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

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friction due to the motion of precession, the velocity

′

decreases. Because

=M0,

from (16.2.16) it results

()

cos 0IIJωωθ

′

+− =

(16.2.29)

In this case,

cosθ

must increase till 1 for

IJ< (prolate spheroid) or to decrease till

1− (oblate spheroid). It results thus 0θ = and θπ= , respectively, the axis of the

gyroscope taking the place of the axis of precession; the motion of the gyroscope is thus

reduced to the proper rotation about its axis.

Fig. 16.24 The influence of a moment of friction

on the gyroscope: case of

a prolate cylinder (a); case of an oblate cylinder (b)

If the velocity

ω is no more maintained constant, then appears a moment of friction

along the axis of the gyroscope and of sense opposite to ω , which leads to a

decrease of

ω ; the action of this couple on the annulus

a is given by the moment

′

(

′

+=MM 0

), the projection of which on the axis of precession is cos

M θ

′

In the case of a prolate spheroid we have

0/2θπ<< , the moment cos

M θ

′

tending

to increase

′

as in Fig. 16.24a – prolate cylinder); cosθ must decrease, hence θ must

reach the value

/2π , the axis of the gyroscope being normal to the axis of precession

(labile position), so that the relation (16.2.29) be verified. If the gyroscope is an oblate

spheroid, then it results

/2πθπ<<, the moment cos

M θ

′

having the tendency to

decrease

′

(Fig. 16.24b – oblate cylinder); cosθ must increase, hence θ must

decrease, tending to zero, the axis of the gyroscope coinciding thus with the axis of

precession (stable position).

Let be now a heavy gyroscope, which is rotating with the

proper velocity of rotation

ω about the fixed point O, situated under the centre of gravity C (Fig. 16.25). If the

initial velocity is great, then we can assume – with a good approximation – that the

moment of momentum

′

=K ω is directed along the

Ox -axis, the velocity of the

extremity

Q of this vector being given by

=vM, where

M is the moment of the

16 Other Considerations on the Dynamics of the Rigid Solid

423

own weight G with respect to the fixed point. Neglecting the pivoting friction around

the fixed axis

′

as well as the friction of nutation, we take into account the moment

of sliding friction

M at the fixed bearing support O; we assume that the moment

is constant and of horizontal direction, being contained in the

Ox x

′

-plane.

Decomposing

along the axes

Ox and

′

, we see that the velocity of proper

rotation

decreases, while the velocity of precession ω

′

increases. In this case, the

velocity of the point

Q will be

QQ O f

′

=+= +vvvMM

, so that

=vM. We

notice that, under the action of the moment of friction

M ,

the axis of proper rotation

of the gyroscope is brought on the shortest way to the vertical line (it is straightened);

Fig. 16.25 The decreasing of the motion of nutation of a heavy gyroscope,

due only to the moment of sliding friction

the point

Q describes, in a horizontal plane, a spiral, attaining the vertical line after a

finite number of nutations in a finite interval of time. Afterwards, the moment

disappears, remaining the action of the pivoting friction, which leads to a decreasing of

the angular velocity

ω and of the moment of momentum

′

, till the gyroscope is

falling. The velocity of nutation



changes always and fast its sense, the friction which

arises changing its sense too; the effect of this friction consists in the decreasing of the

velocity



, hence in the decreasing of the motion of nutation.

16.2.2.3 The Influence of the Rotation of the Earth on the Gyroscope

Let be a Cardanically suspended centred gyroscope, to which was imparted an angular

velocity of proper rotation

ω about its

-axis, situated in the horizontal plane Π of

the position (the

Px x

′

-plane), at the point P on the surface of the Earth, at the latitude

λ (Fig. 16.26a). If the velocity of rotation of the Earth is ω, then we obtain the

MECHANICAL SYSTEMS, CLASSICAL MODELS

424

component

sinωλ

′′

nω along the local vertical (the line of nodes PN) and the

component

cosωλ

′′

iω along the tangent

′

to the local meridian, in the plane Π.

