The influence of earth's flattening on the behavior of a horizontal gyrocompass

THE INFLUENCE OF EARTH'S FLATTENING ON THE BEHAVIOR OF A HORIZONTAL GYROCOMPASS (VLIIANIE

SZHATIA

ZEMLI

PMM Vo1.27,

NA RABOTU

No.2,

OIROGORIZONTKOMPASA)

1963,

D. R.

pp.

203-210

MERKIN

(Leningrad) (Received

January

24,

1963)

This paper presents investigations of a model of a force system proposed by Ishlinskii in [l-41 which acts on a material system moving close to earth’s surface. Simultaneously with the new method Ishlinskii introduced new definitions of a local vertical and of a horizontal plane which Consequently, the results prediffer from the conventional definitions. sented in the above mentioned papers require special explanations. In this paper the flattening of the earth is taken into account, the differential equations of motion of a horizontal gyrocompass are constructed and the conditions under which the instrument will show the true local vertical and the true meridional plane are determined.

1.We shall ordinate

investigate

the motion of a particle

system which translates

in an inertial

in the Ox,y,zl

reference

co-

frame while

the point 0 moves arbitrarily on the earth’s surface. lhe force Q which acts on a particle in the Oxlylzl system can be represented by two equivalent expressions

Q = P - mw*

(1.1) (4.2)

Q=F-mw Here m is the mass of the particle,

P = mg is

the gravity

force,

which is the resultant of the earth’s gravitational attraction force F and the centrifugal force mw,, where we is the translational acceleration

of the point

0 caused by the earth’s

P= F-

rotation.

l’hus

mw,

(1.3)

‘he vector w* is the sum of two vectors wr and wc, where wr is the acceleration

of the point

0 in its

motion relative

303

to earth’s

surface,

304

D.R.

and wc is

the Coriolis

Met-kin

acceleration w* = w, $- WC

and w is

the acceleration

of

the point

(1.4)

0 with

respect

to

the inertial

system

w = we + w, + Direct

measurements

magnitude direction that tion

on earth’s

or the direction does not coincide

(1.5)

WC

surface

cannot

determine

either

the

of the gravitational attraction force F. This with the local vertical and the assumption

it is pointing toward the earth’s center is only a first approximac5-71. The gravity force P = mg = F - mwe is being measured directly

and the

force

F is

not

needed

for

its

determination.

The force

P is

directed along the plane equals zero.

local vertical and its projection on the horizontal For these reasons, when due to conditions of a probmust be referred to the local vertical and the horizontal

lem the motion plane,

it

is

customary

In the years

to

use formula

1956 and 1957 Ishlinskii

formula (1.2) instead of (1.1); earth and that the gravitational the

earth’s

center

along

obtained

many results

behavior

of

navigation authors,

[l-4,81.

In order tained

from

used

by the

the

definitions The local

line with

of

in

[l-41

vertical,

attraction

force

systems

years

properly it

all

these diff er

or the

of

true

Ishlinskii

systems

on the

of

appeared

and the

to examine

inertial by other

is

results

certain

unduly

conventional

vertical,

the

assumptions.

without

papers,

of

toward

pendulum,

papers

the new method

is necessary

use of

directed

assumptions

and on the

similar

the

sphericity

a physical

a number of

from the

[l-41

the F is

Under these

equilibrium

who used

application,

authors

coinciding the normal

geoid*).

l

appreciate

its

in

besides

gyroscopic [9-141,

proposed

he assumed radius.

In recent

example to

its

on the

complicated

for

(1.1).

obideas

stressing

that

ones.

understood

to be the

The local vertical coincides also with the plumb line. to the earth’s surface (assuming that the earth is a

‘lhe horizontal

In most books

plane

is

on theoretical

perpendicular

mechanics

it

to

is

the

local

assumed

for

vertical.

simplicity

that the earth is a sphere whereas the local vertical is always defined as Coinciding both with the plumb line and the direction of the the shape of the gravity force P = mg = F - mw,. Without explaining earth local

this inconsistency vertical does not

face.

