JPRS ID: 10364 TRANSLATION ROTOR VIBRATION GYROSCOPES INNAVIGATION SYSTEMS BY YU. B. VLASOV AND O.M. FILONOV
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4 March 1982
Translation
ROTOR VIBRATION GYROSCOPES
IN NAVIGATION SYSTEMS
By
Yu.B. Vlasov and O.M. Filonov
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JPRS L/10364
4 March 1982
ROTOR VIBRATION GYROSCOPES IN NAVIGATION SYSTEMS
Leningrad ROT4RNYYE VIBRATSIONNYYE GIROSKOPY V SISTEMAKH NAVIGATSII
in Russian 1980 (signed to press 22 Aug 80) pp 1221
[Book "Rotor Vibration Gyroscopes in Navigation Systems", by Yuriy
Borisovich Vlasov and Oleg Mikhaylovich Filonov, Izdatel'stvo
"Sudostroyeniye", 1,200 copies, 221 pages]
~
COhTENTS
Annotation 1
Foreword 1
Introduction 2
Chapter 1. Classification and Mathematical Nbdel of Rc>tor Vibration
Gyroscopes........................................................ 6
1.1. Equations of Mbtion of a Generalized Rotor Vibration Gyroscope
Model 6
1.2. Classificatioa of Layouts of Rotor Vib ration Gyroscopes 12
1.3. Fquations of Motion of Rotor Vibration Gyroscopes 15
1.4. Signal Readin g and Information Processing Syatems 28
1.5. Methods for Solving Jifferential Equations With Periodic
_ Coefficients 33
1.6. ]lynamic Characteristics of Rotor Vibration Gyroscopes 40
A Chapter 2. Defects in Rotor Vibration Gyroscopes
59
: 2.1. Reaction of Rotor Vibration Gyroscopes to Harmonic Vibrations
of the Base at a Frequency Equal to the Doubled Frequency of
_ Rotation of a Rotor..................................................
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2.2.
Reaction of Rotor Vibratfon Gyroscopes to Disturbing Moments.........
67
2.3.
Operating Errors in Rotor Vibration Gyroscopes
.
74
.
2.4.
Errors in a SingleRotor Modulation Gyroscope
.
78
.
Chapter
3. Composite Rotor Vibration Gyroacopes
92
3.1.
Principles of the Construction of Composite Rotor Vibration
Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.2.
Dyiiamic Characteristics and Basic Errors in Composite Rotor
Vibration Gyroscopes
98
3.3.
Synthesizing the Parameters of an Output Filter for Composite
Rntor Vibration Gyroscopes
106
Chapter
4. Stabilization Syatems Utilizing Rotor Vibrations Gyroscopes
as the Basic Sensitive Elements
ill
4.1.
Stabilization System Equatians of Irbtion, Structural Diagrams
and Transfer Functions
111
4.2.
Static (1laracteristics of Stabilization Systems
119
4.3.
QZOOSing the Structure and Parameters of a Uniaxial
Stabilization System
123
4.4.
Dynamic Errors in a Uniaxi al Stabilization System
128
4.5.
Effect of NonSteadyState Feedback on the Operation of a
Uniaxial Stabilization System........................................
134
4.6.
Multidimensional Stabilization Systems Using Rotor Vibration
Gyroscopes
139
4.7.
Effect of Transient Feedback on the Operation of a Multi
dimensional Stabilization System
147
Chapter 5. Rotor Vibration Gyroscopes in the Deflection Correction
 Circuit of a Gyroscopic Stabilization System 156
5.1. Des crip tion o f S tabilization Sys temg With a De flec tion
Correction Circuit 156
5.2. Static Characteristics of Gyroacopic Stabilization Systems With
~ a Deflection Correction Circuit 162
5.3. Selecting the Structure and Parameters of aDeflection
Correction Circuit for a Uniaxial Gyroscopic Stabilizer 166
Conclusion 178
. Bibliography 179
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_ [Text] ANNOTATION
The authors present materials concerning the investigation of the characteristics of
rotor vibration gyroscopes and methods for balancing them, in addition to foxmulat
irig requirements for information extraction and processing devices. They also pre
sent techniques for synthesizing a regulator for stabilization and correction cir
cuits for the purpose of obtaining the required characteristics.
' FOREWuRD
In recent years, rotor vibration gyroscopes (RVG) have been used more and more fre
quently in navigation systems. Since they are small in size and low in weight, they
make it possible to miniaturize the sensitive elements of navigation systems and ob
tain characteristics with a level of accuracy that is no worse than that of gyro
scopic devices constrocted according to the classical method.
, In order to obtain the best characteristics of such systems, it is important to de
scribe correctly not only the static, but also the dynamic characteristics of RVG's
as components of an automatic controZ system. In turn, when planning RVG's it is
_ necessary to take the special features of their operation into consideration in the
composition of the specific control system. All of this requires a detailed inves
tigation of the theory of RVG's aDd gyroscopic systems in which they are used.
This book consists of five chapters. In the first two we discuss thE generalized
model on which most existing RVG systems are based. The analysis of the equations
of motion is carried out with the utilization vf welldeveloped operator methods.
The special features of RVG's as twodimensional measuring uriits make it possible to
use a special apparatus that was developed foz twodimensional automatic control
systems [16,17]. Such an approach to the analysis of the operation of RVG's makes
it possible to perfozm operations ii:ectly with their transfer functions. For the
simplest RVG system:s, the transfer functionsas approximated in the area of the es
sential frequenciesare described by simple analytical expressions. For complex
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systems, the analogous frequency characteristics can be constructed quite easily
with the help of a computer. A knowledge of the frequency characteristics makes it
possible tn select the instrument's basic parameters in such a manner as to provide
it with the required dynamic characteristics. 'i'his is especially important for
RVG's operating in automatic regt:lation and control systems. For the analysis of
RVG's having rotors with different angular velocities, we propose tio use the spec
tral methods of (Khil's) generalized theory of equations [10,16,27,34].
In the second chapter we discuss the most cammon errors in RVG's with a single drive
motor. The basic attention is devoted to errors related to angular vibrations of
the motor's shaft at a frequency equal to twice its frequency of rotation and to
static disbalance of the rotors.
In the third chapter we present one of the most promising (from the authors' view
point) layouts for composite RVG's. In an example of this layout we investigate the
basic errors in this type of instrument, as well as methods of reducing them.
The fourth and fifth chapters are devoted to an investigation of the special fea
tures of tne operation of gyroscopically sta.bilized platforms (GSP) based on RVG's.
The in:estigation is based on the frequency methods of analyzing and synthesizing
automa*_ic control systems that are widely used in engineering ca:culations. Here
tnere is a detailed discussion of the technique for selecting the stabilization
cnannels' basic parameters. There is an analysis of the effect of the nonsteady
state component of an RVG's output signal on the operation of a GSP. Multi
dimensional stabilization systems are discussed from the viewpoint of the effect on
their operation of the specific crosscouplings between the stabi).ization channels.
This book lays no claims to being a complete explication of the theory of RVG's and
systems that utilize them. In it we do not discuss questions of the optimum synthe
sis of systems with RVG's, the effect of basic nonlinearities on the operation of
such systems, the theory of systems using composite RVG's that are selforienting in
the plane of the hor9.zon and the meridian, and others. The investigation of these
questions is necessary for the creation of RVGbased gyroscopic systems operating
effectively in various navigation complexes.
_ The book is intended for engineers and scientific workers specializing in the field
of the development and use of new gyroscopic instruments. .
The authors are deeply grateful to Professor Ye.L. Smirnov, doctor of techn:ical sci
ences, for his valuable advice and the comments he made during the preparation of
the manuscript for publication.
i',+TRODL'rT ION
t�todern navigation systems are constnucted on the basis of the most recent achieve
ments of computer technology and, for all practical purposes, carry out completely
tnose operations that were previously performed by man.
Desoite the presence of modern information processing facilities, the accuracy of
navigation systems is determined primarily by the accuracy of the instrume^ts that
 tney utilize as sources of primary infozznation. A special group among these instru
ments is composed of gyroscopic instruments and systems. They are used as the basis
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 for the construction of inertial navigation (ISN) and orientation (ISO) systems that
insure the independent determination of an object's location and its spatial orien
tation, regardless of the presence of external reference points.
In order to create an ISN, it is necessary to have on board information about the
object's orientation relative to a reference system of coordinates and the instanta
neous absolute velocities or accelerations dicplacing its centPr of mass. When de
signing platformless inertial systems (BIS), the sources of such information are
qyroscopic sensors of absolute angular velocities and accelerometers or gyroscopic
 linear acceleration integrators (GILU). In other ISN design variants, the reference
system of coordinates is created directly on board the object with the help of
GSP's. In this case the accPlerometers or GILU's are installed either on the GSP or
in the object itself.
Kost cf the gyroscepic instruments now in use are designed according to the classic
method of a precession gyroscope in a cardan suspension. The basic element of these
instruments is a massive, rapidly spinning rotor, in which an additional one or two
degrees of freedom is provided because of the suspension's framework. An improve
ment in the accuracy of such gyroscopes is achieved by increasing the rotor's kinet
ic moment and reducing the disturbing moments. Judging by voli:minous data that are
avaiiable, the possibilities for improving gyroscopes built according to the classic
method have been exhausted to a considerable degree, and further progress in this
direction would require significnnt expenditures.
The high degree of saturation of modern transportation facilities with onhoard
instrumeilttype equipment makes a matter of concern the question of the miniaturiza
tion of separate elements of u^,is equipment, with particular emphasis on electro
mechanical devices, among which are included gyroscopic instruments and systems. At
the same time, a reduction in the size and weight of gyroscopic instruments des:gned
according to the classical method entails considerable design and technolcgical dif
ficulties, as well as a reduction in the kinetic moment, which leads to a lowering
of their accur3cy. All of this has forced the developers of ISO's and ISN's to look
for new ways to create gyroscopic instruments that, along with high accuracy, wou13
he snall in size and cost comparatively little.
At the present time we have seen several fundamentally new trends in the creation of
 gyroscopic instruments capable of competing successfully with gyroscopes constructed
according to the classic method [19]. One of the most highly developed directions
is the construction of rotor vibration gyroscopes. The term "vibration gyroscope"
(VG) is understood to mean a device containing special elenents that, gi.ven absolute
_ angular velocities of *_he gyroscope's base, perform induced oscillations. It can be
said that any vibration gyroscope can modulate a constant inpLt angular velocity by
transforming it into an amplitudemodulated gyroscopic moment. If the frequency of
 the cnange in the gyroscopic moment coincides with the natural frequency of the me
chanical system transforming this moment into angular deflectiors of the sensitive
elements, resonance occurs in the instrument that makes it possible to increase its
transmission factor by several orders of magnitude. In connection with this the
value of the kinetic moment still doas not play an eseenti3l rolE, which means that
 hi.qh sensitivity can be achieved along with miniaturization of the instrumerit.
The suspension of a VG's sensitive elements is usually made of elastic elements,
which eliminates such an important source of errors as the "dry" friction that
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occurs in the ballbearing supports of gyroscopes constructed according to the clas
 sic method. The presence of an amplitudemodulated output signal makes it possible,
with the help of welldeveloPed radio engineering methods, to avoid the effects of a
number of sources of random interference in the information extraction and process
ing system.
