Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems
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15 Dynamics of the Rigid Solid with a Fixed Point
123
123
aaa
bbb
≥≥
, ,0
ii
ab> , 1, 2, 3i = ,
we can write also
22 22 22 22
11 22 22 33
13
22 22
11 22 22 33
II II
III
II II
ωω ωω
ωω ωω
++
≥≥≥≥
++
.
(15.1.48')
In this subsection, we assume – at the beginning – that
I differs from the principal
moments of inertia and that the ellipsoid of inertia is not of rotation (
1
II> ,
23
II> ,
2
II≠ ). In this case, from the relations (15.1.47) it results
()
()
()
22 3
222
122
11 3
II I
II I
ωβω
−
=−
−
,
()
()
3
22
2
22 3
II I
II I
βΩ
−
=
−
,
()
()
()
21 2
222
322
31 3
II I
II I
ωβω
−
=−
−
,
()
()
1
22
2
21 2
II I
II I
βΩ
−
=
−
.
(15.1.49)
The differential equation (15.1.47') becomes
()()
()()
1223
22222
22222
13
IIII
II
ωβωβω
−−
=−−
,
(15.1.50)
hence a differential equation of first order for the unknown function
22
()tωω= . We
notice that
()()
()()
13 2
22 2
22
22 3 1 2
II I I I
II I I I
ββ Ω
−−
−=
−−
,
so that
()
()
()
22
22 22 2
sign sign sign IIββ ββ−= −= −, where we assume that
22
,0ββ> too.
To fix the ideas, we suppose that
2
II< ; it results
22
ββ<
, while the relation
(15.1.49) allows to state that
22
ωβ≤ , inequality which holds during the motion. We
also point out that
1
0ω = for
22
ωβ=± , while
3
ω preserves a constant sign (because
2
3
0ω > ). We denote (we take into account that
123
III<+; see the relation (3.1.24)
too)
()
()
()
()
22 3 3
2222
12 2
11 3 11 3
II I II I
II I II I
ββΩβ
−−
==<
−−
,
()
()
()
()
21 2 1
2222
32 2
31 3 31 3
II I II I
II I II I
ββΩβ
−−
==<
−−
.
(15.1.49')
In this case, the relations (15.1.49) correspond to two ellipses in the plane
12
Oωω and
in the plane
32
Oωω
, respectively, of equations (
122
0 βββ<<<,
32
0 ββ<<)
303
MECHANICAL SYSTEMS, CLASSICAL MODELS
22
12
22
12
1
ωω
ββ
+=,
22
32
22
32
1
ωω
ββ
+=
(15.1.49'')
and to a hyperbola in the plane
31
Oωω, of equation (
33
0 ββ<<,
12
0 ββ<<)
22
31
22
31
1
ωω
ββ
−=,
()
()
2
22
1
11 2
II I
II I
βΩ
−
=
−
,
()
()
2
22
3
32 3
II I
II I
βΩ
−
=
−
.
(15.1.49''')
Fig. 15.5 The ellipse
1
E (a), the ellipse
3
E
(b) and the hyperbola
2
H
(c)
in the motion of a rigid solid with a fixed point
We assume that at the initial moment
0
tt= we have
0
22
0ωω=≥; to make a choice,
we suppose, as well, that, at this moment,
1
0ω < and
3
0ω > (if
2
ω begins to grow,
starting from the initial moment, we have
2
0ω >
and the equation (15.1.47') shows
that
13
0ωω < ). The point
()
112
,ωωP describes the whole ellipse
1
E , the interior
remaining at the right (Fig. 15.5a), because, from the first equation (15.1.40), it results
that
1
0ω >
if
23
0ωω > ; correspondingly, the point
()
332
,ωωP describes only an
arc of the ellipse
3
E (Fig. 15.5b), because
222
ωββ≤<, while the point
()
231
,ωωP describes also an arc of the hyperbola
2
H
(Fig. 15.5c), because
11
ωβ≤ ,
33
ωβ≤ (for
2
II= the hyperbola degenerates into its asymptotes, the
point
()
31
,ββ being situated on one of them). By separation of variables, from
(15.1.50), one obtains
()( )
22
0
2
2
/
0
222
/
1d
11
z
tt
p
zkz
ωβ
ωβ
−=
−−
∫
,
(15.1.51)
where we have denoted
304
15 Dynamics of the Rigid Solid with a Fixed Point
()()
231
22
123
IIIII
p
III
Ω
−−
=
,
()()
()( )
2
21312
2
123
2
IIII
k
IIII
β
β
−−
⎛⎞
==
⎜⎟
−−
⎝⎠
.