The best

explanation

may lead to the wrong conclusion that coincide with the normal to the earth’s of

this

is

in Sommerfeld

[151.

the sur-

Behavior

The line will

between

be called

horizontal would geoid

a point

plane.

If

with

plane

is

on the

pseudovertical

coincide

horizontal

of a horizontal

drawn

the earth the normal

would

thetical

spherical

5 ’ is

coincide

earth the

pseudohorizontal

surface

is

plane

with

the

drawn

tangent broken

q is

the

‘In the

line;

first

of

segment

Fig.

hypo-

the

true

plane,

q’

approximation the

is

the

plane

and the

when all

the

vertical

and a horizontal should [23,

the

geographical

in particular,

it

assumed

the

of

principal

represented 1 and

9 is

and the

gyrocompass not

(Fig.

gravitational

and conditions

indicate

the

by the

formulas

(2.3)),

acceleration, latitude. [l-4,81

in

that

the

horizontal component of the gravity force F equals zero, therefore instead true vertical there are used in these

plane

requirements

and

velocity

R, is

curvature

02

is

the

vertical

angular

the

It

the pseudohorizontal

the

g is

1.

the horizontal

In

5 is

rotation,

radius

fixed ship.

1 the

the

equals

U is

earth’s

plane.

In Fig.

whereas

v between

where

instruments

pseudo-

it

pseudovertical and the pseudo-

horizontal

pseudovertical

of

center

to

plane.

angle

papers

earth’s

plane.

curve,

by the

the

then the surface

ellipse-like

pseudovertical,

and

perpendicular

were a sphere to the earth’s

by a continuous

vertical,

earth’s

and the

305

gyrocompass

the is

pseudovertical. obtained

are

in

rigorously

horizontal stated

Consequently, [2-41

but the

that

the

for

satisfied,

a gyrothese

pseudohorizontal

xy-plane

which

is

in the gyroframe remains horizontal during all manoeuvres of a whereas in reality the word “horizontal” should be replaced by

pseudohorizontal. In the ments the concepts

inertial exact has

accelerometer tionary l/2 R,U2

navigation meaning

fundamental is

in the

of

and in the the horizontal

importance

for

pseudohorizontal

theory plane the

of

gyroscopic

and the

following

plane

and the

local

vertical

reasons: base

relative to the earth, then it will show accelerations sin 29 which are being determined from the horizontal

of the centrifugal force caused can be as large as 8’ which for too large. (3) It is known that the angle between the axis of a

instru(1)

remains

If

an

sta-

equalling component

by the earth’s rotation. (2) The angle many gyroscopic systems is prohibitively in the position of dynamic equilibrium gyrocompass and the true horizontal

v

306

D.R.

plane

p,

is

Br vanish. the angle

not

zero.

p,

in

(1.6)

such

a gyrocompass

Ishlinskii’

(see,

peculiar

It

is

of

clear

of

order

avoid

to

from using

their

exactly

El61 ),

by the

plane.

right-hand the

This

angle period,

axis

of

occurs

also

in

therefore these two instruments same function and they may differ only in

the

construction.

the

expressing

M. Schuler

means that

pseudohorizontal

when we use of

to make the

the

given

which

now why we must explain

direction

pedience

is

is

equalling

gyrocompass,

perform to

obtained

the

base

s [21 horizontal

a gyrocompass

period

example,

shows the

must in principle

results

for

of

with

on a stationary

term

errors

The problem

In a gyrocompass

hlerkin

the

the

gravitational the

possible

accepted

assumptions

force

of

force

F,

Q by formula

misunderstandings

such common concepts

terminology of

the

earth.

and chiefly

of

the

ex-

(1.2).

For this

and errors,

as the

and the

sphericity

horizontal

which

plane,

reason,

in

can arise

the

local

vertical, the horizontal velocity component etc., we shall regard in the future the motion of a particle or of a system as being referred to the true

vertical

done

in the

and true classical

Q must be expressed o movZg

We shall

horizontal

plane.

textbooks by formula

consider

on the earth’s

We shall

on mechanics

first

the problem of kinematics

surface.

