MODERN PROBLEMS IN THE TECHNOLOGY OF ASSEMBLY IN INSTRUMENT MAKING

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R002300210011-7
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 145
• Document Creation Date: December 23, 2016
• Document Release Date: September 20, 2013
• Sequence Number: 11
• Publication Date: January 1, 1956
• Content Type: REPORT

Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
STAT
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
???
Ministry of Higher EduCation or the USSR
Moscow Aviation Technological Institute
MODERN PROBLEMS IN THE TECHNOLOGY OF ASSEMBLY
IN INSTRUIENT MAKING
Editor
V. P. Chumakov, Cant. Tech. Sc., Lecturer
Transactions of the Institute
No. 27
State Publishing House for Defense Industries
Moscow 1956
STAT
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
R
STAT
Next 1 Page(s) In Document Denied
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
gr.
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Ministry of Higher EduCation of the USSR
Moscow Aviation Technological Institute
MODERN PROBLEMS IN THE TECHNOLOGY OF ASSEMBLY
IN INSTRUMENT MAKING
Editor
V. P. Chumakov, Cand. Tech. Sci., Lecturer
Transactions of the Institute
No. 27
State Publishing House for Defense Industries
Moscow 1956
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
STAT
?
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
PREFACE
The fulfillment of the goals set by the Communist Party of the Soviet Union for
the Soviet Instrument Builders requires further development of the technology of in-
strument building and specifically of the technology of assembly, regulation and in-
spection of instruments.
The most laborious and complicated processes in the production of modern instru-
ments are assembly and regulation or control. Assembly and testing-control operations,
according to the data of a number of instrument-building plants, represent more than
50 to 60% of the total labor in the production of the instruments. There are many
different assembly operations in instrument building, which is explained by the vari-
ety of operating principles employed in modern instruments. Often one instrument in-
corporates mechanical, electrical, pneumatic, and other elements. Many subassemblies
of the instruments must be selected so as to satisfy the requirements of interchange-
ability not only with respect to geometric parameters, but also with respect to a
number of physical characteristics. The requirement of interchangeability of sub-
assemblies with respect to elastic, magnetic, electrical, and other characteristics
could be mentioned as an example.
At present, the demands on the technology of production of instruments are con-
siderably increased. The technological processes of assembly, regulation, and control
of the instruments should yield a high degree of accuracy and reliability of opera-
tion under difficult operating conditions; at the same time, these processes should
in many cases be planned and designed for large series production, assembly and in-
STAT
Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2013/09/20 CIA RDPFli-ninaqpnno4,4
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
spection of instruments for continuous flow, on assembly lines.
One of the main directions of work of the scientific-research work of the chair
of ?Technology of Aircraft Instrument Making? at the Moscow Aviation Technology In-
stitute ( NATI ) is the study of questions of accuracy and production capacity of
assembly and control processes in instrument making.
A number of investigations performed by members of the scientific staff of the
institute in the foremost instrument plants was reported during the conference on
the technology of instrument assembly, organized by the chair of Technology of Air-
craft Instrument Making of NATI in 1954, in collaboration with representatives of
industry and of Moscow Institutes.
During the conference it was resolved to publish the most pertinent reports and
investigations about the assembly of instruments. That resolution is in some measure
being fulfilled by the release of this collection.
Included in the articles of the collection are the results of investigations,
theoretiGal relationships, announcements, classifications, descriptions of original
designs, and other material which could be useful to engineering-technical and sci-
entific employees of plants and Institutes.
The Chair will be obliged to all readers who will submit to the Institute their
remarks and wishes regarding the articles in this collection.
V. P. CHUMAKOV
candidate in technical science,
Lecturer
Head, Chair of Technology of Aircraft
Instrument Making
STAT
11
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20 ? CIA-RDP81-01041Rnn9gnn9i nnil 7
Declassified in Part - Sanitized Cop Approved for Release
50-Yr 2013/09/20 : CIA-RDP81-01043R002300210011-7
PROD OF ACCURACY OF TECHNOLOGICAL PROCESSES
IN THE WINDING OF ELECTRIC SUBASSEMBLIES
DI
by
V. P. Chumakov, Cand. Tech. Sci., Docent
Many modern instruments and automatiC devices are based on electric or electro-
mechanical principles of operation. Windings of various types find numerous applica-
tions in instrument making.
Subassemblies of current-carrying coils of wires are called windings. Windings
are used in coils, potentiometers, rotors, stators, and other electric cubassem-
blies.
