Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems

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16 Other Considerations on the Dynamics of the Rigid Solid

initial position and the force

F. But one sees that, due to the gyroscopic effect, the axis

is rotating in the sense of the vector product

× Fω

. Obviously, also in this case arises

the gyroscopic

moment

OQ=− ×

JJG

MF.

To can study the problem, it is convenient to take – at the initial moment – the

Ox -axis in the direction of the

′

-axis in its negative sense, the

Ox -axis along the

′

-axis and the

Ox -axis along the

′

-axis (Fig. 16.18b ). If the force

′

=−Fi

is in the negative sense of the

′

-axis (we suppose that it maintains all the time its

direction and its sense) and acts at the point

Q on the axis of rotation, then the moment

()( )

23231

OQ l F Fl Fl

′′′′′

=×=−×− = ×=

JJG

MFiiiii is directed towards the

positive sense of the

′

-axis. If the moment of momentum

′

, situated along the

-axis has a variation

′

ΔK

parallel to

(corresponding to the relation

′

Δ=ΔKM

), in the

Ox x

′′

-plane, then one obtains a moment of momentum

OO O

′′ ′

=+Δ



KK K

along the new position of the axis of the gyroscope. The gyroscopic

effect (the displacement

of the

Ox -axis in the horizontal plane

Ox x

′′

towards the

′

-axis) is thus justified. The initial conditions (at the moment

tt= ) are of the

form

0ψ = ,

/2θπ= ,

0ϕ = ,

0ψ =



0θ =



, wherefrom

0ωω== and

ωϕ=



. The equations of motion lead to first integrals of the form (15.2.1'''),

wherefrom, taking into account the initial conditions, it results

′

()

2Ihω = , obtaining thus 0α = , 0β = . If the spin vanishes (

0ω = ), then the

first integrals (15.2.2) have only the solution ( we replace

Mg by F and

ρ by l)

() 0tψ = , () 0tϕ = ,

2cos 0JFlθθ+=



(16.2.17)

Fig. 16.19 The zeros of the polynomial ()Pu : the case

0uu==

Hence, the axis of the gyroscope has a pendular oscillation about the fixed point

O, in a

vertical plane. If

0ω ≠ , then we are led to

ψω

=−

−



()()

ubuuuuu=− − − ,

10u λλ=− + <,

11u λλ=+ + >,

(16.2.18)

413

() ()

20 20

333

bJFl

ωω

==.

It results that

0uu≤≤

, hence () /2tθπ≥ (Fig. 16.19); one obtains 0ψ >



, for

tt>

, the gyroscopic phenomenon being thus put in evidence. By means of Jacobi’s

elliptic functions, the formulae (15.2.5'), (15.2.6') allow to express

u in the form

()

cnuu ptt=−,

()

111

2 1 1 ...

pb a

λω

⎡

⎤

=+=+−

⎢

⎥

⎢

⎥

⎣

⎦

(16.2.19)

We can use the results obtained concerning the gyroscopic effect to show the

stability character of the motion of a gyroscope subjected to an initial rotation about its

axis of symmetry (corresponding to the initial conditions assumed above). Choosing the

co-ordinate axes as above, the moment of momentum

′

will be directed along the

Ox -axis (Fig. 16.20). We assume a deviation of angle dψ of the axis of the gyroscope

in the horizontal plane

Ox x

′′

; in conformity to the previous considerations, intervenes

the differential

′

of the moment of momentum, parallel to the

Ox -axis,

corresponding to the moment

M , along this axis. Taking into account the theorem of

moment of momentum written in a scalar form (

KMt

′

), we notice that

()

dtand d/ / d

OO OO

KK MK tψψ

′′ ′

== =

, so that

MKψ

′



. The moment

has the tendency to impart to the gyroscope a motion of rotation of angle θ about

the

Ox -axis, governed by the equation

IJKθθ ψ

′

  

