Asymptotical stability of the motion of mechanical systems with partial energy dissipation

Nonlinear Dynamics, Nov 2017

We consider a linear mechanical system under the action of potential, gyroscopic and dissipative (partial) forces. The classical Kelvin–Chetaev theorems are not applicable here, and another approach, which is based on Barbashin–Krasovskii theorem, is suggested. This approach is based on decomposition of the whole system and is convenient for systems of high dimension or with uncertain parameters. Some advantages of the proposed method are demonstrated by examples.

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Asymptotical stability of the motion of mechanical systems with partial energy dissipation

Asymptotical stability of the motion of mechanical systems with partial energy dissipation Volodymyr Puzyrov 0 1 Jan Awrejcewicz 0 1 0 J. Awrejcewicz ( 1 V. Puzyrov Department of Higher Mathematics and Methodology of Teaching Mathematics, Faculty of Mathematics and Information Technology, Vasyl Stus Donetsk National University , Vinnytsia 21021 , Ukraine We consider a linear mechanical system under the action of potential, gyroscopic and dissipative (partial) forces. The classical Kelvin-Chetaev theorems are not applicable here, and another approach, which is based on Barbashin-Krasovskii theorem, is suggested. This approach is based on decomposition of the whole system and is convenient for systems of high dimension or with uncertain parameters. Some advantages of the proposed method are demonstrated by examples. Stability; Dissipative/gyroscopic/potential forces; Kelvin-Chetaev/Barbashin-Krasovskii theorems 1 Introduction The methods of the stability theory are mathematically rigorous and widely used in various applied problems. Later these theorems were generalized in numerous papers, both from theoretical [4–10] and applied [11–14] points of view. Rather complete observations of references on this topic are given in [15–18]. Most of these studies considered the classical case of full energy dissipation when Rayleigh function R is positive definite on all generalized velocities. However, numerous examples of mechanical systems in different areas of technology (celestial mechanics, robotics, seismology, etc.) consider the situation when dissipation is partial, i.e., function R is semi-positive. As Chetaev has noted, “If the dissipation is incomplete, then the stability of equilibrium existing under certain potential forces will not be strengthened from adding such dissipative forces to the asymptotic stability” [2]. Actually, this is not quite true, because the result of the influence of such forces may vary, and whether the system becomes asymptotically stable or not strongly depends on other forces. This dependence is more complicated and less obvious than those addressed in Theorems 1– 3. One of the first researchers, who drew attention to this, was Zajac. He noted that “In the design of attitudecontrol systems, one strives not for a positive definite damping matrix but rather for damping that affects the entire system, so that any motion induces energy dissipation” and introduced the term “pervasiveness” for this kind of damped systems [5]. Some results concerning this problem are presented in papers [19–23]. Bernstein and Bhat [24] formulated necessary and sufficient conditions for Lyapunov stability, semi-stability and asymptotic stability of matrix second-order systems with and without damping. The problem regarding stability of mechanical systems subjected to dissipative, gyroscopic, conservative/nonconservative forces has been reconsidered again by Agafonov [25]. The condition of asymptotic stability under action of the mentioned forces has been proposed as well as the estimation of the attraction domain in phase space has been formulated. The relation between stability of origin and precession systems has been derived, and examples of stabilization of the stationary motion of the balanced gimbal suspension gyro through external moments have been provided. A generalization of the Barbashin–Krasovskii theorem has been proposed by Jiang [26]. More recently, Tonkov [27] addresses the problem devoted to the Barbashin– Krasovskii asymptotic stability theorem in application to control systems on smooth manifolds. Kalyakin [28] studied the autoresonance problem aimed at distinguishing solutions with unboundedly increasing amplitude. The constructed Lyapunov function allowed to investigate stability of the autoresonance with respect to perturbation of the initial data and regarding constantly acting perturbations. In this paper, we confine ourselves to autonomous linear system with potential, gyroscopic and dissipative forces. Such a system can be presented in the following form Mξ¨ + Bξ˙ + K ξ = 0, ( 1 ) M, K , B are square real matrices, two first of them are symmetric and positive, B is semi-positive and always can be separated to symmetric (dissipative) and skewsymmetric (gyroscopic) components B = D + G, ξ ∈ Rn. Our aim is to suggest the constructive approach to distinguish cases whether the stability is asymptotical with respect to all variables or not. We believe it will be effective both for systems of high dimension and systems with uncertain parameters. 2 Preliminaries Moran’s criterion [20] states that system ( 1 ) is asymptotically stable if and only if none of the eigenvectors v of the conservative system M G K lies in the null space of D, that is Dv = 0 for all eigenvectors v. Also Muller [21] has given criterion whether system ( 1 ) is pervasive from the viewpoint of the control theory rank M−1 D, (M−1 K )(M−1 D), . . . , (M−1 K )n−1(M−1 D) = n. Both conditions are effe (...truncated)


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Volodymyr Puzyrov, Jan Awrejcewicz. Asymptotical stability of the motion of mechanical systems with partial energy dissipation, Nonlinear Dynamics, 2017, pp. 1-13, DOI: 10.1007/s11071-017-3872-8