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

where

ω is the mean value of the proper rotation angular velocity, while



()tϕ

represents the oscillation of period

2K/p around the mean value

()

ϕω+−.

Starting from (15.1.59'), we can write

()

12 12

1tan

II II

Ω

−

⎡

⎤

=++−

⎢

⎥

⎣

⎦



()

() ()

12 12

cos sin

ktt tt

II II

ktt tt

ϕϕ

ωβω

Ω

ωβω

′

−− −

⎡⎤

=+−−

⎢⎥

′

−+ −

⎣⎦

()

12 12

()

Ikk

II II Rt

II k

Ωββ

′′

⎧⎫

⎡

⎤

=+−− −

⎨⎬

⎢

⎥

′

−

′

−

⎩⎭

⎣

⎦

in the frame of the order of approximation considered above; by integration, we obtain

the precession in the form

()



() ()tttt

ψψω ψ=+ − + ,

1 II

ωβΩ

⎡

⎤

′

⎢

⎥

′

⎣

⎦



()

() 1 sin2

II I

tAntt

βΩ

ψω

∞

′

−

=−−

′

−

∑

(15.1.62')

where

ω is the mean value of the angular velocity of precession, while



()tψ is the

oscillation of period

2K/p around the mean value

()

ψω+−.

Noting that

()( )

31 1 3 33 0

//cosII I II I I IωΩ θ−−= =, corresponding to the

initial moment

t , the first formula (15.1.59'') and the last formula (15.1.60''') lead to

()

12cos2

cos cos

12cos2

qtt

γθ θ

+−

′

−−

where we have retained only the first power of

q in the expansion into series (15.1.60'), in

conformity to the notations (15.1.60''). Expanding the above ratio into a power series and

proceeding as in the preceding cases, we can represent the nutation in the form

cos ( ) ( )ttθγγ=+



()

14 cosqkγθ

′

(15.1.62'')

() ()

00 0

( ) 4 cos 3 cos 2 cos2

t qk q tt q n tt

ϕϕ

γθω ω

∞

−

⎡

⎤

′

=−+−

⎢

⎥

⎣

⎦

∑



where

γ is a mean value, while ()tγ



represents the oscillation of period 2K/p

around this mean value.

313

MECHANICAL SYSTEMS, CLASSICAL MODELS

We have seen in Sect. 14.1.1.3 that the position of the rigid solid with respect to the

fixed frame of reference

′

can be specified also by means of the Cayley-Klein

parameters

α, β, γβ=− , δα= (the upperlining indicates the complex conjugate);

15.1.2.3 Geometric representation of the motion after Poinsot

Setting up the ellipsoid of inertia

E (Poinsot’s ellipsoid) at the point O, one obtains a

geometric representation of the motion of the rigid solid with a fixed point; the position

vector of a point

P of the ellipsoid is situated along the angular velocity vector ω at this

point, being specified by the formula (15.1.15). The plane

Π tangent to the ellipsoid at

Fig. 15.7 The motion of rolling and pivoting without sliding of

Poinsot’s ellipsoid on one of Laplace’s planes

the point

P, called pole, is normal to the moment of momentum

′

K at the pole O. The

distance

||hOQ=





from the point O to this plane is given by (15.1.16), being constant

in time (Fig. 15.7) (these results hold for any motion of the rigid solid with a fixed point;

see Sect. 15.1.1.2 too). But, in the considered Euler-Poinsot case, the vector

′

constant in time with respect to the fixed frame of reference

′

R , so that the plane Π is

of constant normal, being situated at a constant distance from the point

O; hence, the

plane

Π is fixed with respect to the frame

′

. The instantaneous rotation axis pierces

Poinsot’s ellipsoid at a second point

and the support of

′

K is normal also to the

plane

Π tangent at P to the ellipsoid. Projecting the constant vector

′

K on the

normal to a given plane, one obtains a constant; the planes parallel to this plane are

called Laplace planes. The planes

Π and Π belong to a family of invariable planes of

Laplace. The point

P of contact is on the instantaneous axis of rotation, having thus a

null velocity; we can state

F. Klein showed that in the Euler-Poinsot case these parameters are elliptic functions

of second kind which, both at the numerator and at the denominator, have only one

theta function.

