A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton
For a chain consisting of n heavy Lagrange tops coupled by ideal spherical joints, the existence of a class of nonstationary motions with a non-planar configuration of the chain’s skeleton is proved. Sufficient conditions for existence of these motions are established, and the equations of motion of...
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irk-123456789-715982014-12-07T03:01:51Z A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton Chebanov, D.A. For a chain consisting of n heavy Lagrange tops coupled by ideal spherical joints, the existence of a class of nonstationary motions with a non-planar configuration of the chain’s skeleton is proved. Sufficient conditions for existence of these motions are established, and the equations of motion of the chain are reduced to quadratures. Under the assumption that the mass distribution of the bodies forming the chain is given, it is shown how they have to be coupled so that the motions of interest could be realized. Some properties of the new motions are discussed. Для цепочки n тяжелых гироскопов Лагранжа, соединенных идеальными сферическими шарнирами, установлено существование класса нестационарных движений, при которых остов системы имеет неплоскую конфигурацию. Получены достаточные условия существования таких движений. Найдена зависимость основных переменных от времени. При заданном распределении масс в телах для цепочки, состоящей из четырех тел, определены способы их сочленения, при которых установленные движения возможны. Указаны некоторые свойства новых движений. Для ланцюжка n важких гiроскопiв Лагранжа, з’єднаних iдеальними сферичними шарнiрами, встановлено iснування класу нестацiонарних рухiв, при яких остiв системи має неплоску конфiгурацiю. Отримано достатнi умови iснування таких рухiв. Знайдено залежнiсть основних змiнних вiд часу. При заданому розподiлi мас в тiлах для ланцюжка, що складається з чотирьох тiл, визначено способи їх зчленування, при яких встановленi рухи можливi. Указано деякi властивостi нових рухiв. 2011 Article A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton / D.A. Chebanov // Механика твердого тела: Межвед. сб. науч. тр. — 2011. — Вип 41. — С. 244-254. — Бібліогр.: 4 назв. — англ. 0321-1975 http://dspace.nbuv.gov.ua/handle/123456789/71598 531.38 en Механика твердого тела Інститут прикладної математики і механіки НАН України |
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For a chain consisting of n heavy Lagrange tops coupled by ideal spherical joints, the existence of a class of nonstationary motions with a non-planar configuration of the chain’s skeleton is proved. Sufficient conditions for existence of these motions are established, and the equations of motion of the chain are reduced to quadratures. Under the assumption that the mass distribution of the bodies forming the chain is given, it is shown how they have to be coupled so that the motions of interest could be realized. Some properties of the new motions are discussed. |
| format |
Article |
| author |
Chebanov, D.A. |
| spellingShingle |
Chebanov, D.A. A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton Механика твердого тела |
| author_facet |
Chebanov, D.A. |
| author_sort |
Chebanov, D.A. |
| title |
A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton |
| title_short |
A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton |
| title_full |
A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton |
| title_fullStr |
A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton |
| title_full_unstemmed |
A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton |
| title_sort |
new class of nonstationary motions of a system of heavy lagrange tops with a non-planar configuration of the system's skeleton |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/71598 |
| citation_txt |
A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system's skeleton / D.A. Chebanov // Механика твердого тела: Межвед. сб. науч. тр. — 2011. — Вип 41. — С. 244-254. — Бібліогр.: 4 назв. — англ. |
| series |
Механика твердого тела |
| work_keys_str_mv |
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| first_indexed |
2025-07-05T20:32:36Z |
| last_indexed |
2025-07-05T20:32:36Z |
| _version_ |
1836840446336172032 |
| fulltext |
ISSN 0321-1975. Механика твердого тела. 2011. Вып. 41
УДК 531.38
c©2011. D.A. Chebanov
A NEW CLASS OF NONSTATIONARY MOTIONS
OF A SYSTEM OF HEAVY LAGRANGE TOPS
WITH A NON-PLANAR CONFIGURATION
OF THE SYSTEM’S SKELETON
For a chain consisting of n heavy Lagrange tops coupled by ideal spherical joints, the existence
of a class of nonstationary motions with a non-planar configuration of the chain’s skeleton is
proved. Sufficient conditions for existence of these motions are established, and the equations of
motion of the chain are reduced to quadratures. Under the assumption that the mass distribution
of the bodies forming the chain is given, it is shown how they have to be coupled so that the
motions of interest could be realized. Some properties of the new motions are discussed.