Almost all instrument-making plants concern themselves in one way or another
with the production of windings. The technology of production of windings has its
own peculiarities and is only little explored. The winding process itself is the
most important and most difficult one in the production of windings. The quality of
the windings produced, and consequently the quality of the instruments, depends on
the proper processing practices in winding operations.
The quality of the windings is characterized by the ohmic resistance, the cor-
rectness of the geometric shape and dimensions, high insulation strength and resis-
tance, and the number and arrangement of the coils. For certain windings, the fol-
lowing may be required: a predetermined magnetic flux, a certain space factor in the
slots, rate of change of ohmic resistance, and other parameters.
The requirements for accuracy of the ohmic resistance of the windings vary; for
3.
STAT
Declassified in Part - Sanitized Cop Approved for Release 50-Yr 2013/09/20 CIA_RnDszi_ninAnrn
Declassified in Part- Sanitized Co y Ap roved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
lexample',-the deviatidn-af the-WEI67t6didtAhoe
-Ispecific cases not more than 3 - 1%; therel.are potentiometer applications with non-
llinearities of not more than 1 - 0.5% and even of 0.1 - 0.01%,
The technological process-of winding should be so planned and carried out in
actual production as to guarantee that thci necessary quality of the entire series of
windings is obtained with minimum losses in time and resources.
With any technological process of wiriding it is practically impossible to ob-
tain truly accurate parameters of the windings. Because of detectable errors or de-
viations of dimensions and of electrical properties of the wire and tension forces
during winding, the parameter obtained is in all cases larger or smaller than that
given. Even under uniform conditions during the winding of two or more subassemb-
lies deviations of dimensions and of electric characteristics of the windings are
unavoidable. To obtain completely uniform characteristics of windings consistently
is practically :Impossible.
Deviations in the technological process used from ideal conditions of manufac-
ture of windings, under which the windings in the best case correspond to their
nominal values, cause errors in the windings.
The direct basic reasons causing the appearance of errors in winding are con-
sidered in this article. The possibility of establishing a quantitative relationship
between the production error and its cause is shown for several cases.
As a preliminary step we will briefly review the characteristics of systematic
and random errors occurring in the manufacture of windings.
An error which, within the limits of the problem under consideration, remains
constant or changes according to a law is called systematic. If, for example, the
counter for the turns being wound, which transmits a stop signal to the machine
after the winding of a given number of turns, has an instrument error of one revo-
lution in 200, and if this error is not taken into account in setting up the
machine, then all windings with a number of turns larger than 200 will be wound on
2
Declassified in Part - Sanitized Copy Ap?roved for Release ? 50 -Yr 2013/09/20: Cl
nn,v)nnn
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
the machine with not less than one excessive coil.
An error which, within the limits of the problem under consideration,has dif-
ferent values (Dibl. 1) so that the time of its appearance and its exact value for
each subassembly in the lot cannot previously be determined, is called a random er-
ror. Random errors are caused by the influence of factors whose regularity cannot be
previously determined or by the influence of a large number of errors, which, even
though subject to laws, enter into and disappear from the process at random.
Fig.1 - Two Schematic Representations of Probable Correct Arrange-
ment of Turns in a Winding
It would appear that the turns of wire in the winding are distributed correctly
and evenly (Fig.1). Actually the turns arrange themselves with various, hardly de-
tectable, deviations from the correct arrangement.
As shown in Fig. 2 the turns (1) and (2) of the second layer are located be-
tween the turns of the first layer, while the turns (3) and (4) are lying on top of
the turns of the first layer. This could occur as the result of random factors:
nenuniformity of the wire diameter, deformation of the wire, varying wire tension
during the winding process, and other factors. The turns (3) and (4) are located
higher than the turns (1) and (2) by the value Ar, and their perimeters are differ-
ent. The turn (5) extends slightly above the plane of the second layer of coils;
this may have been due to an elastic deflection of the wire, to insufficient tension
in winding, to adherence of dust particles to the lacquer insulation of the wire,
and to other reasons, For the same reasons, and possibly due to a deviation of the
3
Declassified in Part - Sanitized Copy Approved for Release 50 -Yr 2013/
nr1,1,3nn
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
diameter an4 of the crc.,. s'dction of the wire or nonuniformity of the layer of the
insulation, thek-t*rn (7) protrudes from the first layer of turns, entraining the
turns (6) and (8) with which it is in contact. The coil (9) is raised since the gap
between the preceding turns and the end of the body is too narrow.
tt?
