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)