Keywords: analytical multibody dynamics, Lagrange top, nonstationary motion of a system of
coupled rigid bodies
1. Formulation of the problem. We consider a mechanical system S
consisting of heavy Lagrange tops B1, B2, . . . , Bn. The bodies Bi and Bi+1
(i = 1, 2, . . . , n − 1) are coupled by an ideal spherical joint at a common point
Oi+1 so that the system S constitutes a chain of rigid bodies. One of the chain’s
end links, B1, is absolutely fixed at one of its points O1(6= O2). It is assumed
that the attachment points of the body Bi lie on its axis of symmetry li, i.e.,
O1 ∈ l1, Oi ∈ li−1 ∩ li(i = 2, 3, . . . , n− 1).
Let {O1, e1e2e3} be a Cartesian reference frame whose vectors are fixed in
inertial space so that the vector e3 is vertically directed. Let also {Oi, e
(i)
1 e
(i)
2 e
(i)
3 }
be a Cartesian frame that is rigidly embedded in the body Bi such that e
(i)
3 || li. We
determine the position of the body Bi with respect to the reference frame by Euler
angles θi, ψi, and ϕi. The vector equations of motion for a chain of coupled rigid
bodies is given in [1]. Projecting these equations on the axes of the corresponding
body-fixed frames, one can obtain the following scalar equations of motion of the
system S:
F
(m)
i + ai
i−1∑
j=1
sjG
(m)
ij + si
n∑
j=i+1
ajG
(m)
ij = 0, (1)
ϕ̇i + ψ̇i cos θi = qi, i = 1, 2, . . . , n, (2)
where m = 1, 2,
F
(1)
i = J ′
i
(
θ̈i − ψ̇2
i sin θi cos θi
)
+ Js
i qiψ̇i sin θi − aig sin θi,
G
(1)
ij =
(
θ̈j sin θj + θ̇2j cos θj
)
sin θi +
(
ψ̈j sin θj + 2θ̇jψ̇j cos θj
)
cos θi sin(ψi − ψj)+
+
(
θ̈j cos θj −
(
θ̇2j + ψ̇2
j
)
sin θj
)
cos θi cos(ψi − ψj),
244
A new class of nonstationary motions of a system of heavy Lagrange tops
F
(2)
i = J ′
i
(
ψ̈i sin θi + 2θ̇iψ̇i cos θi
)
− Js
i qiθ̇i,
G
(2)
ij =
(
ψ̈j sin θj + 2θ̇jψ̇j cos θj
)
cos(ψi − ψj)−
−
(
θ̈j cos θj −
(
θ̇2j + ψ̇2
j
)
sin θj
)
sin(ψi − ψj)
and the dots denote differentiation with respect to time. In equations (1), (2),
si = |OiOi+1|, qi is an integration constant, Ji and Js
i are the moments of inertia
of body Bi with respect to Oi about the axes e
(i)
1 (or e
(i)
2 ) and e
(i)
3 , respectively,
and
J ′
i = Ji + m̃is
2
i , ai = mici + m̃isi, (3)
where ci = |OiCi|, Ci is the center of mass of body Bi, m̃i =
n∑
j=i+1
mj, and mi is
the mass of body Bi.
The motion of system S is a superposition of the motion of its skeleton, that
is composed of the segments of axes li bounded by the corresponding suspension
points, and the pure rotation of each body about its axis of dynamic symmetry.
The former motion is completely determined by all angles θi, ψi, while the rotation
of Bi about li is described by the angle ϕi.
When the system S performs P.V. Kharlamov’s motion [2], the skeleton
belongs to a vertical plane Π rotating about the vertical in accordance with a
non-stationary law ψ(t) while its segments change their position with respect to
Π identically in time, i.e., all the bodies move similarly. Therefore, these motions
of the bodies system are called similar motions. For such motions, it is fulfilled
that θi = θ(t), ψi = ψ(t) + δiπ, where δi ∈ {−1, 0, 1} and i = 1, 2, . . . , n. Some
properties of the system’s motion and its generalizations can be found in the
works [2–4].
Let n∗ be a fixed integer with 1 ≤ n∗ < n. This integer partitions the set
I = {1, 2, . . . , n} of indices of all bodies constituting system S into two subsets
I∗ = {1, 2, . . . , n∗} and I∗ = {n∗ + 1, n∗ + 2, . . . , n}. In what follows, we consider
two subsystems of S: S∗ = {Bj |j ∈ I∗} and S∗ = {Bk|k ∈ I∗}. (Clearly, the
subsystems are coupled at the point On∗+1.) In this paper, we seek to find a
new class of nonstationary motions of system S with the following properties: the
bodies forming the subsystem S∗ (S∗) move similar to each other and the planes
Π∗ and Π∗ containing the skeletons of the subsystems S∗ and S∗, respectively, in
general, do not coincide, i.e.