; integrating this equation,

we get

JKθψ

′



, the initial conditions vanishing. To a displacement of angle dθ of

the gyroscope axis in a vertical plane one obtains a differential

′

of the moment of

momentum, parallel to the

-axis; it corresponds a moment

′

of modulus

K θ

′



along this axis, in its negative sense. This moment has the tendency to impart a motion

of rotation of angle

ψ about the

Ox -axis, in conformity to the equation

IJ Kψψ θ

′

==−

  

. Eliminating the angle θ, we obtain the equation

MECHANICAL SYSTEMS, CLASSICAL MODELS

414

16 Other Considerations on the Dynamics of the Rigid Solid

Fig. 16.20 The gyroscope effect: the case

′

()

ψψ

′



(16.2.20)

which leads to an angle

ψ as a harmonic function of time; it results that, in the given

initial conditions, the axis of the gyroscope is a stable axis of rotation.

16.2.1.7 Prandtl’s Wheel. Tendency of Parallelism of the Axes of Rotation

The gyroscopic effect can be put in evidence by means of Prandtl’s wheel. This is a

homogeneous wheel, having a suspension which diminishes the frictions at the centre

O; through this point passes a horizontal axle (along the symmetry axis of the

gyroscope), hanged up so as to make possible the rotation about a horizontal axis,

normal to that one, as well as about a vertical axis (Fig. 16.21). We take the

Ox -axis

along the symmetry axis of the gyroscope. Euler’s equations of motion are of the form

(16.1.1). We give to the wheel a rapid motion of rotation at the initial moment, taking

the horizontal axle fixed (

0ωω==,

0ω ≠ ); then we let free the axle, so that

the wheel does effect a motion about it.

If the gyroscope is acted upon only by its own weight, the centre of gravity being at

O, then we have

123

OOO

MMM===. We get

ωω+

() ()

ωω=+

0= ,

from the first two equations (16.1.1), so that

() () 0ttωω==; the third equation

leads to

()tωω= , the gyroscope continuing its initial motion of rotation about the

Ox -axis. Noting that

0ρ = , from (15.2.1'') we get

()

2Ihω = , so that 0β =

and

0b = ; the second first integral (15.2.2) gives

22 2

sin 0ψθθ+=



, wherefrom

ψψ= ,

θθ= . The third relation (15.2.2) leads then to

ϕω=

Hence, we see that

the axle of rotation

Ox remains fixed during the motion (the

Ox -axis has the director

cosines

sin sinθψ,

sin cosθψ− ,

cos θ with respect to the fixed frame of

reference; we can take

0θψ== too, because

Ox Ox

′

≡

as in Fig. 16.20), the

wheel having a uniform rotation (we neglect the frictions and the resistance of the air)

with respect to this axis; as much the velocity of rotation is greater, as much the parasite

phenomena can be neglected.

415

Fig. 16.21 Prandtl’s wheel

If at a point

′

, at the distance l from the fixed point O, acts a force

11 22

FF=+Fi i, normal to the symmetry axis of the gyroscope (Fig. 16.21), then we

get the moment

21 12

lF lF=− +Mii. Because the initial angular velocity

is great,

we can assume, in a first approximation, that

0ω ≅



0ω ≅



, remaining with the

equations

()

323 2

JI lFωω−=,

()

313 1

JI lFωω−=,

(16.2.21)

which determine

ω and

ω . The velocity v of the point O

′

is given by

311223332112

() ( ) ( )ll lωωω ωω=× = + + × = −vi iiiiiiω

21 12

()

JIω

=−

−

or by (we have

IJ> )

33 33

() ()

IJ IJωω

′

=×=

−−

JJJG

vFM

(16.2.22)

Hence, after the application of the external force

F, normal to the gyroscope axis, the

point

′

will be displaced along a direction normal to the plane formed by the force

and by this axis, in the sense of the vector product

′

=×

JJJG

(the sense of the

′

-axis). This gyroscopic effect is as much greater as the initial angular velocity

is smaller (formula (16.2.22)); on the contrary, if we maintain fixed the axle

Ox , then

its reaction will be given by the force

F, being as much intense as

ω is greater

(formula (16.2.21)).