314

15 Dynamics of the Rigid Solid with a Fixed Point

Theorem 15.1.9 (Poinsot) In the Euler-Poinsot case, the rigid solid with a fixed point

is moving so that Poinsot’s ellipsoid

E corresponding to the fixed point has a

slidingless rolling and pivoting motion on one of Laplace’s planes. The magnitude

ω of

the rotation instantaneous angular velocity is in direct proportion to the magnitude of

the position vector of the point

P at which the instantaneous axis of rotation pierces the

ellipsoid.

We can express the distance

h also in the remarkable form

(15.1.16')

which allows an interesting interpretation of the constant

I; as well, we have

()

vers

2cos,

Ih h

OP h

Ω

===

′′





ωω ω

(15.1.15')

Because

||OP





is contained between the semi-minor axis

/KI

and the semi-major

axis

/KI, we get

()

cos ,

II I I

′

<= <

K ω , justifying once more the

inequality

II≥ . On the other hand, ||hOP≤





and

/hK I≥ (because the point

Q is exterior to the ellipsoid or at the most on it), so that

///KI KIKI≤≤ ; finally, it results

III≤≤.

We have used, in the above exposition, the ellipsoid of inertia

E, represented with

respect to the principal axes of inertia taken as axes of the non-inertial frame of

reference

R in the form

22222

11 22 33

Ix Ix Ix K Ih++==,

(15.1.63)

where

0K > is a constant which specifies the units (K has the dimensional equation

[]

1/2 2

KML= ) and is conveniently determined (see Chap. 3, Sect. 1.2.6 too). Giving

various values to the constant

K, one can use different ellipsoids of Poinsot; for

instance, one can take

1K = (in this case, the unit has dimension), the respective

ellipsoid being obtained from (15.1.63) by a similitude with the ratio

K . One can use

also the second equation (15.1.47), with

KIΩ=

; in this case, the position vector of

the pole

P is just ω. The equation of the tangent plane to the ellipsoid E at the pole P

(

KTω

′

1,2, 3i =

) is given by

111 222 333

2Ix Ix I x KT K IIhωω ω ΩΩ

′

++= = =

(15.1.63')

where

123

,,xxx are the co-ordinates of a point of the Π-plane.

315

MECHANICAL SYSTEMS, CLASSICAL MODELS

Starting from the ellipsoid of gyration

′

E (the ellipsoid reciprocal to the ellipsoid of

inertia), introduced in Chap. 3, Sect. 1.2.6 and used in Sect. 14.1.1.6, J. Mac Cullagh

succeeded to give, in 1840, another intuitive image of the motion of the rigid solid with

a fixed point. We will use the equation (3.1.105) of the ellipsoid of gyration, with

respect to the principal axes of inertia, in the movable frame of reference

R, in the

form

222

123

xxx

IIIM

++=

(15.1.64)

where we take

KRM= (a point P

′

∈

′

E is the inverse of the point Q Π∈ from

the preceding subsection, with respect to the sphere

()

,OR ); we put thus in evidence

the principal moments of inertia, which is more convenient in the following

calculations.