θj = θ1, ψj = ψ1 + δjπ, j ∈ I∗,
θk = θn, ψk = ψn + δkπ, k ∈ I∗,
(4)
where δi ∈ {−1, 0, 1} and θ1, θn, ψ1, and ψn are functions of time to be determined.
We also require that
cos θn = µ cos θ1, (5)
where µ(6= 0) is a constant.
245
D.A. Chebanov
2. Structure of the solution. In this section we establish sufficient condi-
tions for existence of the class of solution to equations (1), (2) with properties
(4), (5). We restrict our study to the case of nonstationary motions of the bodies
system, i.e., θ̇1 6≡ 0, θ̇n 6≡ 0, ψ̇1 6≡ 0, and ψ̇n 6≡ 0. In all notations used below, we
assume that i ∈ I, j ∈ I∗, and k ∈ I∗.
Introducing the notation
εlm = cos [(δl − δm)π] =
{
1, if δl = δm or δl = δm ± 2,
−1, if δl = δm ± 1,
l,m ∈ I, (6)
we derive, by virtue of (4), that
cos(ψl − ψm) =
{
εlm, if l,m ∈ I∗ or l,m ∈ I∗,
εlm cos(ψ1 − ψn), if l ∈ I∗,m ∈ I∗ or l ∈ I∗,m ∈ I∗,
sin(ψl − ψm) =
0, if l,m ∈ I∗ or l,m ∈ I∗,
εlm sin(ψ1 − ψn), if l ∈ I∗,m ∈ I∗,
εlm sin(ψn − ψ1), if l ∈ I∗,m ∈ I∗.
(7)
Substituting (4) into (1) and taking into account formulas (6) and (7), we
obtain
Pj
(
θ̈1−ψ̇2
1 sin θ1 cos θ1
)
+
(
Js
j qjψ̇1 − ajg −Qj (cos θn)
··−R̃j (cos θ1)
··
)
sin θ1+
+ Tj cos θ1
(
f (1)n cos (ψ1 − ψn) + f (2)n sin (ψ1 − ψn)
)
= 0,
Pj
(
ψ̈1 sin θ1 + 2ψ̇1θ̇1 cos θ1
)
− Js
j qj θ̇1−
− Tj
(
f (1)n sin (ψ1 − ψn)− f (2)n cos (ψ1 − ψn)
)
= 0, (8)
Pk
(
θ̈n−ψ̇2
n sin θn cos θn
)
+
(
Js
kqkψ̇n−akg−Qk (cos θ1)
··−R̃k (cos θn)
··
)
sin θn+
+ Tk cos θn
(
f
(1)
1 cos (ψn − ψ1) + f
(2)
1 sin (ψn − ψ1)
)
= 0,
Pk
(
ψ̈n sin θn + 2ψ̇nθ̇n cos θn
)
− Js
kqkθ̇n−
− Tk
(
f
(1)
1 sin (ψn − ψ1)− f
(2)
1 cos (ψn − ψ1)
)
= 0,
where, for m = 1, n,
f (1)m = θ̈m cos θm −
(
θ̇2m + ψ̇2
m
)
sin θm, f (2)m = ψ̈m sin θm + 2ψ̇mθ̇m cos θm,
246
A new class of nonstationary motions of a system of heavy Lagrange tops
and
Pj = J ′
j + aj
j−1∑
l=1
slεjl + sj
n∗∑
l=j+1
alεjl, Qj = sj
n∑
l=n∗+1
al,
Pk = J ′
k + ak
k−1∑
l=n∗+1
slεkl + sk
n∑
l=k+1
alεkl, Qk = ak
n∗∑
l=1
sl, (9)
R̃j = aj
j−1∑
l=1
sl (1− εjl) + sj
n∗∑
l=j+1
al (1− εjl) , Tj = sj
n∑
l=n∗+1
alεjl,
R̃k = ak
k−1∑
l=n∗+1
sl (1− εkl) + sk
n∑
l=k+1
al (1− εkl) , Tk = ak
n∗∑
l=1
slεkl.