MECHANICAL SYSTEMS, CLASSICAL MODELS

416

16 Other Considerations on the Dynamics of the Rigid Solid

If, initially, the wheel is at rest, being applied only the force

F at the point O of the

axle

to the latter one will be imparted a motion of rotation about the

′

-axis

(normal to

′

and to F, in the sense of the vector product OO

′

JJJG

); hence, the

gyroscopic effect, put in evidence by the formula (16.2.22), tends to bring the symmetry

axis of the gyroscope (the

-axis), on the shortest way, in the direction and the sense

of the rotation axis of the rigid solid (the

′

-axis), if this one would be acted upon,

from its state of rest, by the force

F. This represents the parallelism tendency of the

rotation axes, with multiple applications in technics.

Let us suppose that at the point

′′

(specified by

OO l

′′

=−

JJJG

) acts a force −F too;

in this case, the velocity

v given by (16.2.22) doubles its intensity and we will have

()

2/vlFIJω=−. If, independently, we apply to the mechanical system a

rotation of angular velocity

222

′′′

iω about the vertical axis

′

, then the motion

which takes place (the velocity of the point

′

is of magnitude

vlω

′

) is identical to

that corresponding to the gyroscopic effect considered above; it results thus

()

FIJ

ωω

′

=−

(16.2.23)

IJ , then one obtains

332

ωω

′

(16.2.23')

a formula particularly useful in applications. The considered forces, of magnitude

F, are

called gyroscopic reactions; corresponding to the principle of action and reaction, the

forces which have the same direction and the same magnitude, but opposite sense, are

called gyroscopic pressures.

16.2.1.8 Inertial Forces in the Motion of Regular Precession of the Gyroscope

The motion of the gyroscope with respect to the fixed frame of reference (the absolute

motion) can be obtained by composing the proper rotation of it about the symmetry axis

with the constant angular velocity ω (the relative motion), with the motion of

precession about the fixed axis

′

, with the constant angular velocity

′

(the motion

of transportation); we assume that the angle

θ between the two axes is constant

(Fig. 16.22). An element

dm of the gyroscope, situated at the point P, is subjected to

the centrifugal forces (forces of transportation) due to the proper rotation and to the

motion of precession, as well as to the Coriolis force.

417

Fig. 16.22 Inertial forces in the motion of regular precession of the gyroscope

The centrifugal force

mω=Fr

( PP=

JJG

, where P is the projection of P on

the

-axis), due to the proper rotation, leads to a zero torsor with respect to the point

O on the

Ox -axis, because of the properties of symmetry with respect to this axis

(

=F0,

′

M0).

The centrifugal force

mω

′

Fr ( PP

′

JJJG

, where P

′

is the projection of P

on the

′

-axis) due to the motion of precession, leads to the resultant

mω

′

∫

Fr and to the resultant moment

OP mω

′

=×

∫

JJG

.To simplify

the calculation, we choose the

Ox -axis along the line of nodes ON and the

Ox -axis

along the transverse line

′

. Corresponding to the point

()

123

,,Px x x , we obtain

()

33223

cos sin sin cos

OP OP x x xθθ θ θ

′′

=⋅= ⋅ + = +

JJG

iii i , so that the co-

ordinates of the point

′

on the

′

-axis are

0x =



()

sin cos sinxx xθθθ=+



()

sin cos cosxx xθθθ=+



We are thus led to

Fxmω

′

()

[]

2223

d sin cos sin d

Fxxx mωθθθ

′

=− +

()

[]

3323

dsincoscosd

Fxxx mωθθθ

′

=− +

Taking into account the relations which give the static moments and the position of the

mass centre

(0,0, )C ρ on the symmetry axis of the gyroscope, we get

F = ,

sin cos

FMωρ θ θ

′

=−

sin

FMωρ θ

′

=−

(16.2.24)

MECHANICAL SYSTEMS, CLASSICAL MODELS

418

16 Other Considerations on the Dynamics of the Rigid Solid

The resultant

is applied at O being contained in the plane

Ox x

′

and normal to the

′

-axis and having the magnitude

sin

FM Maωρ θ

′′

(16.2.24')

where

′

a is the acceleration of the mass centre with respect to the inertial frame of

reference. If

OC≡ , then we have

=F0.