Noting that

KIω

′

KIω

′

KIω

′

, the equations of the support

D of the vector

′

K , fixed with respect to the frame of reference

′

R , are

222

123 123

22 22 22

11 22 33

x x x xxx

III

ωωω

===

(15.1.65)

Assuming that

x , 1,2, 3i = , are the co-ordinates of one of the points

′

≡∩

′

and eliminating these co-ordinates between the equations (15.1.64),

(15.1.65), we obtain

222

11 22 33

22 22 22

11 22 33

III

ωωω

′

wherefrom

OP i

′

===





(15.1.65')

i being a constant quantity of the nature of a radius of gyration, corresponding to the

constant quantity

I. We notice that ||

′





is contained between the semi-minor axis

and the semi-major axis of the ellipsoid

′

, so that

iii≤≤

(so as

III≤≤

justifying thus once more both inequalities established before. Hence, the points

′

and

′

, situated on the invariable axis D and diametrically opposed in the ellipsoid of

gyration

′

, are fixed with respect to the frame of reference

′

(

||||const

OP OP

′′









) (Fig. 15.8); to fix the ideas, the position vector of the point

′

will be /

JM, of components

111

xJI KMω

′

15.1.2.4 Geometric representation of the motion after Mac Cullagh

316

15 Dynamics of the Rigid Solid with a Fixed Point

222

xJI KMω

′

333

xJI KMω

′

. The equation of the plane Π

′

tangent to the ellipsoid

′

at the point

′

, is given by

11 22 33

xxx

ωωω

′

++=

(15.1.64')

the normal

′

to this plane having the direction parameters

ω ,

1,2, 3i =

; hence,

the angular velocity vector

ω is normal to this plane. Moreover, the distance h

′

from

the point

O to this plane is given by

222

123

ωωω

′′

′

(15.1.64'')

Fig. 15.8 The inertial motion of the rigid solid with a fixed point

in Mac Cullagh’s geometric representation

Hence, we can state

Theorem 15.1.10 (J. Mac Cullagh) The inertial motion of a rigid solid about a fixed

point

O of it, in the Euler-Poinsot case, takes place so that the ellipsoid of gyration

′

corresponding to this point passes through the fixed points

′

and

′

, situated on the

invariable axis

D. The rotation angular velocity ω is normal to the plane tangent to the

ellipsoid of gyration

′

at one of the fixed points, while its magnitude ω is in inverse

proportion to the distance

′

from the fixed point O to the plane Π

′

15.1.2.5 The Polhode

The locus of the point

P at which the instantaneous axis of rotation pierces the ellipsoid

of inertia

E (rigidly linked to the rigid solid S ) is a curve P called polhode,

intersection of the ellipsoid with the polhodic cone

C (the locus of the instantaneous

axes of rotation with respect to the frame of reference

R ), hence a directrix of this

cone. Analogously, the locus of the point

P on the fixed plane Π, tangent to the

317

MECHANICAL SYSTEMS, CLASSICAL MODELS

ellipsoid

E at this point, is a curve H, called herpolhode, intersection of the plane Π

with the herpolhodic cone

C (the locus of the instantaneous axes of rotation with

respect to the frame

′

) (in Fig. 15.9 we represent the polhode and the herpolhode

corresponding to the point

P and the plane Π; for the point P

′

and the plane Π

′

one

obtains analogous results). Obviously, the two curves are tangent one to the other,

measuring equal lengths between corresponding points, because of the slidingless

rolling of the polhodic cone over the herpolhodic one (and of the polhode over the

herpolhode).

Fig. 15.9 The polhode P and the herpolhode H in the motion

of a rigid solid with a fixed point

If we eliminate

Ω between the first integrals (15.1.47), then we find

() () ()

222

11 1 22 2 33 3

0II I II I II Iωωω−+ −+ −=.

(15.1.66)

Assuming that

123

III>>, one cannot have

II> or

II< , because the left

member would be strictly negative or strictly positive, the cone being imaginary; hence,

III≤≤, result previously obtained. Taking into account (15.1.15'), the equation of

the instantaneous axis is written in the form

11 22 33

////xxxhωωωΩ=== in the

frame of reference

R, the equation of the polhodic cone ( in the same frame) being

() () ()

222

11 1 22 2 33 3

0I I Ix I I Ix I I Ix−+ −+ −=.