(15.1.52)
Denoting
22
sinωβ κ= and introducing the elliptic integral of the first kind
()
,Fkκ , given by (7.1.41), where κ is the amplitude, while k is the modulus of the
integral, we may write the relation (15.1.51) in the form
()
()
[]
0
0
1
,,tt F k F k
p
κκ=+ −
,
(15.1.51')
where
00
22
sin /κωβ= . Denoting
()
0
uptt=−, we can write
()
()
0
,,uF k F kκκ=−
(15.1.51'')
too. Without any loss of generality, we assume that
0
2
0ω = (the points
0
1
P and
0
3
P
are at the extremities of the minor diameters of the ellipses
1
E and
3
E , respectively). It
results,
()
00
,0Fkκκ==, so that
()
,uF kκ= , (15.1.51''')
where
argu κ=
, amuκ = . Let us introduce Jacobi’s elliptic functions: the sinus
amplitude (
sn sinu κ= ), the cosinus amplitude (cn cosu κ= ) and the delta
amplitude (
22
dn 1 sinukκ=− ); the signs of cnu and dnu are chosen so that
cn dn 1uu== for 0u = . One can introduce the tangent amplitude
(
tanam sn / cnuuu= ) too. We mention also the differential relations
d
sn cn dn
d
uuu
u
=
,
d
cn sn dn
d
uuu
u
=−
,
2
d
dn sn cn
d
ukuu
u
=−
.
We can express the components of the rotation angular velocity vector in the form (we
notice that
0
11
ωβ=− ,
0
2
0ω = ,
0
33
ωβ= )
()
11 0
() cntpttωβ=− −
,
()
22 0
() sntpttωβ=−,
()
33 0
() dntpttωβ=−,
(15.1.53)
where we took into account the relations (15.1.49'') and the second notation (15.1.52).
To
/2κπ= corresponds
22
ωβ=
and we denote
(
)
()( )
1
222
0
d
() ,
2
11
z
TKk F k
zkz
π
== =
−−
∫
,
(15.1.54)
305
MECHANICAL SYSTEMS, CLASSICAL MODELS
where
()Kk is the complete elliptic integral of the first kind. The component
1
ω varies
between
1
β− and
1
β with the period 4T/p (
0
cn0 cos 1κ==,
()
cn cos /2 0pT π==), while the component
2
ω varies between
2
β
and
2
β−
with
the same period (
0
sn0 sin 0κ==
,
()
sn sin /2 1pT π==); the component
3
ω is
varying between the limits
3
β and
()( )
22
32 22 2 32 3
//II I I I Iββ ββ Ω−=− −
with the period
2T/p (dn0 1= ,
2
dn 1pT k k
′
=−=, where k
′
is the
complementary modulus).
If
2
II>
, then we have
22
ββ> too; an analogous study, in which we assume also
that
0
0t = and
0
2
0ω =
, leads to
()
11 0
() dntpttωβ=− − ,
()
22 0
() sntpttωβ=−,
()
33 0
() cntpttωβ=−,
(15.1.55)
with
()()
12 3
22
123
IIIII
p
III
Ω
−−
=
,
()( )
()( )
123
2
2
312
1
IIII
k
II I I
k
−−
==
−−
.
