Gimbal stabilization

GIMBAL

STABILIZATION.* BY

V. BUSH, Eng.D. ResearchLaboratories, American Radio and Research Corporation. GENERAL. THE following is an analysis of the effectiveness of gimbals of various designs for maintaining a horizontal platform on shipboard, and of gyroscopic stabilizing of such devices. Gimbals can never maintain a platform absolutely horizontal, for there always will be a certain deviation. The motion of the ship in the waves can affect the gimbals in three ways, namely by reason of (I) horizontal motions impressed upon the support, (2) vertical motions impressed upon the support, and (3) tipping of the support. If the friction of the gimbals about the pivots is small, as must always be the case in a good design, the third effect is entirely negligible. The second effect comes in only when the supported If the angle of deviation platform deviates from the horizontal. is small, as it must be in a case of practical value, the second effect is also negligible. There remains to be considered only the first It is suffipoint, the effect of moving the support horizontally. cient to consider a single plane only, as motions at right angles produce independent effects. The to and fro motion of the support, due to rolling or pitching of the ship in a s’ea, is not strictly harmonic. It is usually a rough beat motion which can be considered the sum of two sinusoidal motions of somewhat different frequencies. Each of these may be considered separately if desired. To a first approximation, however, it is sufficient to conObsider a simple harmonic horizontal motion of the support. viously the motion of the gimbals does not react appreciably to affect the motion of the ship, hence we may consider the motion of the support to be an undisturbed sinusoidal oscillation. PEBDULUIU.

The above problem is hencei the problem of considering the motion of a physical pendulum of which, the support is given a sinusoidal horizontal motion. In Fig. 1, we have such a pendulum located by the coijrdinates * Communicated by the Author. I99

[J. F. I.

V. BUSH.

200

x and 8, where expression

x is caused

to vary

3c=A

in accordance

with

the

sin wt

The pendulum has constants m = total mass

b = length from centre of mass to support a = radyus of gyration FIG. I.

x

-

i

\ t-

It will also be convenient to use the coordinate y, which for the small values of B to be considered throughout is given by y =

x-be

ANALOGOUS ELECTRICAL

CIRCUIT.

The formulas for forced harmonic motions of electrical circuits are in much more convenient form than the formulas for mechanical systems. It will accordingly be convenient to solve first the electrical circuit which is analogous to the mechanical system of our problem, and then to interpret the results on the problem itself.

Aug., 1919.1

GIMBAL STABILIZATION.

201

We will consider first the simple pendulum in which a=b

Such a pendulum

when the support

is moved in accordance

with x=

A sin at

is analogous to the series electrical circuit of Fig. 3, containing resistance R, inductance L and capacity C, when independently FIG

2.

xI-

of the circuit a charge is supplied to the condenser in accordance with p = Q sin d This will he true when we have in a pair of leads connected across the condenser, the current

Using the vector notation this would he written I = j Qu

V.

202

[J. F, I.

BUSH.

If I, is the vector current in the inductance, we may write the relation between I, and I by considering that I divides into pa,rts proportional to the admittances of the two branches. That is I

j

I

Co +

I*= I

-

U-.+J=~

jQ

I

R +jLw

I-LW

TRANSFER

+ j R Co = I-L Cd+jRCo

TO MECHAHICAL

SYSTEM.

Considerable care must be used, in interpreting this result on the mechanical system, to obtain exact analogues. We have already set the maximum quantity Q analogous to FIG. 3.

L

R

the maximum analogous

displacement

to +

or i.

A. Hence the current

If a displacement

in the leads is

A is produced, and the

centre of mass is simultaneously prevented from leaving the centre line, there will be, for a small angle 8, a back force produced equal to y

A; as can be seen from Fig. 4.

Analogously

in the

electrical circuit if a charge Q is introduced through the mains, and the branch circuit is open so that this affects the condenser only, there will be a back electromotive C analogous

to &

forceT. Q

We thus have

Aug., w9.1

GIMBAL STABILIZATION.

The kinetic energy of the mechanical pendulum with concentrated bob is m (&Ii,)2

203

system in the case of a

2

and the energy stored in the magnetic field of the electrical system is Lip L -it-

Hence L and m are analogous ; and.& corresponds to (& - b$ or j, that is to the net horizontal speed of the centre of the bob. FIG. 4.

