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

Подождите немного. Документ загружается.

ends of the axle of the flywheel are sustained by a reinforcement which is rotating about

the axle

AOA

′

, which is transversal to the ship (Fig. 16.32); at the bottom of the

reinforcement is put a weight

G, so that – in the normal position – the

-axis be

vertical. If, under the action of the waves, the steamer is inclined and rotates about the

longitudinal axis

′

with the angular velocity

′

(motion of rolling), then arises a

gyroscopic moment

of magnitude

I ωω

′

, which imparts to the flywheel a rotation

about the axle

AOA

′

(motion of pitching); this leads to a new gyroscopic moment

′

which, by means of the bearings

A and A

′

, is in opposition to the motion of pitching,

diminishing thus the inclination of the ship about its longitudinal axis. This gyroscope

proves its utility in the rectilinear motion of the steamer, as well as in the case of the

rotations about the vertical line in the same sense as its rotation; in the case of a rotation

in an opposite sense, take place supplementary motions of pitching, which can be

dangerous. For this reason, there have been invented and built also other anti-pitching

gyroscopes much more perfected, used to stabilize the steamers (e.g., the Sperry

gyroscope); in this order of ideas, we mention the pilot gyroscope too, which

determines the motion of precession.

Fig. 16.32 Schlick’s gyroscope

The stabilization of the cars with only one rail (monorail) is analogously realized.

The gyroscope with a vertical axle considered by Scherl to this purpose is analogous to

Schlick's gyroscope. Brennan has introduced a gyroscope with a horizontal axle, useful

in case of a rectilinear motion. To increase the precession, other auxiliary apparatuses

are used. We mention also the realization of the monorail coach, of the gyroscopic

motorcar on two wheels and of the gyroscopic bicycle with only one wheel.

The motion of the bicycle with two wheels is also stabilized by the gyroscopic

moments which appear. If the linear velocity is

v, then the angular velocity is given by

/vrω = , where r is the radius of the wheel. Assuming that, for some reason, the

bicycle is inclined with an angular velocity

′

, arise the gyroscopic moments

16 Other Considerations on the Dynamics of the Rigid Solid

433

′

=×ωωM , to which correspond the angular velocities

ω and

ω , which lead to

the new gyroscopic moments

′

=×ωωM ,

′′

=×ωωM , of a sense opposite

to that of

′

(Fig. 16.33); these moments tend to bring back the bicycle in the vertical

position, stabilizing its motion. Obviously, this modelling of the phenomenon is

approximate; the problems which arise are multiple and much more difficult.

Fig. 16.33 The bicycle with two wheels

We meet applications of the gyroscopic effect also in case of mills with movable

pulleys, of pendulary mills or in case of other mills used to grind cereals, grains or other

materials.

As we have seen in Sect. 16.2.2.3, the gyroscope can be used successfully also as an

orientation device. Thus, the gyroscopic compass, with the mobile axle in the local

horizontal plane is used to determine the direction north-south in the motion of ships

and, especially, of submarines. The azimuth gyroscope (with the mobile axle in the local

meridian plane) is used to determine the latitude at the respective position on

the surface

of the Earth. As well, a heavy gyroscope, with the fixed point over the centre of gravity,

allows to determine the gyroscopic horizon (the artificial horizon), even when – on the

sea – this one is in a thick fog; the gyroscopic inclination compass determines the

inclination with respect to the horizon of the longitudinal axis of an aircraft, as well as

of its transverse axis.