Equations (8) form an overdetermined system of 2n second-order differential
equations with respect to four unknowns θ1, ψ1, θn, and ψn. (Note that θ1 and θn
are not independent due to (5).) In the rest of this section, we shall examine the
compatibility of the system (8) in the case when
Ti = 0. (10)
Using (5) and (10), we rewrite the system (8) as follows:
Pj
(
θ̈1−ψ̇2
1 sin θ1 cos θ1
)
+
[
Js
j qjψ̇1 − ajg−Rj (cos θ1)
··
]
sin θ1 = 0,
Pj
(
ψ̈1 sin θ1 + 2ψ̇1θ̇1 cos θ1
)
− Js
j qj θ̇1 = 0, (11)
Pk
(
θ̈n−ψ̇2
n sin θn cos θn
)
+
[
Js
kqkψ̇n−akg−Rk (cos θn)
··
]
sin θn = 0,
Pk
(
ψ̈n sin θn + 2ψ̇nθ̇n cos θn
)
− Js
kqkθ̇n = 0, (12)
where
Rj = R̃j + µQj , Rk = µR̃k +Qk. (13)
For each index j ∈ I∗, the pair of equations (11) has the following first integrals
Pj
(
θ̇21 + ψ̇2
1 sin
2 θ1
)
+Rj θ̇
2
1 sin
2 θ1 + 2ajg cos θ1 = hj ,
Pjψ̇1 sin
2 θ1 + Js
j qj cos θ1 = pj,
where hj and pj are constants of integration. Solving the above equations for θ̇21
and ψ̇1 yields
θ̇21 = Θj(θ1), ψ̇1 = Ψj(θ1), (14)
247
D.A. Chebanov
where Θj (θ1) =
[
Pj sin
2 θ1 (hj − 2ajg cos θ1)−
(
pj − Js
j qj cos θ1
)2]
/[Pj sin
2 θ1×
×
(
Pj +Rj sin
2 θ1
)
],Ψj (θ1) =
(
pj − Js
j qj cos θ1
)
/
(
Pj sin
2 θ1
)
.
Similarly, for each index k ∈ I∗, the pair of equations (12) leads to the equati-
ons
θ̇2n = Θk(θn), ψ̇n = Ψk(θn), (15)
where Θk (θn) =
[
Pk sin
2 θn (hk − 2akg cos θn)−(pk − Js
kqk cos θn)
2
]
/[Pk sin
2 θn×
×
(
Pk +Rk sin
2 θn
)
],Ψk (θn) = (pk − Js
kqk cos θn) /
(
Pk sin
2 θn
)
, hk and pk are
constants of integration.
One can check that if the conditions
Pj
P1
=
Rj
R1
=
Js
j qj
Js
1q1
=
aj
a1
=
hj
h1
=
pj
p1
,
Pk
Pn
=
Rk
Rn
=
Js
kqk
Js
nqn
=
ak
an
=
hk
hn
=
pk
pn
(16)
are fulfilled, then Θ1(θ1) ≡ Θ2(θ1) ≡ . . . ≡ Θn∗
(θ1),Ψ1(θ1) ≡ Ψ2(θ1) ≡ . . . ≡
≡ Ψn∗
(θ1),Θn∗+1(θn) ≡ Θn∗+2(θn) ≡ . . . ≡ Θn(θn), and Ψn∗+1(θn) ≡
≡ Ψn∗+2(θn) ≡ . . . ≡ Ψn(θn). Hence, in this case, the system of equations (11),
(12) reduces to the four equations
θ̇21 = Θ1 (θ1) , θ̇2n = Θn (θn) , (17)
ψ̇1 = Ψ1 (θ1) , ψ̇n = Ψn (θn) = Ψ̃n (θ1) , (18)
where Ψ̃n (θ1) = (pn − Js
nqnµ cos θ1) /
[
Pn
(
1− µ2 cos2 θ1
)]
.