‘Ihe earth

as is the

force

of a particle

is assumed to be a geoid

coincide

force P = mg. ‘Ihe parameters

which determine

with

the direction

on

of the gravity

the dimensions

and the

are

a = 6378.4 where a is

that

(1.1).

which normals to the surface shape of the geoid

assume also,

[15,17-191,

the equatorial

km,

radius

1 = 296.3

a-b

a = - a and b is

the half

(2-l)

distance

between the

poles. To first

order

accuracy

mated by the ellipsoid meridian equals

in a,

of Clairaut,

the surface

of the geoid

where the eccentricity

can be approxie of a

(2.2) From now on we shall Clairaut ellipsoid a (or e2) we shall The principal

regard

the surface

of the earth

as being

a

and in all expansion in powers of the small parameter keep only first order terms. normal sections

of an ellipsoid

the point 0 are the plane of the first vertical ‘Ihe corresponding principal radii of curvature

R2 =

of revolution

a (1 -

(1 -

through

and the meridional plane. are given by the equations e2)

e2sin* cp)”

(2.3)

Rehauior

of

where 0 is the geographical

a

horizontal

latitude.

307

gyrocompass

The first

principal

radius

of

curvature (Fig. 1) is represented by the segment OC and the second one is the radius of curvature of the meridian. Let us mention the easily derived and useful identity

a {RI

cp)

co.5

--=--Rssincp arp

(2.4)

We shall

introduce now the geoGraphically oriented coordinate system l-axis is along the true vertical pointing up, the i_ and q $n!G (the axes are horizontal, the $-axis is directed east and the q-axis directed north), and denote by uB, uN, UC the 5, q, 3 components of the velocity of the point 0 with respect to the earth. Since the point 0 moves on the surface of the earth, the components of the velocity vector V of the point 0 with respect to the inertial frame of [161 are

V, = The time rates

vE

of

+

RIU

the latitude

cp_z The angular

velocity

v, =

cosq),

V&I’

v, =

cp and of the longitude

0 h equal

j$=-..FE.-

I

(2.6)

Rx cos cp

of the triad

&j

is given

(2.5)

by the equations (2.7)

lhe simplest way to calculate the components of the velocity vector of the point 0 with respect to the inertial reference frame [18] is by using the formulas

Substituting

(2.7)

in the above equations

2L’cz..zV,-v$,tJJ,

w,=~‘l+~ql

Tf we used the equations WE

=7

w, =

;,

(2.4)

"EVN RI un9

-

-

iNfg29

WC=-R1-Ra

and (2.5)

2uv,

GUI rp -j- 2Uv, VN$

-

vE2

2vv,

we obtain WL=r -

(“”RI + 2)

(2.8)

we would obtain

sinQ, sin cp + cos cp -

lJeR,

sin cp cos cp

(2.9)

UV?, cossqi

In the above expressions the terms containing the angular velocity of the earth’s rotation in the second degree are components of the transport acceleration we which is caused by the earth’s rotation; the terms

308

D.R.

II in the first

containing ration the

WC; finally

the

acceleration

wr of

Thus the <, q,

degree

terms

Let

us denote

v,2

and V2 when the

of

t6e

earth’s

VE*z =

are

r Ez +

us introduce

with

motion

relative

to the

earth.

vector

the

higher

small,

order

determined

W*

+

by (1.4)

equal

2Uv,sincp

(2.10)

respectively

the

squares

of

approximate

values

the

velocity

angular

neglected

l.k = ‘lhen,

of

0 in its

containing

2R,VEU

the

accele-

components

-2uv,coscp

terms

rotation

the Coriolis U are

by V *’ and V*‘,

t

of

contain

= -pg

of

and let

of

components

do not

N +!$--cp

yJ*=; 4 q*

are

which

the point

5 components

Merkin

coscp,

v*2 = -v-t*2 + v,2

(2.11)

parameter

variable

(2.12)

e2 cos2 cp in ~1 than the

accuracy

we have

first,

(2.13) We shall specified

write

down the

us mention

that

which

the point

0 has vertical

3.

theory

In the

the y”-axis

is

within

the

will

all

the

follow

(2.14)

formulas

which

can be easily

we have

generalized

already for

cases

displacements.

of

gyroscopic

instru-

we often use a coordinate system in the x”yO-plane is horizontal and

component triad

(valid

+2p&6+I

1$-p,

and those

ments which

two formulas

accuracy)

Rl -= RZ Let

following

coincides of

the


determined

with

angular 2).

the horizontal

velocity

By (2.7)

of

the

the angle

ti

through

(3.1)

Fig.