(15.1.55')
The component
1
ω varies between
1
β− and
22
12 2 2
/ββ β β−−
()( )
2112
/II I I I I Ω=− − − with the period 2/Tp; the component
2
ω varies
between
2
β and
2
β− with the period 4/Tp, while the component
3
ω varies
between
3
β and
3
β− with the same period. In this case
()
TKk= .
If we associate the relation
222 2
123
ωωωω++=
(15.1.47'')
to the first integrals (15.1.47), then we can calculate the components of the vector
ω as
functions of
ω in the form (the determinant of the coefficients is of Vandermonde type)
()()
()
23
222
11
1213
II
IIII
ωωγ=−
−−
,
()()
()
31
222
22
2321
II
IIII
ωωγ=−
−−
,
()()
()
12
222
33
3132
II
IIII
ωωγ=−
−−
,
(15.1.56)
where we have introduced the notations
()
23
22
1
23
II I I
II
γΩ
+−
=
,
()
31
22
2
31
II I I
II
γΩ
+−
=
,
()
12
22
3
12
II I I
II
γΩ
+−
=
.
(15.1.56')
306
15 Dynamics of the Rigid Solid with a Fixed Point
Multiplying the first equation (15.1.40) by
11
/Iω , the second one by
22
/Iω and the
third one by
33
/Iω and summing, it results
23 3112
11 22 33 123
123
II IIII
III
ωω ωω ωω ωωω
−−−
⎛⎞
++= + +
⎜⎟
⎝⎠
or
()()()
122313
123
123
IIIIII
III
ωω ω ω ω
−−−
=
.
(15.1.57)
Taking into account (15.1.56), we obtain, finally,
()
()()()
2
222222
123
d
22
d
t
ω
ωω ωγωγωγ
==±−−−
,
(15.1.57')
where we take the sign
+ or the sign − as
2
ω increases or decreases in time. The
magnitude of the angular velocity vector
ω can be thus expressed by means of
Weierstrass’s elliptic function
()tP , having thus a real period; we notice that
2
ω has
values contained between
()
22
13
min ,γγ and
2
2
γ . Unlike F. Lindemann, who uses
Jacobi’s functions, J. Haug presents thus another approach (with a certain symmetry
character) of Euler’s problem for the rigid solid with a fixed point, which – obviously –
leads to the same results.
15.1.2.2 Determination of the Position of the Rigid Solid
To specify the position of the movable frame of reference
R (hence, of the rigid solid)
with respect to the fixed frame
′
R it is sufficient to determine Euler’s angles ψ, θ and
ϕ. We notice that, during the motion, the moment of momentum with respect to the
pole
O, in the inertial frame of reference, is conserved in time ( const
O
′
=
K ); without
any loss of generality, we can choose the
3
Ox
′
-axis along the direction of this vector, so
that
3
OO
K
′′′
=Ki. The direction cosines of the unit vector
3
′
i with respect to the frame
R are thus /
i
Oi O
KKα
′′
= , 1, 2, 3i = ; taking into account the relations (5.2.36) and
the relations
11
1
O
KIω
′
= ,
22
2
O
KIω
′
= ,
33
3
O
KIω
′
= ,
O
KIΩ
′
= , we may write
11
sin sinIIωΩθϕ= ,
22
sin cosIIωΩθϕ= ,
33
cosIIωΩθ= .
(15.1.58)
The angles
θ and ϕ are obtained easily, being given by
22 22
11 22
33
tan
II
I
ωω
θ
ω
+
=
,
11
22
tan
I
I
ω
ϕ
ω
= .
(15.1.59)
307
MECHANICAL SYSTEMS, CLASSICAL MODELS
To calculate the angle of precession
ψ, we use the first relation (14.1.15); taking into
account the previous relations, we can write
22
22
12 1122
2
12
sin cos
sin cos
sin
sin
II
I
II
I
ωϕω ϕ ω ω
ϕϕ
ψΩ
θ
Ωθ
++
⎛⎞
==+=
⎜⎟
⎝⎠
()()
()
22 22
2
11 22 33
3
22 22 2 2 22 2 2 22
3
11 22 33 33
10
II I II I
III
I
I
II II II
Ωω ω ΩΩ ω
Ω
Ω
ωω Ωω Ωω
+−
−
⎡⎤
===−>
⎢⎥
+− −
⎣⎦
(15.1.59')
and the angle
ψ is obtained by a quadrature.
