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

as well as

() consttωω== ,

ωωω= ,

200

ωωω= ;

ω , 1, 2,3i = , are the

components of the rotation angular velocity vector at the initial moment

tt= .

As a matter of fact, the first integrals (15.1.47) take the form

()

22 2 22 22

12 33

JIIωω ω Ω++ =

()

22 2

12 33

JIIωω ω Ω++ =,

wherefrom

()

() ()

22 2 2 0 0

12 1 1 2

II I

JJ I

ωω Ω β ω ω

−

+= == +

−

()

2220

333

IJ I

ωΩβω

−

===

−

(15.1.79''')

We are thus led to the same result. We can write

()

JIJI

JII

Ω

−−

(15.1.79

)

in this case; we have

0k = too, so that the elliptic functions in (15.1.53) become

circular functions (

dn 1pt = ,

cn sn 1pt pt+=). We notice also that

()

22 222

120

()

JI I

ω Ω γγω

+−

= ===,

()

()( )

33 0

cos , const

JJ I

JI JI I

ωω

−

== ==

−+−

i ;

hence, the instantaneous angular velocity ω is constant in magnitude and makes a

constant angle with the axis of symmetry

Ox of the ellipsoid of inertia relative to the

fixed point

O. As well, assuming that the

′

-axis is along the fixed direction of the

moment of momentum

′

K , the formula (15.1.46) shows that the angle formed by the

vector

ω with the fixed axis

′

is constant too. Hence, the polhodic cone

C is a

cone of rotation of equation

()

22 2

12 3 33

JJ I x x I I I x−+=−,

(15.1.78')

the axis of symmetry being

, while the herpolhodic cone

C is a cone of rotation

too, for which the axis of symmetry is

′

(Fig. 15.15a). As a consequence, the

polhodes are circles, intersections of the cone (15.1.78') by the planes

constx =

. As

well, the herpolhodes are circles too, specified by the radius (we notice that

max

min 1 2

ρρδδ===

, the circular annulus being reduced to a circle, which is just the

herpolhode)

()

/KJIII IJIρ =−−

. From (15.1.72') one observes that

δδ< , constδ = , while the formula (15.1.71) leads to

333

MECHANICAL SYSTEMS, CLASSICAL MODELS

()

22 2

//constIJκΩρ δ ρ Ω=− = =



, so that the motion of the pole P on the

herpolhode is uniform; obviously, the motion of the pole

P on the polhode has the same

property, because the polhode rolls without sliding over the herpolhode.

Fig. 15.15 The polhodic and herpolhodic exterior tangent cones (a).

The canonical decomposition of the vector ω

The relations (15.1.58) become

() ()

[

]

11 02 0

cos sin sin sinJJ ptt ptt Iωω ω Ωθϕ=−+−=,

() ()

[

]

22 01 0

cos sin sin cosJJ ptt ptt Iωω ω Ωθϕ=−−−=,

33 33

cosIIIωωΩθ== .

Hence, it results

()

33 3

cos ( ) cos

IIJI

θθ

Ω

−

== =

−

(15.1.80)

the angle of nutation

θ between the axes of the two cones being thus constant

(

()tθθ= ). Because the plane which passes through OP and is tangent to the two

cones is normal to the meridian planes formed by

OP with the

′

-axis and with the

-axis, respectively, it results that these meridian planes coincide; consequently, in

its motion, the

-axis describes a cone having as axis the support of the vector

′

(Fig. 15.15a).

Using the notations

() ()

00 0

11 2 1

sin sinωω ωγβγ=+ =

() ()

00 0

21 2 1

cos cosωω ω γβγ=+ =

tan

= ,

(15.1.81)

334

15 Dynamics of the Rigid Solid with a Fixed Point

we get

()

tan tan pt tϕγ=−+, wherefrom

()

()tpttϕϕ=−+,

arctan

(15.1.81')

the motion of proper rotation (about the

Ox -axis) being uniform. If we make

IIJ==

in (15.1.59'), then we obtain /IJψΩ=



; we notice that ψκ=



, which

was to be expected. We are thus led to

()

ttt

ψΩ ψ=−+,

(15.1.81'')

corresponding to the uniform motion of precession (about the

′

-axis), called also

regular precession. These results allow the canonical decomposition of the angular

velocity vector

ω in the form (we use another notation, different from that

corresponding to the decomposition in the frames of reference

′

and R and we

notice that

0θ =



; as a matter of fact, the relations

cosωωωθ

′

cosωωωθ

′′

)

33 33

ωωψϕ

′′′ ′

=+= +=+



ωω ω ii ii

(15.1.81''')

where the vector

′

is constant, while the vector ω is constant only in magnitude (Fig.