2.

obtained when

Behavior

Within

the specified

of

a horizontal

accuracy

gyrocompass

309

we obtain

(3.2) Let

us now derive

formulas

for

the

x0,

y”,

z” components

of

the velo-

V

city

V X0 = TIEcos 6 + V, sin fi, V,o = - V, sin 2) + V, cos 6, With higher

order

accuracy

(it

is

useful

the

to mention

x0-axis,

Similarly,

as it

we obtain

the

first

we find

I+_+,

v,o = v,

with

than

that does

the in

the x0,

vtO = velocity

the case y”

Vp = V,

the

(3 -3)

V does not coincide

vector of

0

spherical

and z” components

of

earth). the acceleration

w*

In these as if

the

formulas earth’s

6,

is

flattening

the time

derivative

of

the angle

fi obtained

were absent

(3.5) The x ‘, tion

of

have

the

oe of the rotaand z” components of the angular velocity with respect to the inertial frame of reference triad x”yozo

y”

the

form

where 0 = 00 + p Here

o,, is

the

value

of

(

6, -

2 v+ 00

)

(3.7)

o when ~1 = 0 V. oo=60++Mcp

(3.3)

310

D.R.

‘Ihe parameter the limits some cases

~1 is

small

Merkin

when compared

to unity

and varies

within

0<~<0.00675, with p = 0 on a pole. For this reason in the x0, y” and z” components of the acceleration w* can be sufficiently accurately for practical purposes by the

calculated formulas

wp*=~_I& (E ve -

vE**) tea (p

1

(B,+ g$qD)

* wYO = Finally,

if

obtain

the

point

moving

in the

x0,

y”

on a spherical

We shall

(2 determined along

If

the

earth’s

Q is obtained

-

mwxO =

=

(in

this axes

case are

In order (4.2)

the

It the

y”

to avoid latter

we the

the

mixing

of

the

force

P = mg is directed

force

we have

QP = -

be calculated

from the

m (g + w*)

by formulas

equation

(1.2)

(4.1)

(3.4).

as in

[l-4,81

then

equal -

is

(3.10)

--

and z” components

mwyo*,

m$‘,

along

the pseudohorizontal

QUO’ =

-

mwyO =

-

~(,JJ

- mf)

the pseudovertical,

(4.2) and the x0 and

plane).

up the components

(4.1)

with

the components

are marked by primes.

and Q o of Q xo (! may differ substantially from its pseudohorizontal comio’ and Q o’. For example when uE = 0 and vN = const we have

can be easily

force

nents

z” -axis

the in

*, of

R

Qp' = - (8'+ mwp) = -(F

y”

superscript

acceleration

&.a

Since

surface

and z” components Qxo’ =

[21

now the x0, by (1.1).

w * and w * should z” Y0

force

IIO, y”

its

surface

(3.9)

Rl

V”

Qvo= -

Qp = - mwxo*,

total

P2

* =

the

the

wllo = OJ,

calculate to the

WP

we delete of

earth’s

v.

previously

a normal

where TU *, x0

formulas

and z” components

w.y = 4.

above

v,

shown that

the horizontal

components

0 -xo 0 very little and 0 ‘, however, differ : 0 ’ = 2. IlTe components i) ‘to ‘X0 ‘X0 ‘ZO because F = mg and at relatively small velocities (say u Q 50 knots)

5.