Fig. 16.34 Motion of a projectile. Constant direction of it (a). Screw motion of its vertex (b)

We meet gyroscopic effects which must be taken into consideration in ballistics too;

thus, a projectile, besides the motion of its centre of gravity

C, has also a motion of

rotation of angular velocity

about its axis of symmetry

Ox . If the motion takes

MECHANICAL SYSTEMS, CLASSICAL MODELS

434

place in vacuum or in the air of resistance

R , passing always through C, we have

=M0

, so that the

-axis does not change its direction (Fig. 16.34a); the axis of

the projectile moves away from the

tangent to the trajectory and comes up against a

resistance of the air more and more greater, while the target is reached with the inferior

part of the projectile. In reality, the resistance of the air

R does not pass through the

centre

C, but through a point P at the distance l from C; we assume that this resistance

is approximately parallel to the velocity

v of the centre of gravity. There appears a

gyroscopic moment

applied at C, normal to the plane of the trajectory, which leads

to a motion of precession of the projectile about the vector

v with an angular velocity of

magnitude

sin

ωθ ω

′

(16.2.40)

Fig. 16.35 The direction of the vertex of a projectile: upwards (a) or downwards (b)

Thus, the vertex

V of the projectile is directed upwards (Fig. 16.35a) and then

downwards (Fig. 16.35b); the vertex of the projectile tends to the tangent to the

trajectory (the projectile runs after its tangent). In the real motion, the projectile vertex

describes a screw motion around the trajectory of its centre of mass (Fig. 16.34b). By a

convenient choice of the form and of the dimensions of the projectile, the point

P will

be situated between the points

C and V, the angle remaining sufficiently small, so that

the projectile be falling with its vertex.

An important rôle is played by the directional gyroscopes, e.g., by those used to

maintain the direction of the automobile torpedo; we mention also the automatic

directional gyroscopes (e.g., the gyropilot). We put in evidence the navigation by

inertia and the gyroscopic directing of the rockets, useful for the guided rockets, the

intercontinental rockets and the interplanetary rockets.

16 Other Considerations on the Dynamics of the Rigid Solid

435

16.3 Dynamics of the Rigid Solid of Variable Mass

We take again the problem of dynamics of the mechanical systems of variable mass by

some considerations concerning

the application of variational methods of calculation;

we apply then these methods to some particular cases of motion. As well, we consider

the case of the aircraft fitted with jet propulsion units.

16.3.1 Variational Methods of Calculation

In the following we present firstly the approximate variational methods of R. Goddard

and H. Oberth, passing then to considerations concerning the general methods of

calculation with a variational character. We will assume a particle model of the rigid

solid, using the results obtained in Chap. 10, Sect. 3.

16.3.1.1 Goddard’s Approximate Method

Let us consider the rectilinear motion of a particle of variable mass in a gravitational

field, with a resistance of the medium

(, )vx=QQ , where v is the velocity along the

Ox-axis; the equation of motion, corresponding to the equations obtained in Chap. 10,

Sects. 3.1.2 and 3.2.3, reads

sin ( , )mv mg Q v x mwθ=− − −



(16.3.1)

Fig. 16.36 Goddard’s approximate method of calculation

where

w is the relative velocity of the emitted mass with respect to a non-inertial frame

of reference rigidly linked to the particle in motion while

θ is the angle made by the

trajectory with the horizontal line (Fig. 16.36). Taking the mass of the form (10.3.11')

and putting

(, ) (, )Qvx m vxϕ= , we may write

sin

vfg gfwθϕ=− − −



(16.3.1')

finding thus solutions of the form

(; , ; )vvtffC=



(; , ; , )xxtffCC=



, where

,CC are integration constants; these solutions can be taken as functional equations of

some problems of optimization, where various characteristics of the motion (the

distance travelled through, the time in which a certain position can be reached, the work

and the resistance of the medium) take extreme values. In 1919, R. Goddard proposed

an approximate solution to determine the function

()vvx= , so as to reach a given

MECHANICAL SYSTEMS, CLASSICAL MODELS

436

height with a minimal mass; the total height is divided in

n parts and on each part the

resistance of the medium is considered to be constant (

constϕ =

). As well, one

assumes that

/2θπ= ,

constva==



, obtaining an equation of the form (10.3.13')

with constant coefficients

++=



(16.3.2)