15.15b). If we make

IIJ== in the formulae (15.1.62)–(15.1.62''), then we find

0k = , 1k β

′

, hence 0ε = , /2K π= , so that p

ω = , 0q = ; in this case, the

proper rotation, the precession and the nutation are reduced to their mean values, which

correspond to the values given by (15.1.80), (15.1.81'), (15.1.81''). As closer we are

from the case of symmetry considered above (or as the motion of the rigid solid is

closer to a stable rotation about the

′

-axis) as quicker tends to zero the parameter q

(hence, as better is the approximation made by deducing the formulae (15.1.62)–

(15.1.62'')). In exchange, the error growth as we are closer to the particular case

1k = ,

hence as the motion is closer to a (labile) rotation about the mean principal axis of

inertia. In the case considered by us, the ellipsoid of rotation is a prolate spheroid (the

major axis of the ellipsoid is axis of symmetry); we notice that, in this case,

0pϕ =>



and

0ψ >



, the polhodic and the herpolhodic cones being exterior tangent. The sense

of motion of the pole

P on the polhode is indicated by the relation (15.1.77'), which

leads to

pχϕ=− =−



(result which had to be expected), while the sense of the

motion of the same pole on the herpolhode is specified by the relation

ψκ=



(Fig.

15.15a).

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MECHANICAL SYSTEMS, CLASSICAL MODELS

Fig. 15.16 Permanent rotation about the stable major axis

Ox and about an arbitrary

principal axis in the equatorial plane (b) of the prolate spheroid

II J=<, then we obtain

ωΩ=

()

/pJI JΩ=− ,

0ωω==

, so

that

() () 0ttωω==,

() ()ttωωΩ==; then, cos 1θ = , hence 0θ = and the

axes

Ox and

′

coincide. The angle ϕ is non-determinate (it can be taken equal to

zero), while the angle of precession is given by

()

0 30

() /t IttJψ ψΩ−= − , the

precession being uniform. The polhode and the herpolhode are reduced to the point

(we have

0ρ = ). The motion is a permanent rotation about the major axis

Ox of the

prolate spheroid, which is a stable axis of rotation (Fig. 15.16a); this is the case of the

gyroscope, which will be studied in the next chapter. If

IJI=>, then we get

0ω =

() ()

0022

120

ωωωΩ+== and

0p =

; as well,

sinωΩϕ=

cosωΩϕ= and

ϕϕ= . Then,

()

()tttψΩ ψ=−+, the precession being

uniform; we notice that

cos 0θ = , hence /2θπ= , as well as 0p = too. The motion

is a permanent rotation about an arbitrary principal axis in the equatorial plane of the

prolate spheroid, which coincides with the

′

-axis, being thus fixed in space

(Fig. 15.16b); the polhode and the herpolhode are reduced to a point

A on the equator

of the prolate spheroid and in the plane

Π tangent to the ellipsoid at this point. At a

small perturbation of the position of the instantaneous axis of rotation, the herpolhode

will be a curve in the vicinity of the point

A with

max

ρ of the same order of magnitude;

in exchange, the polhode will be a circle parallel to the equatorial circle and close to

this one. Because, in the rigid solid, the axis moves much away from the initial position,

it results that this one is a labile axis of rotation.

In the case in which

123

IIIIJ>> = =, the axis of symmetry of the ellipsoid

being its minor axis, the latter one is an oblate spheroid of equation

()

2222

11 2 3

Ix J x x K++=.

(15.1.82)

We can make a study analogue to that above, obtaining

336

15 Dynamics of the Rigid Solid with a Fixed Point

()

II J

ωω Ω

−

() ()

()

22 0 0 2

23 2 3

() ()

II I

JI J

ωω ω ω Ω

−

+= + =

−

(15.1.82')

and also

() ()

22 03 0

() cos sintptt pttωω ω=−−−,

() ()

33 02 0

() cos sintptt pttωω ω=−+−,

(15.1.82'')

where

()

IJ IJIJ

JJ II

ωΩ

−−−

(15.1.82''')

Fig. 15.17 The polhodic and herpolhodic interior tangent cones (a).