We shall

derive

the

differential

equations

of the

precessional

Behavior

motion

of

a gyroframe

To construct zontal

these

a horizontal

of

using

the

equations

we can replace

plane,

results

while

the components of the acceleration components of w* which are given and o,, of

V/R

to the

the

inertial

angular

frame

differential

o,

reference

by Ishlinskii

the motion

w which by (3.4);

velocity

of

obtained

relating

in the

311

gyrocompass

of

to

equations

the triad

hori-

of

(40)

OX~Y~Z~ with

must be replaced

by the

2Bcos e [(cl + 0) c0s p cos 7 + $sin r + 01 (sin a sin T w,.*

[21.

true

[21

are given in (3.10) by the besides, the two components

expressions ol and o as given by (3.7) and (3.8). After indicated substitutions we obtain the desired differential

= ml [-

in

the

respect

corresponding

performing the equations

cos a sin p cos r)] =

sin a cos p + wy”* cos a cos p + (g + w,.*) sin /3] -

M,’

(2B cos e) = ml [w,“* (cos a cos r - sin a sin p sin 7) +

$

(5.1) + wyol (sin a cos T + cos a sin j3sin r) -

(g + w,.*) cos p sin T] +

M,'

2Bcos e [-

(i + o) cos p sin T + E cos T + 01 (sin a cos T + cos a sin, /3sin T)] = M,

In these

equations

bl,‘,

h!

can be generated

by a special

the

as in

same meaning

corresponding

angles

[21,

in

In order

and M ’ are arrangzment;

while

does

not

to make our gyroframe

to select

such

would

permit

(5.1)

we obtain

values

the motion

for

they

behave

mlw

--f& (2B cos e) = mlw,.*

with

u”+-M

moments which

like

in the

the

true

it

but

is

neces-

M ’ and MZ’ which

Substituting

j a-9

system

vertical

a gyrocompass

moments N, U,‘,

a = p = y = 0.

2Bocose=

are defined

coincide

the

supplementary

the remaining symbols have angles a, p, y differ from the

the

[21 because

x ’ Y’ 2’ where the 2’ -axis with the pseudovertical.

sary

’

i = p = y = 0 in

O=M,

2Bol sin e = -

+ My’,

(5.2)

N

From where 2B cos e =

mlwyO *-_M

x

I

0

I

mlwyD*-

M&

M,’

0

- mlw,,+

(5.3)

3

N=-2Bolsine

The moment $1 ’ is x

arbitrary,

besides,

if

AI ’ =

x

hl '(E)

x

then

N can

also

312

D.R.

be regarded

as a function

angle

related

E are

If

the

place true

local

are

respect

and the

equalling

From the

E because

first

the

equation

satisfied

while

the

system

meridional

plane

it

to

t and the

(5.3).

the motion

a = p = y = 0, to

time

of

then

is

the

xO~‘Z’,

taking

gyroframe

showing

(accurate

will

always

within

the

the

course

6).

equations

To achieve

angle

instant

with

vertical

correction

tions.

(5.3)

initial

be in equilibrium

the

as shown in the

conditions

and at the

of

Merkin

(5.3)

is

easy

obtain

the

Ishlinskii

condi-

J1 ’ = 0 and replace the components w l x xo and w o* by w o = k and w o = o,,V; besides. we must take into account

that

in the

this

c&e

of

(3.10)).

Substituting

(compare

with

we set

a sphzre

Let

us repeat

that

components

of

vertical

by the

gyroframe up the written

2B cos e’ = mlV. replacing the

total

pseudovertical.

will

not

The author of

(3.7) (5.3)

and we find

be showing

the

the

is

grateful

this

4Bz sin e’ cos e’ RmL

N=-

components

acceleration Therefore

of

Lur’e

(5.4)

acceleration

replacing conditions (in

plane

corresponding a prime).

to A. I.

the

w means under the

horizontal

angles a, p, y, E with the the angle E in (5.4) with

cussions

and o = o,, (formulas in the conditions

[21) M,’ SE 0.

by the

01 = V/R expressions

these

angles

order

not to mix [21 we have

in

Dzanelidze

and G.Iu.

w*

the true (5.4) the

for

dis-

work.

BIBLIOGRAPHY 1.

Ishlinskii,

A. Iu.,

s podvizhnoi physical No. 2.