Fig.2 - Schematic Presentation of Possible Arrangement of
Turns in a Winding
b) Second layer
There can be an infinite number of combinations of various arrangements and de-
viations of turns. Deformations of the cable and insulation are of great influence;
of importance are also the angle of incidence of the lay of the turns, frictional
forces between the turns and also frictional forces between the turns and the body,
and other factors.
The transition to the second layer of turns during winding causes a change in
the direction of lay of the turns and an increase in the length of the turn (Fig.3).
A
in the
and in
change in the arrangement and in the density of lay of the turns is reflected
overall size of the winding, the length of the wire, the ohmic resistance,
other electric characteristics of the windings. Below, a typical example will
be given for demonstrating the influence of nonuniformity of the lay on single-
layer windings with a definite pitch.
The wiper of the potentiometer (Fig.4), after traveling past the first turn
STAT
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20 ? CIA-RDP81-01043R00230071 oni 1-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
_iimmediateiy contacts turn (4), bypassing turns (2) and (3). tiis-caus-es an error in
,the change of the ohmic resistance.
The position of the turns and their density of lay depends mainly on the method
of winding. It wae found that the density of lay of the turns in the slots of rotor
windings of small electric machines is slightly higher
if wound on winding machines with wire stays than on
winding machines without wire stay although the dif-
ference is insignificant. Rotors wound by hand usually
result in the highest density of lay of the turns.
Therefore, due to various reasons influencing the
Ex- quality of the windings during the process of winding,
Fig.3 - Position
of the
treme Turn at the Point of the windings obtained are not exactly uniform in terns
Transition to the Next Layer of resistance, geometric Oimensions, nonlinearity, and
other characteristics.
of Turns
Random errors occurring in different windings of
'
a series being produced depend on,the winding process and the combination of struc-
tural features of the windings; the method seie'cted and the operating conditions
during winding; nonuniformity of the diameter and of the mechanical properties of
the wire; the quality of the insulation; and many other factors.
Direct basic reasons causing errors in the manufacture of windings are sub-
divided into two groups:
Group I - Errors caused by the winding materials;
Group II - Errors inherent in the winding process.
Of the causes of Group I, the following have a great influence on the accuracy
of manufacture of windings:
1. Nonuniformity of diameter and of physical properties of the wire
used in windings;
2. Low quality of the wire insulation;
5
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
3. Inaccuracies due to manufacture of the bodies.
Reasons for errors caused by the winding process are:
1. Inaccuracies in the kinematic scheme adopted for obtaining the windings.
2. Geometric inaccuracy of equipment and apparatus.
3. Deformations of the wire, body, and machine parts.
4. Measurement inaccuracies during the winding process.
5. Inaccuracies in the construction of the winding machine.
ERRORS CAUSED BY TNE WINDING MATERIALS
1. Nonuniformity of Diameter and Physical Properties of the Wire.
The experience of the winding departments of instrument-making plants shows
that one of the basic conditions for obtaining windings which fully satisfy techni-
cal requirements and drawings in series
production, is accuracy of dimensions
and high quality of the wire. This is
confirmed on the example for determin-
ing the influence of the variation in
the parameters of the wire on its ohmic
resistance
?' - Touching of Turns of the Andinc
on Displacement of the Potenti-
ometer Wiper
a) Wiper; b) Wire; c) 11ody
where R is the ohmic resistance;
d is the wire diameter;
P is the resistivity;
/ is the length of the wire.
Declassified in Part - Sanitized Cop Approved for Release 50-Yr 2013/09/20: CIA_RnDszi_ninAnn
STAT
Declassified in Pad- Sanitized Cop Ap roved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
For purposes of calculating the absolute errorAn of the ohmic resistance we
disregard small quantities of higher orders; setting the incrementAR equal to
differential function of R, we obtain
to
AR= 4/ 6e+ 4p Ai _Sptd.
7cd= vd2 1.:d3
(2)
The value of the relative error for the ohmic resistance of the wire is equal
AR
+2
Ad
(3)
a) A considerable influence on the relative error of ohmic resistance of the
wire is exerted by nonuniformity of its diameter. This is of especially great im-
portance in the case of thin and very thin wire*.
The relative error of the ohmic resistance due only to nonuniformity of the
wire diameter is equal to
AIR 2Ad
(4)
Equation (4) indicates that the relative error of the ohmic resistance for
each rated wire dimension is different for the same absolute deviation of the di-
ameter.
For copper wire of d = 0.1 ? 0.005 mm (according to COST - Soviet State Stan-
dard - 2112-46)
AIR 2.0,005
R 0,1
Diameter variations of the wire within the limits of tolerance permitted by
COST may cause a c1ian7e in the ohmic resistance of ? lOs, i. e., simificantly more
than the tolerance permitted for the overall ohmic resistance of many windings.