The canonical decomposition of the vector ω

The

Ox -axis is an axis of symmetry for the polhodic cone, while the

′

-axis is an

axis of symmetry for the herpolhodic cone (Fig. 15.17,a); the fixed angle made by the

two axes is given by

()

31 11

cos , sin cos /IIθϕ ω Ω

′

==ii

()

/II J II J=− −.

The angular velocity vector

ω is decomposed in the form (Fig. 15.17b)

3131

kpωω

′′′ ′

=+= + = −



ωω ω iiii,

(15.1.83)

where

/0IJκΩ=>



, and, if we take into account (15.1.77), then

0p χ=>



; thus,

the motion of the point

P on the herpolhode and on the polhode is specified

(Fig. 15.17a). From (15.1.82), (15.1.82'), it results

()

() /tI IJIIJωΩ=+−

ω= . The relation (15.1.46) allows to write

()

cos , / / constΩω Ωω

′

== =ω i ; as

well, we have

()

11 10

cos , / / constωω ω ω

′

== =ω

i . Noting that

ωΩ< , it results

()

cos , cos ,

′

<ωωii, wherefrom

()

′

>ωωii. In conclusion, the

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MECHANICAL SYSTEMS, CLASSICAL MODELS

herpolhodic cone and the polhodic cone are interior tangent, the first one being interior

to the second one. The minor axis of the spheroid (axis of symmetry) is stable in case of

a permanent rotation about it; in exchange, the axes in the equatorial plane are labile

axes of rotation.

123

IIIIJ====, then the ellipsoid of inertia is a sphere, any axis of it

being a principal axis of inertia, as well as an indifferent axis of rotation.

15.1.2.9 Another Geometric Representation of the Motion after Poinsot

The geometric interpretation given by Poinsot to the problem of Euler is remarkable by

its simplicity and clarity, as well because it is an intuitive method; in exchange, this

interpretation does not put time in evidence, and there is no one element to vary in

direct proportion to the variable

t. To eliminate partially this lack, Poinsot imagines a

new geometric representation, starting from the decomposition of the angular velocity

vector

ω in two components: ω

along the direction of the moment of momentum

′

and

⊥

ω along a direction normal to this one; the vector ω and its components are

contained in a plane specified by the

Ox-axis, normal to this plane (Fig. 15.18). We

notice that

const=





(we have Ω=ω

too), while

⊥

ω varies both in direction

and in modulus, being contained, for any

t, in the fixed plane

, normal to the vector

′

K (its locus with respect to the frame of reference

′

), which will be called

herpolhodic plane (degenerate herpolhodic cone). With respect to the frame

R, the

locus of the component

⊥

ω will be a second polhodic cone

⊥

C of Poinsot.

Fig. 15.18 The herpolhodic plane

Let

D be a point on the support of the vector ω, while A and B are its projections on

the supports of the vectors

and

⊥

ω , respectively. We denote the co-ordinates of the

point

B by

x and the co-ordinates of the point D by

λω

, constλ = , 1, 2, 3i = , with

respect to the frame of reference

R; we notice that the quantities

I ω

may be used as direction parameters of the straight line

OA, so that we can write

338

15 Dynamics of the Rigid Solid with a Fixed Point

()

11 1

xIωλ λ

′

=+,

()

22 2

xIωλ λ

′

=+,

()

33 3

xIωλλ

′

=+, constλ

′

= . The

equation of the plane

Π is written in the form (orthogonality condition)

111 222 333

0Ix Ix I xωω ω++=. (15.1.84)

Eliminating the co-ordinates

123

,,xxx and taking into account (15.1.47), we get

0Iλλ

′

+=. Whence, it results

()

111

xIIλω

′

=−,

()

222

xIIλω

′

=−,

()

333

xIIλω

′

=−; the relation (15.1.66), which is verified by the components of the

vector

ω, allows now to write the equation of the second polhodic cone

⊥

C of Poinsot

in the form

123

222

123

III

xxx

II II II

++=

−−−

(15.1.84')

this one being thus a cone of second degree. In the motion of the rigid solid, the cone

⊥

C is rolling with sliding on the plane

Π , because it is rotating together with the

rigid solid about the

′

-axis with the constant velocity Ω; assuming that the plane

Π has a uniform motion of rotation, of angular velocity Ω, about the

′

-axis, the

rolling of the cone

⊥

C is slidingless.

