Haiping studied stability problems of the equilibrium state manifold of ... Manifold of Nonholonomic Svstems ... at the origin of the noninertial reference frame.
Mechanics Research Communications,
Vol. 28, No. 4, pp. 46%469,2CKll
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Pergamon
0 2001 Elsevier Science Ltd
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I/$-see front
PII: SOO93-6413(01)00196-3
Stability Theorems for the Equilibrium State Manifold of Nonholonomic Systems in a Noninertial Reference Frame Luo Shaokai
Chen Xiangwei
Fu Jingli
Department of Physics Shangqiu Teachers College Shangqiu, 476000 Henan Province, P.R.CHINA
1. Introduction In 1904, Whittaker ET first studied the stability problems of the equilibrium position of nonholonomic
systems”].
Since then, Bottema 0t2], Aiserman MAt3], Karapetyan
AVt4t, Neimark UItsl, Rumyantsev
VVt4,6*71,and Mikhailov GKtS1 have done much
research on this work and obtained a series of important results, but these results are confined to linear. homogeneous and steady systems with nonholonomic constraints, and some of them are confined to Chaplygin systems. In recent years, Mei Fengxiang and Zhu Haiping
studied
nonholonomic
stability problems systems,
and
of the equilibrium
obtained
some
state manifold
important
results
of
of nonlinear considerable
generalityt9-‘31. It should be noted that all these previous works are confined to inertial reference frames, but the stability of mechanical systems in noninertial reference frames is of great significance both in theory and in practice. In this paper the stability problems of the relative equilibrium state manifold of nonlinear systems with nonholonomic
constaints in a noninertial reference frame are
studied. Firstly, the Routh equations of relative motion of these systems are constructed, and are regarded
as the relative motion of the corresponding
holonomic
systems;
moreover their relative equilibrium equations and relative equilibrium state manifold are obtained. Secondly, stability criteria for the relative equilibrium state manifold of the nonlinear nonholonomic
systems are obtained by the method of Lyapunov functions.
Finally, one example is presented to illustrate the results.
463
matter
S LUO, X CHEN and J FU
2.
Relative Eauilibrium Eauations and Relativk Eauilibrium Manifold of Nonholonomic Svstems
State
In a noninertial reference frame let us investigate the motion of a mechanical system of N particles. The configuration of the system can be specified by the generalized coordinates q,(s = l:..,n) of the noninertial reference frame attached to a massive rigid body having acceleration
translational 4r)
a,(r) , angular velocity
acceleration
independent
O(r) and angular
of the motion of each particle. For the potential and
inertial force field and inertial centrifugal force field, introduce the corresponding
force
functions
.ar:
u/o = _p = _p&,fa
u/”
=-vu
=i,.i,.,
I/, =p+c.~
o %,, ’
(1)
The nonpotential inertial forces are N
QF =-Cm,(&r;),-, ,=I
ar’
N
I,
< =-cc2
a=,,=I
ah
%’ ar: qm m,o. -xt 34,. aq,,1
where r,’ is the relative position vector of the i-th particle. and i,
(2)
is the inertia tensor
at the origin of the noninertial reference frame. Suppose that g ideal and steady nonholonomic constraints of Chetaev type are exerted on the motion of the system, i.e. fp(9,,4,)=0 where q,- =(4,,,q2~,...,q,,,)Tand
(3)
(P=J,2,.~.,g)
4? =(9,,,&,,.‘.,
Q,,,)r The relative motion equations of
the system can be expressed in Routh form as d ar, dtae,, where
ar, _a(u+u,) ----+Q,, aq.,,
a9,,
+Qp’+ ‘:+&$ a=,
(s = 1,2;..,n)
(4)
.M
T, and U are respectively the relative motion kinetic energy and the potential
function of the active force, and u = U(q, ) $~,(%)~.~,q,, s.k.1 Q, is the nonpotential active force, and d, (p = l;..,g) are undetermined T, =;
(5) multipliers.
The relative motion of the system is determined by constraints (3) and Eqs.(4). Differentiating Eqs(3) with respect to f, we have (P=l,2 ;.., g)
(6)
Substituting the expression for &, obtained from Eqs.(4) into Eqs.(6), we may obtain A,, then the A, can be obtained by solving
the equations determining the multipliers the equations and are denoted as 1.p = $&W$,)
(P = 1,2;..,g)
(7)
STABILITY
Substituting
Eqs(7)
d ar,
aq
dr %W
%.”
OF NONHOLONOMIC
SYSTEMS
into Eqs.(4), we obtain _a(u+u,) -+Q,+Q,;+
state manifold
L,, is the set of all
state manifold is t, =O]
(11)
Now suppose 0=((9,,il)(~~a(9~,~,)=0. Obviously
D
is a closed manifold
in
system. From Eqs.(S), it follows that motion
(8) corresponding
equilibrium
state manifold
P=1,2;-$1
R*“, and is called the constraint manifold D
of the
is an invariant set of the equations of relative
to the holonomic
system. Thus, the stability
L of relative motion of the nonholonomic
is equivalent
to the stability
corresponding
holonomic system (8) in the constraint manifold
3.
(12)
of the relative
eq’uilibrium
of the relative
system (3) and (4)
state manifold
L
of the
D.
