Spherical and Planar Ball Bearings a Study of Integrable Cases Vladimir Dragovi c Borislav Gaji c Bo zidar Jovanovi c Abstract. We consider the nonholonomic systems of nhomogeneous balls B1 Bnwith the

2025-05-03 0 0 641.43KB 14 页 10玖币
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Spherical and Planar Ball Bearings – a Study of Integrable Cases
Vladimir Dragovi´c, Borislav Gaji´c, Boˇzidar Jovanovi´c
Abstract. We consider the nonholonomic systems of nhomogeneous balls B1,...,Bnwith the
same radius rthat are rolling without slipping about a fixed sphere S0with center Oand radius
R. In addition, it is assumed that a dynamically nonsymmetric sphere Swith the center that
coincides with the center Oof the fixed sphere S0rolls without slipping in contact to the moving
balls B1,...,Bn. The problem is considered in four different configurations. We derive the
equations of motion and prove that these systems possess an invariant measure. As the main
result, for n= 1 we found two cases that are integrable in quadratures according to the Euler-
Jacobi theorem. The obtained integrable nonholonomic models are natural extensions of the well-
known Chaplygin ball integrable problems. Further, we explicitly integrate the planar problem
consisting of nhomogeneous balls of the same radius, but with different masses, that roll without
slipping over a fixed plane Σ0with a plane Σ that moves without slipping over these balls.
1. Introduction
We continue our study the spherical and the planar ball bearing problems, which we introduced
in [11]. Here we focus on four different configurations of the spherical ball bearing problem. In [11]
we dealt with the first configuration: nhomogeneous balls B1,...,Bnwith centers O1, ..., Onand
the same radius rroll without slipping around a fixed sphere S0with center Oand radius R. A
dynamically nonsymmetric sphere Sof radius ρ=R+ 2rwith the center that coincides with the
center Oof the fixed sphere S0rolls without slipping over the moving balls B1,...,Bn(case I,
Figure 1).
As the second configuration (case II), we consider homogeneous balls of radius rwithin a fixed
sphere S0of radius R. The balls support a moving, dynamically nonsymmetric sphere Sof radius
ρ=R−2r(see Figure 1).
Proposition 2.1 implies that the centers O1, ..., Onof the balls are in rest in relation to each
other. Thus, there are no collisions of the balls B1,...,Bn. For n≥4 there are initial positions of
the balls B1,...,Bnthat imply the condition that the centre of the moving sphere Scoincides with
the centre Oof the fixed sphere S0. In order to include all possible initial positions for arbitrary
n, the condition that Ocoincides with the centre of the sphere Sis assumed to be a holonomic
constraint.
For n= 1 we introduce two additional configurations assuming that B1is not a homogeneous
ball but a sphere (spherical shell). The first one is when the sphere B1is within the moving sphere
Sand the fixed sphere S0is within B1(case III,ρ= 2r−R,ρ > R, Figure 2). The second one is
when the sphere B1is within the fixed sphere S0and the moving sphere S0is within B1(case IV,
ρ= 2r−R,ρ<R, Figure 2).
In Section 2 we present the equations of motion of the spherical ball bearing systems for all four
configurations. The kinetic energy and the distribution are invariant with respect to an appropriate
action of the Lie group SO(3)n+1, and the system can be reduced to M=D/SO(3)n+1 and to
N=R3×(S2)n(the second reduced space) (Theorem 2.1), where D ⊂ T Q is the nonholonomic
distribution and Q=SO(3)n+1 ×Snis the configuration space of the problem. The system also has
an invariant measure (Theorem 3.1, Section 3). The proofs of Theorems 2.1 and 3.1 are similar to
the proofs of the corresponding statements given for the configuration I in [11] and they are omitted.
2010 Mathematics Subject Classification. 37J60, 37J35, 70E40, 70F25.
Key words and phrases. Nonholonimic dynamics; rolling without slipping, invariant measure; integrability.
1
arXiv:2210.11586v1 [math-ph] 20 Oct 2022
2 V. DRAGOVI´
C, B. GAJI´
C, B. JOVANOVI´
C
In this paper we consider the integrability of the spherical balls bearing problem in the case of
n= 1. The system can be reduced to N=R3{~
Ω} × S2{~
Γ}and takes the form (see Section 3)
(1.1) ˙
~
M=~
M×~
Ω,˙
~
Γ = ε~
Γ×~
Ω,
where ~
M=I~
Ω + d~
Γ and
I=I+DE−D~
Γ⊗~
Γ,E= diag(1,1,1).
Here ~
Ω is the angular velocity of the sphere S,I= diag(A, B, C) is its inertial tensor, ~
Γ is the
unit vector determining the position of the homogeneous ball B1and ε,d,Dare parameters of the
problem that are described in Sections 2 and 3.