(15.1.66')

Because

P ≡∩EC, the polhode will have the equations (15.1.63), (15.1.66); hence,

this is an algebraic curve of the fourth degree with two distinct closed branches and

with properties of central symmetry (with respect to the co-ordinate planes too), the

polhodic cone having the same axes of symmetry as the ellipsoid

E. Multiplying the

equation (15.1.63) by

I and summing with the equation (15.1.66'), we obtain the

equation of an ellipsoid

E , called kinetic ellipsoid (due to the signification of the first

integral (15.1.47)),

318

15 Dynamics of the Rigid Solid with a Fixed Point

22 22 22 22 2

11 22 33

Ix Ix Ix Ih IK++==.

(15.1.67)

The ellipsoid

E is coaxial with the ellipsoid E, the two ellipsoids having as

intersection the polhodic curve.

Fig. 15.10 Polhodes drawn on the ellipsoid of inertia E

Concerning the polhodic cone, we notice that the first coefficient of the equation

(15.1.66) is always positive, while the last one is always negative; in what concerns the

second coefficient, this one is positive or negative as

II< or

II> , respectively,

depending thus on the initial conditions. In the first case,

hKI> and the polhodic

cone contains the

-axis; in the second case,

hKI< and the polhodic cone

contains the

Ox -axis. If

II= , hence if

/hK I=

, then the polhodic cone

degenerates in two planes (we use the notations (14.1.49') and notice that

ββΩ==)

()

32 3 1

13 3

11 2

II I

xx x

II I

−

=± =±

−

(15.1.68)

which pass through the mean axis (the axis

′

) of the ellipsoid E, the corresponding

polhode being constituted of two ellipses (

ε and ε

′

) for which this axis is a common

one (Fig. 15.10) (the only case in which the two branches of the polhode have common

points). If

II= or

II= we have

0xx== or

0xx==, respectively, the

polhode being reduced to the points

A and

′

(extremities of the minor axis of the

ellipsoid) or to the points

A and

′

(extremities of the major axis of the ellipsoid),

respectively. If

III<<, then the polhode is formed of two closed curves

γ and

319

MECHANICAL SYSTEMS, CLASSICAL MODELS

′

, which surround the points

A and

′

, respectively; as well, if

III<<, then

the polhode is formed of two closed curves

γ and

′

, which surround the points

and

′

, respectively (Fig. 15.10). In the hypothesis considered in Sect. 15.1.2.1

(

()

20 2

0tωω=>,

()

0tω < ,

()

0tω > ), the pole P travels through the

polhodes in the sense indicated in Fig. 15.10 (on the curves on which the sense has not

been indicated, that one is obtained by symmetry). To have a clearer image of the

polhode, we consider also its projection on the three planes of co-ordinates, in the frame

of reference

R. Thus, projecting on the plane

Ox x (we eliminate

x between the

equations (15.1.63) and (15.1.66')) for various values of

I (hence, for various initial

conditions), we obtain a family of coaxial ellipses of equations (cylinders of elliptic

section which pierce the ellipsoid of inertia after polhodes)

()()()

222

11 31 22 33 3

II Ix II Ix II Ih−+−=−.

Taking into account the notations (15.1.49), (15.1.49'), we can write these equations

also in the form

22 2 2

ββΩ Ω

+= = .

(15.1.68')

Moreover, starting from the first equation (15.1.49'') and using the relation (15.1.15'),

we find again these equations, the respective ellipses being equivalent. We notice that

for

II> one obtains arcs of ellipse, bounded by the ellipse of inertia

222

11 22

Ix Ix K+=, while for

II≤ there result complete ellipses (Fig. 15.11a).

Projecting the polhode on the plane

Ox x (we eliminate

x between the equations

(15.1.63) and (15.1.66')) and using the same notations, we get – analogously – a family

of coaxial ellipses (cylinders of elliptic section, which pierce the ellipsoid of inertia

after polhodes)

22 2 2

ββΩ Ω

+= = .