The frictional constant F, which is the force which occurs when the bob moves at unit net constant speed, corresponds to the resistance constant R which is the force arising when unit steady current flows in the branch circuit. We may make the analogy more apparent by writing the differential equations for the two systems. For the mechanical system, equating the forces on the bob, we have my+

Fy=mgfl

which may be arranged mj+Fy+yy=yA

sin

of

204

[J. F. I.

V. BUSH.

For circuit

the electrical

system,

summing

L i, + R i, +

;

I

(i,-ij

which becomes upon differentiating

the voltages

about

the

dt = o

and substituting

L 1 +Ri+;i,=

.&. c Qw cos cot

Place side of this the derivative of the equation cal system mg mY + Fy + T y = mjAocoswt and we may read off immediately

for the tiechani-

the analogies

Inserting these quantities in the expression above for the electrical circuit, we obtain for the mechanical system jAw Y=

I - 7

+j Fbo m&T

or Y= *-~!+jE$!

and

since y = .%--be, and x = A sin w t, A-

fi=+

A (I-!!!!)

+jE$

C

We are interested in the maximum value to which 6’attains, and hence in the maximum value of the above vector expression, which becomes when simplified

J[(I-

timax = A - _ b

7) *_(I_ 7) + (!5)2] *+ (E2) (I-g+

(!F)

This formula checks dimensionally, thus affording cation that the substitutions have been made correctly.

an indi-

.4ug.,

GIMBAL STXBILIZ.ATIOIC.

1919.1

When

the frictional

2%

force is zero it reduces to il

These formulas hold only for small values of 8, and the absolute value of the result should be considered irrespective of sign. When there is resonance, that is when the period of oscillation is the same as the natural period of the pendulum, which will occur jvhen \ve have and the formula shows that a large angle may result from even a small amplitude of oscillation of the support. On the other hand, when b is very long, the angle of deviation is nearly zero. In the above the frictional force is assumed proportional to -1, that is to the net velocity of the bob through the medium; and the friction of the pivot is considered negligible, as must be the In case if tipping of the support is not to affect the pendulum. many practical cases it is nearer the truth to consider the frictional force to be proportional simply to be, that is to consider that the medium moves with the support. If this assumption is made, the differential equation for the mechanical system becomes my+

Fy+yy

= m~A~cos~~t-I;Aal’sin~t

In Fig. 3, if we alter the current leads to be connected across the coil alone, instead of the coil and resistance, the equation for the circuit becomes L;~fR&t+=f-

c Qu cm d - RQd sin wf

The same analogy thus holds as before, and if this new circuit be solved, and the substitutions made, we obtain for the maximum deviation for the assumption that the medium moves with the pivot :

which reduces, upon setting for the frictionless case.

F = o, to the same simple

formula

[J. F. I.

V. BUSH.

206

PHYSICAL

PENDULUM.

We will now corAder the case where all of the mass cannot be considered as concentrated at a point, as is the case for the systems shown in Figs. I and 2. The peculiarity of this case is that there may be kinetic energy stored in the system even when the centre of mass is not moving. If x and 0 are both increasing in such a manner that ; =i-bti is zero, there still is a kinetic energy of rotation about the centre of mass. If “a” is the radius of gyration about the pivot, the radius of gyration about the centre mass at a distance “b” from the pivot will be and the above kinetic energy will hence be equal to m (a2- b=) iit 2

Consider now the circuit of Fig. 5, in which there is added The circuit the inductance “ 1” in series with the condenser. may store energy in a magnetic field even when there is no current in the branch circuit, that is when i, = 0. An analogy with the mechanical system may be set up for this circuit exactly as before, and by the same reasoning as above used we have Mechanical

Electrical

A

Q

i

X ,-

il

C

Now

bti’

bo‘ b/w

i2

when i1 = o, the energy in the magnetic 1;

field is

and we see that if this is to correspond to the kinetic energy above when i, is set equal to b8 we must have Mechanical

Electrical I

I

?Tt-

az - b1 ba

If now be’is taken as zero so that we have simply translation,

the

Aug., 1919.1

Also, if i, is zero so that i = iI, the magnetic

kinetic energy is mT. energy

207

GIMBAL STABILIZATION.

LZ=

is :,

and since i and x are analogous,

we have

Mechanical

Electrical

m

L

as before. FIG. 5. I

I

c-

I

i2

T

i

1

.