Equation (4) is based on a substitution of the differential dR for the incre-
ment Art, and the error caused by this is of practical significance in calculating
AB
for thin and very thin wire. Nore accurate results are given by a formula
*Wire of a diameter up to 0.02 mm is referred to as very thin or microwire.
7
' Declassified in Part- Sanitized Copy Approved for Release @ 50-Yr 2013/09/20 ? C R Rl_ni no,)nnn nn A
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
whose use requires in every case that the direction of the diameter deviation from
the rated value is taken into account; in the case of a positive deviation, the
quantity Ad has to be taken with a plus sign and, in case of a negative deviation,
with a minus sign.
Ad
2 71-
AIR
Ad
1+2 --
d
In some cases, it is more convenient to use equation (5) in the arrangement
AIR == 9Ad
d+2A11 *
(5)
(5a)
For wires of a diameter of 0.05 ? 0.003 mm, the relative error of the ohmic
resistance computed according to equation (4) equals ? 12 4, then in determining the displacement of the various points of the core
it is necessary to take torsional deformation into account in addition to bending.
The angle of twist (in radians) of the cores (Fig.9) in any given cross section
23
Declassified in Part- Sanitized Copy Approved for Release @50-Yr 2013/09/20 ? CIA-RDP81-01041Rnn7Inn91nn11 7
STAT
ueciassified in Part - Sanitized Co
y Approved for Release
50-Yr 2013/09/20 : CIA-RDP81-01043R002300210011-7
perpendicular to the axis may be determined from the formula
where M is the torque in kg-cm. M = P ?
7"
G is the modulus of shear in kg/cm2;
IK is the geometric modulus of rigidity in torsion, in cm4.
where
The angle of twist of any given cross section of the core to the left of the
section CC' may be determined on the basis of similiarity of the triangles ACC' and
ADD' and, correspondingly for cross sections to the right of the section, from the
triangles BCC' and BEE'.
Fig. 8 - Diagram of Action of Forces during Winding onto a Flat Core
with Reinforced Ends. a) Section through HM
Declassified in Part - Sanitized Co .y Approved for Release
50-Yr 2013/09/2n ? ('IA ID run r,
Declassified in Part - Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
.During the action of a torque appliei at he centii of thiliourciii?Fig:81--
the twist of the center section is deterntned from the formula
4-1
?????-?
MI
cP"'"= 401K
U-
10.
-4
For determining the linear displacemrnt of points located at
the axis of the core we Use eq.(11)? obtaining
c.cpz= Mbizz
ICJIK
The linear displacement of the points on the fins of the core
making the substitution z
= ?
2
Mb;zt
C
P 2/G/K
(13)-
a distance z from
(14)
is
determined
(15)
For determining the total displacement (Fig.10) of the points of a given
section of the core, the bending determined according to eq.(7) or
1
be added to the linear displacement computed according to eqs.(14)
cross
eq.(8), has to
or (15)*.
The maximum total displacement depenlois on the ratios of the bending and twist-.
1,_J ing moments acting on the body. In orderl to determine the cross section at which
4
; the total deflection is at a maximum, it is in every case necessary to perform cal-
culations and to construct deformation diagrams for bending and twisting.
12_J
It should be mentioned that the deftmation of the cores obtained by means of
!I_ eqS.(7) - (15) are approximate, since the dynamic effects during winding and certain
(always possible) deviations of actual conditions in fastening of the parts from
!, theoretical ones are not considered in the calculation.
c) Deformation of Machine Parts. thider the action of forces during winding,
such deformations may occur as a consequence of insufficient rigidity of the parts
* For simplicity in the given case, we do not consider bending of the core along
its plane cross section.
STAT
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20 ? CIA-RDP81 0104:1Pnn9qnnol nr-111 7
Declassified in Part - Sanitized Co .y Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
and assemblies of the machine, and also dito- defori--ta
___tween_the_parts (contact deformations). Calculation of deformations of parts or
_
2
? a
rs
R DCE
Fig.9 - Diagram of the Angles of Twisti
of Cross Sections of Plane Cores
in Winding
Fig.10 - Total Displacement of the
Cross Section of a Plane Core due to
Bending and Twisting
assemblies of the machines may be performed according to formulas of the strength
, of materials and elastic theory.
t, ! In a winding machine, deformation of sleeves and of face plates used for
strengthening its frame, and bending of the spreader attachments was observed.
