's theorem in some special cases of rheonomic systems. Mechanics Research Communications 32 (3), 265–280], the original mechanical system is substituted by an equivalent one whose Lagrangian function, nontransformed with respect to nonholonomic constraints, does not depend on time explicitly. Using the properties of the equivalent system, which, in contrast to the original one, moves in a stationary field of potential forces and in a nonstationary field of gyroscopic forces, the definition of cyclic coordinates is generalized, as well as sufficient conditions for the existence of (cyclic) first integrals, corresponding to coordinates mentioned and linear in velocities are established. Further, the conditions for the existence of steady motion of the system considered are found. In the case of existence of such a motion of the system, the Theorem of Routh's type on stability of that motion, based on the minimum of reduced potential for which it is shown that, in contrast to known cases (see, for example, [Gantmacher, F., 1975. Lectures in Analytical Mechanics. Mir Publisher, Moscow; Neimark, J., Fufaev, N., 1972. Dynamics of Nonholonomic Systems. Amer. Math. Soc., Providence, RI; Pars, L., 1962. An Introduction to Calculus of Variations. Heinemann, London; Karapetyan, A., Rumyantsev, V., 1983. Stability of conservative and dissipative systems. In: Itogi Nauki I Tekhniki: Obschaya Mekh., vol. 6, VINITI, Moscow, pp. 3–128 (in Russian)]), it includes the influence of the positional forces field, is formulated. Thus, the Routh's Theorem on stability of steady motion of a conservative mechanical system is extended to the case of a nonconservative system.     

On the stability of motion of uncertain systems

The basic theorem of the direct Lyapunov method on the stability of solutions of uncertain systems with respect to a moving invariant set is generalized. A corollary of general assertion is cited in terms of scalar Lyapunov functions. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 105–109, February, 1999.     

On the exponential stability of the permanent rotational motion of a gyrostat

This paper is devoted to the study of the problem of exponential asymptotic stability of the rotational motion of a gyrostat using servo-control moments which are applied to the internal rotors. The servo-control moments which impose the rotational motion are obtained. The stabilizing servo-control moments are obtained from the conditions to ensure exponential asymptotic stability of the desired motion. Estimations of the phase coordinations as exponential functions are presented. The method based on a choice of the structural form of the servo-control moments such that the equations of motion reduce to a system of differential equations with exponential asymptotic stability of an special solution.     

On the stability of an equilibrium position and rotational motion of a gyrostat

This article is devoted to study the compulsory stability of equilibrium position and rotational motion of a rigid body containing fluid with the help of three rotors carried on the body. The control moments on the rotors using that condition which impose the stabilization of equilibrium position of the rigid body and rotational motion are obtained.     

A criterion for the stability of motion of nonlinear system

As to an autonomous nonlinear system, the stability of the equilibrium state in a sufficiently small neighborhood of the equilibrium state can be determined by eigenvalues of the linear part of the nonlinear system provided that the eigenvalues are not in a critical case. Many methods may be used to detect the stability for a linear system. A lot of researches for determining the stability of a nonlinear system are completed by mathematicians and mechanicians but most of them are methods for the special forms of nonlinear systems. Till now, none of these methods can be conveniently applied to all nonlinear systems. The method introduced by this paper gives the necessary and sufficient conditions of the stability of a nonlinear system. The familiar Krasovski's method is a special case of this method.     

On the motion stability of an airplane on a runway under wind loading

A nonlinear mathematical model is constructed for an airplane in high-speed plane-parallel motion along a runway when the airplane's weight exceeds slightly the lift of its wings in the presence of a cross wind. The airplane is considered a two-weight mechanical object. A system of second-order equations is obtained that describes the airplane's behavior. A system of three phase variables is suggested in which the dynamics of transverse motion is described by a set of three second-order equations. A stationary solution of this system is obtained. A stability criterion for the plane-parallel motion of the airplane is established using the Routh-Hurwitz criterion. Analysis of the data of other authors indicates that the mathematical model is adequate for some objects of aviation technology. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 101–107, October, 1999.     