Because of the rotations of velocities

ω and

′

arises a gyroscopic moment

′′

M of

magnitude

sin

MI Iωω ω ω λ

′′ ′′

, contained in the plane Π (

′′

⊥ωω

); imposing,

by construction, that the symmetry axis of the gyroscope does oscillate only in the plane

Π, the moment

′′

M has not one effect on it (it cannot take it out of this plane). The

rotations of angular velocities

ω and

′

lead to a gyroscopic moment

′

M , normal to

the plane

Π, in the negative sense of the ON-axis, its magnitude being

sin cos sin

MI Iωω θ ω ω λ θ

′′

; this moment leads to a decreasing of the angle θ

between the axes

′

and

Ox , hence to a rotation of the

Ox -axis in the Π-plane

(Fig. 16.26b), in conformity to the equation of motion

cos sin 0JIϕωωλθ+=

. (16.2.30)

Fig. 16.26 The gyroscopic compass: position (a); rotation of

the

Ox -axis in the Π-plane (b)

If we

can take

sinθθ≅

then the axis of the gyroscope will perform a harmonic

oscillation about the fixed axis

′

, of period

cos

ωω λ

(16.2.30')

the position of equilibrium (stable) corresponding to

0θ = , hence to the situation in

which the axis of the gyroscope is directed along the local meridian. Hence, by a

convenient system of damping, the

Px -axis will stop on the direction

′

, property

put in evidence by Foucault’s experiments and used to build up the gyroscopic compass

for the determination of the north pole.

Let us suppose now that the gyroscope is built up so as to allow the rotation of the

16 Other Considerations on the Dynamics of the Rigid Solid

425

Ox -axis of it only in the meridian plane which passes through P (Fig. 16.27a). In this

case, the rotation

ω of the Earth leads to a rotation of the same angular velocity about

the

′

-axis. The angular velocities ω and ω determine a gyroscopic moment

M in

the negative sense of the

PN-axis, its magnitude being

sinI ωω θ ; the equation of

motion

sin 0JIθωωθ+=



(16.2.31)

puts in evidence a harmonic oscillation of the

Px -axis

about the

′

-axis with the

period

ωω

= .

(16.2.31')

The position of equilibrium (stable) of the gyroscope takes place for

0θ = ,

corresponding to the situation in which its axis of symmetry is along the direction of the

Earth’s poles. By a convenient system of damping, the axis of the gyroscope will stop

on the same direction (so as, Foucault too, showed experimentally); one can thus build

up the azimuthal gyroscope, which allows to determine the latitude of the point

P at the

surface of the Earth (one determines the azimuth, that is the angle

/2πλ− ).

Fig. 16.27 The gyroscope the axis of which rotates only in the

meridian plane (a); the free gyroscope (b)

Let be, in general, a free gyroscope for which the motion of rotation is not at all

hindered; the gyroscope has a motion of rotation of angular velocity

ω about the

Ox -axis, which makes an angle θ

with the

′

-axis, parallel to the velocity of rotation

ω of the Earth (Fig. 16.27b). Supposing the Earth fixed, we must introduce the

gyroscopic moment

I=×M ωω, of magnitude

sin

MIωω θ= , as an external

loading (corresponding to the Coriolis force in the relative motion); the gyroscope has

MECHANICAL SYSTEMS, CLASSICAL MODELS

426

thus a motion of precession of angular velocity

−ω. The symmetry axis of the

gyroscope describes thus a cone of precession in 24 sidereal hours, having a sense of

rotation opposite to the sense of rotation of the Earth. Thus, the effect of rotation of the

Earth is nullified by the rotation of precession of the gyroscope; in a fixed frame of

reference, the axis of the free gyroscope remains fixed in space and the rotation of the

Earth has not one influence on its motion.

16.2.2.4 The Gyroscopic Pendulum

The gyroscopic pendulum is a heavy gyroscope for which the fixed point O (situated on

the symmetry axis of the gyroscope) is over the centre of gravity

C. If the

Ox -axis

about which the gyroscope has a rapid rotation is along the local vertical

′

, then we

are in the case of the sleeping gyroscope, while the

-axis is a stable axis of rotation

(see

Sect. 16.2.1.5); thus, the gyroscopic pendulum allows to determine the local

vertical. Besides its own weight, this pendulum is subjected

, in general, also to the

action of perturbing vibrations; because of its importance, many researchers dealt with

this problem (e.g., M. Schuler, R. Wieblitz and Y.A. Ishlinskiĭ), the gyroscopic

pendulum being, thus, an self-excited rigid solid (see

Sect. 15.2.3.8).