Assuming that
2
II< , we use the solution (15.1.53); there result G.R. Kirchhoff’s
formulae
()
()
()
31
0
13
cos dn
II I
pt t
II I
θ
−
=−
−
,
()
()
()
()
12 3 0
21 3 0
cn
tan
sn
II I ptt
II I ptt
ϕ
−−
=−
−−
,
()()
[
]
()()()
2
23 12 0
2
12 3 31 2 0
sn
sn
II I I I pt t
II I II I ptt
ψΩ
−+ − −
=
−+ − −
(15.1.59'')
()()
()()()
1323
2
3
12 3 31 2 0
10
sn
IIII
I
I
II I II I ptt
Ω
−−
⎡⎤
=− >
⎢⎥
−+ − −
⎣⎦
.
To be precise, we notice that
cosθ
is varying between the inferior limit
()( )
31 1 3
/kII I II I
′
−−
()( )
32 2 3
/II I II I=− −
and the superior one
()( )
31 1 3
/II I II I−− (obviously, both subunitary), tan ϕ varies on the whole
real axis, while
ψ
is varying between the inferior limit
1
/IIΩ and the superior one
2
/IIΩ . Such considerations are due to N. Lindskog and W. von Tannenberg. From the
second relation (14.1.15), we get
() ()
1212 1212
22 22
11 22
sin
II II
I
II
ωω ωω
θ
Ωθ
ωω
−−
=− =−
+
.
Hence,
()
12
sign signθωω=−
and (4)()tT tθθ+=
. It results thus that
(4)()consttT tθθ+=+ ; but cosθ is a periodic function of time, of period 2T (the
period of the function
dnpt), which never vanishes (
dn 0pt ≠
). Hence, the angle of
nutation is periodic too (
(2)()tT tθθ+=). In what concerns the function tanϕ , this
one is periodic with respect to time, of period
4T (the period of the functions cnpt and
snpt), and can take the value zero too; from (15.1.58) it results that ()tϕ can vary by
±2π. But, from the third relation (14.1.15) we obtain
308
15 Dynamics of the Rigid Solid with a Fixed Point
()
() ()
[
]
22
22
11 22
11 2 1 22 3 2 3
3
22 22
11 22
cot
0
sin
II
II I II I
I
II
ωωθ
ωωω
ϕω
Ωθ
ωω
+
−+−
=− = >
+
;
hence,
3
sign signϕω=
; we have (4) ()tT tϕϕ+=
, wherefrom
(4) ()2tT tϕϕπ+= + (because
3
0ω > , the function ()tϕ is an increasing one),
the angle of proper rotation growing continuously with
2π. In what concerns the angle
of precession, we notice that
(2) ()tT tψψ+=
, so that
0
(2) () 0tT tψψψ+− =>,
0
constψ = ( 0ψ >
, hence ψ is an increasing function); the angle of precession is no
more a periodic function of time (as a matter of fact, as well as the angle of proper
rotation).
Fig. 15.6 Euler’s angles on a unit sphere of centre O
Let
Q be the trace of the
3
Ox -axis on a sphere of centre O and unit radius (to fix the
ideas); if
12
Ox x
′′
is the equatorial plane, then the angle θ is the colatitude, while the
angle
/2ψπ− is the longitude with respect to the meridian plane
13
Ox x
′′
(Fig. 15.6).
During the motion of the rigid solid, the point
Q (hence, the axis
3
Ox ) has a motion of
nutation, coming back on the same parallel circle, after a period
4T, as well as a motion
of precession, which brings no more
Q on the same meridian (after a period 2T or 4T).