4. Inaccuracy of Measurements in the Process of Windinic
Inaccuracies in measurements of the parameters of the windings, creating the
wrong impression as to the quality of their production, may appear as a consequencei
of the fact that the testing instruments used during the production of the windings:
have inherent errors and that during the use of these instruments certain errors
, caused by subjective evaluation of the readings may appear.
The number of turns wound is read on a counter in which instrument errors are
, inherent. The number of divisions in counters with circular scales is insufficient'.
STAT
26
Declassified in Part- Sanitized Co.y Approved for Release ? 50-Yr 2013/09/20 ? CIA RDPRi-ninaqpnn-, rw-v14,1,1.14
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20 : CIA-RDP81-01043R002300210011-7
Many winding operations are not performed with methods which automatically re-
sult in the required parameter of the windings. For example, during the production
of certain types of windings, the worker stops the process of winding only after a
_., trial measurement of the wound assembly has indicated that the winding has the nec-
essary ohmic resistance. The accuracy of measurement of the ohmic resistance of the
winding depends on the class of accuracy of the control instruments and the method
of measurement.
The density and overall shape of the windings, and sometimes their ohmic resis-
tance, are largely determined by the tension of the wire during winding. As a rule,
the tension of the wire is determined "by eye, by experience, and by trial measure-
ment of the first windings in the series produced first piece inspection.
On certain winding. machines, special gram-meters for control of the tension of
the wire are attached. They have considerable instrument errors, and their readings
are being made difficult by strong fluctuations of the pointer, caused by inertia
loads and vibrations.
5. Inaccuracies in Design of the Machine
Work on tooled,-up machines is the most popular way of obtaining certain parame-
ters. The method of trial winding is only used in cases where the method of work
of the setup is, for some reason, not usable, for example, where the zone of scatter
of the ohmic resistance of the wire within the limits of the spool exceeds the per-
missible region of ohmic resistance of the winding.
Winding machines are set up simultaneously for several parameters: for pitch
or density of lay of the winding, for the number of turns, etc.
During setting up of winding machines it is necessary:
To regulate the tension mechanism ot the machine for a given wire;
To correctly select the speed of winding and set the gears of the machine for
that speed. Winding speeds which are too high cause rupture of the wire or unaccep-
Declassified in Part - Sanitized Copy Approved for Release
27
50-Yr 2013/09/20
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
tably excessive elongations of the wire. Low speeds affect the productivity of the
machine. The speed depends upon the type and diameter of the wire and upon the
shape and dimensions of the core;
To regulate the feed of the wire along the axis of the core for each turn of
the spindle of the machine;
To regulate the starting and braking mechanisms of the machine;
To regulate the interaction of auxiliary mechanisms of the machine, which turn
the part at the pitch for winding the following sections, which eject the loop for ,
soldering of the wire to the collector and which serve other functions.
Despite preparatory setup of the machine the operator is required to regulate
the work of the machine in a number of cases. For example, if the starting mechan-
ism does not give a sufficiently smooth start, the operator is required to brake the
flywheel of the spindle by hand during starting of the machine. With existing, not
completely satisfactory, tension attachments the operator is often required to
change the setting of this attachment during the process of winding, i.e., to change
the magnitude of the braking moment according to the amount of wire unwound from the
spool.
Inaccuracies in setups of the winding machines (similarly as in mechanical
work) influence the quality of the produced assemblies. For example, our investi-
gations of winding processes for PEL copper wire of a diameter of 0.05 mm on a core
of square cross section indicated that, in cases of unsatisfactory regulation of the
tension attachment (which could- be seen on an oscillogram), the elongation of the
wire and its ohmic resistance increased sharply.
The magnitude of tension on the wire is in most cases established on the basis
of experience and Ilfeeln of the foreman and operator. To date, there are no accur-
,
? ate and objective methods and instruments used in the factories to determine the
' magnitude of tension on the wire during the process of its winding. No methodology
, of setting up winding machines is used.
28 STAT
Declassified in Part - Sanitized Copy Approved for Release 50 -Yr 2
Declassified in Part- Sanitized Co y Ap roved for Release ?50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
-. ;31;y1,:*
-
10
12
' 14
16
18
20
2'1
2.6
Ilhe:derilopment oflauch-zlia e al
ro,
-10
. ?
Very important in Setting up winding
lishment of the teneion of the wire and o
*,
chines is the proper selection and estab
the speed of winding, and also that the
center of the-fiiid of diiEFEERI-676-6t ea Iiiparameter sHoUia-be so re-lit-0-361??th?e-
1
??-????4
^
entire lot of windings produced, as to co
erances of the corresponding parameter.