The solution (10.3.13") of this equation leads to

{

}

[( )/ ] [( )/ ]

() e e

ag wt ag wt

ft C

mag

−+ +

== −

(16.3.2')

with

1/()Cgagϕ=+ + (we notice that (0) 1

= ). If the mass used up at a given

moment is

mm− , hence if

1( )/

mmm=− − , then it results

{}

[( )/ ]

11e

ag wt

mag

−+

−

⎛⎞

=+ −

⎜⎟

⎝⎠

(16.3.2'')

Imposing, together with Goddard, that that the final mass be equal to unity (

1m = ),

we get

{}

[( )/ ] [( )/ ]

e1e

ag wt ag wt

=−−

(16.3.3)

0Q = and 0g = corresponds

(/ )

awt

∗

= , so that the problem of extremum

of the mass, in the given conditions, consists in minimizing the ratio

(/ )

00 0

awt

mm m

∗−

= .

The calculation is made successively for each of the

n intervals, till the final mass

equates the unity. Thus, we obtain also the time in which the displacement takes place

and which corresponds to the minimum of the ratio of the masses.

16.3.1.2 Oberth’s Approximate Method

For the same problem, H. Oberth assumes in 1929 that the velocity of the particle is

negligible with respect to the relative velocity of emission of the mass of combustible

(

vw

); denoting m=+RQ g, it results the equation of motion along the

ascendent vertical (Fig. 16.37)

mv R mw=− −



 . (16.3.4)

We notice that

(d /d )vvxv=



, (d /d )mmxv= , so that we can also write

vR m

xv x

++ =

(16.3.4')

16 Other Considerations on the Dynamics of the Rigid Solid

437

If at the height

x we impose a condition of minimal consumption of combustible

(

d/d 0mx= ) and if we differentiate the equation (16.3.4') with respect to v, then we

get

()()

vxvv

∂∂

Fig. 16.37 Oberth’s approximate method of calculation

In the hypothesis

dconstmv= , it results

() ()

RmgQ

vv v v

∂∂+

∂∂

(16.3.5)

where the force

R is opposed to the motion ( /Rv is the resistance on unit of path and

time). One obtains thus the optimal velocity to raise a particle of variable mass (for

which the unit loss caused by the force

R is minimal). We have seen in Chap. 10,

Sect. 3.2.5 that the resistance can be taken in the form

() /2Qv bAvρ= , where

()bbv= is the aerodynamic coefficient, ρ is the density of the air, while A is an area

characteristic for the rigid solid modelled as a particle (in case of the aircraft, the area of

the wing). If we assume that

constm = on a stratum of width dx, then the relation

(16.3.5) leads to

11d

22d

mg b

bA Av

ρρ−+ + =,

wherefrom

[(/)]

Ab v db dvρ

(16.3.6)

Taking

constb = , it results

2mg

= .

(16.3.6')

MECHANICAL SYSTEMS, CLASSICAL MODELS

438

Hence, if the resistance of the medium

()Qv is in direct proportion to the square of

the velocity, then it will be equal to the weight of the particle (

Qmg=

); in this case,

the optimal velocity of the rigid solid of variable mass is the velocity which must have

the particle of mass

m in free falling in a homogeneous medium of given density ρ.

From the above results, one obtains easily all the kinematic and dynamic characteristics

of the motion.

16.3.1.3 General Considerations on the Application of Variational

Methods of Calculation

The methods of calculation of Goddard and Oberth have an approximate character; we

give a formulation of the same problem, in what follows, in a rigorous variational

calculus. We have seen in Chap. 7,

Sect. 2.1.4 that, in case of a functional

( , ,..., )

Iy y y of the form (7.2.13) for the functions ()

yx, 1,2,...,kn= , of the

same independent variable

x, we are led to the Euler-Lagrange equations (7.2.13').

