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On the reduction of the equations of motion of a gyrohorizoncompass and two-gyroscope vertical
ON THE REDUCTION OF THE EQUATIONS OF MOTION OF A CYRODORIZONCOMPASS AND TWO-GYROSCOPEVERTICAL (0 PRIVODIWOSTI UKAVNBNII DVIZRRNIIA BIPOQOBIZONTKOKPASA I DVUKtl6IR0SKOPICIBSKOI VEBTICALI) PMM Vo1.26, No.2, 1962, pp.
369-372
V. F. LIASHENKO (Yoscor) (Receivad
Some results Seckeler for on the earth An analogous sidered [31.
22,
1961)
of [ll are generalized. The use of simplified equations of arbitrary motion of a gyrohorizoncompass suspension point sphere Is justified with the aid of a theorem by Erugin [21. uuestion applicable to a two-gyroscope vertical is also con-
1. In the absence of damping gyrohor isoncompass are [41
mlv$+nl~a &-
November
the
equations
of perturbed
- mgl$ - Q2B sin e”&= 0
motion
of a
(1.1)
2B sin 3 &B 6 +JG = 0, -$- (2B sin sod) -mgl~+mlvPa=O
v=~~RUcOscp+vg)*+vN~
,O =usinp++uncp-&,
a+ = -
VN Ru cos q + vu.
Here B is the kinetic moment of the gyroscope rotor; E’ Is some equilibrium position for the separation angle of the gyroscopes; a is the deviation angle for the gyrosphere axis in azimuth; p is the lift angle for the north end of the gyrosphere axis above the surface tangent to the earth sphere at the point of gyrosphere suspension; y is the gyrosphere rotation angle about the line north-south; 6 is the gyroscopes’ rotation angle about their frames, defining the perturbed location of gyro rotor axes relative to the gyrosphere; I = P/g is the mass of the gyrosphere (P is the weight of the gyrosphere, g the acceleration of gravity): 1 is the metacentric height of the gyrosphere; R is the earth radius; U is the earth angular velocity: Q is ship’s location latitude (geocentric); VI, vN are east and north velocity components, respectively,
531
538
V. F. Liashenko
of the gyrosphere
suspension
Let the ship perform We will
introduce
new variables
for
V2 u cos cp 2B sin q@
%+
pi
[II
Koshliakov
P =
Xl,
dt
----x2-h
qcose
zlsin
v
1
.54,=-c0tqxlsin8hl This substitution system --6
dt
?-==s,
2, 3, 4) by formulas
+
x2cose
8 +
6=
sin cp sin
(1.a
x4
[II
becomes
V -x5sin8-_~ #y cos cp
UCOSqI
.Ea=
=-A,
cp.
the substitution
xacose+
ucosq
= I,
surface.
tan@Q=O,
V
-y-
latitude
xi (j
new variables
suggested
~~=xlcose-~co,~
E2=
to earth
maneuvers on a given
V
system (1.1)
d=1
relative
RU cos cp
Cl= The
point
5)
-x3
sin 8
sine-
(e(t)==
v2B sin cp +x3
case
+
dr)
Pl
v2B sin dp x3c0se+x4c0se
the system (1.3)
&-E2=%
\Q(r) 0
X~COS~
p1
(i-4)
t
v2Bshcp 2 sin8 B1 4
Pl x2sin8v2B sin cp
reduces
tan.qx4sin8
to the Schuler-Geckeler
2B sin qv2 Pl
%+
E4=0 (1.5) .g3=0.
2. Substitution (1.4) is applicable for any form of the function S(t). [I] presents its substantiation only for a special case Nevertheless, when Q(t) is a periodic function of time. Equations (1.5), however. can be justified for any form of the function Q(t) by means of a theorem due to Erugin [21. The system (1.3) is solved in a similar way to that suggested Let us represent the system (1.3) in the form
-
x1 - vx2 -
2Bsincpv ax _. PI
4-
9
in 141.
Equations
of motion of e gyrohorizonconpass
2B sin TV
d zr+Qxa=O,
Introducing
xt
two complex-valued
u cos cp
x (t) = 7
The system (2.1)
zr+ k4,
the following
-$(x + PL)+ i PJThese are easily
t by z4(i=m)
p1
to two equations
-$ + ivX = which yield
of
2B sin ‘pv
IrP)=z3-i
is reduced
-skq=o
p1
functions
539
of the form
$+ivp=iBX
iQp’,
(2.3)
a&rations $(X
Q)(x + 14= 0.
integrated.
- P) + i (v + Q) (x -
C1exp -i (
5
SW)
.
constants.