In this last case, in the motion of the rigid solid, the angle of rotation in the plane

is in direct proportion with the interval of time

tt−

; associating to the plane

a hand along the

Ox-axis, we can measure the time by means of the motion of this hand

on a fixed dial. If, by a clock mechanism, the plane

Π would have a uniform rotation

about the

′

-axis (with the angular velocity Ω), transferring a motion to the cone

C (by

friction or by some gearing), the rigid solid (rigidly linked to this cone) would have an

inertial motion about the fixed point

O. The building up of the Darboux-Koenig

herpolhodograph, which allows to obtain a complete kinematic image of the motion of the

rigid solid, is based on these considerations. But the practical sensibility of the apparatus is

relatively small, the ellipsoid of inertia

E being not an arbitrary one, because its semi-axes

must verify the relations (3.1.96); indeed, one can show that the ratio of the smallest to the

greatest magnitude of the angular velocity varies thus between

2/2

and 1. As a matter of

fact, the magnitude of the angular velocity has thus a certain stability and the herpolhode

has not points of inflection.

15.1.2.10 Poinsot Type Motions. The Sylvester Theorems. Volterra’s Problem

/bKI= ,

/bKI= are the squares of the semi-axes of the

ellipsoid of inertia, then the equation (15.1.63) reads

222

123

xxx

bbb

++=,

(15.1.85)

where

123

bbb<<; in this case, Euler’s equations (15.1.40) take the form

339

MECHANICAL SYSTEMS, CLASSICAL MODELS

11 23

ωωω

⎛⎞

+− =

⎜⎟

⎝⎠



22 31

ωωω

⎛⎞

+− =

⎜⎟

⎝⎠



33 12

ωωω

⎛⎞

+− =

⎜⎟

⎝⎠



(15.1.86)

while the two first integrals (15.1.47) read

222

22 2

123

222 2 2

123

bbb K b

ωωω

ΩΩ

′

++= = =

222

123

2TI

bbb b

ωωω

ΩΩ

′

++= = =

(15.1.86')

where

b is the square of the distance from the centre O to the fixed plane Π over which

is rolling the ellipsoid of inertia

E, given by (15.1.16'). The conditions

123

111

bbb

231

111

bbb

312

111

bbb

(15.1.86'')

corresponding to the conditions (3.1.103), must be verified, as well as the condition

bbb<< ; only two of the three differences

bb− ,

bb− can have the same

sign. We notice that the relation (15.1.15') allows to pass from the equation (15.1.85) to

the second first integral (15.1.86'). The loci of the point

P of tangency of the ellipsoid E

with the plane

Π on the ellipsoid and on the plane are, obviously, the polhode and the

herpolhode, respectively.

Let us suppose now that the magnitudes

b ,

b and

b do not fulfil the conditions

(15.1.86''), possibly being negative too (the ellipsoid (15.1.85) may be replaced by a

unparted or by a parted hyperboloid, being, in general, a quadric with centre). In this

case, the equations (15.1.86) lose their dynamical signification, no more representing

the inertial motion of the rigid solid; but they keep their kinematical signification,

corresponding to a motion of the rigid solid, the geometric image of which is

represented by the rolling without sliding of a quadric

Γ with centre over one of its

tangent planes

Π, fixed in space. A relation of the form OP μ=





ω , constμ = takes

further place. The polhode

P will be the locus of the points which are on the quadric Γ

and for which the planes

Π, tangent to the quadric at various points of this curve, are at

a constant distance from the centre

O of the quadric. Such a motion is called a Poinsot

type motion; it takes place only if two of the products

()()

bb bb−−,

()()

bb bb−−,

()()

bb bb−− are negative, the third one being positive, as it

results from the above inequalities. One observes that we have introduced the quantities

340

15 Dynamics of the Rigid Solid with a Fixed Point

123

,,bbb (instead of the semi-axes

123

,,aaa) to can pass from Poinsot’s motion to a

motion of Poinsot type. Considerations in this direction are due to A. Clebsch, E.J.

Routh and P. Appell.

Let be two motions of Poinsot type with the same centre and with the same principal

axes of inertia, characterized by the principal moments of inertia

123

,,III and

123

,,III and by the angular velocities ω and ω , respectively; two such motions for

which takes place the relation

+ωω=0 are called conjugate motions in the sense of

Darboux. Obviously, two such motions have the same polhodes. One can show that to

any motion of Poinsot type kinetically possible corresponds always a conjugate motion

in the sense of Darboux, also kinetically possible.