Fig. 16.28 The gyroscopic pendulum

We represent the motion with respect to the inertial frame of reference

123

Ox x x

′′′

unit vectors

′

i , the gyroscopic pendulum being rigidly linked to the non-inertial frame

123

Ox x x

of unit vectors

i , 1,2, 3j = ; for the sake of simplicity, we denote

=im

=in,

231

=×=×iiimn, these axes being principal axes of inertia of the rigid

solid ( Fig. 16.28). The moment of momentum vector with respect to the fixed frame is

thus expressed in the form

33 1

IJωω

′

== +

KI m nω ; indeed, because of the

symmetry with respect to the

Ox -axis, we can make the decomposition of this vector

in the plane determined by the vector

ω and by the axis of the gyroscope. But

16 Other Considerations on the Dynamics of the Rigid Solid

427

() ωω×=×× =− =



mm m m m nωω , so that

()

IJω

′

=+×



Kmmm,

intervening thus only one unit vector. Let us assume that upon the solid act the own

weight

′

−

i , at the centre of gravity

(0,0, )C ρ ,

0ρ < , and the perturbing elliptic

forces

111

cosktν

′

=±

Fi (the sense is chosen corresponding to the sense of rotation

with respect to the axis of the gyroscope), at the point

(0,0, )Pl, and

222

sinktν

′

Fi , at the point

(0,0, )Pl,

12 1 2

,, , , constllkk ν = ; if

kk= , then the

force is circular, while if

0k = (or

0k = ), then the force is linear. The theorem of

moment of momentum leads to

()

33 11 1

cosIJ kl tων

′

+×=± ×



mmm mi

() ()

22 2 3 3

sinkl t Mgνρ

′′

+× − ×

mi mi,

(16.2.32)

wherefrom, in components with respect to the inertial frame of reference, we obtain the

non-linear system

13223 223

sinmmmmm m m tΩΩγν

′′′ ′′

−+=−

 ,

21331 113

cosmmmmm m m tΩΩγν

′′′ ′′

−+=−±



3211221 12

sin cosmmmmm m t m tΩγνγν

′′′′ ′

−+=

 ∓ ,

(16.2.32')

where the constants

MJΩρ=− and

/IJΩω= are of the nature of an

angular velocity, while the constants

111

/kl Jγ = ,

222

/kl Jγ = are of the nature of

an angular acceleration.

In the case of the circular force (

kkk==,

lll==, hence

γγγ==),

Wieblitz obtains

cosmCtν=± ,

sinmC tν= ,

mC= ,

Ωνγν

−±

(16.2.33)

where the constants

C and C are determined by the relation

′′

If the spin is great, then we can write, approximately,

I ω

′

≅

Km, and the

differential system (16.2.32') reads

()

123

sin

OOO

KKKtΩλλν

′′′

′′′ ′

=−



()

213

cos

OOO

KKKtΩλν

′′′

′′′′

=−



∓

()

312

sin cos

OOO

KKtKtλΩ λ ν ν

′′′

′′′ ′



∓

(16.2.34)

in the normal form, where the constant

333

//Mg IΩΩΩ ρ ω

′

==−

is an angular

velocity, while the constants

1113

//kl MgλγΩ ρ==−

and

21 2211

//kl klλγγ==

are numbers; the same Wieblitz solves this system by successive approximations,

Ω

′

being a small parameter.

In the case in which the axis of the

gyroscope is not very far from the vertical line,

MECHANICAL SYSTEMS, CLASSICAL MODELS

428

the differential system (16.2.32') can be linearized and we can write

m β

′

≅ ,

m β

′

≅− ,

′

≅ , where

β and

β correspond to the rotations by which the fixed

frame of reference can attain the position of the movable frame. Indeed, starting from

the frame

123

Ox x x

′′′

, by a rotation of angle

β about the

′

-axis, we obtain a frame

123

Ox ξξ

′

, then, by a rotation of angle

β about the

Oξ -axis, we obtain a frame

12 3

Oηξη and, finally, by a rotation of angle

β about the

Oη -axis, we get the frame

123

Ox x x ; we obtain

[]