The first serious attempts to create vibration gy_oscopes resulted in the realiza
tion of designs of socalled oscillator VG's, among which the tuningfork gyrosopce
[3,4] is included. Oscillator VG's lack rotating masses, and in order to create
Coriolis acceleration in the presence of absolute angular velocities of the base,
forced vibrations of special elastic elements (tuning fork blads, strings, rods and
 so forth) are used. The Coriolis forces' amplitudemodulated moment acts on an
elastic suspension that is tuned to resonance on a carrier frequency in order to in
crease the instrument's transmission factor.
Oscillator VG's have a whole series of advantages: small size and low energy con
sumption, high reliability with a fundamentally achievable high sensitivity to abso
lute angular velocities of the base, and so on. However, the tran,mission factor of
oscillator VG's is very sensitive to even insignificant changes in the instrument's
parameters. The high sensitivity of oscillator VG's ia achieved by tuning the elas
tic system's natural oscillations into resonance, with the minimum possible damping
factor. Keeping this performance sta.ble �reguires that the frequency of the elastic
system's natural oscillations and the frequency of the forced vibrations be kept
constant during operation, which is zardly achievable at the present time without
the use of complicated and cumbersome special equipment. The result of the effect
of these facts is that instruments built according to the plan of oscillator VG's
are in extremely limited practical use.
Dynamically tiined RVG's are, to a considerable degree, free of the basic flaw ir.her
ent in oscillator VG's. In them, the amplitudemodulated gyroscopic moment is cre
ated ~)ecause of one or several rotating bodies. In connection with this, the posi
tional moments, which try to bring the system into a position of equilibrium, are
determined not only by static rigidi*_y, but also by dynamic rigidity (nr the so
called centrifugalpendulu~n rigidity). An RVG's parameters are usuallv selected so
that in the resonance operating mode, the static rigidity is much less than the dy
namic. In connection witn this, a change in the elastic system's characteristics
changes only the static rigidity and has an insignificant effect on the system's to
_ tal rigidity. However, a change in tne drive motor's frequency of rotation, which
causes a change in the frequency of the gyroscopic moment acting on the elastic sys
tem, results in a corresponding change in the dynamic rigidity, which means a change
in the elastic system's total rigidity. Detuning from resoance obviously has a con
siderably smaller effect in this case than under analogous co:iditions for oscillator
VG's, the rigidity of the elastic suspension of which does not depend on the fre
quency of the inducing force. Therefore, the most recent developments and successes
in the field of the creation of VG's are basically related to the realization of
 various plans for dynamically tuned RVG's [15).
`I'he theoretical principles of the operation of RVG's have been created primarily
through the efforts of Soviet and American scientists. Among the numerous investi
gations, the basic ones are the works of Ye.L. Smirnov and L.I. Brozgul' [3,4] and
A.I. Sukov, (Dzh. N'yuton), E. Howe, R. Craig and P. Savet [20,33,42,46]. These
�aorks (wzth the excention of [42]) are devoted to the theory of single an3
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doublerotor R4G's, which are the easiest to realize. However, the systems dis
_ cussed in them do not give a complete picture of the potential capabilities of
RVG's.
The existence of basic KVG 3efects, which are related to angular vibrations in the
supports of the drive motor's shaft at a frequency twice that of the frequency of
rotation, as well as static disbalance of the rotors under conditions of linear ac
celeration of the base, forces us to look for ways of reducing these defects.
Tightening of the tolerances in the production process for the separate elements
and assemblies of an instrument leads to the same problems encountered by the devel
opers of gyroscopes built according to the traditional method. ~i'herefore, some in
vestigators are looking into the possibility of improving RVG accuracy by creating
multirotor VG's [42] and VG's having different angular velocities of the rotors.
A great deal of interest is being shown in systems of composite RVG's, which make it
possible to combine a twocomponent measurer of absolute angular velocities and lin
ear accelerations of the base in a single instrument (43,48]. The theory of multi
rotor VG's, allowing for the possibility of imparting differEnt angular velocities
_ to tne rotors, as well as composite RVG's, is not yet sufficiently developed.
 Previously, ths development and use of RVG's was held back by technological problems
tnat were difficult to solve. The present state of the technology is such that we
can speak as boldly about RVG's as about today's gyroscopes. Their widespread use
will make it possible to create a new generation of gyroscopic instrumants and sys
tems distinguished by high accuracy, sma11 size and relatively low cost.
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.M
CHAPTER 1. CLASSIFICATION AND MATHEMATICAL MODEL OF ROTOR VIBRATIGN GYROSCOPES
1.1. Equations of Motion of a Genera?ized Rotor Vibration Gyroscope Model
1!0,12
In accordance with the data in the intro
�
duction, in the general case we will under
stand an RVG to be a system consisting of n
material bodies that are connected to each
other sequentially and have different mo
3P1
ments of inertia relative to their main ax
(2
es, which are rotating around a common axis
.yp'
at different rates of speed. Relative to
~Z
each other., these bodies must have degrees
'
of freedom of angular displacement around
(1
a
axes lying in a plane that is perpendicular
`
s
to the axis of rotation. The system must
also contain units that measure the bodies'
angular rotations relative to each other or
p~
to the housing. A functional diagram of
such a generalized RVG model is depicted in
=p%' ~
Figure 1, while the system of coordinates
is shown in Figure 2.
Let us derive the equations of motion of
the generalized RVG model, and introduce
the concept of a stage of the generalized
model. We will understand a stage of the
generalized model to mean the drive motor's
rotor PDi, which has axial moment of iner
tia JW and equatorial moment of inertia
'o
JeW and rotates at a relative velocity of
Figure 1. Layout of generalized RVG
Oi; inner rotor VRi, which is attached to
model.
it, has moments of inertia J~~ , J~~ and
Key: 1. VR,
2. NR
~i~
JZN (sic] and rotates at an angle ~i rela
3. PD,
tive to an axis that is perpendicular to
the axis of rotation of rotor PD�; outer
rotor NRi, which is attached to inner
W
rotor VRi, has moments of inertia JXN), JY
N
and JZN) and rotates at rn angle 9i re
lative to an axis that is perpendicular to the
plane in which the axis of rotation of
rotor PDi and the axis of rc;tation of VRi
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~N ,
~
zN;
n.,)
N
H It `
~
~ \ ~I \
Vlt) ' '
z,~ n
Figure 2. Systems of coordinates.
M
,fU y(t)
N�'II
relative to rotor PDi are located. Subscript "i" indicates that the structural ele
 ments and their par.ameters belong to the ith stage. Thus, the proposed generalized .
RVG model can be regarded as a set of elementary stages in which the stator (PD) of
the preceding stage is coupled rigidly with the NR of the following one. We will
_ stipulate that tne numbering of tne stages begins with the stage that is farthest
from the ba.se.
In formulating the equations of motion, let us introduce the following basic assump
tion: we will assume the angles of rotation of NRi and VRi relative to the base to
be quite small. This, naturally,, means that the relative angles ~2 and 6i are also
required to be snall. Considering the diversity of inethods for taking readings from
an RVG, wel will formulate the generalized model's equations of motion in the i.ner
tial system of coordinates OiXYZ. In order to do this in accordance with Figure 2,
we will introduce angles ai, si, Yi of rotation of NR� in the i.nertial system of
reference, having related system of coardi.nates 0(l)XH1)YHl)ZHl) to NRi and system
0(`)X~1)YBi)ZB1) to VRi. In order to determine the characteristic features of mo
ticn of motion of the RVG elements a,zd obtain visible results, we will limit our
selves to a discussion of linearized equations of motion only and take into consid
eration only tezzns of the first order of smallness when the PD is in a steadystate
operating mode.
The equations of motion of the generalized model's first stage were derived in [8].
Using these equations, and allowing ior the presence of the gyroscopic moments gen
erated by rotor PD in the subsequent stages' equations of motion, we write the equa
cions of motion of the ger.eralized RVG model in the following form:
1Cj Sln rl =
cos
,
 =1c, (a, ~in 2 u cos _
u8t'6, sin 4 .llC' cos .698'sin ,bfzl) ;
_os 2�l;  x,sin 2�?;),
COS 2'l ,
~lfl 2~~1 ~l =
n ' �
_  Ri i sin Yi _  602
T
_ ,q~ a[ ('l=  rDi) 
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a
(cc, cc5 2y;  di sin `_',ti) ('l2  d),)  uCsin
; uBAcos yi  .til~ ) sin ~l, = :V1g ~ cos y,
~.NR;1'~r,�.,Mit,y1 (~2sin 21 x.,cos��1,):
;li
~
a, sin ,lj  cos ~11 ('rz  ml) 1~1 r
 ~:cos (Y=  (D,)  a�., sin (�l2  (D,);
a, cos 1171  c, sin  (7z  (DI) 9, 
 02 sin (,,1:  (D,)  a., cos (y_  (D;);
{Ic") I Ca COS 2'1,R~ nn  sin 2'1rt h'n  HR_I ~n =
_ c~R`rt1 ~.n CO$ tQ(:' ~niil)

~ ^
 21(cfll) (CLn Slil 2'jz  ~R COS 2yn) (il0 ~.1C ~~rt l".OS'jn 
inl '
ug 8R sin .:!'C' cos .;tB' sin
Vf2 )  Rln ynF":l R(In) ,
((DypCOS 2yn (OZp sin L'j,
[fC(" COS 2`,.a] ~n  I~' sIfl `~`T~sxa  H.at an =
sifly.1WJ . n~ 'x;Gk) 
IR)
_ ?1
Cn (an CaS 2;'n  ~a sin s lTl
= uB 8., cos .~f sin y� =.14a' c:os y;, 
~i) i.zl ' In~ �
M1.
A  Ri 'f n2n_~ (lil y, Sifl 2`~rt  WZ~ COS 2'~ rt1~
Ocn Sitl Y,a  I~n l'OS Yn  wuTn +
(1),,o cos (1)~  wz,) sin
4;n Ccn COS Yn + K SI(1 Yn + IJoAn I
u,Y, sin w�t 1 wZO cos
wnere
l(n) i,ii trit (11) c111>\.
C I~ZN ~}'u'f IZM + f~. Jr
(n) ( f (nl (n) ~1 (n) �
~Cn 1~'Le l(n) ~i ~Zn~~
1~~) ~ ( (n) l(n) (n)1.
~~I  2 ~~.Ya  Yu  J'LeJ+
R(a) _ Ilnl + /(n).
' .\u Lu+
(]..i)
J(n1) = equatorial moment of inertia of rotor PD of the (n  1)th stage; H =
e
J(n1)Y1  kinetic moment of rotor FD of the (n  1)th stage; u~n) , u Bn~ 1 = mo
n
ments of viscous friction alang the axes of suspension of the nth stage; M~n),
a
M(n) = moments acting on NRn in the inertial system of c:oordinates, MCn) MBn) 
moments acting on the axes of suspension of the nth stage.
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_ It is obvious that in the first part of equations (1.1) there must be tenns allowing
_ fox the disturbing moments, the elastic moments in the suspension axes and the reac
tion moments superimposed from the (n  1)th stage onto the nth stage. In the
 simplest case, the moments of the eiastic forces are described by the expressions
tn
Cc CB 0� p (1.2)
where C`n), CBn~  torsional rigidity of the elastic moments along the axes of sus
pension of the nth stage.