Imposing the supplementary conditions

12 12

( ; , ,..., ; , ,..., ) 0

xy y y y y y

′′ ′

= , 1,2,...,jh= ,

(16.3.7)

too, we introduce the auxiliary function

FF fλ

∗

∑

(16.3.7')

where

12 12

( ; , ,..., ; , ,..., )

FFxyy yyy y

′′ ′

= is the function under the integral operator

in the functional (7.2.13), while

()

xλλ= , 1,2,...,jh= , hn< , are Lagrange’s

multipliers which must be determined; the equations (7.2.13') lead to

()

∗∗

′

−=, 1,2,...,kn= ,

(16.3.7'')

′

being the partial derivatives with respect to the corresponding arguments (

and

yyx

′

=∂ ∂ ). We obtain thus a system of n differential equations (16.3.7") and a

system of

h non-holonomic constraint relations (16.3.7) for the n functions

, ,...,

yy y and for the h parameters

, ,....,

λλ λ.

In our case, the problem of determination of the law of variation of the particle mass

is put, so that the path travelled through

xvt=

∫

have a maximum, T being the

time in which the motion of the particle takes place, while

X is the space travelled

through. In this variational problem the function

v under the integral must verify the

differential equation of the particle of variable mass (16.3.l') on the active segment

[

]

0,t (till the end of the process of emission, hence of combustion of a rocket) and the

differential equation of the particle of constant mass (

()mmt= ,

()

ft= )

16 Other Considerations on the Dynamics of the Rigid Solid

439

sin

vfg gθϕ=− −



, (16.3.8)

on the passive segment

[

]

,tT. To apply the general method of calculation, we

assume, after A.A. Kosmodemyanski, some simplifying hypotheses. Thus, we consider

that, on the active segment, the consumption of mass on a second is a finite magnitude,

being negligible on the passive segment; thus, the function

f, partially continuous on the

interval

[

]

0,T

, is replaced by a continuous function, sufficiently close to that one. In

what concerns the condition (16.3.8), we can have any velocity at the initial moment, as

well as any value for

> ; hence, this condition may be overlooked in the

variational problem which was put, remaining with the condition (16.3.1'), written in

the form (

d/dvvvx=



, d/d

vf x=



)

sin ( , ) 0

vv fg g v x vwfθϕ

′′

++ +=

(16.3.8')

In our case,

1F = , while

()yfx= ,

()yvx= , so that

[

]

1sin(,)Ffvvfggvxvwfλθϕ

∗

′′

=+ + + +

The equations (16.3.7") lead to

()()0

fv g wf vw

λϕ λ

′′ ′

++ − =

(sin)()0

vv g vw

λθλ

′

+− =

Developing and eliminating

λ and

′

between these equations, we get

() sin0

vgg

wfw

ϕθ

′′

⎛⎞

′

+−−+ =

⎜⎟

⎝⎠

Eliminating the sum

fvw

′′

+ between this relation and (16.3.8'), we obtain

[]

()

sin

fvwvw

ϕϕ

′

=−+

(16.3.9)

Taking

() () /2vbvAv mgϕρ= , as in aerodynamics, we can write

[]

()

2sin

mmf vwbvwb

′

== ++

(16.3.9')

where we assume, usually, that the gravity acceleration does not

vary with the altitude;

we have thus determined the relation between the mass and the velocity of the particle

in case of an optimal regime. In the hypothesis in which

constb =

, the relation

(16.3.9') is reduced to

MECHANICAL SYSTEMS, CLASSICAL MODELS

440

(

)

2sin

Abv

(16.3.10)

while, in case of the motion on the vertical, it results

(

)

mg Abv

ρ=+

(16.3.10')

vw , then we find again Oberth’s formula.

16.3.2 Applications to Dynamics of the Rigid Solid of Variable Mass

Using again the modelling as a particle of the rigid solid of variable mass, we apply the

above results to the study of the motion in a homogeneous atmosphere and of the

motion by simultaneous capture and emission; as well, we make some considerations

concerning

the motion of the rocket.