33 1 1

22 11

cos sin 0 1 0 0

cos 0 sin

sin cos 0 0 1 0 0 cos sin

001

sin 0 cos 0 sin cos

ββ

ββ β β

ββ ββ

⎡

⎤⎡ ⎤

⎡⎤

−

⎢

⎥⎢ ⎥

⎢⎥

′

⋅=−

⎢

⎥⎢ ⎥

⎢⎥

⎢

⎥⎢ ⎥

⎢⎥

−

⎢⎥

⎢

⎥⎢ ⎥

⎣⎦

⎣

⎦⎣ ⎦

23 123 13 123 13

212 12

cos cos sin sin cos cos sin cos sin cos sin sin

cos sin sin sin sin cos cos cos sin sin sin cos

sin sin cos cos cos

ββ βββ ββ βββ ββ

βββ ββ

+− +

−− + +

−

⎡ ⎤

⎢ ⎥

⎣ ⎦

wherefrom

sinm β

′

= ,

212

sin cosm ββ

′

=− ,

312

cos cosm ββ

′

= , the

approximation made being thus justified. The first two equations (16.2.32') read

12 12

sin tβΩβ Ωβγ ν+=− −

 

21 21

cos tβΩβ Ωβγ ν−=− ±

 

(16.2.35)

the third equation (16.2.32') becoming a linear consequence of the system (16.2.35), in

the frame of the approximation thus made; Wieblitz has given the solution of this

system in case of a linear force.

Because of the properties with respect to the gyroscope axis, the

Ox -axis can be

chosen arbitrarily in the plane normal to

Ox ;

in particular, the

Ox -axis can be even

the line of nodes

ON. The respective frame of reference, called the frame of Résal, is

not involved in the proper rotation of the rigid solid, so that the matric relation (3.2.11''')

reads (we make

0θ = ).

[]

cos sin 0

cos sin cos cos sin

sin sin sin cos cos

ψψ

θψ θ ψ θ

θψ θψ θ

⎡

⎤

⎢

⎥

′

⋅=−

⎢

⎥

⎢

⎥

−

⎢

⎥

⎣

⎦

Hence,

3123

sin cosψθθψθψθ

′

=+=+ +

 

ini i i

and

(sin)

J θψθ

′



Ki i

(cos )I ψθϕ++



i . With these data, we can write the theorem of moment of

momentum (

OOO

′′

∂∂+×=KKMω

, where we have introduced the derivative with

respect to time in the non-inertial frame of reference) in the form (in the frame

16 Other Considerations on the Dynamics of the Rigid Solid

429

123

Ox x x )

()

1 sin cos sin sinaaθψθθψϕθΩθ+− + =−

  



cos sin cos cos cos sinttγθψνγθψν±− ,

()

sin 2 cos cos cos sin sinaa ttψθ θψθ θϕ γ ψ νγ ψν−− − =± +

   



cos constψθϕ+=



(16.2.36)

where

/aIJ= . This differential system is sensibly simplified if ϕ is great, while θ

is small (the gyroscope axis, which is rotating rapidly, is close to the vertical line).

16.2.2.5 Technical Applications

The considerations made till now can be applied to a great number of problems with a

technical character, as well as

to the construction of many apparatuses useful in various

domains. Let be thus a rigid solid with the symmetry axis

Ox inclined by an angle θ

with respect to the

′′′

-axis, about which it rotates with a constant angular velocity

′

(Fig. 16.29); one obtains thus a motion of regular precession, arising the gyroscopic

moment (we make

0ω =

in the formula (16.2.16))

Fig. 16.29 A regular motion of precession – the gyroscopic moment

()

333

cos

IJωθ

′′

=− ×Mii,

(16.2.37)

of magnitude

(/2)sin2

MIJωθ

′

=− . On the bearings at O

′

and O

′′

act the

gyroscopic forces

−N and N, respectively, of magnitude

sin 2

ωθ

−

′

= ,

(16.2.37')

where

lOO

′′′

; if

IJ> , then the sense of these forces is that in Fig. 16.29 and

opposite if

IJ< . The forces −N and N are contained in the movable plane

Ox x

′

and rotate together with the rigid solid with the period

2/T πω

′

= ; thus, they load and

unload successively the bearings

′

and O

′′

, which leads to their wear. Hence, the

assembling of wheels and fly-wheels on axles must be made very carefully, so that their

axis of symmetry coincide with their axis of rotation (to have

0θ = , hence 0N = ). In

MECHANICAL SYSTEMS, CLASSICAL MODELS

430

particular, in the case of a full wheel of radius

R and weight G, which is assembled

with the axle inclined by the angle

θ, obtaining thus

2/2IJGRg== , so that

()