The reaction moments on the nth stage are composed of the inertial and gyroscopic
_ moments generated by the (n  1)th stage's VR, as well as the moments of the elas
 tic �orces and forces of viscous friction along the suspension axes of the
(a  1)tn stage. Using Euler's method, let us derive the expressions for the reac
,:ion moments, which have the following forr,i when the equations are written in the
ir.ertial system of coordinates:
(nII� (nI) _L /(nI) IrtI)\ I
21~'~n =  ~n p'iI  ~~.Ce i 4'u  ~Zn 1 YnlunI 
CUJ
n  ~Zn ~ i&tnI l'OS2'Yn_j  lift _1 SI(1 2}'rtI)
 CUS , ~(n I)
I_
'nI COS}1nI I_
} u riI ~r~I E 1J ~ ~
+ nI)Sl(l~~~i_~ ~~Cy COS
(nIl ( (nI) 11\ �
2CC 1})~~_~ J111 l~Xn ' ~Zn 1 I~),~_1 CUS
~y
y I~ 1j~tl Il  ~Z~ It';i l ~
~~I1~n) I \1~t1
~Ye (17.n, CU$ Zy,i._, Slil
(nI) (n~1 frtI)
 Xy ~}�y  /Zn ~ Y11I(C1R_I SI[l 2yn 1 f
rt  I
COS 2Y,i~~  Z/i'u ) Slll Yn~ 
(itI)' (n
~ue U�_, sin y,ll  ~E~ccos ynI
 stn}~
d n~ ,i, 
( InI1 InI)
 l1Xa  ~Zn ~ ~~nI~)~~_~ SI(1 ~~l~1 � )
Thus, when e:cpressions (1.2) and (1.3) are taken inta consideration, equations (1.1.)
are a linearized, gene.ralized mathematical model of the RVG represented by the func
tional diagram in Figure 1.
Equations (1.1) contain the absolute coordinates of rotation of the rotors in iner
tial space and the relative coordinates of their rotation in the suspension axes.
Let us eliminate tne relative coordinates from the equations of motion. In order to
' do this, we will determine the solution of the three kinematic equations for each
stage. The first equation has the obvious solution
~I~rtl I i)n1 'n, (1.4)
 where ~n = initial phase of rotation of tne NR.
In ordzr to solve the two remaining equations, let us write them in complex form:
i7nlx1i�l =  Xnle tt xn(1.5)
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where an_1 = en1 + l~n1' Xn1 ~n_1 + ian_1, while angle ~n is assumed to be zero.
It is obvious that the solution of equation (1.5) has the form
i
vn
~n..l = ~nle ' Yr~lt  7(n_lCfYntr Xnt, 1~ (1.6)
where an_1 = A~~1 + i~n_1 = vector of the initial angles of rotation in the suspen
 sion of the (n  1) th stage.
Henceforth, in order to shorten and simplify the computations we will make use of
the complex coordinates we have introduced in order to write the equations of motion
of tne generalized RVG model. First, let us write in compiex form the expressions
 for the moments of the elastic forces acting along the suspension axes and the reac
tion moments of the (n  1)th stage on the nth stage. Considering (1.2) and
(1.6), the vector of the moment of the elastic .forces is deterntined by the expres
sion
l~~(n) =  CjJ 'f CnXit + a,O�L~CnE~~YnI  xitACae'~nt 
(1.7)
i4ynt
 Cn;(fj 11 + ACnXn+le ~
where
C(n) + C(n) C(n)  C(fj)
A'IIn) _ (n) ~ ~ . C B C d
d r 11u ~ C, n = 2 , ACn = 2 1
while the vector of the reactive moment is
;I~~n  I fn~> ~�I) ' 
2 ~1'" xn ya YnI
~ (iiI) ' ' ,
f~ yn~  ~1ft~~~;(n 
aC,,,) X,I e~.v,~1` i
(i � L
1 RI }~r~I r L ~ni ) xnI r
' L~ ( \~)1~2
r I"Y111 +Cn1) xnI +
(11I)'
+ ~1/~~ Yn1 + DEAnI) Xn1 _f'
i ylli AC�i) Xniie`2y"'1 
b _i (i/Y~ ~~y�~
ini) �
  (R, Y,l I  iynI �n1
.
/`u 1) !~~~'i t/~,1i> � o�
 Z r (Yn Ynt  A�ijI ) inI
/ (nI)' 1 U.
lRl ACnI) ~n1I ei2yn_1!
~
Let us write, in system (1.1), the equations of motion of each stage in complex form
and, considering expressions (1.7) and (1.8), in operator form we will obtain
+
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 ~'~I(v) X' A r,' t ziecsv,r = A1~�  ;'erl'v~~
i ~P
+ 11%';" (P)X,}1V(pi2yi);tlei''"+
I  I Znei:V~i~ 
IV~i (P) 1v, (P i2
 Yn)
 . = ~it~i~  �`~rta'ri ~ T
(P  i2y,i5 xne
f W~ (P) W. (P  i2Yn) Xn~.le'.yni + Bn ti ~_I
+ (p) XnI
i
Vnf
f
l~~,'(rt) l~~~uuY
I~n (~J  l`~YnI) X',Rg!2 ynt
+
T
~ = r ~ ~oto \p
"
(p) l c + 1
+ ( i (R!,")1~n_ Y~(nn ~),Yn f f/rtI) + ELn + !inI I l
I p f
~Ri`i)YR +R~n1)Yr
i_I _
 i (finYn + tinI + Crt +Cn1
AWn ~P)  ICnP2 ~`~lcri Yn AN�) f~
+ (RIn)Yn  lL1~LnYn + AL'n);
~ i I(n 1)p2;(il(;, 1)y,.i
(P) + AEtni) P
c1W�_~
(n1l ' ?
(f~~ YnI i4., Y111  ACa1)+
WR (P) = iRi,~~ Y�I i �111 +
~ (n_i) . .
i p (R~ 'rCrs1);
(P) _  (iR;") Y,,i f' L\F1 o)0)2 i c1 �i ((p l + ow)  AC i ] .
Let us first turn to the equations of motion of layouts with single modulation of
*he signal. There are two varieties of such layouts, namely when the modulation is
provided by the second stage's drive motor PU2 and when it is provided by the first
stage's drive motor PD1. Let us er.amine these two cases in sequence.
Tn the first case, in equations (1.15) it is necessary to set ;1 = 0, I(2) = Ie2)
= 0. In connection with this, the first and second stages' suspension axes are ar
bitrarily oriented relative to each other in the plane of rotation. In order to al
low for this orientation, we will introduce angle ~12, which characterizes the rota
tion of the first stage's suspension axis relative to the second, from an initial
position where the suspension axes of the corresponding rotors in each stage coin
cide, in the direction of rotatiqn at velocity wo. In equations (1.15), it is then
necessary to set, additionally, 'Dlt =~12. 'I'hus, the equations of motion of a four
rotor VG with singlE modulation realized by the rotation of PD2 have the form:
~  I ~i~,i�1r.t e i_w,r =
Wi ~P) xI Jll%i (p f i2w,)
 AiVI  fl,X;'e`2Nj I e`�~W.~ w ;"(P) X�i
 i':~ue
i"wrt
(f7 l?(il,) x�e ~ i u
1 I '
(D) x AW1 (p i?w,) ~(_e
 t i_wi~ i"w,r
IV, (P i z(.,U)
 A.?~�  .~�.3~1'e''~'e (P) w
o u� 1^.~Di~ei?w.[
+ i~"u~) W�~'_;_W' + Ql~l I L
f W 2 (n)x t~ 2` w)xj�ei:t{'ie
f t .
e`2w�1 + ~til"
(1.16)
Three, two anC singlerotor MRG's are special cases of this layout. In order to
obtain the equations of motion for a threerotor MRG in which each subsequent rotor
has only cne degree of freedom relative to the preceding one, it is sufficient to
set I~B) = I~B) = IZB) = 0 and CB2) in equations (1.16). If the suspension of
the third (counting from the instrument's base) rotor has two degrees of freedom in
the plane o� rotation relative to the second rotor, in equations (1.16) we should
set IXB) = IYB) = IZB~ = 0. If the second rotor has two degrees of freedom relative
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to the first, we should set 1(2) = I(2) = I(2) = 0, and if the first rotor has two
XB YB ZB
 degrees of freedom relative to the PD's shaft, we should set
XB YB ZB
A tworotor MRG with one degree of freedom of the suspension of each rotor is well
 known [3] and is a setup that has already been realized in practice. Its equation
corresponds to the first equation in (1.16), for X2 = w and ~12 = 0:
~ xi  ~ xle`~~,�` _~~~i  A i~~'E,_'_W.'
 W i (P) A lL"; (p i?u),)
(1.17)
+ W i~(P) W}' ll'/ i(P r i2wo) w`eizm,e + h1") TV1;,0etW .r~
where
~'~I(P) = Mp f (iRli)~~ l~i~ P h'i11,hi fti~u Ci;
11t"' _ ]caN2 + (i21c..u~o Diii)P Ri~)u~o ic~fti~ ~1Ci;
,i (P)
A , = R;"wu i�,wo + C,;
Ai = R,1~�~ i~1 AC1;
2 Wi (P) il~i,~~ui fii f_ p (Rii)~ + i�io~, C,); .
~ (/ZiI)ut;  icl�iwu ,~Ci)�
The equation of motion of the most widely used type of twQrotor MRGthe socalled
iKhaui) gyroscopeis obtained by substituting into formula (1.17) the equalities
I(1) = I(1), I(1) = I(1), C= C, u= u .
ZB JB ZH JH B C B C
r^inally, the equation of motion of a singlerotor VG, which is sometimes called a
(Seyvet) gyroscope, allowing for the flexural as well as the torsional rigidity of
 che torsion bars, is obtained by substituting the equality I(1) = IM = IM = p
into (1.17). ZB XB YB
Tne equations of motion of a fourrotor VG with single modulation realized with PD1
are obtained from (1.15) by substituting tne equalities wo = 0, ~21 = 0, and has the
form I _ I
1
W'i (P)X, ,11Y'i (N t''2'Di) Xie
W ~ _;:,~D t
,F (P) X 2 W +1 (P  I~ ~24)1) I
W7: (P)X" X
(P) (p i2cD1)
L:11  A_~.��,~ W:~ (P) w lY/~ (P) f
~
F (P) Xi L W _ (P f i2(~,) x ie_ i_,t1'1 f
Iyl'=') F N~;;".
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(1.18)
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If the f~~~' t sta e has a single rotor, in equations (1.18 it is necessary to set
i(l) = I = I( = 0. If, in connection with this, C(2~ = C(2) = 0, from (1.18)
XB YB ZB B C
we will obtain the equations of motion of an OMG with NP, or a socalled precession
vibration astatic gyroscope (PVAG) [16], allowing for the finite rigidity of the
shaft of the gyroscope's rotor. The equations of motion of a gyrotachometer with a
rotor having unequal equatorial moments of inertia also follow from e~uations (1.18)
[45J. In order to derive these equations it is sufficient to set IXH IYH,
= I (2) = 2 (1) = I (1) = I (1) = I (l) = I (2) = 0.
ZH hB YB ZB e
It is interesting to note that equations (1.18) also describe the motion of a bi
axial gyrostabilized platform when its suspension axes are in a position that is
close to orthogonal and the sensitive elements are two or onerotor (IXB) = I,~B) _
IZB) = 0) RVG's with single modulation. In connection with this, the relationship
CB2) = C~2) = 0 must be fulfilled.