It follows from (5) that θ̇2n sin
2 θn = µ2θ̇21 sin
2 θ1. Therefore, the right-hand
sides of equations (17) are related to each other by the formula
µ2Θ1 (θ1) sin
2 θ1 −Θn (θn) sin
2 θn = 0. (19)
Since Θm (θm) sin2 θm (m = 1, n) is a polynomial in cos θm, we can eliminate
cos θn from (19) by means of (5). This results in a polynomial in cos θ1 which
needs to be satisfied identically in cos θ1. Equating the coefficients of all powers
of cos θ1 to zero leads to the following conditions:
ãnR̃1 = µã1R̃n,[
h̃n +
(
J̃s
n
)2
q2n
]
R̃1 = µ2
[
h̃1 +
(
J̃s
1
)2
q21
]
R̃n,
µ2ãng − p̃nJ̃
s
nqnR̃1 = µ
(
ã1g − µ2p̃1J̃
s
1q1R̃n
)
,
µ2
[
h̃n +
(
J̃s
n
)2
q2n
](
1 + R̃1
)
+
(
h̃n − p̃2n
)
R̃1 = (20)
= µ2
{[
h̃1 +
(
J̃s
1
)2
q21
](
1 + R̃n
)
+ µ2
(
h̃1 − p̃21
)
R̃n
}
,
ãng − p̃nJ̃
s
nqn
(
1 + R̃1
)
= µ
[
ã1g − p̃1J̃
s
1q1
(
1 + R̃n
)]
,
248
A new class of nonstationary motions of a system of heavy Lagrange tops
(
h̃n − p̃2n
)(
1 + R̃1
)
= µ2
(
h̃1 − p̃21
)(
1 + R̃n
)
,
where ãm = am/Pm, h̃m = hm/Pm, J̃
s
m = Js
m/Pm, p̃m = pm/Pm, and R̃m =
= Rm/Pm (m = 1, N).
We can now state the following:
Proposition 1. If the conditions (10), (16), and (20) are fulfilled, the system
of equations (1), (2) has a class of exact solutions with properties (4), (5).
Proof. Indeed, we infer from the previous discussion that, under the assumpti-
on of the Claim, the system (8) is compatible. To find the dependence of the vari-
ables θi, ψi, and ϕi, i ∈ I, on time, we proceed as follows. We find θ1 as a function
of time by integrating the first equation in (17). Next, we determine ψ1 (t) and
ψn (t) from (18). We can now obtain θn(t) from (5) (or the second equation in
(17)) and θ2(t), θ3(t), . . . , θn−1(t), ψ2(t), ψ3(t), . . . , ψn−1(t) from (4). Finally, the
remaining variables ϕi can be found from (2). This competes the proof of the
proposition.
Based on the quadratures (17) and (18), geometry of the motion of each body
in the system can be analyzed by means of the methods that are usually used for
studying the motion of a symmetric top.
3. On Compatibility of the Conditions (10), (16), and (20) . As
stated in Proposition 1, the system of equations (1), (2) has exact solutions with
properties (4), (5) if the conditions (10), (16), and (20) are fulfilled. In this section
we show that there exist physically meaningful values of the multibody chain
parameters making these conditions compatible in the case when
si 6= 0, ai 6= 0. (21)
In this case, relations (10) are equivalent to n∗ conditions
n∑
l=n∗+1
alεjl = 0 and
n − n∗ conditions
n∗∑
l=1
slεkl = 0. Using (6), one can verify that, for i, l,m ∈ I,
εli = εlmεmi and
n∑
l=n∗+1
alεjl =
n∑
l=n∗+1
alεjnεnl = εjn
n∑
l=n∗+1
alεnl,
n∗∑
l=1
slεkl =
n∗∑
l=1
slεk1ε1l = εk1
n∗∑
l=1
slε1l.
Hence, conditions (10) can be replaced with the two equalities:
n∑
l=n∗+1
alεnl = 0,
n∗∑
l=1
slε1l = 0. (22)
249
D.A. Chebanov
We also note that, if either of the subsystems S∗ and S∗ consists of a single
top only, one of the above equalities contradicts one of the assumptions (21).
Therefore, the system S can perform the motion of interest only when each of its
subsystems S∗ and S∗ consists of at least two tops, making the total number of
bodies in S not less than four.
The case of a four-body system. Below we consider the simplest possible case
of a four-body system assuming that S∗ = {B1, B2}, S∗ = {B3, B4}, δ2 = δ3 = 1,
and R1 = R2 = R3 = R4 = 0. Then, the last conditions and relations (22) imply
s1 = s2, s1 + µs3 = 0, a3 = a4, a2 + µa4 = 0, (23)
the conditions (16) become
J ′
2 − s1a2
J ′
1 − s1a2
=
Js
2q2
Js
1q1
=
a2
a1
=
h2
h1
=
p2
p1
, (24)
J ′
3 = J ′
4, Js
3q3 = Js
4q4, p3 = p4, h3 = h4, (25)
and the relations (20) reduce to
ã1 = µã4, (26)
h̃4 +
(
J̃s
4
)2
q24 = h̃1 +
(
J̃s
1
)2
q21, (27)
ã4g − p̃4J̃
s
4q4 = µ
(
ã1g − p̃1J̃
s
1q1
)
, (28)
h̃4 − p̃24 = µ2
(
h̃1 − p̃21
)
. (29)
The relations (23)–(29) form an algebraic system of 16 equations with
respect to 32 unknowns J0
i (> 0), Js
i (> 0),mi(> 0), ci, pi, hi, qi (i = 1, 2, 3, 4),
si (i = 1, 2, 3), and µ. Here J0
i is the central equatorial moment of inertia of the
body Bi and
Ji = J0
i +mic
2
i . (30)
In the rest of this section we solve the following problem: if the chain parameters
defining the mass distribution of its bodies and the parameter µ are known, find
possible ways for coupling the bodies as well as the initial conditions of their
motion. In other words, given the values of J0
i , J
s
i , mi, and µ, we seek to find
the quantities ci, si, pi, hi, and qi so that the system of relations (23)–(29) is
compatible.