ExAriples of determining relative err
meter, caused by inaccuracy of preparatio
the density of the lay of the turns, are
tiometer (Fig.11) is equal. to
11_
cide with the center of the field of tol-
rs of ohmic resistance of a flat potentio-
of the core and wire and by deviations in
ited below. The resistance of the potenr.
R ,
where p is the specific resistivity;
1 is the length of 4ire;
S is the cross section Of-the Wire
12=
d is the diameter of the metal in th-a Wire.
Substituting 1 = nw, we obtain
R.p nw p 2n (b-l-y+2di)
where n is the nnmber of turns,
w is the length of the turn*
tv=2(b+y+2d1)k1;
(16)
(17)
(18)
--4
? 10_4
--i* The increase in length of the turn, due to the inclination of the turns is omit-
ted, since in our case (dense winding of thin wire) this increase represents hund-
r) A
iredths or thousandths of a percent.
_ 29
Declassified in Part - Sanitized Copy Apa?roved for Release Y 2013/09/20: Cl
nn,v)nnn
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
is the diaseter of the wire across,the
Ito
kl is a coefficient allowing for
is a
ulation
?
Coefficient allowing for ineulatici thickness;
loose fit
-1
t _
36_
1,2, where k2 is the
(19)
between the turns and the cores (Fig.12)
and dependent on the diameter and the
elastic properties of the wire and its
insulation, the tension force during
winding, the dimensions and relationship
between the sides of the cross section
of the core, the radii of curvature of
its fins and other factors.
Piga' - Diagram of the Lay of Turns
on a Flat Potentiometer
n= Ln,
(20)
where no is the number of turns per unit 3jerigth of the winding.
1
no=
(21)
dik2
to_
coefficient of density of lay of the turns.
For our case we assume: k2 = 1.03 to 1.08. Performing the substitution of n,
12 _.S, and d1 according to eqs.(20), (21), anti (19), we obtain the following relation
.from eq.(17):
8 (b?y 4-2di) Lki
R? p
ird8kok2
(22)
For calculating the absolute error of the ohmic resistance A R., we disregard
e of small quantities of a higher order; equating the increment KR.
? 1
STAT
30
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20 CIA-RDP81 nin.anpnn9qnnoi nni .1 7
ueciassified in Part - Sanitized Cop
Approved for Release
?
,,_ - ?
4- tne differential function It, we obtain
- - - __
1
1
, 1
AR Ap Ab-11-11jr+24di +AL +ilk! 3Ad Ako Aks
----? lIat - + ? (23)
.., -
i? p bi-y1-2d1 L ki d Ito k2
? , ....., The difference between 2 A di and 2 A d is small in comparison with Ab + Ay p
-1
_Jand the difference between 2d1 and 2d is imall in comparison with b + y; therefore,
i
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
- in the second term, d may be substituted tor dl' For more accurate determination of
14 -
:the influence of variations in the diamettir of thin wire on the magnitude of the
1
) 1
ohmic resistance, the following has to be used in place of the term - 3'6 d 2
_
3
1
d-1-3Ad
1+3
( 24 )
,
In eq.(23), this term is obtained with a negative sign if the diameter of the
-wire is less than nominal. We then obtain
1
AR Ap +Ay+2Ad -FAL d-Akt 3Ad Ako Aks
- R p b L ki d+3.1d ko k2 (25) -*
;
The maximum relative error of ohmic resistance of the potentiometer is deter-
AR=IAp1+1Abi+11A+12.1d1 +1AL1+1Akil+
/Z p b 1- y-1-2d L ki
N3arip 1.1kol lak21
-F +-F , .
d+3.1d k0 K2 )
(26)
3A d
--Here, A d in the denominator of the term has a negative sign if the diameter of
dt3Ad
-7 the wire is less than nominal.
As was shown in the beginning of the article, many primary errors of winding
_
processes are random and depend upon the *ariation of one or several factors. These
Declassified in Part - Sanitized Cop Approved for Release
?
50-Yr 2013/09/2n ? r-IA
?-? y
STAT
?
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
!errori min be characterized by means of corresponding distribution:Curl/6i: -
-1
- For determining the resultant error of the controlling characteristic for wire
_;of a certain type and dimensions, the character and magnitude of the constant errors
has to be known and, for random errors, also the 114 of their
Compared to the mechanical working of metals, the field of production of wind-
lings still has insufficient empirical data for the establishing of guide lines on the
calculation of winding processes for accuracy.
For practical calculations of the accuracy of technological processes in the
Fig.12
- Section
of the Core of a Po-
tentiometer with Applied Turns
ability.