The equations of motion of an RVG with single modulation realized by PD2 are non
stataonary and have periodically changing coefficients. The nonstationary nature of
the equations for this type of gyroscope is caused by the choice of the inertial
reading system when describing their motion, and can be eliminated with the help of
the metnod proposed in [16]. Actually, let us examine equations (1.16). They con
tain four unknowns: Xl, X2, Xle2wot, X2e2wOt, To each of these equations we will
apply operator e"'12c~ot, and then the operation of complex conjugation to the left
and right sides of the ensuing equations. As a result, we obtain two equations:
~i e_ e+'2w11 _ A,,�~_r:,w.r _
W i(P i 20),) ~W i(P) X i t i
~ ie t^w.. + W; ' (p i2w~,) x~etz~,r +
+ W I � (p) ei21Di. ~ IV~~I ~'C~?cJD! + IV1~I'eilDuei4J.1'
I ~e_;2W.r l  ei='ro,.  
lG'., (P + i2wo) x OWz (P) xn AWi (V) x: _
I
= ~l "2w 11  .~.~,lz I~/1 ~ (p I2c,~) wei
+ * (P) ~ + I
 � r:~, r �o� ;sw r
(P 1 2~v) X i e � !3 i ~ i e �
(P) X ~er.~~v� rYl (2) 'e'""d M;,' )*e'w�l,
(1.19)
which, together with equations (1.16), form a system of four algebraic equations
relative to the four indicated unknowns. Having solved system of equations (1.16)
and (1.19) for X1 and x2, we obtain stationary equations of motion for a fourrotor
VG with single modulation from PD2, in the form
 I
(P) '4 iWii (P)  Aie`''v~I Wia (P) + QtWi:.~(P) +
Bie,.(vis Wii (P)] tii ~ 1,+_W1. (P) 
 (P)) ~z I (~,eIVII (P) +
A i IY/ 13 (P) + Bie~~z~v~ ~ W il (P) + 8 iW w (P) ~ �.er:~,,,r
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rqJK urrlkiwL u,r. U1vLY
. . . . . . . . .
i A.,W,�_ (P) A.',W ia (P) J ~i I [Wl (P) Wil (P) +
. i (P)~ w + IWz (P i2u)o) W (P) I
+ Wi ~ (P i2wo) W1a (P) I w~etr.,r +Wit (P) +
I W13(P) M~
n ~
FV(P{t2(IP, }wu))x_,}iV (P}i2(4P, f
� r21~~tW.l~ (1.26)
+ wo)) 'X,e  Ws (P + i2tA,)
 A.*, io�erl(,~,+Wo)t  A�;1~�e`~'~'` + LO(P + i2(61 +
+
wo)) (f~ 't i2 (4)i w,,)) ii
f IK/; ' (P f i? (4), + wu)) w~e+ W 1 (P F i2(~,) C.)e`~'~'~
+ Iy~(') .e~ ~11~2)'e`(_~1v,+W.) t f M~ (P + i2 (~ I w f~ ~P) t
L L
F N (P) 1" ~ (n + i2 (n), + ~,~~i) 1 ~~~�er~~~u,.W.~~ + .
N(P 12 (,~j t W,,)) ~
1 1.' (P f� i2wo) iu�
I AW'i (P f' i?o),) N� (p i2w�) I e +
(P f i2uj,)
(P i2wu) N' (P i2wu) I [W.+ , (A) +
0." '(P) I W
+ c11~(P I i2o)~) A/* (P i2wu) J [W ; ~P l2co~~
(P f i2wu) 1 . +
+ JWz (P L i2wu) tP i2mu1 ~ ~ ~it~,~
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+ N ~P) W.# (p i2tD) We_~2~v,~
N' (P f t2 (~i F ~u))
f' N (P) (P + i2 (m, + w�)) ~�~_r.~~~~,+~.~r Njt~~� (1.27)
N' (P i2 ((Ui wu))
I I
+ TIT� (P + i2coo) N' (P + i2cu~) M~2~�~~~~~f +
+ . N (P) ~,~(~=1�~_~~('1'~+u~o)f ' * 1~~~~~C'i~4t
N (P i2 (iDt cou)) n
1
+~1W1 ~P f i2wu) N' (P i?w~) Mn:)� ~,~~�t +
f' N(p) , Mn(_^1'e~=(~irwu)t +
 N' (P 4 i2 ((Di f wu))
r~1~1(N) f N (P) W (n + r2 ((u1 + (10) 7V1"'
L N' (P i'? (Ril 1 wu)) ]
Ir(P) + N (P) ~N~ (P i2 (iD, I wn)) ~~,~~~~�~_;.s~,ti~~~.)t + I hl� (P f i?,o�) n,~c~�~t:wof
` ' (P J i2 (,DL + i2w,,) N' (P d i2wu)
, I .11' (p + i?~~o) 1~~)
eM� (p i2 ((Dl wo})
~i~
~AIG.', (P + i2uo�) N' (P + i2w ) ~t' M (P) N(P) I n e rt
� . N' (P 1 i2 (111, F w.))
1)�~,iob~+w,l
4 M (P) + N (P) `11  (P I i2 ((liI I 4.?) ~til~
,~.cp+i2cm,+~a� I M, (P 4' i?I),) M* (p + lL(,)
(P 't i2c~o) N(P f i2(0) e
� f i2w n
n + D IY'~ (P u) N' (P i2t~,)
where
I l l
(DI~ (P) = N (P)  W_ (p) AW' (P I i?w,) N' (P + i2wo) ~
N' (v + i2 (,n, + we)) .
  N (p) ,
N� (P + i2 ((Dl F w,,))
(P) _ I N' (P F i2(d�) .
AMY/_ (P i2co,) N* (P 4 2wu) '
1 N
~i'�r�: (p) (p)
_ �
dWi (P r2(1)1) +V' (P r i2 l(I)i 1 ~u))
Thus, in order to solve system (1.15) it is necessary to solve equation (1.27) with
hannonic coefficients that change with frequency 2$1 and then substitute the solu
tion tnat has been found with respect to X2 into equation (1.24) and integrate the
derived stationarf equation with respect to X1�
1.4. Signal Reading and Information Processing Systems
One of the most important components of an RVG, and one that has much to do with de
termining its accuracy and sensitivity, is the system for reading the angle of
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rotation of the RVG's rotors relative to the base: or each other and t1i'n processing
the obtained infonnation. Depending on the reading method that is usEd, tne useful
infornation about the angular velocities of the base's rotation re'Lative to the in
strument's axes of sensicivity consists either of constant angular deflections of
_ the rotors in a system of cooz3inates that is coupled with the base or of amplitude
modulated oscillations of the rotors at frequencies that are equal to or multiples
 of the PD's frequencies of rotation or are composite frequencies. For any signal
reading method, however, the range of working angles for the unit that measures a
rotor's angle of rotation should range from approximately 0.01 to tens of angular
 seconds. The measurement of such small angular movements is a quite complicated
technical problem. In principle it can be solved with the help of anale measurers
of different types [38]. Zn connection with this, the general requirements for them
are.
1) nigh static accuracy; that is, the absence at the angle measurer's output of a
zero signal that is synchronous with the useful signal and exceeds a given level;
2) high dynamic accuracy, wnich means that signal formation must take place with
msnimum distortions within the limits of the instrument's band of operating frequen
cies;
3) high sensitivity and a low sensitivity threshold;
4) minimum reactive effect on the RVG's rotors;
5) sufficiently high output signal power;
6) niyh reliability and resistance to interference when operating under conditions
determined by the tactical and technical requirements.
 Angle measurers can be divided, according to their operating principle, into:
 1) passive measurers, requiring an external power source;
2) active measurers, which generate a signal proportional to the value being meas
ured.
Passive measurers include those of the capacitive and inductive types, while active
ones include those of the induction and piezoelectric types. In order to eliminate
the erfect of linear oscillations of the rotors on the instrument's operation and
 provide L.rlC waximum possible sensitivity, all of these measurers are built with dif
ferential circuitry.
Let us examine several features of passive angle measurers. Their operating princi
ple is based on the measurement of the change in the reactance of the gap between a
sensitive element and the sensor's eiements. In the case of a capacitive measurer,
tne sensitive element is one of the capacitor's plates. A secon,~ ~apacitor plate is
mounted on the housing or rotating part of the instrument. When the rotor is de
flected tne size of the gap in the capacitor changes and, consequenbly, there is a
change in its capacitance. For an inductive measurer, the rotor acts as an armature
chat completes the magnetic current of the sensitive coil.
A comparative analysis of capacitive and inductive measurers showed that, all other
conditions being equal, the sharpness of a capacitive measurer's signal is 50100
times higher than that of an inductive measurer. Some comparative characteristics
of these sensors are given in Table 1. From the table it follows that witn respect
to all the basic parameters, a capacitive sensor is considerably better than an in
ductive one.
r^igure 4 is a diagram of a capacitive measurer. Rotor P is located between four
plates 0, ahichdepending on the choice of the measuring system of coordinatesare
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Table 1.
I 71M AJT411KJ yrne (2)
KTCI I Mi7NK11
eMicocruu~t( 3) I
NMJ{YKTIIBIILI'(A~
Y
Kpyr113Ha, B/rpen (5)
I10
10'210'3
Yponeab wytita, nIB (6)
0.54
5
fa6apnT, o. e. (7)
I
3
Macca, o. c. I
1
4
Key :
1. Characteristics
2. Type of angle sensor
3. Capacitive
4. Inductive
5. Transconductance, V/deg
6. Noise level, mV
7. Size, rel. units
8. Weight, rel. units
attached to the base, or rotate together
with the PD's shaft, or are mounted on
another of the RVG's rotors. The rotor,
together with the plates, forms four capa
. citances C1, C2, C3 and C4. When the rotor
turns through angle a, capacitances C1 and
C3 will be reduced, while C2 and C4 will
increase, and vice versa. Capacitances C1,
C2, C3 and Cq are connected in parallel, in
Figure 4. Diagram of capacitive angle pairs, and can be connected to the arms of
sensor. a bridge (Figure 5). If plates C1C4 are
mounted on the rotating part of an instru
ment, an inductive or capacitive cor.unutator is used to transmir the signal to the
base. Tr.e former consists of a transforner, one of the windings L1, L2 (which is
mounted on the PD's shaft) and another winding L3 that is on the instrument's hous
ing (Figure Sb). A capacitive commutator consists of two pairs of concentric rings,
Q) b)
c,
~y
cs
~1 La
u3, 5
p6u
us.a
Figure 5. Electrical connection diagrams far capacitive
angle sensors.
one of which in each pair is mounted on the rotating part, while the other is mount
?d on the nousing. These =ings forn capacitances CS and C6 (Figure 5a). Since the
commutator's capacitances are connect2d in series with the capacitances that are
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changing with angle a, in order to obtain nigh reading system sensitivity it is ne
cessary that C5 " C1 + C3 and C6 � C2 + C4. Variable power voltage upoW is fed
into one of the bridge's dia5onals. In accordance with the recommendations fr~r this
type of ineasurer, the power frequency is chosen in the 0.11 MHz band. Signal volt
age usi9 is read from the bridge's cther diagonal. The signal voltage can be calcu
lated w.tth the formula
28rup~u (1.28)
u 6b'  u=r' a'
8 = 4rt1L f =eeOS,
where L= inductance of the measuring bridge's choke coils (L = L1 = L2); f= fre
quency of the power voltage; eE0 =absolute permittivity of the air gap; S= area of
the capacitors' plates'
L16 = U  U,r LIO ~ YtuS .