We start our analysis of the abovementioned system with relations (23). From
the first two relation in (23), we have
s2 = s1 and s3 = −s1/µ. (31)
By virtue of (3) and (31), the remaining pair of equations in (23) can be solved
for c2 and c3 as follows:
c2 = − (µm4c4 + m̃2s1) /m2, (32)
c3 = m4 (µc4 + s1) / (µm3) . (33)
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A new class of nonstationary motions of a system of heavy Lagrange tops
Next, we observe that equation (26) can be transformed, by means of (24),
into
µa4
(
J ′
2 − s1a2
)
= a2
(
J ′
4 − s3a4
)
. (34)
After successive substitutions of the expressions for J ′
i , ai, s2, s3, c2, and c3 from
(3) and (30)–(32) into (34) and the first equation in (25), we arrive at the system
of two equations with respect to c4 and s1:
µ2m4 (m4 −m3) c
2
4 + 2µm2
4c4s1 + m̃2m4s
2
1 + µ2m3
(
J0
3 − J0
4
)
= 0, (35)
µm4
(
m2 + µ2m4
)
c24 +m4
[
m2 + µ2 (m̃1 + m̃2)
]
c4s1 + µm̃1m̃2s
2
1+
+ µm2
(
J0
2 + J0
4
)
= 0. (36)
For the sake of brevity, below we consider a special case when
J0
3 = J0
4 . (37)
Then, equation (35) decomposes into the following two cases: s1 = −µc4 or
s1 = µc4(m3 −m4)/m̃2. (38)
In the first case, equation (36) reduces to the form c24 = −m2
(
J0
2 + J0
4
)
×
×
[
µ2m3 (m2 +m3)
]
−1
and cannot be satisfied by the acceptable values of the
quantities it contains.
In the second case, when s1 and c4 are related by (38), equation (36) assumes
the form
c24 = α1, (39)
where α1 = m2m̃2
(
J0
2 + J0
4
)
/ (m3α2) , α2 =
(
µ2 − 2
)
m2m4 − µ2m̃1m3. Clearly,
one can find a real-valued c4 from (39) only when α2 > 0. This inequality is true
if the parameters µ, m3, and m2 are selected such that
µ2 > 2, (40)
m3 <
(
1− 2/µ2
)
m4, m2 > m3 (m3 +m4) /
[(
1− 2/µ2
)
m4 −m3
]
. (41)
(Note that it follows from (41) that
m3 −m4 < 0 and m̃1m3 −m2m4 < 0, (42)
respectively.)
We now consider the equation (see (24)) a1 (J
′
2 − s1a2) = a2 (J
′
1 − s1a2) . Usi-
ng (3), (30)–(32), (38), and (39), we can write it in the form β1c
2
1+β2c1+β3 = 0,
where
β1 = −α2µm1m̃2m3c4,
β2 = m1m̃2m3
[
2J0
2m2m4 − µ2J0
4 (m̃1m3 −m2m4)
]
,
β3 = µc4
[
m̃2m3m4
(
µ2m̃1m3 +
(
2− µ2
)
m2m4
)
J0
1+
+m2m4(m3 −m4)
((
2− µ2
)
m̃1m3 + µ2m2m4
)
J0
2+
+ µ2
(
m2
4 −m2
3
)
(m2 +m3) (m̃1m3 −m2m4) J
0
4
]
.