? In addition, it is necessary to determine the character of the relationship of
,the centers of the groups of these errors of their systematic elements with the
factors to be controlled.
production of windings, it is necessary toi
,observe a large number of typical winding
loperations and to evaluate the laws of dis4
tribution of individual type of errors
under various conditions of winding, on
the basis of observation data.
This, in turn, permits a determination
of individual deviation limits of given
errors, at a certain selected level reli-
The assembled statistical material permits the establishment of. norms of ac-
curacy for typical winding operations.
We believe that statistical and mathematical-analytical methods for the deter-
,
1
imination of resultant errors in the produCtion of windings will soon find wide op-
plication.
STAT
)10
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Pad- Sanitized Cop Ap roved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
_BIBLI
1 Sokolovskiy,A.P. - Calculations of Acc acy of Work in Mstal-Cutting Machines.
16
18
"Present-Day Problems of the Tech ology of Instruments". Transactions NATI,
No. 22, Oborongiz (1954)
Yevteyev,Fae. and ZhUkov,V.A. - Techn logy of Radio Equipment.
Gosenergoizdat (1952)
6. Chumakov,V.P. and Nikolayev,N.Ye. - Sc entific Investigation of the Chair of
"Technology of Aircraft Instrumen Manufacture". MATI, 1954 (unpublished).
Mashgiz (1952)
Bulovalay,P.I. and Pova1yayev,A.V. - T chnology of Production of Parts and
10
Assemblies of ilectric Measuring struments. Mashgiz (1952)
3. Lakteyev,V.S. - Investigation of the T chnological Process of Production of
14
Windings for Small Motors. Candi atels dissertation, NATI (1954)
4. Chumaksi,V.P. - Two Basic Conditions o the Windihg Operation. Volume on
20
22
21
o
36_
12
50_4
r
33
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20 ? C R Ri_ni r v-v)
nn A
STAT
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
mli????1
?
8--
I 0 ?
12 --
14 _
18_
20 _
ACCURACY AND PHYSICAL INTERC GEABILITY OF PICKUP ELEMENTS
Engineer D. A. Hraslavskiy
Modern remote-indicating instruments ? automatic devices contain a large
number of pickup elements, utilized in the conversion of a measured or regulated
)
(input) physical quantity into another (output), which is more convenient for trans-
2 1
_ mission over a distance, for amplification, or for starting motion of a recording
'o
device.
The most common elements are represented in Table 1, where the horizontal
_ columns show the input parameters, and the vertical ones the output parameters*.
One of the most effective means of decreasing the cost of mass production and
{4
jof simplification in the use of complex iniptruments and automatic device is maximum
1
linterchangeability of their parts, subassemblies, and pickup elements.
1.)
-
While a sufficient specification as to interchangeability of parts and mechan-,
-;
ical subassemblies is the retention of their geometric shape and dimensions within
,the limits of permissible tolerances, the 1interchangeability of pickup elements Also
,
,requires maintenance of a given accuracy un its characteristics, expressing the func-
;tional dependence between input and outpu in physical quantities (Fig.1).
The term "physical interchangeability!! originates from this, and is understood ;
-to represent an interchangeability of elements according to their physical charact-
_
*,For---more detailed information on the function of_ pickup elements shown
see the book by D. A. Hraslavskiy, S. S. Logunov and D. S. Pelpor
_ 34
Declassified in Part- Sanitized Copy Approved for Release @50-Yr2013/09/20:CIA-RDP81-01043R002300210011-7
STAT
Declassified in Part - Sanitized Copy Approved for Release . 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release . 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Pad- Sanitized Co .y Approved for Release ? 50-Yr 2013/09/20 : CIA-RDp81-01043R002300210011-7
eristics.
In an analysis of the accuracy and of the physical interchangeability of a
pickup element, not only its shape and dimensions but also the physical parameters of
the materials from which this element is produced must be taken into consideration:
specific gravity, modulus of elasticity, specific electric conductivity, temperature
coefficients, etc., which have an influence upon its physical characteristics.
In the following article, general methods of calculating the errors of a separ-
ate pickup element (see Chapter 1) are considered, and a relationship between the
error of the measuring instruments and its elements (see Chapter II) is established.
Only statistical errors of the elements are discussed.
As a basis for the calculation of errors a method developed by N. Bruyevitch* is
used; in this method a small movement of a system, caused by a deflection of the
actual system from the ideal one, is linearized and studied separately from the main
movement.
I. CALCULATION OF ERRORS OF A PICKUP ELEMENT
1. Determination and Classification of Errors
?