11

We may check the consistency of our analogy as follows: In the electrical system assume i = O, and hence i, = i,. The magnetic energy is CL -I- 1) 2 2

Similarly in the mechanical system, assume the pivot at rest, and The kinetic energy is then the pendulum simply rotating. a% ml ma2 Hence we must have the analogy Lfl -iF But we already have

1

so

L

that we must have

but this reduces to simply

L

a2 - b2 p a2- b2 aa *p-m-p-

*

??a

208

[J. F. I.

BUSH.

V.

as before, giving us a check on the analogy as far as this part of the circuit is concerned. Now in the electrical system we have I I

f-J-R + jb

.ilw+* I, :

I

Hence by analogy

RtjLw

in the mechanical y -_ $0

system we have

(I- 02)

w2

a%? f j Fbw ‘-bg

fig

and from this by the same process of reduction as used before, we may obtain for the case of the medium at rest :

It will be noted that this formula reduces to the one derived for a simple pendulum when a= b, and also that it is dimensionally correct. When F = o it reduces to -__.4 emax =(L? _- - _ b

5

This formula is particularly interesting in showing that when f is large, that is with a pendulum of large radius of gyration, mounted only slightly away from its centre of mass, the angle of deflection wi!l be very small. Also resonance may be obtained exactly as before, the frequency of oscillation in this case being I 2*

GYROSCOPIC

g6 d

-2

STABILIZATION.

The effect of a very long pendulum may be obtained in an instrument of small dimensions by the use of the gyroscope. An arrangement of this sort is shown in Fig. 6. The size and

Aug., 1919.1

GIMBAL

STABILIZATIOX.

209

speed of gyro, and the strength of the restoring spring, determine the constants of the equivalent pendulum. Assume first that the weight of the frame can be neglected in comparison with the gyro. Let the gyro be of such size and speed that a unit torque about the pivot causes precession at the rate of p radians per second. Let the spring be of such strength that a displacement of the gyro of one radian brings to bear on the gyro a restoring torque s. Let the centre of gravity of the system be a distance c below the pivot, and let the mass of the system be M. FIG. 6.

Consider the system initially displaced through a small angle +, the gyro being in mid position. A torque Mgc+ acts on the gyro, and it precesses at a rate Mgcpe radians per second. The

spring then exerts

a restoring SJ AJgcpedt

torque

and under its influence the gyro precesses about the pivot at a rate MgscpzJqd2 radians per second. That

is &

dt =

- Mgscp2(9dl

or

The pendulum will then oscillate in accordance 9 = $JO sin pl/Mgscl

with the formula

[J. F. I.

V. BUSH.

210

If b is the length of the centre of gravity of the equivalent cal pendulum, and a is its radius of gyration we have

physi-

from which I

a2 -=b

MS@

Disregarding friction we may then write for the maximum deviation of the pendulum from the vertical when the support is moved harmonically a distance A either side of the centre position A

8max = --a2

g cd2

b

or 8max =

The rate of precession

A

I

MscP

--

5

of a gyroscope II, =

is given by

T Tv

where Q is

the angular velocity

T is

the applied torque.

of precession,

I is the polar moment of inertia of the wheel, V is the angular velocity of the wheel,

and Letting T=r I

we have

P = Ti

from which EZ = PP MSC b

and thus hax

=

A Ixv2 --Msc

2

In order to find a, assume the pendulum to be hanging vertically, with the gyro in its mid position. Suppose now we apply a torque T tockflect the pendulum from its central position. The gyro first precesses at a rate

$J = I;

radians per second,

Aug., rgrg.1 whereupon

GIMBAL the restoring

211

STABILIZATION. torque

spring exerts a restoring

equal to

STt I2112

which in turn results in a precession torque

in the direction

of the applied

at a rate STt radians per second. IF@ PLATE I.

QU

b

OX

0

4*

4 d

rrE

A

04 - 2.3

.3

1.7

.2

1.1

a

‘M-

I

I

I

I

I

I

I

I

I

I

2

4 10

6

8

10

12

14

I6 40

The angular

15

20

acceleration

25 30 FEET

35

I

I

“b” Case 45 “Vcase

.6

1

I I

II

is thus ST p-j72

If we consider a physical pendulum of mass M and radius of gyration a acted upon by torque T, the angular acceleration will be T Ma2 Hence we have $=

I2 - 7,‘2 SM

[J. F. I.