On the unsteady motion and stability of a heaving airfoil in ground effect

This study explores the fluid mechanics and force generation capabilities of an inverted heaving airfoil placed close to a moving ground using a URANS solver with the Spalart-Allmaras turbulence model. By varying the mean ground clearance and motion frequency of the airfoil, it was possible to construct a frequency-height diagram of the various forces acting on the airfoil. The ground was found to enhance the downforce and reduce the drag with respect to freestream. The unsteady motion induces hysteresis in the forces’ behaviour. At moderate ground clearance, the hysteresis increases with frequency and the airfoil loses energy to the flow, resulting in a stabilizing motion. By analogy with a pitching motion, the airfoil stalls in close proximity to the ground. At low frequencies, the motion is unstable and could lead to stall flutter. A stall flutter analysis was undertaken. At higher frequencies, inviscid effects overcome the large separation and the motion becomes stable. Forced trailing edge vortex shedding appears at high frequencies. The shedding mechanism seems to be independent of ground proximity. However, the wake is altered at low heights as a result of an interaction between the vortices and the ground.     

On the stability of relative programmed motion of satellite- gyrostat

This paper is devoted to study the asymptotic stability of the relative programmed motion of a satellite-gyrostat with the help of the three rotors attached to the principal axes of inertia of the satellite. The programmed control moments are obtained. The control moments on the rotors using the condition which impose the asymptotic stabilization of the programmed motion are obtained.     

On the motion of a nonholonomically constrained system in the nonresonance case

The motion of a nonlinearly nonholonomically constrained system comprised of two material points connected by a “fork” is investigated in the nonresonance case. This leads to two equations of motion; one of which is nonlinear in the system velocities. The system is shown to be integrable in the nonresonance case, and the motion is described analytically and also computed numerically for several parameter values yielding results that conform to the analytical predictions.     

ON THE GLOEAL STABILITY OF THE ROTATIONAL MOTION AND THE QUALITATIVE ANALYSIS OF THE BEHAVIOUR OF DISTURBED MOTION OF A RIGID BODY HAVING A LIQUID-FILLED-CAVITY

This paper is a continuation of[1].In this paper we inves-tigate the distribution of steady motions of the liquid-filled-cavity body,decide the stability of each steady motion and findout the corresponding regions of stability and instability.Be-sides.the behaviour of disturbed motion is analysed qualitatively.     

On the Ljapunov stability of a system with known first integrals

Summary The differential equations of the perturbed motion with known independent first integrals are considered, and it is shown that a sufficient condition for the stability of the imperturbed motion of the complete system is given by the total stability of the corresponding solution of a certain reduced system, of lower rank. This theorem finds an interesting application in the study of the stability of the merostatic motions of a holonomic system with ignorable coordinates and acted upon by dissipative forces.

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Sommario Si considerano sistemi di equazioni differenziali del moto perturbato per i quali siano noti alcuni integrali primi indipendenti, e si dimostra che una condizione sufficiente per la stabilità del moto non perturbato del sistema completo è data dalla stabilità totale della corrispondente soluzione di un certo sistema ridotto, di rango inferiore. Questo teorema trova una significativa applicazione nello studio della stabilità dei moti merostatici di un sistema olonomo con coordinate ignorabili e soggetto a sollecitazioni dissipative.

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Research supported by the Gruppo di Ricerca No. 7 of the C. N. R.     

ON THE STABILITY OF FORCED DISSIPATIVE NONLINEAR SYSTE

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On the stability of a circular system subjected to nonlinear dissipative forces

A circular system is a mechanical system subjected to potential forces and positional nonconservative forces (circular forces). The latter linearly depend on the coordinates and are characterized by a skew-symmetric matrix. The influence of linear dissipative forces on the stability of a circular system is ambiguous: on the one hand, they can stabilize a stable circular system (making it asymptotically stable); on the other hand, they can destabilize it [1–4]. The action of linear dissipative forces on a circular system results in the so-called destabilization paradox: the stability threshold decreases by a finite value.A detailed survey of this phenomenon can be found in [5]. The destabilization effect is also preserved under the action of nonlinear dissipative forces. The influence of these forces on the stability of the Ziegler pendulum with a tracking force was studied in [6]. It was shown that the critical value of the tracking force decreases by a finite value. A similar effect was discovered in the analysis of a continual system in [7].In the present paper, we study how nonlinear dissipative forces affect the stability of the equilibrium of a circular mechanical system with two degrees of freedom. The stability problem is solved without any references to specific mechanical systems. The results are used to analyze the stability of a gimbal gyro with allowance for dry friction in the rotor bearings.