On the other hand, the motion of proper rotation about the
3
Ox
-axis contributes to the
growing of the corresponding angle by
2π. Finally, we see that after the period 4T
(which is period for all the three components
i
ω , 1, 2, 3i = , with respect to the
movable frame of reference) the rotation angular velocity vector
ω takes the same
position with respect to the rigid solid (with respect to the movable frame
R ), but not with
respect to the fixed frame
′
R
, because neither the frame R (hence neither the rigid
solid) does not come back at the same position with respect to the frame
′
R .
309
MECHANICAL SYSTEMS, CLASSICAL MODELS
We may put in evidence the poles of the function ψ
in the form (15.1.59''),
introducing a constant argument
ic, i1=−, c ∈ , specified by one of the
equivalent relations
()
()
12 3
2
31 2
sn i
II I
c
II I
−
=−
−
,
()
()
21 3
2
31 2
cn i
II I
c
II I
−
=
−
,
()
()
13
2
31
dn i
II I
c
II I
−
=
−
.
Taking into account the notation (15.1.52), we obtain (we can choose the sign of the
argument
ic so that to take the sign + in the right member)
()()
()
1323
3312
isni cni dni
IIII
I
ccc
pI I I I
Ω
−−
=
−
.
It results (we denote
ptτ = )
22
3
d
isni cni dni
d
sn i sn
Iccc
pI
c
ψ
Ω
τ
τ
=+
−
.
To can integrate this equation with separate variables, we introduce the theta functions
()
,
jj
vϑϑ χ= , 1,2, 3, 4j = , as solutions of the partial differential equations
2
2
4
j
j
v
∂ϑ ∂ϑ
π
∂χ
∂
=
, 1,2, 3, 4j = ,
(15.1.60)
in the form
()
()
()
()
2
21/4
1
1
1
,21 sin21
n
n
n
vqnvϑχ π
∞
−
−
=
=− −
∑
,
()
()
()
2
21/4
2
1
,2 sin21
n
n
vq nvϑχ π
∞
−
=
=−
∑
,
()
2
3
1
,12 cos2
n
n
vqnvϑχ π
∞
=
=+
∑
,
()
()
2
4
1
,121 cos2
n
n
n
vqnvϑχ π
∞
=
=+ −
∑
,
(15.1.60')
where
eq
πχ−
= , while /KKχ
′
= , ()Kk being the complete elliptic integral of first
kind (15.1.54) and
() ( )Kk Kk
′′
= ,
2
1kk
′
=−. We notice that ()kχχ= , being
thus constant for a fixed
k; in this case, ()
ii
vϑϑ= , 1,2,3,4i = . We introduce also
the functions
4
() ()uvΘϑ= ,
1
() ()Hu vϑ= ,
13
() ()uvΘϑ= ,
12
() ()Hu vϑ= , /2vuK= ;
(15.1.60'')
310
15 Dynamics of the Rigid Solid with a Fixed Point
()
1
sn
()
Hu
u
u
k
Θ
=
,
1
()
cn
()
Hu
k
u
k
uΘ
′
=
,
1
()
dn
()
u
uk
u
Θ
Θ
′
=
.
(15.1.60''')
One can show that
()()
() ()
2
12 12
22
12
22
12
(0)
sn sn
Hu u Hu u
uu
k
uu
Θ
ΘΘ
−+
−=
.
Calculating the logarithmic derivative with respect to
1
u , we can write
111
22
12
sn cn dn
2
sn sn
uuu
uu−
()
()
1
11
d
2
d
u
uu
Θ
Θ
=−
()
()
12
12 1
d
1
d
Hu u
Hu u u
−
+
−
()
()
12
12 1
d
1
d
Hu u
Hu u u
+
+
+
.
Putting
1
iuc=
,
2
u τ= , introducing the real constant
3
d(i)
i
d
(i )
c
I
pI u
c
Θ
Ω
λ
Θ
=−
(15.1.61)
and integrating with respect to
τ (we assume that the axes are chosen so that
()
00ψ = ), we obtain the angle of precession in the form (
()
0
pt tτ =−)
()
()()
()()
0
0
0
i
i
() ln
2i
Hpt t c
tptt
Hpt t c
ψλ
−+
=−+
−−
.