Cu '
do = initial size of the gap betcaeen the capacitors' plates when the rotor is in the
equilibrium position; Co = initial capacitance of the capacitors.
rrom fo r�nula (1.28) it follows that the dependence of usig on angle a is, in the
general case, of a nonlinear nature. Therefore, it is extremely important to take
into consideration the required range of ineasured frequencies and the allowable non
linearity of the instrument's static characteristic wnen selecting the initial value
of the gap between the sensitive element and the capacitors' plates.
In order to create an instrument with a linear static characteristic, it is possible
to use a transfornertype bridge connection circuit. Such a circuit corresponds to
tne one depicted in r^igure 5b if upow is
fed into winding L3 and the signal is read
from the bridge's other diagonal. ^uisre
gardir.g the bridge's reactance, the output
signal's dependence on angle a then zas
the fornt
rupo;, (1.29)
j`~
~X. '
u
However, such a circuit has a nigh ^oise
level and a high sensitivity threshold.
rP
It is feasible to use a piezoelectric sen
HM(1)
sor [8] only when the amplitude of the
RVG's sensitive elements' oscillations that
~ 2~
is being measured is of a sufficiently high
T f""' "~7
yD
frequency. The connection diagram of such
77
t
a sensor is shown in Figure 6. It provides
,~y ~
for tne measurement of angular vibrations,
P
with selfcompensation for linear vibra
tions, with accuracy up to that of the
Figure 6. Connection diagram of
identity of the piezocrystals' (PK) parame
piezoelectric sensor.
ters. The transcDnductance of a piezo
Key: 1. IK
electric sensor is computed with the for;nu
2. PK
la
Jr~ J!lllrr ~
a, (1.30)
u'.
J C,; fC,
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rvrc UPFII lAL t�r, uIVLY
where d33 = piezomodule of the crystal; mIN = inertial mass (IM); Ck = capacitance
_ of the piezocrystal; Co = input capacitance of the signal processing unit.
The signal voltage's dependence on angular acceleration a is linear. Consequently,
_ the dependence of usig on the base's angular veloity will also be linear.
The shortcomings of a piezoelectric sensor .include the effect of the processinq cir
cuit's capacitance and the cable's capacitance on the output signal's value, as well
_ as the necessity of deriving the signal from the rotating part of t;.e RVG.
. r 0A (4
I u ~m
a Ny u5i� ~ u y I CC1 , ro
(1 2 ic5j
i ~
(Lun CC`5 2 ~2 `wau
r ~
 r cev ~9) L  
~ ~ I~oni IWI
LU__Ej_H T
w0 ~ ~
u
tUN2 Y2 'T 2
I (7) 1 L~I$ll I 111.0
1 ~
L J
Figure 7. Functional diagram of information process'_ng system.
Key.
1.
Ili
6.
G
2.
D
7.
GOI.
3.
U
8.
U.
4�
FD
9.
SVO
5.
SS.
10.
T.
when the useful signal is read in the tor.n of amplitudemodulated oscillations, the
processiny system solves the problen of obtaining two directcurrent signals that
are proportional to the angular v2locities of the base's motion relative to the in
strur.lent's axes of sensitivity. Figure 7 is a functional diagram of such a system.
Tha signal arriving from the bridge of the :neasurer of anqular rotor displacements
(Ili) consists of oscillations with the frequency of the change in the power voltage
and balancemodulated oscillations with the usefiil signal's frequency. The ampii
`,.de and phase of the latter contain the information about the base's angular velo
cities.
_ ':he first stage of the signal's processing is its demodulation on frequency fPoW.
In order for this to take place, the signal usig from the bridge enters demodulator
D, into ahich a signal from generator G(which generates the bridge's power voltage)
is sent as a reference signal. Oscillations with an amplitude proportional to the
RvG rotcr's angle of rotation are obtained at D's output. This part of the circuit
is abser.t in active measurers.
After preliminar1 amplification by amplifier U, the signal uD from D's output is
sent into prasesensitive demoduiator FD. Thz reference signal for FA is generated
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with the help of reference pulse generators GOI. A GOI consists of a rnagnet that is
 press��fitted into the rotating part of the instrument and a coil mounted on the
_ housing. When the magnet passes by the coil, an EDS [electromotive force] in the
form cf a short pulse is induced in it. The magnets in the rotating part are situ
 ated at an angle of 180� to each other, while the two coils in the housing are at ari
angla of 90� to each other. When reading on intermediate frequencies, the magnets
and coils can be situated on the appropriate ro.*_ating parts. After amplification
cvith the help of amplifiers iJl and U2, the pulses from the GOI's enter reference
 signal processing circuit SVO. When necessary, this circuit can contain a frequency
conversion circuit. On receiving signals from GOI1 and G0I2, triggers Tl and T2, '
which are operating in a waiting mode, generate rectangular reference voltage pulses
 that are shifted 901 relative to each other.
 T:e piiasesensitive demodulator is a comparison circuit (SS1 and SS2). The refer
eace viltages from the triggers and the amplified signal from the demodulator enter
it and constant voltages ul and u21 which are proportional to the base's angular ve
, locities, are obtained at its output.
The extremely simple circuits ror reading and processing information from an RVG are
not the only ones possible. Inductive angle sensors can be used successfully in in
struments of tnis type. In order to increase tneir sensitivity, capacitive or in
ductive sensors can be built into the driving crystal oscillator's resonant circuit.
Preliminary processing of the useful signal, by placing electronic units directly on
the rotating part of an instrument, is used quite frequently. Conversion from amp
litude modulation to frequency modulation or to amplitude modulation on some inter
mediate frequency is another practice that is used.
This brief analysis of RVG inrormation reading and proces5ing systems snows that, in
_ contrast to gyroscooes constructed according to the traditional plan, RVG's are a
symbiosis of inechanical an3 electrical parts, with the latter having a substantial
effect on the design and basic characteristics of these instruMents. This fact com
pels us to develop a new approach to the process of designing such gyroscopes that
 involves a more nearly complete consideration of the demands made oii the mechanical
part by the information rzading and processing system.
1.5, yethods for Solving Differential Equations With Periodic Coefficients
_ Az was shawn in Section 1.3, in the general case RVG's are described by linear dif
ferential equations with harmonically changing coefficients. In connection with
tnisand regardless of tne information reading and processing methoda character
istic of all RVG's is tne presence in the output signal of a harmonic component that
changes witn the doubled frequency of rotation of the instrument's rotor, while for
RVG's with multiole modulation it also changes with mt:ltiples of this frequency and
composite frequencies. Therefore, when there is inadequate filtration of the output
signal ar.d a broad transmission band, systems containing RVG's will (in the general
ca.se) bl described by differential equations with periodically changing coeffi
cieats.
Accord'_ng to Lyapunov's funda.iental theorem [11), any differential equation with
periodic coef�icients can be reduced to an equivalent linenr differential equation
with constant coefficients. At the oresent time, however, no universal algorithm
for this reduction has yet been found. Therefore, the analysis of the properties of
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solutions of equations with periodic coefficient usually involves considerable dif
ficulties.
The development in recent years of electronic computer technology makes it possible
to use the methods of numerical integration of differential equations to analy2e
equations of m.)tion. When these methods are used, however, the presenc2 in an RVG's
output signal of a slowly changing comgonent and a harmonic component having a rath
er high frequency should be taken into consideration. The requirement that both
components be allowed for means that the amount of machine time needed to integrate
the equations of motions is increased sharply. At the same time, the necessity of
solving the synthesis problems that are related to the numerous possible structures
of systems and their parameters, as well as the parameters of the RVG's themselves,
stipulates the use of analytical methods to investigate the properties of the dif
ferential equations with harmonic coefficients that describe the operation of RVG's
and systems containing them.
For the investigation of periodically nonstationary systems, the most fruitful
ideas proved to be those related to the use of a Laplace transform [14]. The com
plete theory of such linear systems, as presented in [281, makes it possible to de
rive the narametric transfer functions and the pulsefrequency characteristics of a
nonstationary system. However, the use of these characteristics in engineering cal
culations is considerably more complicated than the use of normal transfer functions
and the frequency characteristics corresponding to them. 'I'he spectral theory of
dirferential equations with periodic coefficients is explained in the works of V.A.
Taft [34,35] and modified in [16] for the case of twodimensional systems with amp
litude modulation. There the result of the solution is presented in the form of an
infinite sequence of determinants, for the formation of which it is required to know
the specific parameters of a system already, which narrows the possibilities of us
ing this approach. For the purpose of developing the spectral theory, in [27] the
author proposes an algorithm of a rather simple method that is free from the defects
enumerated above and, in principle, makes it possible to obtain stationary transfer
functions of twodimensional linear systems with periodically changing parameters.
Let us discuss the features of the use of this algorithm in the investigation of
linear differential equations with periodic coefficients and the possibility of
writing their solution analytically (10].
It is not difficult to demonstrate that any linear differential equation with mono
harmonic coefficients can be represented, with the help of Euler's formula, in the
iorm
.r F Iv., (P) Xer(6r f'f'j (p) xerWr _ R (P) u, (1.31)
wnere P2(p), 'A3(p), R(p) = operators representing rational functions of differentia
i.ion operator p and having complex coefficients (in the general case); W= frequency
of change of the coefficients of the nonstationary part of equation (1.31).
Ns tne first step in the solution (n = 1), let us perform the following operations.
To the lert and right sides of equation (1.31) we will first apply operator elwt and
then operator elwt. As a result, we obtain two equations from which the values of
;`iwt and xelwt can be found. Substituting these values into equation (1.31), we
_ obt:ain tne firststep equation
,
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11 _ 'n_ (P) (pa (P  iw)  (I)i (P) (D.: (P + i w) J .r 
 (Dz (P) (P  iw) xe+'"(0r  cps (P) (
Ds (P I iW) xe`_'we = (1.32)
= R (P) �  q)s (P) R (p  iw) tce�t _ cp3 (P) R (P + iw) uerwr,
which is equivalent to the original equation (1.31).
Let us redesignate the operators in equation (1.32):
(U(,') (P) = I  (D;., (P) (Dj (P  io))  tU;s (P)'U�~ (P + i(A);
and as our second step
and e12Wt to equation
xei2wt and xei2wt tha
step equation
(p (P) _ (D2 (P) (D~_ (p  iw);
'ps') (P) = (Da(n3 (P + iW).