(15.1.68'')

Taking into account the relation (15.1.15'), one observes that these ellipses are

equivalent with the second ellipse (15.1.49''). For

II< there result arcs of ellipse,

bounded by the ellipse of inertia

222

22 33

Ix Ix K+=, while for

II≥ there result

complete ellipses (Fig. 15.11b). Finally, the projection of the polhode on the plane

Ox x (one eliminates

x between the equations (15.1.63) and (15.1.66')), with the

same notations, leads to two conjugate coaxial hyperbolae (cylinders of hyperbolic

section, the traces of which on the ellipsoid of inertia are the polhodes)

22 2 2

βΩ Ω

−= = ,

(15.1.68''')

320

15 Dynamics of the Rigid Solid with a Fixed Point

which degenerate in two straight lines, specified by the equations (15.1.68), for

II= .

II< , then the polhode is formed of two arcs of hyperbola, the real axis of which is

the axis

′

(Fig. 15.11c). Taking into account the relation (15.1.15'), one observes

that these hyperbolae are equivalent to the hyperbola (15.1.49''').

Fig. 15.11 Projections of a polhode on the planes

Ox x (a),

Ox x (b) and

Ox x (c)

Concluding, there are two families of polhodes, separated by the singular polhode

corresponding to the case

II= (ε and ε

′

); a family of polhodes (

γ and

′

)

surround the extremities of the minor axis of the ellipsoid of inertia, while the other

family of polhodes (

γ and

′

) surround the extremities of the major axis of this

ellipsoid. Through each point of the ellipsoid passes a polhode and only one. The

polhodes being plotted on the ellipsoid, to can determine that one which corresponds to

given initial conditions it is sufficient to know the piercing point

P of the support of

the angular velocity vector

ω on the ellipsoid of inertia at the initial moment; the

searched polhode is that which passes through

P , while the plane tangent to the

ellipsoid at

P is the plane Π.

15.1.2.6 The Herpolhode

Studies to determine the herpolhode have been made by W. Hess, Sparre, G. Darboux,

A.G. Greenhill, G. Halphen, J.N. Franke, A. Mannheim, A. de St. Germain, A.H. Résal,

P. Barbarin, E. Lacour, A. Petrus etc. In what follows, we use polar co-ordinates in the

plane

Π, taking as pole of reference the point Q, projection of the fixed point O on the

fixed plane

Π (Fig. 15.9). The vector radius is given by

321

MECHANICAL SYSTEMS, CLASSICAL MODELS

222 22

QP OP h

ρωΩωΩ

Ω

== − = − = − .

(15.1.69)

Fig. 15.12 The herpolhode drawn in a circular annulus

Taking into account that

/||/KI OPKI≤ ≤



, it results that ρ is inferior and

superior bounded, so that the herpolhode is a curve contained in a circular annulus

(

max

min

ρρρ≤≤ ) (Fig. 15.12). We get

22 2

ωΩ Ω

(15.1.69')

from (15.1.69), while by replacing in (15.1.57') we may write

()

[]

()

[]

22 2222

12 2 3 2

2 sign sign

II II

Ω

ρδ δρρδ

=± − − − − − ,

(15.1.70)

where we have introduced the notations

()

||IIII

III

−−

()()

II I I

III

−−

()

||IIII

III

−−

(15.1.70')

II< , then we have

min 1

ρδ= , while if

II> , then we have

min 3

ρδ= ; in both

cases,

max

ρδ= . Integrating the differential equation (15.1.70), one can express the

vector radius

ρ by means of elliptic functions.

We get the same result noting that

2 2 2222

123

OPhxxxhρ = −=++−; using the

equations of the cylinders (15.1.68')–(15.1.68''') and eliminating the variables

x and

x or the variables

x and

x , we obtain

322