Assuming that the motion of Poinsot type is a direct motion, we can give a geometric

interpretation to the inverse motion of Poinsot type, considered as a slidingless motion

of a movable plane

Π over a quadric with centre Γ, fixed in space, to which it is

tangent, remaining at a constant distance from this centre. Obviously, we can represent

this motion as a rolling without sliding of the herpolhodic cone

C (movable now)

over the polhodic cone

C (fixed now). In case of conjugate motions in the Darboux

sense, the two quadrics

Γ and Γ are pierced along the same polhode. The

corresponding inverse motions of Poinsot type can be represented by the rolling of the

herpolhodic cones

C and

C , rigidly connected to these quadrics, over the same

polhodic cone

C ; the cones

C and

are tangent to the cone

C along the same

generatrix (support of the vector

ω), hence they are tangent one to the other. The motion

of a tangent plane

Π with respect to a tangent plane Π is thus reduced to the rolling of

the herpolhodic cone

C over the herpolhodic cone

C ; the angular velocities of the

planes

Π and Π with respect to the inertial frame of reference are ω and −ω,

respectively, so that – in the relative motion of the two planes – the angular velocity

will be

±2ω. One can state

Theorem 15.1.13 (J.J. Sylvester) If at all the points of the polhode P we take equal

segments of a line along the normals to the quadric

Γ, their extremities will be on a new

polhode

′

P , on another quadric Γ

′

, homofocal and homothetic to Γ and orthogonal

to the set up normals.

As a kinetic consequence of this theorem, one can state

Theorem 15.1.13' (J.J. Sylvester) Let be a rigid solid which has a Poinsot type motion

and to which has been imparted an angular velocity about the normal to the fixed plane

Π over which the quadric Γ is rolling; the compound motion will be a motion of

Poinsot type too, the new plane of rolling

′

being parallel to the plane Π.

The corresponding quadric

′

differs, obviously, from the quadric Γ.

Another generalization of the Euler-Poinsot problem is due to V. Volterra, which

considered, in 1895, the system of equations

341

MECHANICAL SYSTEMS, CLASSICAL MODELS

()

11 3 2 23 32 23

0III ccωωωωω+− + − =



()

22 1 3 31 13 31

0III ccωωωωω+− + − =



()

33 2 1 12 21 12

0III ccωωωωω+− + − =



(15.1.87)

where

c ,

c and

c are constant quantities. One obtains the first integrals

222

11 22 33

2III Tωωω

′

++=,

()( )( )

222

11 1 22 2 33 3

Ic Ic I ccωωω++ ++ +=,

(15.1.87')

c being an integration constant. Starting from these results and from one of the

equations (15.1.87), the integration of this differential system is reduced to quadratures,

as in Sect. 15.1.2.1.

As well, in case of a moment

M along the direction of the moment of momentum

′

K , so that

′

=MK, where λ is a constant scalar (

MIλω= ,

MIλω= ), A.G. Greenhill and E. Padova performed a change of independent

variable and of unknown functions of the form

()

ln 1ttλλ=+,

()

tωλω=+ ,

1,2, 3

i = , finding again the homogeneous equations of the Euler-Poinsot case for the

unknown functions

()

tωω= ,

1,2, 3

i =

15.2 The Case in which the Ellipsoid of Inertia is of Rotation.

Other Cases of Integrability

In what follows, we study the Lagrange-Poisson case of integrability and the Sonya

Kovalevsky one, for which the ellipsoid of inertia is of rotation (it is a prolate spheroid,

having

IIJ==). We consider then also other cases of integrability, corresponding

to particular initial conditions, as well as other cases of loading in the dynamics of the

rigid solid with a fixed point.

15.2.1 The Lagrange-Poisson Case

Returning to the problem of the rigid solid with a fixed point, subjected to its own

weight

G, we will consider the research made by J.-L. Lagrange in 1788; in this case,

the ellipsoid of inertia is of rotation, the centre of mass

C being situated on its axis of

symmetry (

12 3

IIJI==>, hence the case of a prolate spheroid, and

0ρρ==,

0ρ > ). The problem has been considered again, in 1815, by S.-D. Poisson, without

taking into account the results due to Lagrange and without quoting him; this case of

integrability (considered to be the IInd case of integrability) is called the Lagrange-

Poisson case. Such a situation is encountered, for instance, in case of a homogeneous

rigid solid of rotation around the

Ox -axis; as well, we are situated in such a case if the

solid is non-homogeneous, having a geometric as well as a mechanical axial symmetry

(the density is of the form

()

123

,xxxμμ=+ with respect to the

Ox -axis.

342