V. BUSH.

212

and since from the above a2 - E b

I2 v2 SMC

it follows that b==c

Thus the effect of the gyroscope is simply to cause the equivalent radius of gyration of the system to be increased to the value a_Usually

with gyroscopic

JSM

stabilizing+

only the gyroscopic effect need be considered, small we may write approximately Bmax = AMsc~2 =

will be so small that and unless w is very

AMsc 121.‘y

EXAMPLES.

In the following illustrative examples we will consider two cases. Case No. I is intended to be typical of what will be found on smaller boats. The roll is considered regular and 30” each side of the vertical, occurring with a period of five seconds. The instrument is assumed mounted at a distance of 4 feet above the axis about which the craft rolls. We then have in terms of feet and seconds A-2

w=zn-.2

and x : 2 sin 1.2561 Case No. 2 is intended to represent a large ship. The roll is considered to be I 5 ’ either side, with a period of I 5 seconds, the instrument mounted 20 feet above the centre of roll. Then A = 5.2

, w _-- 2

and x

5.2

sin .4181

SIMPLE PENDULUM.

Consider first a simp’e pendulum. and neglect friction. maximum angle of deviation is given by Hmax--

-

b-

A 5

The

Aug..IgIg. 1

GIMBAL

STABILIZATION.

213

The results of computation for various values of b are plotted on Plate I for the two cases above, The curves show that in Case No. I a pendulum of reasonably short length will deflect about 5 degrees while a pendulum 15 feet long will deviate an amount greater than twenty degrees, so much in fact that the formulas no longer hold even approximately. In Case No. 2 the deflection for a short pendulum will be about I .5 degrees, this deflection becoming large as the length of pendulum approaches 183 feet. PLATE

II.

.2 .4 .6 .8 1.0 1.2 14 ;",,';B&O 2.2 2.4 2.6 2.8 30 3.2 3.4 3.6 'b"Case I. .I .2 .3 .4 .5 .F 7 .8 .9 1.01.1 1.2 1.3 1.4 1.5 1.6 1.718 'b"CaselI

This example shows conclusively that a simple pendulum of reasonable length swung in frictionless gimbals on board ship cannot be expected to remain vertical within several degrees. PBYSICAL

PENDULUM.

Xs a second example consider a fly wheel with heavy rim swung slightly off centre. Suppose the radius of gyration, which will be approximately the mean radius of the rim, to be I foot, and disregard friction. The formula now is Hmax :Y

\~oI.. 188, KCI. 1124-16

a4 -b

A

w’? e

2r4

V. BUSH.

[J. F. I.

On curve sheet 2, are plotted the results with this fly wheel suspended various distances off centre. Considering Case No. I we see that when the distance off centre is’several inches, the deviation is several degrees, as with PLATE

III.

1

1000

I 2000

3000

4000

5000 “.V” R.P.M.

the simple pendulum. If the eccentricity is shortened the angle becomes larger, and at one inch is about 15 degrees. If “b” is made very small, however, conditions improve, and when b is .12 inches the deflection is 1.4 degrees. If b is made .OI inches

GIMB;\L

Aug., 1919.1

STABILIZATION.

215

the deflection will be in the neighborhood of . I degree.. In this latter case, however, there would not be probably enough eccenFor Case No. 2 conditions tricity to properly overcome friction. It will thus be seen that even using a are even less favorable. massive pendulum it would be difficult to design in such a manner that very small deflections would result. USE OF GYROSCOPE

Let us now consider the use of the gyroscope for stabilizing. Suppose the gyro wheel to weigh I pound, with a radius of gyration of .25 feet, revolving at speeds up to 5000 Y. p. m. Suppose the mass of the whole system to be two pounds, with a centre of Suppose the restoring spring gravity .5 feet below the support. to be of such strength that when the gyro is deflected through r radian or 57 degrees it exerts a force of .05 pounds at a lever arm of .I foot. Then c =- .5 M= 2 s =.005

The

formula

is fhax =

A

12v2 --Msc

and

for the values Case No. I

of the example

fhzax=

Case

No.

,“2

: 2

.78 v - 20.4

2 Hmar =

5.2 .78P - 183.6

On Plate III are plotted curves to 5000 r. p. m., that is up to v=

‘T

for this example

for speeds

up

= 524 radians per second.

It will be noted that at this highest speed the maximum deflection in the two cases is .0005 and .0014 degrees, respectively. Thus practically complete stabilization is obtained. MEDFORDHILLSIDE, MASS., Nov. 8, 1918.