Figure 1. Spherical ball bearings for n= 3, case I (left, ρ=R+ 2r) and case II
(right, ρ=R−2r)
According to Theorem 3.1, the flow of (1.1) in variables {~
Ω,~
Γ}preserves the measure with
density pdet(I). Also, it always has the first integrals F1=1
2hI~
Ω,~
Ωiand F2=h~
M,~
Mi(see
Proposition 3.1). Thus, for the integrability, according to the Euler-Jacobi theorem, we need one
additional first integral.
Figure 2. Spherical ball bearing, case III (left, ρ= 2r−R) and case IV (right,
ρ= 2r−R)
As the main results of the paper, in Section 3 we prove
Main result 1.The spherical ball bearings problem (1.1) in the configuration III, when 2r=
3R, i.e., ε=−1, is integrable. The third first integral is
F3= (B+C−A+D)M1Γ1+ (A+C−B+D)M2Γ2+ (A+B−C+D)M3Γ3.
INTEGRABLE SPHERICAL AND PLANAR BALL BEARINGS 3
Main result 2.The spherical ball bearings problem (1.1) for B=Cis integrable for any εin
all configurations. Along with F1and F2, the system has a nonalgebraic first integral1
F±
3=±pD(A−C)F+DG −dCexp(±(1 −ε)pD(A−C)Φ).
For d= 0, the equations (1.1) coincide with the equations of motion of a Chaplygin ball with
the inertia tensor Ion a sphere (ε6= 1) and the plane (ε= 1) with slightly different definitions
of parameters εand D. Thus, the above results can be seen as natural extensions of well-known
integrable Chaplygin ball problems.
In [11] we also considered the associated planar problem. This is the case I in the notation of
the present paper, when the radius of the fixed sphere S0tends to infinity. We found an invariant
measure and proved the integrability by means of the Euler–Jacobi theorem [11]. Here, in Section
4, we perform an explicit integration of the reduced problem.
2. Rolling of a dynamically nonsymmetric sphere over nmoving homogeneous balls
and a fixed sphere
2.1. Kinematics. Let O~
e0
1,~
e0
2,~
e0
3,O~
e1,~
e2,~
e3,Oi~
ei
1,~
ei
2,~
ei
3be positively oriented reference frames
rigidly attached to the spheres S0,S, and the balls Bi,i= 1, . . . , n, respectively. By g,gi∈SO(3)
we denote the matrices that map the moving frames O~
e1,~
e2,~
e3and Oi~
ei
1,~
ei
2,~
ei
3to the fixed frame
O~
e0
1,~
e0
2,~
e0
3. Using the standard isomorphism between the Lie algebras (so(3),[·,·]) and (R3,×) given
by
(2.1) aij =−εijkak, i, j, k = 1,2,3,
the skew-symmetric matrices ω=˙
gg−1,ωi=˙
gig−1
icorrespond to the angular velocities ~ω,~ωiof
the sphere Sand the i-th ball Biin the fixed reference frame O~
e0
1,~
e0
2,~
e0
3attached to the sphere S0.
The matrices Ω = g−1˙
g=g−1ωg,Wi=g−1
i˙
gi=g−1
iωigicorrespond to the angular velocities
~
Ω, ~
Wiof Sand Biin the frames O~
e1,~
e2,~
e3and Oi~
ei
1,~
ei
2,~
ei
3attached to the sphere Sand the balls
Bi, respectively. We have ~ω =g~
Ω, ~ωi=gi~
Wi.
Then the configuration space of the problem is
Q=SO(3)n+1 ×(S2)n{g,g1,...,gn, ~γ1, . . . ,~γn},
where ~γiis the unit vector
~γi=−−→
OOi
|−−→
OOi|
determining the position of the centre of i-th ball Bi,i= 1, . . . , n. In the cases I and II, the velocity
of the centre of the i-th ball is ~vOi= (R±r)˙
~γi, while for the cases III and IV (n= 1) we have
~vO1=±(r−R)˙
~γ1. We will show in Proposition 2.1 that if the initial conditions are chosen such
that the distances between Oiand Ojare all greater than 2r, 1 ≤i < j ≤n, then the balls will
not have collisions along the course of motion. This is the reason why we do not assume additional
one-side constraints
(2.2) |~γi−~γj| ≥ 2r
R±r,1≤i<j≤n(cases I and II, n≥2).
Let A1, ..., Anand B1, B2, ..., Bnbe the contact points of the balls B1,...,Bnwith the spheres
S0and S, respectively. The condition that the rolling of the balls B1,...,Bnand the sphere Sare
without slipping leads to the nonholonomic constraints:
~vOi+~ωi×−−−→
OiAi= 0, ~vOi+~ωi×−−−→
OiBi=~ω ×−−→
OBi, i = 1, ..., n,
that is,
(2.3) ~vOi=±r~ωi×~γi, ~vOi= (R±2r)~ω ×~γi±r~γi×~ωi(cases I and II)
and
(2.4) ~vO1=±r~ω1×~γ1, ~vO1=±(2r−R)~ω ×~γ1±r~γ1×~ω1(cases III and IV).
The dimension of the configuration space Qis 5n+ 3. There are 4nindependent constraints in
(2.3), defining a nonintegrable distribution D ⊂ T Q of rank n+ 3. The phase space of the system,
Dconsidered as a submanifold of T Q, has the dimension 6n+ 6.
1Through the paper, the sign ±denotes + for cases I and III, and - for the cases II and IV.
摘要：

SphericalandPlanarBallBearings{aStudyofIntegrableCasesVladimirDragovic,BorislavGajic,BozidarJovanovicAbstract.WeconsiderthenonholonomicsystemsofnhomogeneousballsB1;:::;Bnwiththesameradiusrthatarerollingwithoutslippingabouta xedsphereS0withcenterOandradiusR.Inaddition,itisassumedthatadynamicallyn...

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