On
stability of steady motion
of non-holonomic systems
with non-uniform constraints
α.Ya.Krasinskiy
33, Talalikhin,
B.Atazhanov
Yunus-Abad, Tashkent-114, 100114,
The stability of steady motions of non-holonomic
systems with non-uniform constraints at cyclic coordinates is investigated. It
is supposed that a system, apart from potential forces, can be under any
exponential generalized forces. The main differences of the considered problem
from comprehensively investigated problem of stability of steady motion of
non-holonomic systems with homogeneous constraints are established. As is
known, the steady motion of such systems are located on diversities whose
dimensionalities for equilibrium positions are not less than a number of
non-holonomic constraints of a general form, and for stationary motions (in
some cases) not less than a sum of constraints number of a general form and a number
of cyclic coordinates. But in general case the dimensionality of stationary
motion diversity is not directly associated with a number of constraints and
cyclic coordinates. Owing to non-isolated (in general case) steady motions the
number of zero roots of a secular equation of the
system of the first approximation in the appropriate stability problems is not
less than dimensionality of diversity. The problem of stability of researched
motion is solved using Lyapunov-Malkin theorem on stability in special case of
critical case of several zero roots.
The systems with non-uniform constraints considered in the given paper,
in contrast to the systems with uniform constraints, can have both isolated and
non-isolated steady motions in general case. In the first case the stability
problem can be solved using Lyapunov theorems on stability in the first
approximation. In the second case (for non-isolated motions) the theory of
critical cases should be applied which represents far larger (in comparison
with the similar problems for systems with uniform constraints) difficulties
because of an essentially complicated structure, in particular, non-holonomicity members. Therefore it is particularly important to choose a
type of variables (Lagrange, Routh or Hamilton) and form of the motion
equations for the examination. It is necessary to mark that stability (and
stabilization) research of steady motions of non-holonomic systems, except for
independent theoretical interest, has, especially recently, a large number of
practical applications in the problems of yaw stability, stability of wheeled
carriage motion, and robotics. Keeping the initial structure of forces and
coefficient matrices of kinetic energy and coefficients in constraints equations
is highly important for the choice of the form of motion equations which are
the most suitable for research of stability and stabilization of non-holonomic
systems motions. And such choice is connected to a simplicity of application of
mathematical stability theory (particularly in critical cases theory) and (for
stabilization problem) mathematical control theory. Here Voronets
vector-matrix equations in Routh variables are used provided that a vector of
generalized coordinates according to the considered problem is divided into 5
vectors.
The situations are allocated when the equations of the first
approximation admit an integral. Accepting this integral as a new variable the
problem of stability may in some cases be reduced to a special case of several
zero roots.
As an application of obtained results one can consider a problem of
stability and stabilization of steady motions of non-uniform sphere with a
rotor in a horizontal plane rotating with a constant angular velocity.
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