Stabilitv Theorems for the Relative Eauilibrium State Manifold of a Nonlinear Nonholonomic Svstem
S LUO. X CHEN and J FU
166
Suppose that the nonlinear nonholonomic constraint equations (3) can be expressed as -9,(9,,$)=0 fa = 4(r+B,r where
(P=1,2,...,g;&=n-g)
4: = (&,,4,, ;‘., 4,)T. Suppose that the functions
(13)
9’a satisfy (14)
9)B(9, >O)= 0
and the system is subject not only to the potential force, but also to the active force and the dissipative force of the generalized force of inertia of rotation as follows I,
F, = -2 k.\,,,(9,)G”,, .I=,
F,, = -c
.,=I
k,,,, (9, )(i,
Let F,‘=F,+F,,
kl, = km + k,,,,,,
From the relative equilibrium equations (9). we know that the relative equilibrium states of relative motion of the system comprise a manifold of dimension not less than the number g of constraint equations. Let V = T, -Cl -II, From the equations
of motion (8) of the corresponding
holonomic system, we obtain (15) where the 9p can be expressed as (16) Obviously,
9;l satisfies Qb/,,;;, = 0,
F
=o a satisfy
-Z$&)=
~G(9A)%~~, 0.h=, CW K&(L) = 0 Substituting Eq.( 19) into Eq.( 17), we obtain r; = - T(z;
-K&)&&
V.kl
where His at least of order three in ;3:, and zk is
+H
(19)
(20)
STABILITY
OF NONHOLONOMIC
SYSTEMS
467
(21)
If
the dissipative
generalized infer that
force
velocity ti
f,’
is negative
negative definite for
is completely
9:) then the matrix
9:
definite
for
dissipative
(6)
9:
with
respect to the isolated
is positive definite.
in some neighborhood
From Eq.(20),
of L
we
. Hence ri is
in the constrained manifold as follows
D=((9,,9,)/4,c+fi,,
P = 1,2,...,gJ
-~jA9,,4:)=0,
(22)
So we have constraints (13) U(q,) + O,(q,) is negative definite and bounded with respect to L,, , and grad(U + U, ) = 0 holds at every point of L,, , then
Theorem 1
For the conservative mechanical system with nonholonomic
and satisfying condition
the
relative
asymptotically
equilibrium
of (14), if its force function
state
manifold
L
of
relative
motion
of
stable with respect to a dissipative force that is completely
the
system
is
dissipative with
respect to the isolated generalized velocities.
Theorem 2
For the conservative
mechanical system with nonholonomic
and satisfying condition of (14), if its force function
constraints (13)
U(q,.)+ U, (4,) is non-negative and
bounded with respect to L,, , and grad(U + U, ) = 0 holds at every point of relative equilibrium
state manifold
L,, , then the
L of relative motion of the system is unstable with
respect forces that are completely
dissipative
with respect to the isolated generalized
velocities.
4.
An Examtde
Suppose the carrier of the reference frame rotates with an uniform
angular velocity
w
around a fixed axis. The kinetic energy of relative motion of the system is r, = $4;
+ 9:, + 4;; )
(23)
and its force function is U’=lJ+U.
+q,,
The system is subject to the nonholonomic
+qzr +9,,)2wz
(24)
constraint as follows
93, + 9, (9, )4:, + 92 (9, )&, = 0 where
a, and n2 are functions of the generalized coordinates
(25)
q, = (9,,,qZ, ,q,r)T. The
system is acted upon by a dissipative force with dissipation function (26)
368
S LUO,
where p, >O,
pz>O.
X
CHEN
and .l. FU
p,bO.
The equations of relative motion of the corresponding holonomic system are 91, = -(9,r + 92, + 93, )a2 - P,4,, + 2% 92, = -(9,, + 92r-+ 93, )J
A
- P*92, + 2%9,,~
4;, = -(9,, + 92, + 9,#.W -PA
(27)
+a
From the constraint equation (25) and Eqs.(27), we obtain R = (9,, + qzr + 9,r)(l +29,9,,
+ 29,9,,W
+ 2P,9,9:
I+ 4af&
+ 2j+G:,
+ PA,,
-&Z,
- G,
+ 4a:$, (28)
From the relative equilibrium equation of relative motion (9), we obtain the relative equilibrium state manifold of the system as L = {(909, )19,, + 92, + 93, = 0.
9, = 01
(29)
Obviously, the force function LI’ is negative definite on L,, = and
gradCI’ = 0
holds at every point of
L,,
{q,[q,, + 9tr + 9,, = 0) ,
From Theorem
I,
we know L is
asymptotically stable. If the force function (24) equilibrium
is changed to
II’ = i(9,r
state manifold of the system is still Eq.(29).
positive definite for L,, , and C/‘(L,,) = 0
+ q2, + q3, )202
, then the
The force function U’
is
; moreover gradll’ = 0 holds at every point of
L,, From Theorem 2, we know that L is unstable.
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E T. A treatise on the analytical
dynamics of particles and rigid bodies with an
introduction to the problem of three bodies. Eng: Cambridge, 1904.221-225
2.
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I951.37( I-2): 74-15
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