The general
\(v+P)dt)
f
y
(2.6)
e-ie
II et8 *e
e
solution
p = + ,-iv* (cleie
e-*“t(cle*e+ C,e_‘“),
It follows, therefore, [2], the system (2.3) is The substitution
(2.5)
_
_ Cae4)
that the integral
I
e-is
of system
(2.6)
matrix
of the
- Liapunov type matrix
(3.1)
that on the basis of a theorem due to Erugin reducible for any form of the function Q(t).
Y = z-lx,
transforms
= C3 exp (-i
0
3. It follows from the solution system (2.3) has the structure p = e-ivtz,
X-P
0
z 0) =
(2.4)
t (v-
Here C,, C2 are arbitrary (2.3) is of the form
x’$
P) = 0
We have
t x+p=
(2.2)
)
the system (2.3)
z-1 (t) =
into
$ + ivy1 =
‘-‘* et9
_f”
11,
x(t)_li:
the system with constant +-
0,
Inverse transformation from the variables p according to (3.2) is of the form X=ZY
1
coefficients
+ ivys = 0
yl,
(3.2)
yg to the variables
(3.3) x,
(3.4)
Y. F.
540
Lioshenko
Letting Ya= ‘Is + iq4
y1=11+iqrt we have on the basis ables qi
From (3.2) a non-singular of the form
of
(3.3)
and considering transformation UCOScp
r)l' 7
qr=-
the Schuler-Qeckeler
x~cos%+xesin8+xacos6-
ucosg, -xztsin8+x~cos8-xqgin8v
inverse
the vari-
28 sin cpv pl x4sin 8 2B sin cpv pr x4co9 6
U COSq -xisin8+~~cos@-xVsin8+~ v
The formulas for variables xj are
system for
(2.2) and (3.5), we obtain the formulas for from the variables qj xi to the variables
u cosa, xlcosB--xesin0-xacosBqa= y q=
(3.5)
2B sin qtv
p1
transformation
(3.8
23 sin ‘pv p1 x4sin 0 x4 cos 8
from the variables
qj to the
(3.5)
4. The preceding theory is applicable virtually without any alteration to the equations of a two-gyroscope vertical as well, given in [33:
Function
9*(t)
satisfies
the (+* (t)
The remaining notation o having the same meaning f ormula8 a=: RLrcosI, V
condition =
sin-‘gy
in system (4.1) is the same as in (1.1). with as 1. Let us introduce new variables t j bS P = La,
r=q,
a==$&
(4.2)
Equations
The systele
of
will.
(4.1)
motion
of
a gyrohorisonconpass
541
become
(4.3) VO dz1 -F.&F----zzP-j-h&~=O, U coscp
2B cos cpv’ pa 4 + Qtp = 0
dSS
-$-!g+
Pith
analogous
non-singular
reasoning
sP=-
above,
one
S--~h=O
can
show that
u cos cp
system
ucosrp
-
II sin
v
e + 1sco9e - g sin e +
sr=
u cos p 2rsine+bc0se-2asine7
is
reducible
for
Pa
qtv
pa
2B cos ipv
-z1cose-_qsinG--_scos9+ V
(4.3)
co9
2B cos rpv
9,
68 =
Schuler-Geckeler
by means
of
the
Pa
any
2B cos qv pa
form
of
the
4 sin
e
t4 cos e
(4.4)
se sin e 24 cos e
function
Q(t)
to
the
system
&
0,
yg--VCa= formulas
variables
28
-zlcose+z,sine+~;lcOse+ V
u co9
The
Pa* 2BcosT
2B9 POU
substitution
F;1=
the
as
As=---
I
for
4s
inverse
++v+o
G
~-v~=O,
~+v61=0,
transformation
from
the
variables
5j
to
(4.5) the
t. are 3
(4.6)
I Pa =‘ = -? v 2B cos cp &sine+fjcose+bsine-_ccose)
BIBLIOGRAPRY 1.
Koshliakov, kompasa
V. N., (On the
horizoncompass).
a.
Erugin,
N. P.
0 ptivodimosti reduction PMM Vol.
t Privodimye
Nauk SSSR, 1948.
of 25,
sisteuy
uravnenii the No.
equations 5,
drisheniia of
motion
girogorizontof
a Gyro-
1961.
(Reducible
Systems).
Izd-vo
Akad.
542
3.
Y.F.
Ishlinskii,
A. Iu.,
a two-gyroscope 4.
Ishlinskii,
A. Iu.,
gyrohorizoncompass).