(n = 2) applyanalogously to the first stepoperators el2Wt
(1.32). As a result of the substitution of the values of
t we have found into equation (1.32), we obtain the second
~ U" (P) mk 1) (P  i2w) m~' I (n) (D;" (n + i2w)
(p)   iD~i i (p  i2w) ^ 1111 " (p k i2w) , x 
_ (I);1) 0~=~' (P  i?~) x~tawr _ n(i) (v~~ ~ (P f i2w) xera~,r _
(P)~~i , (p i2w) ~ 3 (P) 01 (p + i2w)
= R (P) � = R(p  iw) [~As(P) + (A2" (P) (n'�~P i22~;)] rier~ar _
 R~P+ iW) ['I) 3 (P) (p3i)(A) 'p~,(P i2w) , ue r~r + (1.33)
cpt (P + i2w)
f R (P  i2w) iL~" (p) ~ uer�:We + R (P + i~~) (P) tte~~:Wr
(DS~ (p  i..a,) (L, (P + i2w)
Q~0" (P) R)s (P  i2w)  R (p  i3w) (Vi" (P !2w)
 R (P 4 i3w) iDS' ' (P) 'pa (P I i2w)
(Di (P + i2w)
If the har,nonic coefficients change with frequency w in the original equation, in
the firststep equation they change only with frequency 2w, and in the secondstep
equation with frequency 4w. After n steps we obtain an equation, equivalent to the
original one, in which the.harmonic coefficients change with frequency 2nw. In con
nection with this, the recurrent relationships for determining the operators in the
left side of the equation are detezznined by the following expressions:
/It) ,n~) (11 (P) 't~(uU (p + 11nlW) .
in~ (P) _ ~[i ~P)  (I)(,�i) i211iW) Ipinii (P f (1.34)
�nI_I
~Uln) t/~~ nI) rp _ ~,~n_~w~  n' 1I)t (P '~k~)
t
(P i2"1w) n~ �11 ~ (1.35)
n( il)(11)(p  i2J( lI 2k)W)
r=il k_u I
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 2n~~
�1)  n my (P j i2k~)
IP) _ ~3"u (P) ~3ni) (P l2 ~
k0
~inI) ~P i2"1ni �t"!!_1 � (1.36)
tl [n m,"(P + i2`(1 + 2k)W)~
1=1 k=0
_ The right side of the nth step's equation can be derived in the following manner.
_ It is obvious that at the (n  1)th step, the right side of the equation has the
form
2n1_,
E [Ak"')(P) t
k=0
In the nth step, the right side will
lAo (P) f Bo (P)I
k1
terkWr + Bknt> (P) cie ikwe
(1.37)
then be found from the expression
c,11) (p)
(P)  Bc,I~~ ~
(p`'
~ (U~nI) (P !`n1(~1 \ ynIk ~ 
~(nU
~nI j kw! ' (P)
_ i_ w) Ue  ~,(,~i, (P _ i~,,~i(,) lAu ~P I
(p~" 1) B, (P  t2"1w)j ties :snlwr _ ~ (P) A("~i
,V1ln1) /P  IZn1(u) 'n_k (p
~
 !2`1W)llei(:"lrk)wt + I QlnI1 (p) (ns( n  1) (P) alrtU
k ~~n') (1) i?,i1w) /c (P + (1.38)
1 ~3"i~ (P) /
!~~�I(U) J ll2ikwl rP'{' t2rtIW) Bk~I ~ lP ~
1 ~
tDIn~)
!2nI(U) lIt''i2`lwl  (DI~I) (P t2nI.~ Bk1_~~
_ + i2t, i(1)) Uer(2^I +k) wr I .
Using expressions (1.37) and (1.38), for any n it is possible to obtain the values
of the operators standing before each disturbance harmonic in the right side of the
equation. Let us present the expressions for the operators on the first three har
monics:
Aini (P) ~R (P  iu)) I1I),: (P)
Ai (P  iu)
11 1 k 'p,(i11) (P _i2'w)
 I )k~(U(k) (N) ~ r
m(11) (P _ i21w) i
8~,~ ~P) R (p f i(o) ~cn ~P) 
q>, cP + ~ ~iRR~2*)~; (2.27)
 R3i~
2 Rii, (R}uWJ .)C(l 1)
(21.27) :nto dy^a.aic tur.in:, cond'_tion (1.62) , we obtain an exrression
t.:e r=_qttired relaticnsai: between the rigidities of the second and
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tnird rings' suspensions:
(qRii~ ~(,93_iU3~+C~')(2(R1i) ~2Rii) )cil;;~C~~~
~ Cc ~ ~`~Ri~~ L ,~R3~~ R32)) W~1 + lRJl)~J + CC)) [ R~nR3')c~a
2C( ' (R3" T Rs'' Ca' R~~' ~,o I R; Z R~' (i)~ r
)Jf ~ ~ ~ ~ C~ (2.28)
i, 'r Rsu) Wo f 2Cc') Cc ) (Rs" ~R1i)) Wo +
}R~')~,o2Cc))]}R32) (R3l)~o+ CCP )C`i)~CB(1) ;
 + R1 1 ~ ~R~u W~)2 [ �,4~ "Ra2'Wo ~c ~ ~R3'' 4I~i ) W~ = 0.
_ ne observance between the TMG's parameters of the relationships satisfying condi
aons (2.27) an@ (2.28) makes it possible, during ooeration in the resonance tuning
mode, to practically elininate the error caused 'ay angular vibfation of the base.
. The ar,alogous conditions for the case where ~12 = 7/2 have the form
R(Ii) (R(3i)(A) ~ ~ ?~8~~ ~R~2)6'~ 2~c)) T'
CB) (R3i) + R3:>) CO)C8) CIR31) R3'=) .
(2.29)
? R~u (Rju~,J+,~dB9) R~ii ~~i ~
(,,R~n (R3:) WO ~'C c')+l ; R33"~o+~~S"+R3~')C~'ICc'}Ce+ 12
_ + R3>>W0 l3R~u (R~')~~ ~ )~C~~)  R~')Rs2)~o
~1) ~ r ci~ ^ (2.30)
 ~ R3 2 IZa` ).C~c ~ C(e1) ~(Ra~~~ w~i)"�~ ~Ra' c�~Ri wo ~
(2Rj') + R3P) CC! _ 0.
T_f the realization of the dynamic tuning mode is achieved by fulfillment of rela
tionsnips (1.131) and (1.132) among the instrument's parameters, an additional con
ditior. that makes it possible to achieve a substantial raduction in the effect on
its accuracy of angular vibrations at frequency 2wo is
R31, _ R31>
(2.31)
Fulfillment of this condition entails some structural complications, but in princi
o1e it is possible.
Thus, for modulation RVG's having two oz more rotors, when the signal is read in a
 nonrotating system o� coordinates on the zero carrier frequency, it is possible to
select the parameters such that in t:e resonance tuning mode, a significant reduc
*_ion ot the instrument's sensitivity to angular vibrations of the base with a fre
_ quen cv oi 2wo is insured.
r^or an :�L4G witn signal reading en a frequency equal to the doubied frequency of ro
*_ation oL the rotor 2wo, angular vibrations of the base in the iastrument's plane of
 sensitivity on this same frequency a1'. result in a signal t.zat is equivalent *_o a
signal caused, by the base's constant angular velocity. Actually, supoose that the
base's a:.qular vibrations are described by expression (2.1).
'I'her;, ailowing for the de:aodulator, the slowly changing component of the signal at
the i:.strument's outlet has the fonn
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W (P  i2w,)  (p  i2~,) (W~ (P  i2~u) W 1_ (P  2iwu) t
It~ _ F lY/; (P  i_m~~) 1~i4 (P  i2o),,)1 (2.32)
(P  i 2wu) IV (p  ,?wa) G)A'
. In connection with this, signal (2.1) from the base's angular vibrations will be to
tally identical to the signal from some slowly changing angular velocity Zye, which
is determined by the expression
W (p  i:w,)  (P  i2wa) (W t (p  i:wu) lV,: (P  i2w�) I
w� + Wt (Pi2wo) IV� (P i2wu)) W (2.33)
(p  i2~u) I~~~ ~P) W ~P  ~~o) W i ~ ~P) W i~ ~P  ~~~~)I ~
_ In order for there to be no error it is.necessary that in the instrument there be
_ realized that relationship among the basic parameters so that the right side of
(2.33) becomes zezo.
As an example, let us examine in some detail several specific setups of MRG's witn
signal reading on frequency 2wo. P,n OMG's error when there are angular vibrations
of the base is~found from the relationship
YY)vc+(~ xZ)v'B 
~ + "Y)"cFxz)vsO)A. (2.34)
This error can be reduced only by reducing the values of 1:4y and 1 itZ. The
closer to each other the values along the suspension axes are, the greater the ad
' duced instrument error, which is e:cplained by the zeduction in their transfer factor
wi th respect to the useful signal.
�or a DMG, the suspension of each of the rings of which has only one degree of free
dom in the plane of rotation, the expression determining the equivalent velocity we
,,,hen there are angular vibrations of the base at frequency 2wp has the fonn
2 R~ (vg I xY) ; v~~l~cy) ve~lxZn)
ZR .a� (2.35)
ry ~Zn ~vdIx}�~v~+xY) vB+'xZn)
_ In contrast to a DMG with sic,nal reading on the zero frequency, in this case the er
ror does r.o t depend on the angle ~�12 between the suspension axes in the plane of ro
ta*_ion. A joint exami.nation of the condition of low instrtunent sensitivity with re
_ snect to the base's constant angular vibrations (2.1) (ge ti 0) and dynamic tuning
ccndition (1.66) shows that the simultaneous fulfillment of these conditions is pos
;i':1e only when a nonelastic suspensi.on is realized for one of the rings (CB = 0 or
C. = 0).
 ~~:en the signal is read in a rotating system of coordinates on frequency,wo, two
cases are possible, depending on the type of reference function used in the demvdu
3tor. In one of them, where the reference function has the form elWot, the ex
=r?ssion for the instrument error for angular vibrations of the base on frequency
_ 2~jo coincides with expression (2.3). In the second case, where the reference func
=:on ei'Ot is used, che instn:ment error will be deterstined by expression (2,33).
Startin5 wi*_h the generalized mathematical model of an RVG, in a manner analogous to
_ the oreviousll discussed cases it is easy to derive expressions for the computation
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of modulation RVG errors caused by angular vibrations of the base at frequency 2wo
when the signal is read from the intermediate suspension rings in any measuring sys
tem of coordinates.
2.2. Reaction of Rotor Vibration Gyroscopes to Disturbing Moments
No actually existing instrument can be :aanufactured with ideal accuracy. During the
building of instrument assemblies and the assembly and regulation of the instrument
as a whole, some deviations from the given rated parameters are usually allowed.
Sucn deviations also apoear when the instrument is used. One of the consequences of
this can be the aopearance of disturbing moments that affect a gyroscope and result
in additional errors.
_ Let us investigate the errors caused in modulation RVG's by the effect on them of
_ the mechanical part of disturbing moments. From the generalized mathematical model
of ari RVG (1.20) it follows that when disturbing moments are acting on a gyroscope,
the motion in inertial space of the first stage's (for example) rotor is described
by tht~ e:cpression
+ W13(P) ,Yl~~>�e~=~.t+e~~v~,,1f;,~~~erW.r~+ (2.36)
 i W 1: (P) N) + iV1 ~21'eiw~l) /l4 ~;~�~12)~g'i2wo[ + (2)~g'iwll)]
From (2.36) it is obvious that the creation ot an MRG that is totally invariant when
acted upon by disturbing moments is impossible in principle, since the requirement
that all the terms in the nuraerator on the right side of the equation equal zero
automatically results in a requirement that its denominator also equals zero.