251
D.A. Chebanov
The discriminant of the last equation D = m1m̃
2
2α3 is nonnegative only when
α3 = −α4J
0
1 + α5 ≥ 0. (43)
Here α4 = 4α2µ
2m2m̃2m3m
2
4
(
J0
2 + J0
4
)
,
α5 =
(
J0
4
)2 [
α6
(
J0
2 /J
0
4
)2
+ α7
(
J0
2 /J
0
4
)
+ α8
]
, (44)
α6 = 4m2
2m
2
4
[
m1m
2
3 + µ2(m3 −m4)
((
2− µ2
)
m̃1m3 + µ2m2m4
)]
,
α7 = 4µ2m2m4
[
m1m
2
3 (m2m4 − m̃1m3) + 2m̃1m2m3m4(m3 −m4)+
+ µ2(m4 −m3)
(
m̃1m̃2m3(m2 +m3)− 2m2
2m
2
4
)]
,
α8 = µ4 (m̃1m3 −m2m4)
[
4m2m4(m2 +m3)
(
m2
4 −m2
3
)
+
+m1m
2
3 (m̃1m3 −m2m4)
]
.
Note that α4 > 0 for all acceptable values of the mechanical parameters.
Therefore, in order to satisfy (43), one should require that
J0
1 ≤ α5/α4. (45)
A meaningful value of J0
1 satisfying (45) can only be selected when α5 > 0.
The sign of α5 coincides with the sign of the quadratic polynomial (in J0
2/J
0
4 ) in
(44). The leading coefficient of this polynomial, α6, is positive when
m1 > µ2(m3 −m4)
[
(µ2 − 2)m̃1m3 − µ2m2m4
]
/m2
3 (46)
and, in accordance with (42), its discriminant D∗ = 16α2
2µ
4m2
2m
2
3m
2
4 (m3 −m4)×
×
[
m1(m̃1m3−m2m4)+ m̃
2
1 (m3 −m4)
]
is always positive. Hence, the polynomial
in J0
2 /J
0
4 in (44) has two real roots and, by choosing
J0
2 > J0
4
(
−α7 +
√
D∗
)
/(2α6), (47)
we obtain α5 > 0.
Based on the above analysis, we can suggest the following algorithm for selecti-
ng the multibody system parameters satisfying the system of equations (23)–(29).
First, we assume that the moments Jc
i (> 0) (i = 1, 2, 3, 4), J0
4 and the mass
m4 are chosen arbitrarily. Then, J0
3 = J0
4 (see (37)). We also assume that µ >
√
2
or µ < −
√
2 (see (40)) and the masses m3, m2, and m1 as well as the moments
J0
2 and J0
1 are successively selected to satisfy the inequalities (41), (46), (47), and
(45). Now, one can find c4 from (39): c4 = ±√
α1. Selecting one of the two possible
values of c4, we obtain s1 from (38) and, then, s2, s3, c2, and c3 from (31)–(33).
We can also compute c1 by the formula c1 =
(
−β2 ±
√
D
)
/(2β1).
Note that, by this moment, we have already obtained the values of all
parameters of interest, except for pi, qi, and hi (i = 1, 2, 3, 4). To find the remai-
ning parameters, we assign arbitrary values to p1, p4, and q4. This immedi-
ately gives us the value of p3 (see (25)). We can also find p2 from (24):
252
A new class of nonstationary motions of a system of heavy Lagrange tops
p2 = a2p1/a1. Next, by means of (28), (24), and (25), we derive that q1 =(
p4J̃
s
4q4 − ã4g + µã1g
)
/
(
µp̃1J̃
s
1
)
, q2 = a2J
s
1q1/ (a1J
s
2 ) , and q3 = Js
4q4/J
s
3 . Sol-
ving the system of equations (27) and (29) with respect to h1 and h4, we obtain
h1 =
(
J ′
1 − s1a2
) [
p̃24 +
(
J̃s
4
)2
q24 −
(
J̃s
1
)2
q21 − µ2p̃21
]
/(1 − µ2),
h4 =
(
J ′
4 − s3a4
) [
p̃24 + µ2
((
J̃s
4
)2
q24 −
(
J̃s
1
)2
q21 − p̃21
)]
/(1 − µ2).
Finally, according to (25), h3 = h4 and h2 can be found from (24): h2 = a2h1/a1.
Knowing the values of the integration constants, one can determine the initial
conditions of motion, using (2), (4), (5), (17), and (18).
Thus, we have proven that it is possible to select physically meaningful values
of parameters characterizing the chain of rigid bodies under consideration so that
the relations (10), (16), and (20) will be fulfilled. This completes our proof on the
existence of the motions of interest for the chain of Lagrange tops.
4. Some properties of the motions of interest. In this section we give
a mechanical interpretation of conditions (22). We recall that On∗+1 is the point
where the subsystems S∗ and S∗ are coupled to each other. Let also C∗ denote
the center of mass of S∗.