Liashenko
Teoria dvukhgiroskopicheskoi vertical). PM Vol. 21, No.
K teorii
2,
girogorizontkompasa
PMM Vol.
20,
No.
4,
vertikali 1957. (On the
(Theory
theory
of
of
the
1956.
Translated
by V.C.
369-372
V. F. LIASHENKO (Yoscor) (Receivad
Some results Seckeler for on the earth An analogous sidered [31.
22,
1961)
of [ll are generalized. The use of simplified equations of arbitrary motion of a gyrohorizoncompass suspension point sphere Is justified with the aid of a theorem by Erugin [21. uuestion applicable to a two-gyroscope vertical is also con-
1. In the absence of damping gyrohor isoncompass are [41
mlv$+nl~a &-
November
the
equations
of perturbed
- mgl$ - Q2B sin e”&= 0
motion
of a
(1.1)
2B sin 3 &B 6 +JG = 0, -$- (2B sin sod) -mgl~+mlvPa=O
v=~~RUcOscp+vg)*+vN~
,O =usinp++uncp-&,
a+ = -
VN Ru cos q + vu.
Here B is the kinetic moment of the gyroscope rotor; E’ Is some equilibrium position for the separation angle of the gyroscopes; a is the deviation angle for the gyrosphere axis in azimuth; p is the lift angle for the north end of the gyrosphere axis above the surface tangent to the earth sphere at the point of gyrosphere suspension; y is the gyrosphere rotation angle about the line north-south; 6 is the gyroscopes’ rotation angle about their frames, defining the perturbed location of gyro rotor axes relative to the gyrosphere; I = P/g is the mass of the gyrosphere (P is the weight of the gyrosphere, g the acceleration of gravity): 1 is the metacentric height of the gyrosphere; R is the earth radius; U is the earth angular velocity: Q is ship’s location latitude (geocentric); VI, vN are east and north velocity components, respectively,
531
538
V. F. Liashenko
of the gyrosphere
suspension
Let the ship perform We will
introduce
new variables
for
V2 u cos cp 2B sin q@
%+
pi
[II
Koshliakov
P =
Xl,
dt
----x2-h
qcose
zlsin
v
1
.54,=-c0tqxlsin8hl This substitution system --6
dt
?-==s,
2, 3, 4) by formulas
+
x2cose
8 +
6=
sin cp sin
(1.a
x4
[II
becomes
V -x5sin8-_~ #y cos cp
UCOSqI
.Ea=
=-A,
cp.
the substitution
xacose+
ucosq
= I,
surface.
tan@Q=O,
V
-y-
latitude
xi (j
new variables
suggested
~~=xlcose-~co,~
E2=
to earth
maneuvers on a given
V
system (1.1)
d=1
relative
RU cos cp
Cl= The
point
5)
-x3
sin 8
sine-
(e(t)==
v2B sin cp +x3
case
+
dr)
Pl
v2B sin dp x3c0se+x4c0se
the system (1.3)
&-E2=%
\Q(r) 0
X~COS~
p1
(i-4)
t
v2Bshcp 2 sin8 B1 4
Pl x2sin8v2B sin cp
reduces
tan.qx4sin8
to the Schuler-Geckeler
2B sin qv2 Pl
%+
E4=0 (1.5) .g3=0.
2. Substitution (1.4) is applicable for any form of the function S(t). [I] presents its substantiation only for a special case Nevertheless, when Q(t) is a periodic function of time. Equations (1.5), however. can be justified for any form of the function Q(t) by means of a theorem due to Erugin [21. The system (1.3) is solved in a similar way to that suggested Let us represent the system (1.3) in the form
-
x1 - vx2 -
2Bsincpv ax _. PI
4-
9
in 141.
Equations
of motion of e gyrohorizonconpass
2B sin TV
d zr+Qxa=O,
Introducing
xt
two complex-valued
u cos cp
x (t) = 7
The system (2.1)
zr+ k4,
the following
-$(x + PL)+ i PJThese are easily
t by z4(i=m)
p1
to two equations
-$ + ivX = which yield
of
2B sin ‘pv
IrP)=z3-i
is reduced
-skq=o
p1
functions
539
of the form
$+ivp=iBX
iQp’,
(2.3)
a&rations $(X
Q)(x + 14= 0.
integrated.
- P) + i (v + Q) (x -
C1exp -i (
5
SW)
.
constants.