Let us discuss, in sequence, the errors caused in different types of MRG setups by
the effect of different types of disturbing moments. Let the firststage rotor be
subjected to the e=fect of a constant or slowly changing moment M(1). The reason
for the appearance of this moment can be displacement of tre rotor's center of mass,
relative to the suspension axes, along the axis of :otation and the presence of con
stant or slowly changing linear accelerations of the base, the vector of which lies
in a plane perpendicular to the rotor's axis of rotation. In this case, in the non
rotating system of coordinates the rotor can be deflected through some angle and
gerror.n oscillations with a=requency of 2wo around this angle. '+Jhen the signal is
read in the nonrotating system o� coordinates on the zero frequency, the instru
:nent's error will be 3eter,nined by the expression
IrL , ru1 (2.37)
It is obvioLS :.nat in this case, as i.r, the case when angular vibrations of the base
act on an instrunent, it is convenient to represent the MRG's error in the fo r.n oi
an equiv.slent angular velocity, the signal. from which is completely identical to the
signal irom an acting disturbing moment:
PWtl (P) ~
~ W(P)(W.~(P)x~':(P) ~(P)Wi.;P)IP (...38)
For an OMG witc a single degree of freedom in the ?lane of rotation, the eYaression
for the equivalent angular veiocity in the resonance tuning mode nas the fo rn
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i
We f Y + XY) 0)0 M (2.39)
The larger the rotor's moment of inertia relative to the axis of the torsion bars
and the higher the rotor's frequency of rotation, the smaller this error will be.
In the case of two deqrees of freedom in the OMG, its error when affected by moment
g(l) will be found from the expression
[L,2 y e [(I x~,)iEgJ }IZ ~ v~(I xZ) }i;~
WC iti1
 
~y/ Z (i 2 ~l xy) v~ i 2 ~I yZ) v"g  (2.40)
i(1;c`) 2 ~Y (1 ~xy,) ~"'~z)u
Thus, the OMG error caused by disturbing moment MM cannot be eliminated. It can
be reduced by the proper selection of the instrument's parameters and by reducing
the disturbing moment itself.
The error in a DMG in which the suspension of each of the rings has one degree of
treedam is described by the expression '
t v8} ~Y"N vC 2 Rf' ~l xy) (I +COS~~{'1�:~
v' (I Tzzit)+ vC+ xY) +Va !zn + (2.41)
R~ (S~Cy3)  D_ R'lYiRt (1 xzn)
In order to build an instrument that is not sensitive to mvment M(1) it is necessary
that its parameters have values for which expression (2.41) becomes zero. The con
= dition of equality of the numerator of (2.41) to zero is fulfilled for the following
relationship between the rigidities of the rings' suspension:
Cc = (f2e ; ki) Lh;  C2. (2.42)
It is not difficult to see that equality (2.42) is satisfied for the rigidity values
dete rnined from expressions (2.10), and in coruiection with this the denominator of
(2.41) also becomes zero; that is, the instrument operates in a pseudoresonance
mode. In order to find the error in this case, we should take into consideration
the viscous friction acting on the suspension axes. The expression for equivalent
angular velocity we then takes on the form
e
~ v�*a Znr
we 7rH1z�W0 R (2.43)
~y ( I  %~Ili 3 IZn >
Thus, wnen operating in aoseudoresonance mode, a DMG realized according to the dia
gram snown in rigure 12, with a rotor that is acted upon by constant disturbing mo
ment K(1) nas an error determined by expression (2.43).
Zf the vR's suspension has two degrees of freedom (C~2) m) and ?12 = 0, the condi
tion cf nonsensitivity of t.he instrument to disturbing moment M( 1 has the form
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CB" I? (R3 + RI) WJ T LC~ ~ Ca)~ 2RI~o (2R3(Jp CC
(2.44)
+c'B' (2r3W~+c'') o.
 It is also fulfilled when conditions (2.16) and (2.19) are observed, since they make
' it possible to reduce the instrument's sensitivity to angular vibrations with a fre
quency of 2wo. If the VR's suspension from the PD's shaft has two degrees of free
dome and ~12 = 0, the nontrivial conditions for DMG invariance to the effect of mo
_ ment H(1~ in t.he resonance tuning mode is written in the following form:
. C~) _ �Ce1) (2Ri f Ra) � R, , / R,c) ~CBI
(2.45)
2Ric~~ CB
R,W~~~ ) 1 (R, R,) c~;~'B ) ! c(2)c(2) 2R ~'R
CB2 c a~' ~ o ~ ~ (2.46)
(R3 + 4Ri) + 2 (C~ ) CB')~ .
These conditions turn out to be incompatible with the conditions of low sensitivity
to angular vibrations of the base with a::requency of 2w0. Therefore, and depending
on the instruments' ooerating conditions, their parameters must satisfy either rela
tionships (2.21) and (2.22) or relationships (2.45) and (2.46).
For a DMG in which each rotor has two deqrees of freedom,� the condition for the ab
sence of error caused by disturbing :noment M(1) is determined by the expression
h'3wo (Cc ~Ce' CC')C'~~')) r 2R,t~~ (CiC�,  ACi AC2 cos (Pi:) I
4Riwo (CIC2; + C~ )C'~') ~Ra~~ R~W~ (Ci f e~.) ~ (2. 47)
r C2C''Ca' C,C~''C'' = 0.
When each of the rotars has a unifornly rigid suspension, this condition is ful
filled when the condition of dynamic tuning of the instrument is.
Let us examine a TMG in which the suspension of each of the rotors has only one de
gree of freedom. The condition for nonsensitivity of sucn an instnunent to constant
disturbing moment M(1) has the form
~(R311 ~ R~1,) wo L Ci) + Rii~ Cos 2(Qi�:)) 40
2 +
L~
Ci  ~C~ Cos 2cpiz CB'~l _ 2 ((RiI )a,o AC.) (2.48)
1
f 2 (R(I "Uiul r ACi) (RiucL1o f CI) cos 2(Pr: + (Riuwo f Ci)21 = 0.
Fron (2.48) it is possible to derive the dependence of the rigidity of the first ro
r.or's suspension to the PD's snaft on the rigidities of the suspensions of the other
two rotor;, whichwhen the suspension axes axe orthogonalis de5cribed by the fol
lawing expressions:
LOr ~12  0, (
Ri") CiI 2 Rj(2) +2RI 1))+C8
.,Ri1)~,=Cii~ :
)  _ 2 i o e ~ (2.49)
Ce (R.sI , + R1, 1 W+s ; C I
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for ~12  7/2'
[(R3'+ R~l))c~oF'C i) ( 2 R32)(+~J'~Cc1))  2 C~) z
(2.50)
Ce (R31 1) + Ri(,)i; + Ci
The joint realization of relationships (2.49) or (2.50) and (1.62) makes it possible
to build an instnmment that is not sensitive to disturbing moment M(1) when operat
ing in the dynam.ic tuning mode. For the simultaneous fulfillment of the condition
for Iow sensitivity to angular vibrations of the base on frequency 2wo, relationship
(2.26) or (2.28) must be considered jointly with these expressions. In this case we
have a system of three nonlinear equations relative to the three rigidities of the
the RVG's rotors' suspensions. The selection of the optimum instnunent parameters,
with due consideration for the mentioned conditions, can be done with the help of a
digital computer.
Let us discuss the MRG errors caused by a constant disturbing moment M(1) when the
useful signal is read on frequency 2wo in a nonrotati.ng system of coordinates. A1
lowing for the signal demodulator at the instrument's outlet, in this case it will
have the form
ul = I~1:, (v  i2wu) ~~1~1)', (2. 51}
Ir' (P  i2wu)
wnile the equivalent angular velocity is deterntined by the expression
 C _ _ t1",:~ (r  i?wn ) ~t1 ~ i ~ � . (2.52)
Cr (P) W i~ (p  i2o>u) l lG't' (P) Wi, (P i A)u)
=oIr a DMG with a single degree of freedom, in the resonance tuning mode this expres
 si.on talces on the form
� ` (2.53)
F x} rVt
~z  I~� (1
'T^.zis relationship coincides with expression (2.39); that is, the method used to read
the use=ui signal has no effect on the error of an OMG with a single degree of free
aome that is caused by moment M(1).
The error in a DMG in which the suspension of each rotor has a sinqle degree of
freedom can be found from the relationship
2 I R1 + i v, (
_ < >n Cc1S wnt 
 2lzr.~~n sin w�l + 2(17r
1.rr  I I.(,) c1),w7,) SIn (i),l 
 2(Izc 'r l.tr,  I}r) W�c,) ti �COS cnd (ntP.ux.t  ntPi(lxi) 1 v y
A cos ro�! + frrplaxi) Iz oIn w�t:
 17Cixn i" f~Cfln 1' [(1 Xl:  IyC) (nn + Cc 2Cn I GLn lznfZr +
+ Itn6C f W.Yn  l1'n) ~~n r Ciipi aC = 217.n67.uCOSO)nl 
 2IZ,,6)1�nSlil (,),I + 2 (17.n t /Xn cu~co~,51n (di~l 
2 (lz,, l.rn  II n) (j)�cotio COS o001 (nrP2clr,. nrpi(l,rf) ;
jy Cos Wnl + (ntP:Rxa I ntr, iaXt) 1z s Ift
where
lzr. _ �zx I 17i . 1Iti~x /}�i .
~ ir, = l ,
2
~ /.rs /xf
xc = 2 +
2
102
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~.rn = f 'r" ~ _ Ct f Ct.
. 2 2
Figure 23. Structural diagram of a composite RVG with common sus
pension of the rotors.
On the basis of equations (3.21) and allowing for the presence of a phasesensitive
demodulator at the instrument's outlet, the structural diagram for the signals' en
velopes has the form presented in Figure 23. Tn connection with this, we introduce
the following definitions:
Kii  2 (Izc + lxc  It�c) wn:
K21 " 2 (lZn + IXn  lYrt),,)n; �
K221  nt.zaxz F m,ia.ei; K,Z = mpzaxz  mplaxl;
W, i (P) _ JzcP7 + ItCp + [(I.rC  jjC) wn Cc 2C. 1;
W ~t ~P)'= 117iz ~P) Iz.P2 ~F �.P [(IXn  IY'n) ~o Cnp];
Wz: (P) _ IzcP2 f ftcP + f (I,ec  l rr,) G)n + CC l; (3.22)
_ Wo (P) _ {lzcP2 + F(cP + ( (Ixc  lrc) Wo + CcII 117cP2}
ltcP F [(Ixc  lrC) 0)2
0 CC 2Gn ~ ~ 
 ~ I7.nP2 4' �nP + I (I.Cn  IYn) (,)n C"p 1f 2.
In accordance with the structural diagram, the signals at the instrument's outlet
are described by the following expressions:  ac  ~ 11 Wo (G Kii117ii (p f K21W12 (n w
J  i~do)
[ K12Wii (P i(0n) t' I`,i_Ibtz(p  iwn)
(3.23)
4I K::W:z (P  icn,,) Ki:Wxi 1I
a"  Wo (P I tu,o)
+(h:iW:z (P  iu,o) I k 1tW:i (I)  iwn)I @1�
Equations (3.22) and (3.23) make it possible to derive quite easily the expressions
for equivaient angular velocity and linear acceleration when there is an input value
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~~n = F'' 2 }tC�P  CZ 2 Cl.
IVn = ' 2 /}~1 , IZn  ~72 ~ ~ ;
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in the other measuring channel:
Ki:Wii (n  ia,o) Kx_Wi, (P  r,uo)
Ktilt''11 (p  iwn) 1 K21W1: (P i(')o)
KsiWxx (P  ioj + KtIWsi (P  ic)�) I
(J.