16.3.2.1 Motion in a Homogeneous Atmosphere

In 1946, A.A. Kosmodemyanski dealt with the motion of the particle of variable mass

in a homogeneous atmosphere (

constρ = ), using the above considerations. Noting

that

d/d

vf x=



, the equation (16.3.1') reads

()

[

]

sin ( )

wf v g f vθϕ

′

+=− +



(16.3.11)

In this case, the length of the path travelled through the rectilinear trajectory is given by

[]

()d

d(;,)d

sin ( )

TV V

wf v v

vt Fvf f v

gf vθϕ

′

==−

∫∫ ∫

(16.3.12)

where

the initial velocity.

Writing the Euler-Lagrange equation corresponding to this functional, we get

( sin ) ( ) sin

dsin

(sin )

vf f wf v

θϕ θ

θϕ

′

+−+

−=

being thus led to the relation (16.3.9) in the particular case in which

constg = and

()vϕϕ= .

The relation (16.3.11) allows to write

[]

sin ( )

gf vθϕ

′

∫

(16.3.13)

where

()

v is given by the relation (16.3.9); we obtain thus the velocity ()vvt= .

Associating the relation

()

fv= too, we determine the variation law of the particle

mass, hence the optimal regime of work of the motor, in case of a rocket.

16 Other Considerations on the Dynamics of the Rigid Solid

441

16.3.2.2 Motion by Simultaneous Capture and Emission

Let be a particle of variable mass which is moving in the same conditions as above in a

uniform gravitational field, with the initial condition

(0)vv= . In the case of a

phenomenon of simultaneous capture and emission, the differential equation of motion

is of the form

sin ( ) ( ) ( )mv mg m g v m u v m u vθϕ

−+

−

=− − + − + −





(16.3.14)

where

−

< and 0m

> correspond to the variations of the emission and of the

capture of mass, while

−

and

are the absolute velocities, respectively. Denoting by

wuw

−−

=−

the relative velocity – considered to be constant – of the emitted masses,

with respect to the non-inertial frame of reference, assuming that

= (case

considered by Levi-Civita) and that

/mmγ

+−

=−, we also can write

[]

sin ( )

mv mg v m wθϕ

−

⎛⎞

=− + + +

⎜⎟

⎝⎠



(16.3.14')

Because

constw

−

= at the end of the active segment (

tt= ), when 0u

−

= , we

have

−

= , where

v is the velocity at the respective moment.

If we denote

() ()mt mft

−

= , with (0) 1

= ,

(0)mm

−

= , then it results

(1 1/ ) (1 1/ )mm m m mfγγ

−+ −

=+= − = −



  

, because

() ()mt m m t

−

()mt

+ ; we get thus

[

]

() ()mt mbft a

=− + and

[

]

() ()mt m af t b=+,

11/a γ=− , 1/b γ= , 1ab+=. Replacing in the equation (16.3.14'), we obtain

[

]

() ()sin ( )af b v g af b f v bvθϕ+=− + +− −



(16.3.14'')

Choosing

v as independent variable, we can write

[

]

[]

( ) () ()

(;, )d d

() sin ()

vbvfvafvbv

gx F v f f v v

af v b vθϕ

′

−++

′

∫∫

(16.3.15)

We impose a variation of mass given by

()

v , so that the displacement x have a

maximum. Writing the corresponding Euler-Lagrange equation, we get

[

]

111

()()(2) ()(2)sin

()

(2)sin

vv bv v a bv v v bv bv

av bv

ϕϕθ

′

−++−−−

−

(16.3.16)

As well, starting from the same relation (16.3.14"), we can write

[]

( ) () ()

() sin ()

vbvfvafvb

gt v

af v b vθϕ

′

−++

∫

(16.3.15')

MECHANICAL SYSTEMS, CLASSICAL MODELS

442