The general
\(v+P)dt)
f
y
(2.6)
e-ie
II et8 *e
e
solution
p = + ,-iv* (cleie
e-*“t(cle*e+ C,e_‘“),
It follows, therefore, [2], the system (2.3) is The substitution
(2.5)
_
_ Cae4)
that the integral
I
e-is
of system
(2.6)
matrix
of the
- Liapunov type matrix
(3.1)
that on the basis of a theorem due to Erugin reducible for any form of the function Q(t).
Y = z-lx,
transforms
= C3 exp (-i
0
3. It follows from the solution system (2.3) has the structure p = e-ivtz,
X-P
0
z 0) =
(2.4)
t (v-
Here C,, C2 are arbitrary (2.3) is of the form
x’$
P) = 0
We have
t x+p=
(2.2)
)
the system (2.3)
z-1 (t) =
into
$ + ivy1 =
‘-‘* et9
_f”
11,
x(t)_li:
the system with constant +-
0,
Inverse transformation from the variables p according to (3.2) is of the form X=ZY
1
coefficients
+ ivys = 0
yl,
(3.2)
yg to the variables
(3.3) x,
(3.4)
Y. F.
540
Lioshenko
Letting Ya= ‘Is + iq4
y1=11+iqrt we have on the basis ables qi
From (3.2) a non-singular of the form
of
(3.3)
and considering transformation UCOScp
r)l' 7
qr=-
the Schuler-Qeckeler
x~cos%+xesin8+xacos6-
ucosg, -xztsin8+x~cos8-xqgin8v
inverse
the vari-
28 sin cpv pl x4sin 8 2B sin cpv pr x4co9 6
U COSq -xisin8+~~cos@-xVsin8+~ v
The formulas for variables xj are
system for
(2.2) and (3.5), we obtain the formulas for from the variables qj xi to the variables
u cosa, xlcosB--xesin0-xacosBqa= y q=
(3.5)
2B sin qtv
p1
transformation
(3.8
23 sin ‘pv p1 x4sin 0 x4 cos 8
from the variables
qj to the
(3.5)
4. The preceding theory is applicable virtually without any alteration to the equations of a two-gyroscope vertical as well, given in [33:
Function
9*(t)
satisfies
the (+* (t)
The remaining notation o having the same meaning f ormula8 a=: RLrcosI, V
condition =
sin-‘gy
in system (4.1) is the same as in (1.1). with as 1. Let us introduce new variables t j bS P = La,
r=q,
a==$&
(4.2)
Equations
The systele
of
will.
(4.1)
motion
of
a gyrohorisonconpass
541
become
(4.3) VO dz1 -F.&F----zzP-j-h&~=O, U coscp
2B cos cpv’ pa 4 + Qtp = 0
dSS
-$-!g+
Pith
analogous
non-singular
reasoning
sP=-
above,
one
S--~h=O
can
show that
u cos cp
system
ucosrp
-
II sin
v
e + 1sco9e - g sin e +
sr=
u cos p 2rsine+bc0se-2asine7
is
reducible
for
Pa
qtv
pa
2B cos ipv
-z1cose-_qsinG--_scos9+ V
(4.3)
co9
2B cos rpv
9,
68 =
Schuler-Geckeler
by means
of
the
Pa
any
2B cos qv pa
form
of
the
4 sin
e
t4 cos e
(4.4)
se sin e 24 cos e
function
Q(t)
to
the
system
&
0,
yg--VCa= formulas
variables
28
-zlcose+z,sine+~;lcOse+ V
u co9
The
Pa* 2BcosT
2B9 POU
substitution
F;1=
the
as
As=---
I
for
4s
inverse
++v+o
G
~-v~=O,
~+v61=0,
transformation
from
the
variables
5j
to
(4.5) the
t. are 3
(4.6)
I Pa =‘ = -? v 2B cos cp &sine+fjcose+bsine-_ccose)
BIBLIOGRAPRY 1.
Koshliakov, kompasa
V. N., (On the
horizoncompass).
a.
Erugin,
N. P.
0 ptivodimosti reduction PMM Vol.
t Privodimye
Nauk SSSR, 1948.
of 25,
sisteuy
uravnenii the No.
equations 5,
drisheniia of
motion
girogorizontof
a Gyro-
1961.
(Reducible
Systems).
Izd-vo
Akad.
542
3.
Y.F.
Ishlinskii,
A. Iu.,
a two-gyroscope 4.
Ishlinskii,
A. Iu.,
gyrohorizoncompass).
Liashenko
Teoria dvukhgiroskopicheskoi vertical). PM Vol. 21, No.
K teorii
2,
girogorizontkompasa
PMM Vol.
20,
No.
4,
vertikali 1957. (On the
(Theory
theory
of
of
the
1956.
Translated
by V.C.