(15.1.61')
The position of the rigid solid with respect to the inertial (fixed) frame of reference
′
R is thus entirely specified by Euler’s angles ψ, θ and ϕ. The sines and the cosines of
these angles are uniform functions of time or square roots of such functions; e.g.,
()()()()
00
sin i i
()
v
Hpt t cHpt t c
pt
θ
Θ
=−+−−
.
(15.1.61'')
These results have been obtained by O.I. Somov in a form appropriate to that above.
Jacobi calculated the direction cosines of the axes of the non-inertial frame of reference
R with respect to the inertial frame
′
R
as uniform functions of time. Other results
have been given by A. Cayley and A.G. Greenhill.
We calculate, after R. Grammel, the mean values of Euler’s angles, as well as their
variations around these values. Thus, the second relation (15.1.59'') can be written in
the form (we denote
22 22
22 11
/IIβββ= )
()
()
()
()
()
()
()
()
()
()
0
10
21 3 0 0
11 3 0 0
sn sn
cot
cn cn
Hpt t
Hptt
II I ptt ptt
II I ptt ptt k
β
ϕβ
−
−
−− −
=− =− =−
′
−− −
,
in this case, we have
311
MECHANICAL SYSTEMS, CLASSICAL MODELS
() () ()
() () ()
26
000
26
000
sin sin 3 sin 5 ...
cot
cos cos 3 cos5 ...
tt q tt q tt
k
tt q tt q tt
ϕϕϕ
ϕϕϕ
ωωω
β
ϕ
ωωω
−− −+ −−
=−
′
−+ −+ −+
,
with
/2pK
ϕ
ωπ= ; neglecting the powers of q, we obtain the approximate formula
()
0
cot tan tt
k
ϕ
β
ϕω
=− −
′
.
By differentiation, we get
()
()
[
]
22
0
1cot / 1tanktt
ϕϕ
ϕϕ ω β ω
′
+= + −
,
wherefrom
2()
k
Rt
k
ϕ
β
ϕω
β
′
=
′
+
,
()
[]
()
()
1
00
1
() 1 cos2 1 1 cos 2
nn
n
n
Rt tt tt
ϕϕ
εω ε ω
∞
−
=
=+ − =+ − −
∑
,
because
1ε < ,
()()
/kkεββ
′′
=− +
; using the formulae
()
[]
1
21 2
21
0
cos 2 cos 2 1
n
j
nn
n
j
Cnjαα
−
−−
−
=
=−−
∑
,
()
1
212
2
2
0
1
cos 2 cos2
2
n
j
nnn
n
n
j
CC njαα
−
−
=
⎡
⎤
=+ −
⎢
⎥
⎣
⎦
∑
,
where the symbol
p
m
C represents the number of combinations of m things p at a time,
we obtain
()
()
()
0
00 0
1
1
()d 1 sin2
2
t
n
n
t
n
RAtt Antt
ϕ
ϕ
ττ ω
ω
∞
=
=−+ − −
∑
∫
,
the coefficients of the expansion into a series being given by
2
02
2
0
1
2
AC
νν
ν
ν
ν
ε
∞
=
=
∑
,
2
2
12
0
11
22
n
n
n
n
AC
n
νν
ν
ν
ν
ε
∞
+
+
−
=
=
∑
, 0n > .
For the first coefficients it results
0
2
1
2
1
k
A
k
β
β
ε
′
+
==
′
−
,
()
10
2
1
AA
ε
=−,
210
1
AAA
ε
=−,
so that one can use a recurrence formula for the other coefficients. Finally, the proper
rotation is given by
()
00
() ()tttt
ϕ
ϕϕω ϕ=+ − + ,
()
()
0
1
() 1 sin2
n
n
n
k
tAntt
k
ϕ
β
ϕω
β
∞
=
′
=− −
′
+
∑
,
so that
(15.1.62)
312