KxzWs: (p  i(0o) I x1:Wz] (P  iwo) ~
(3.24)
 Assuming, as before, that the instrument's parameters can differ from the nominal
oneswhich are identical for botiz interrelated oscillatory systemsby only small
amounts that are determined by the variations in these parameters, for the area of
essential frequencies we obtain the following expressions for we and je in the first
_ approximation:
X
p  i ( I 
~
_ X Omr
2 1 w; ( mp
x+ v' + Cn 2 VI92 , coo ~w� y~ x{ Vx Cn 1
/zw,~
X
na,r ~,Z ` ~ 2 r/ e�~ nu.r ait.r Z
a ~ / ) + cu, L l m  ~
r ~ ax tt ~
 i ( Llmr~' ~ r ~iZ 1 1 p + Ov.  (I x) ~7 L
~ z
(3.25)
' mm Ac nr
x+ V2 C', r x ~ :
~ 17Wj ~ ( InP ~y Rr ) C'~ Cn
c1� 2 2 ,1nr i'
rn~, f n
.r
where v= (C + CTr)/YZwO is _he ratio of the frequency of one rotor's natural oscil
 lation^ to angular velocity wo when w0 = 0 and the second rotor has lost a degree of
freedom relative to the torsion bars' axis;
p C" l~ I  5= I tno i ~mu
\ ~Z(JJ ~
7C 1 V1  v n
/Zw,l
;i 2 ) r=  < < ~ (AX + (2 + ~ r~ ) P
A x I  x L. V'  c"1, 1 f C~X (I x) e/, ~ +
i, ~ ~ )
+ vl. AC _ l~'l c~x E (I L x~ :117
C Cn I t / Z
(3.26)
From expressions (3.25) and (3.26) it follows that, as was the case for the first
setup, the greatest influence on the level of the crosscouplings between the meas
uring channels is exerted by the nonidentical natural of the rotors with respect to
parameter 'A. In contrast to the first setup, howeve�r, the level of the cross
couplings depends on which of the measuring channels is used to realize resonance
tuning. If resonance tuning takes place along the angular velocity measurement
104
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m aX
/Z (I x) X
I
/7. 0 + K)
mnax
(02
1)
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c::annel, tne crosscoupling `rom the first channel to the second 'nas a high 1eve1,
and vice versa. The level of the crosscoupling from the channel not having reso
nance tt:ning to the other is detenniaed by the magnitude of this channel's detuning
from resonance, which is regulated by the intermediate torsion bar's rigidity Ci.
Thus, by selecting the appropriate value of Ci, it is nossible to change the chan
nels' dynamic characteristics, their measurement range and the level of the cross
couplings.
From the expressions for the transfer functions of an OMG (1.86) and a KOMG it fol
lows that during the creation of a highly sensitive instrument (when resonance tun
ing is realized and the relative damaing factor C is extremely small), its time con
stant Tg can be quite large. P,s a result of this, in a KOMG there can arise large
errors in the detezznination of angular velocities and linear accelerations. These
errors are especially large when the instrument is operating in an open system (for
example, the acceleration measurement cnar:nel's dynamic errors for an inertial navi
gation system and so on). Taking (1.81) into consideration, the expression for the
acceleration measurement channel's d,mar.iic error can be written, with a sufficient
degree of accuracy, in the form
X,L' l~ Ts ~  ~ ) I , (3.27)
~l~r
P`~l
where
~rn,a.r
11~ = i , ,
V 17s1 ( Two I)
L1 = operator of an inverse Laplace transform.
Suppose that the 1aw governing the change in acceleration can be approximated by the
first n terms of the exponential series
j = an + Q,t t... an1rt.
The expression for the dynamic error then taices on the for*.n
(3.2i3)
7'2 Q,~p" i QiP"' _4 nl an 1
;~d(n) _ tt; T~ i +t= p� c3.29~
By breaking the right side of (3.29) dourn into elementary fractions and applying *he
inverse Laplare transform to both sides, we obtain an expression for an instrument,s
dynamic error siqnal when acceleration (3.28) is acting on it:
!
ud(t)   K, D,T e r' Sin 7~,
�T (3.30)
+ B. sln r_ cos T:
The unknown coefficients Ai and Bi can 5e found from the following equalities:
pRP BZ ~ P" Re B, r 1
+ (P""' Re A, iP`2 Re ._4~ . . . + (it  I Re .a,, 1 ,
r P` + P 2
+ I \ _
~ T r= ~
!
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(T:P f �z) E P" ` R e a, E. p""' I m a; J;
i,0 , =J
p"`1 I m.32 T P" 1 m B, +
f [P""I I m A, + p"I I m A~ + . (n  I I m A� )
x ( p 2 p L+F`
T T;
~
n n
_(T:l~ !tx) u Pni
Im a; p,~' Re a
; J) ~
i=0
(3.31)
By equating in (3.31) the coefficients for identical powers of p in the left and
 right sides, we obtain a system of 2(n + 2) linear algebraic equations in order to
find the 2( n+ 2) unknowns ReAi, IatAi, Regi and Imgi,
A7hen the instrument is acted upon by a discontinuous disturbance and the measuring
_ channel is tuned into the resonance mode, the dynamic error will be described by the
expression
r
u(1) irnpa.r 2~ a� (Cos 2t T i sin 25 T.. ~ e T J 1
~
~  T � (3.32)
intraX 2y a�r~ q.
_M Mus, the iarger the value of Tg, the qreater the instrument's dynam?c erzor. At
the same time, as was shown in Section 1.6, the higher the instrument's sensitivity
(the smaller y is), the greater the value of Tg, Therefore, the realization of an
i.zstrument that has both high sensitivity and small dynamic errors is fundamentallp
impossible within the framework of the layout we are discussing.
In addition to the specific errors we have discussed, a KOMG also has all the errors
that are typical of an OMG (see Section 2.4). In this case the errors related to
3.*igular motion of the instrument's base and synchronous interference are doubled in
_ the velecity measurement channel. In the accelezation measurement channel they are
_ 4 eter.nined ~y the nonidentity of the oscillatory systems' parameters. 'rhus, the
iinear acceleration measurement channel's errors are much smaller than those of the
a.n,ular acceleration measurement channel. Thanks to *_his, the instrument's second
channel's sensitivity threshold is lower and the crosscouplinqs from the seco:.d
=hannel to the first can be reduced. The second channel's dynamic errors can be re
iuced by introducing detuning from the resonance mode.
3�3� Synthesizing the Parameters of an Output Filter for Composite Rotor Vibrata.on
Gyroscopes
=n 3ection 3.2 it was shown that composite :4RG's have significan t dynamic errors
that cannot be reduced without lowering the instrument's sensitivity. These errors
zave the greatest effect on the instrument's functioning in the linear acceleration
measurament channel if the signals from the channel are used for an inertial naviga
_ ~ion system. Integration of these signals results in inadmissibly larqe errors in
deter.ni.ning velocity and the path that has been covered. Into the channel's input
*_here also enter disturbances related to angular and linear oscillations of the
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base, which are a source of additional errors. Besides this, both the instrument
itself and the signal extraction and processing devices have internal noise, on
which the instrument's sensitivity threshold depends to a considerable extent.
Thus, there arises the Problem of optimum signal processing at the linear accelera
tion measurement channel's outlet for the purpose of obtaining the most reliable in
formation about the linear accelerations acting on the object, with due considera
tion for the factors listed above. Sucn processing can be carried out with the help
of a special filter installed at the instrument's output. Let us deternune the pa
rameters of this filter.
fn (t) Ifn (t)
J Xj
Figure 24. Diagram of formation of composite RVG errors.
r^igure 24 is a structural diagram of the formation of instrument errors in the ac
celeration measurement channel. Let us introduce the following definitions:
fT(t) = the disturbance caused by angular and linear oscil.lations of the base, re
duced to an equivalent linear acceleration; fn(t) = the instrument's internal
noises, as heard at its outlet; Up) = transfer function of the filter at the in
strument's outlet. Disturbance T,(t) is of a random nature and depends on the char
acteristics of the specific object in which the instrument is installed, as well as
the conditions of its motion. Therefore, we will ignore this disturbance in our
further discussion. An instrument's interr.al noises are also of a random nature and
are usually approximated by "wnite noise." we will also assume fn(t) to be "white
noise" with the same power B2 at both channel outputs.
We will deternune transfer function 4(p) on the basis of the minimum meansquare er
ror in the measurement of 'the linear acceleration's absolute value and the given
limitation on the dispersion of the r.oise at the instrument's outlet. This means
that it is necessary to bring to a minimum the functional
~ .
 ~ = J J�d(r)JZ dt ~ k2Du (1). (3.33)
u
where k= an indeter.ninate Lagrange multiplier; Du(t) = dispersion of the modulus of
the linear acceleration measurement charlnel`s output signal.
An instrt:ment has the greatest dynamic errors when operating in the resonance tuning
mode. in connection with this, the crosscoupli.Zgs between the channels can be ig
nored with a sufficient degree of accurac1, while the instrument's transfer function
is written in the form of (1.86). Ia this case the problem reduces to the synthesis
o. a filter that is optimum, in the sense of criterion (3.33), for each linear ac
celeration measurement channel. We will use the method of syn thesis in a frecuenc1
area, the :nasic ideas of which are e:crlained in [41]. According to (Parseval"s)
theore.n, in the area of images the expression for functional (3.33) can be wri`ten
allowing fcr the asst:.nptions we have madein the following forn:
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rurc urHUAi, USE UNLY
r~
[41~,~(1~) u �
2ni J d (P) kz&(n (P) (1) (P)I dp,
r w
(3.34)
where ud(p) = error signal at one of the linear acceleration measurement channel's
outputs.
Using the structural diaqram presented in Figure 24 and introducing error factars Ci
(23], we will rewrite expressions (3.34) ; substituting the dynamic error's value in
terms of the acting disturbances and the instnunent's parameters:
1= 2i j I Tk3 Tkl ~ Tk2� In connection with this it can be assumed that the
amplification factor increases by a factor of Tti3/Tk4 while the open system's LA1Cz's
cutoff frequency remains unchanged. The required value of the open system's ampli
_ =ication factor also determines the relationship between the time constants Tk3 and
Tk4. Time constant Tk3 should be selected in such a fasnion that relationsnip
Tk3 Tkl is satisfied. In this case the tec;uiique for selecting time constants
'Ik1 and Tk2 and allowing for the effect of the stabilizing engine's time constant TP
is analogous to the one described above. Let us mention, however, that the given
system will nave two sta.bility reserves with respect to amplitude; tnat is, it is
provisionally stable.
As follows from an analysis of the equations of motion, the dynamic characteristics
of a DMG differ substantially from the characteristics of an OMG. Let us discuss an
OSP based on the use of one of the most highly developed types of RVG: a DMG with
single modulation by motor PD2 and signal readiag in a nonrotating system of coord
inates on the zero trequency (a iCnaui gyroscope). we wi11 make use of its transfer
functions, as approximated in the band of essential frequencies by expressions
(1.109). An oscillatory component witn time constant Tn describes the p=ecession
motion and deter,nines the gyroscope's deflection in inertial space. In connection
with the discussion of a system's stability, this oscillatory component and the com
ponent witn the introduction of a derivative in the area of its frequencl cutoff can
be aporoximated by the expression (T/T2)/(1/p). An oscillatory component with time
constan t Tfi describes the gyroscope's nutational oscillations and affects the sys
tem's dynamic characteristics.
One of the simplest correcting circuits in a stabilization circuit, which ;nakes it
possible to increase the regulator's amplification factor K., is a norunini:nalFnase,
seriesconnected circuit with a trans�er functicn in the form
I l T,,:n = I r~~0  I
I wo) or less
(wH