Analogously,

23 32

ddd

MxFxF=−

()

[]

{

23 2 3

sin cos cosxx x xωθθθ

′

=−+

()

[

]

}

32 2 3

sin cos sin dxx x x mθθθ−− + ,

()

[]

{}

31 1 3 2 3

dsincoscosd

Mxxxxxx mωθθθ

′

=−−+

()

[]

{}

12 2 3 21

d sin cos sin d

Mxxxx xxmωθθθ

′

=−+ −

By means of the relations which give the principal moments of inertia (the centrifugal

moments of inertia vanish), we obtain

()

sin cos

MJIωθθ

′

=−

Op Op

MM==;

(16.2.25)

hence, the moment

() ()

3333

cos cos

JI JI

ωθ θ

′

′′ ′

=− ×=− ×

Miiωω

(16.2.25')

is directed along the line of nodes.

To calculate the Coriolis force

d2 d

′

=×

Fvω , we notice that the relative

velocity is given by

=×vrω ; by considerations analogous to those above, we get a

vanishing resultant

()

2d2d

′′

=××=−×× =

∫∫

Fr r0ωω ωω .

The resultant moment is given by

()

[]

OP m

′

=×××

∫

JJG

Mrωω

and has the differential components

()()

[

]

23 22 32

132

ddd 2sin2cosd

OC C C

MxFxFx x xx mωω θ ωω θ

′′

=−=− −

()()

[

]

31 1 2

d2cos2sind

Mxx x x mωω θ ωω θ

′′

=−−

()()

[

]

12 21

d2cos2cosd0

Mxx xx mωω θ ωω θ

′′

=− =

where we took into account the relations

419

()

()()

′′ ′

×× =− ⋅ + ⋅

rrrωωωωωω,

()

sin sin sinrxωθ ϕω θ

′′ ′

⋅= =

rω ,

cosωω θ

′′

⋅=ωω

with

()

,Oxϕ = ) r . Using the known results concerning the tensor of inertia, we

obtain

()

222

212

2 sin d sin d

Mxmxxmωω θ ωω θ

′′

=− =− +

∫∫

so that

sin

MIωω θ

′

=−

OC OC

MM==

(16.2.26)

the moment

333 3

MI Iωω

′′ ′

=− × =− ×

ii ωω,

(16.2.26')

being directed along the line of nodes too.

Comparing with the relation (16.2.16), we get

OOpOC

=− = +MMMM. (16.2.27)

Hence, the gyroscopic moment corresponds to the influence of the inertial forces. We

notice that, by passing from the formula (16.2.16) to the approximate formula

(16.2.16'),we neglect

M , hence the effect of the centrifugal forces in the motion of

precession, remaining only the effect of the Coriolis forces.

We can replace the torsor

{

}

FM by a resultant

F (gyroscopic pressure) in the

Ox x

′

-plane at a distance /

MF from the fixed point O.

16.2.2 Applications

In what follows, we present firstly some applications with a theoretical character, as

well as the Cardanic suspension and the gyroscopic pendulum, introducing the

influence of the friction forces too. We deal then with some technical applications of

the theory of the gyroscope.

MECHANICAL SYSTEMS, CLASSICAL MODELS

420

16.2.2.1 The Gyroscope with a Cardanic Suspension

The centre of gravity of a gyroscope may be practically fixed with the aid of the

suspension discovered in 1545 by Gerolamo Cardano. The gyroscope has the

Ox -axis

hinged in the interior annulus

a , which is hinged in the exterior annulus

by the

horizontal axis ON, which can rotate about the fixed vertical axis

′

; because the

Ox -axis can rotate about both the horizontal and the vertical axes, while the gyroscope

can have a proper rotation about this axis, it results that the gyroscope can take any

position around its centre of gravity (Fig. 16.23). As a matter of fact, by the rotation of