/8 sin2NGRglωθ

′

= , where g is the gravity acceleration. Numerically, for

16 cmRl== , a revolution of 6000 rot/min (hence, 2 6000/60ωπ

′

=⋅

200 rad/sπ= ) and 1θ =° we get 28.1NG= ; we see thus that for an assembling

deviation of only

1° one obtains a significant gyroscopic effect, which cannot be

neglected.

Fig. 16.30 The motion of a railway car in a curve

An analogous study can be made for the motion of vehicles in a curve. Let be thus a

pair of wheels of radius

r, which form a gyroscope of symmetry axis

, inclined

with the angle

α with respect to the horizontal line, and which rotate with an angular

velocity

/vrω = (v is the linear velocity); the velocity of precession about the fixed

axis

′

is /vRω

′

= where R is the radius of curvature (Fig. 16.30). Because

Rr , it results ωω

′

 . The system formed by the two wheels acts (e.g., in case of

a railway car) on the rails with the gyroscopic moment (16.2.16); in the given

conditions,

/2θπ α=+ and we can write, with a good approximation,

cos

α=

(16.2.38)

Thus, arise the gyroscopic forces

N and −N, of magnitude

cos

rRd

α= ,

(16.2.38')

where

d is the rail gauge (the distance between the rails); these forces load the external

rail

r and unload the internal rail

r .

16 Other Considerations on the Dynamics of the Rigid Solid

431

Let us consider now a steamer; the axle of rotation

Ox (with the angular velocity

vector

ω ) of the turbine is along the ship and has the bearings O

′

and O

′′

as supports

(Fig. 16.31a). If, at a certain moment, the steamer has a motion of rotation about the

vertical axis

′

with the angular velocity

′

, then appears the gyroscopic moment

(16.2.16') (approximate value), which leads to the gyroscopic reactions

N and −N of

magnitude (

OO l

′′′

)

Fig. 16.31 The motion of a steamer. The

Ox -axis along the axis of the

ship (a) or the

′

-axis normal to this axis (b)

ωω

′

(16.2.39)

acting on the axle of the turbine; hence, it results a loading of the bearing at

′

and an

unloading of that at

′′

(the gyroscopic forces are of sense opposite to the sense of the

gyroscopic reactions; in Fig. 16.31a are specified the gyroscopic forces). Besides this

phenomenon, takes place a rotation about the horizontal axis in the transverse plane of

the ship (motion-of pitching). To put in evidence the effect of this motion (e.g., due to

the waves), we assume that the fixed axis

′

is horizontal and normal to the axis of

the ship (Fig. 16.31b); the gyroscopic moment

M is directed along the descendent

vertical at

O, while the gyroscopic reactions are given by the same formula (16.2.39),

the gyroscopic forces leading to loadings and unloadings too. The tendency of the

steamer to rotate about the vertical line is thus explained. The cases in which the axle of

rotation of the turbine is vertical or horizontal, in a transverse plane of the ship, can be

studied analogously.

These phenomena can be put in evidence, in the same way, also in the case of

aircraft. But we mention that, in this case, the ratio of the weight of the propeller or of

the rotary engine to the whole weight of the aircraft (built of a material as light as

possible) is much more greater; because of this, by a sharp turning takes place a pitch of

the aircraft which can lead to unexpected damages. This effect can be eliminated in case

of aircraft with two airscrews, the rotations of which are of opposite sense.

Besides the indirect actions mentioned above, the gyroscope can have also a direct

action of stabilizing the ships. Such a gyroscope, conceived by Schlick, is formed by a

flywheel

F which rotates with a very great angular velocity ω about the

Ox -axis. The

MECHANICAL SYSTEMS, CLASSICAL MODELS

432