3.

pendulum

Ishlinskii,

A. Iu.,

Ishl inskii, theory

4.

Ishlinskii, of

5.

with

fizicheskogo

ravnovesii

(On the

a moving

A. Iu., of

K teorii

gyrocompass).

point

of

maiatnika

equilibrium

relative

support).

A. Iu.,

N. I.,

girogorizontkompass PMMVol.

K teorii

a gyroscopic

Teoria

20,

of

PMM Vol.

(On the

No. 4,

giroskopicheskogo pendulum).

Teoria

a two-gyroscope

Idelson,

opory

a 20,

1956.

horizontal 3.

Ob otnositelnom

tochkoi

maiatnika

PMM Vol.

21,

dvukhgiroskopicheskoi

vertical). potentsiala

PMMVol. (Theory

theory

21,

of

a

1956.

No.

1,

verticali No.

2,

of Potential).

(On the 1957. (Theory

1957. ONTI, 1936.

Rehavior

6.

Lur’

e,

A. I.,

Duboshin, giz,

8.

G. N. , Teoria

Ishlinskii, kogo

giroskopa

of

napravleniia

gyrocompass,

p,Mf Vol.

No.

23.

1,

V. N.,

K teorii

passes).

PMM Vol.

23,

Iu.K.,

nogo

opredeleniia

Koshliakov,

Merkin,

Liashenko,

girokompasov

(On the

No.

5.

reduction

kolebanii

uravnenii

of

the

25,

Ob ustoichivosti of

Iu.K.,

16.

Bulgakov.

No.

5,

PMM Vol.

0 privodimosti

26.

K teorii

A., B.V.,

18.

Appel’,

P.,

2.

Fizmatgiz,

Suslov,

19.

G.K.,

Levi-Civita, of

Theoretical

No.

2,

avtonom-

(Investiga-

of

girogorizont-

motion

of

a gyro-

1961. giroramy

25,

No.

(On the

6,

stability

1961.

dvizheniia

girogorizont-

(On the reduction of and a two-gyroscope

the

1962.

girogorizontkompasa PMM Vol.

26,

(Mechanics).

Prikladnaia

teoriia

No.

(On the

6,

theory

of

the

1962.

GIIL,

1947.

giroskopov

(Applied

Theory

of

1955.

Teoreticheskaia

Gostekhizdat,

gyrocom-

v sisteme

dvizheniia

uravnenii

Mekhanika

GITTL,

Gyroscopes).

of

ob’ekta

equation

dvizheniia

a gyroframe).

gyrohorizoncompass). Sommerfeld,

theory

dvizhushchegosia

0 privodimosti

PMM Vol.

15.

object

and an integrator).

1959.

svobodnykh

PMM Vol.

V. F.,

Zhbanov,

ustroist-

a moving

vibrations in the system of autonomous determination of a moving object). PMM Vol. 24, No. 6, 1960.

V.N.,

D.R.,

of

gyroscope

koordinat

(On the

motion

position

a direction

Issledovanie

Zhbanov,

i integriruiushchego of

dvi-

giroskopiches-

1959.

Koshl iakov,

vertical).

17.

Fizmat-

Attraction).

mestopolozheniia

kompasa i dvukhgiroskopicheskoi verticali equations of motion of a gyrohorizoncompass

14.

!Iechanics).

prostranstvennogo

determination

by a spatial

of

(Theory

posredstvom

autonomous

horizoncompass).

13.

pritiazheniia

ob’ekta

kompasa,

kompasa

12.

(Analytical

Ob avtonomnom opredelenii

A. Iu.,

tion of free of coordinates 11.

mekhanika

1961.

va (On the

10.

313

gyrocompass

1961.

zhushchegosia

9.

a horizontal

Analiticheskaia

Fizmatgiz, 7.

of

mekhanika

(Theoretical

Vol.

Mechanics),

1960. Teoreticheskaia

mekhanika

(Theoretical

Mechanics).

1946. T. and Amaldi, Mechanics),

II.,

Kurs

Vo1.2.

teoreticheskoi

Part

1.

GIIL,

eekhaniki

(Course

1951. Translated

by

T.L.