Proposition 2. When the system S performs the motion of interest, the poi-
nts On∗+1 and C∗ move along the vertical line L passing through O1.
We denote s∗ = O1On∗+1, c
∗ = O1C
∗,m∗ = m̃n∗
and observe that
s∗ =
n∗∑
j=1
sj =
n∗∑
j=1
sje
(j)
3 ,
m∗c∗ =
n∑
k=n∗+1
mkO1Ck =
n∑
k=n∗+1
mk
(
ck + s∗ +
k−1∑
i=n∗+1
si
)
=
= m∗s∗ +
n∑
k=n∗+1
(mkck + m̃ksk) = m∗s∗ +
n∑
k=n∗+1
ake
(k)
3 . (48)
As was mentioned in section 1, when the system S moves so that the properties
(4) are fulfilled, the skeleton of the subsystem S∗(S
∗) remains in the vertical
plane Π∗(Π
∗). Clearly, the point On∗+1 belongs to the plane Π∗ which rotates
about L with the speed ψ̇1(t). Introducing the Cartesian frame {O1, e3e∗} rigidly
embedded in Π∗ (the unit vector e∗ is chosen such that e∗ · s1 = s1 sin θ1), we
obtain, using the second relation in (22), that s∗ · e∗ =
n∗∑
j=1
sjε1j sin θ1 = 0, i.e.
s∗|| e3 ||L.
Since the attachment point of the subsystem S∗ moves along L, we conclude
that the plane Π∗, containing the point C∗, rotates about L with the speed
253
D.A. Chebanov
ψ̇n(t). Now, in order to prove that c∗||L, it is sufficient to show that the second
term in (48) is parallel to L. Introducing the Cartesian frame {On∗+1, e3e
∗}
rigidly embedded in Π∗ (the unit vector e∗ is chosen such that e∗ · sn =
= sn sin θn), we obtain, using the first relation in (22), that
(
n∑
k=n∗+1
ake
(k)
3
)
·e∗ =
=
n∑
k=n∗+1
akεnk sin θn = 0, which implies c∗|| e3 ||L.
Thus, we have proven that both points On∗+1 and C∗ move along L.
1. Kharlamov P.V. On the equations of motion of a system of rigid bodies // Mekhanika
Tverdogo Tela. – 1972. – 4. – P. 52–73.
2. Kharlamov P.V. Some classes of exact solutions for the problem of the motion of a system
of Lagrange tops // Matematicheskaya Physika. – 1982. – 32. – P. 63–76.
3. Chebanov D.A. On a generalization of the similar motions problem for a system of Lagrange
tops // Mekhanika Tverdogo Tela. – 1995. – 27. – P. 57–63.
4. Chebanov D.A. Exact solutions for motion equations of symmetric gyros system // Multi-
body System Dynamics. – 2001. – 6. – P. 30–57.
Д.А. Чебанов
Новый класс нестационарных движений системы тяжелых гироскопов
Лагранжа с неплоской конфигурацией остова системы
Для цепочки n тяжелых гироскопов Лагранжа, соединенных идеальными сферическими
шарнирами, установлено существование класса нестационарных движений, при которых
остов системы имеет неплоскую конфигурацию. Получены достаточные условия существо-
вания таких движений. Найдена зависимость основных переменных от времени. При за-
данном распределении масс в телах для цепочки, состоящей из четырех тел, определены
способы их сочленения, при которых установленные движения возможны. Указаны неко-
торые свойства новых движений.
Ключевые слова: аналитическая динамика систем тел, гироскоп Лагранжа, нестаци-
онарное движение системы твердых тел
Д.О. Чебанов
Новий клас нестацiонарних рухiв системи важких гiроскопiв Лагранжа
з неплоскою конфiгурацiєю остова системи
Для ланцюжка n важких гiроскопiв Лагранжа, з’єднаних iдеальними сферичними шарнi-
рами, встановлено iснування класу нестацiонарних рухiв, при яких остiв системи має не-
плоску конфiгурацiю. Отримано достатнi умови iснування таких рухiв. Знайдено зале-
жнiсть основних змiнних вiд часу. При заданому розподiлi мас в тiлах для ланцюжка, що
складається з чотирьох тiл, визначено способи їх зчленування, при яких встановленi рухи
можливi. Указано деякi властивостi нових рухiв.
Ключовi слова: аналiтична динамiка систем тiл, гiроскоп Лагранжа, нестацiонарний
рух системи твердих тiл
The City University of New York/LaGCC, New York, USA
dchebanov@lagcc.cuny.edu
Получено 14.11.11
254
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