the annulus

a about the

′

-axis one determines the angle ψ, while by the rotation of

the annulus

a about the ON-axis is specified the angle θ; as well, by the rotation of the

gyroscope about the

Ox -axis one obtains the angle ϕ. Thus, the position of the

gyroscope is given by Euler's angles, which can be measured. To obtain a Cardanic

suspension of good quality, it is necessary an as good as possible centring of the

rotation axes (to eliminate the parasite moments which may appear), eventually with the

aid of some adjustable wedging screws; as well, it is necessary to have minimal

frictions in the hinges (to eliminate the effect of the moments of friction), while the

weights of all accessories be negligible with respect to the own weight

of the gyroscope

(for analogous reasons).

Fig. 16.23 The Gyroscope with a Cardanic Suspension

Let us assume that the gyroscope with Cardanic suspension is acted upon by external

forces for which

=M0; but if to this gyroscope is imparted, in a certain manner, a

proper angular velocity

ω , about the

Ox -axis, and then an angular velocity of

precession

′

, about the

′

-axis ( which must not be necessarily vertical), then arises

a gyroscopic moment

M , given by (16.2.16) or – approximately – by (16.2.16').

16 Other Considerations on the Dynamics of the Rigid Solid

421

Taking into account the sense of this moment, it results that the angle of nutation cannot

remain constant, becoming smaller. The

Ox -axis of the gyroscope tends to the fixed

axis

′

on the shortest way, so that the senses of the vectors ω and

′

do coincide;

this is, after F. Klein and A. Sommerfeld, the parallelism tendency of the rotation axes

of the gyroscope, put in evidence by Prandtl’s wheel too (see Sect. 16.2.1.7).

If, in the preceding case, we wish to maintain the angle θ constant, then we must

annihilate the effect of the gyroscopic moment

M , by introducing a moment

=−MM, given by the external forces, along the line of nodes ON. To do this, we

act upon the

Ox -axis of the gyroscope with the force F at A and with the force −F at

′

, with the lever arm AA d

′

= ; these forces are gyroscopic reactions (the reactions

of the annulus

a on the

-axis). Noting that

MFd=

, we obtain

sin

ωω θ

′

= ,

(16.2.28)

where we have used the approximate formula (16.2.16'). The force −F applied at A and

the force F applied at

′

represent the gyroscopic pressures (the pressures of the

gyroscopic couple upon the annulus

a ); these pressures induce the tendency of

parallelism mentioned above (see Sect. 16.2.1.7 too).

Let be a gyroscope with a Cardanic suspension at the centre of gravity, subjected to a

proper rotation about its axis; we assume that upon this gyroscope acts also a moment

M , given by external forces. If this moment is directed along the

Ox -axis of the

gyroscope, then it will lead to the increasing or to the decreasing of the proper rotation

velocity; if the respective moment is directed along the ON-axis of the interior annulus

a , then its effect will be a precession about the fixed axis

′

, situated in the normal

to ON plane, which passes through

Ox . If the moment

M is directed along the

′

-axis of the exterior annulus

, then we make a decomposition of it along the

Ox -axis (the respective effect has been already mentioned) and along the transverse

axis

′

; this second component leads to a motion of precession of the

Ox -axis of

the gyroscope about the

ON-axis. If the moment

directed along the

′

-axis is

constant, then one has a tendency of parallelism of the axes (the

Ox -axis tends, on the

shortest way, to the

′

-axis). These considerations are particularly useful in various

technical applications of the gyroscope.

16.2.2.2 The Influence of the Friction Forces on the Gyroscope

It is quite difficult to introduce the influence of the forces of friction in a mathematical

model of the gyroscope. Let us consider, e.g., a heavy gyroscope Cardanically

suspended at its centre of gravity. If we assume that the proper velocity of rotation

ω is

maintained constant by external means, then the effect of friction of the axle

′

the gyroscope on the interior annulus

a is annihilated (Fig. 16.23); because of the

MECHANICAL SYSTEMS, CLASSICAL MODELS

422