We will consider the pickup element as the converter of the general input coord-
inate X into the output coordinate Y (Fig.2).
The absolute error of the pickup elements will be used for denoting the differ-
ence between the actual and ideal (required) values of the output coordinates, at a
certain constant value of the input coordinate.
The absolute error may be found analytically or graphically.
If the absolute expression of the characteristic of the ideal element is of the'
type
* For data, see Bruyevitch (Bib1.4)
Declassified in Part - Sanitized Coq Approved for Releas
110=40 , (3.)
? 50-Y
CI - n
? STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
).
If the ideal and actual characteristls are graphically represented
Fig.1 - Characteristics of Pickup Elements a - Elastic pickup element;
6 - Gyroscope with two degrees of freedom; b- Thermocouple. p - Pres-
sure; s - Displacement; w - Angular velocity; M - Moment of forces;
t - Temperature; E - Electromotive force.
STA
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
'74
f f ?
tt:
0
and171iiii377iFe?in he error crirrifiaiiiidairiFihe difference ortbi-iidinates of
2-4
curves 2 and 1.
The absolute error of the element maybe also expressed in dimensions of the
iiput Obordinate AI if; oh-the'graph-in F3rthidifferoflCe of 1i?iiiii of
curves 2 and 1 is used.
6-1 In view of the relatively small magn4ude of the errors, their transformation
from one dimension to another may be perfdlmed by utilizing a transformation coe-
fficient*.
where
AY=KAX or AY
K
ay)
i(=( my
? .tg 9,
ax mx
mx and my are the scales of the graph on the x and y axes;
is the angle between the tangent and the ideal curve 1.
The subscript zero of the partial derivative indicates that the ideal charact-
eristic is being differentiated.
We further decide to express the absolute error of
the element in dimensions of the output cOordinate.
The relative error of the element will be referred
Fig.2 - Generalized Pickup to as the relation of its absolute error to-the ideal
Element value of the output coordinate at a given point:
x - Generalized input co-
f(X)
ordinate; Y - Generalized
Yo .to (X) Pi (X). (4)
output coordinate.
11 Together with the term "transformation coefficient" the literature also uses the
following terms: transformation ratio, sensitivity, transconductance, steepness of
characteristics, amplification factor.
STAT
37
Declassified in Part- Sanitized Copy Approved for Release @ 50-Yr2013/09/20 : CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
The total relative error of the element will be referred to as the relationship
of its absolute error to the largest ideal value of the output coordinate
A Y ( X)
=.- Ch (X)*
YO max ./.0 (Xmax)
(4')
From eqs.(3), (4), and (4') it follow that the errors of the elements are func-
tions of the input coordinate.
The general error of the element consists of individual (partial) errors which,
depending upon the reasons causing them, may be divided into the following basic
types:
a) Methodical errors, appearing in the design of the element in cases where, for
simplification of constrUction of the element, the exact (required) function of trans-
formation is substituted by an approximate one;
b) Manufacturing errors, produced in the manufacture of the element, due to in-
accurate execution of the given geometric and physical parameters which have an in-
fluence upon the characteristics of the elements (for the sake of brevity we will
call these the main parameters);
? c) Temperatures errors, caused by a change in the main parameters of the element
during a change in its temperature;
d) Errors caused by friction, in the case where the element does work of which
a part is expended in overcoming frictional forces;
e) Hysteresis errors, caused by internal friction in the material of the element;
f) Errors due to clearance, caused by additional (parasitic) displacements of
the moving system of the element within the limits of the clearance in supports or
guides.
The manner of summarizing partial errors of the element depends on whether they
:are systematic or random errors.
?
Systematic errors are added algebraically. Random errors are added with con-
Isideration of the probability of compounding these random errors.
38
Declassified in Part - Sanitized Copy Approved for Release 50 -Yr 2013/
nr1,1,3nn
STAT
Declassified in Part - Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
When the overall error of one complete sample element is determined, then the
partial errors (see paragraphs a, b, and c) should be related to the systematic ones
Fig.3 - Graphic Determination of Error of the Element
1 - Ideal characteristic; 2 - Actual characteristic.
and the remainder to random ones*. If, however, the overall error of a large lot
(series) of elements of the same type is to be found, then the manufacturing errors
(paragraph b) also pass into the category of random errors since the distribution of
the main parameters of the elements during their serial production is mainly due to
random factors.
We will consider methods of determining partial errors of the 'element.
2. Methodic errors
During the design of the element given by a function (1), some approximate func-
tion may be substituted for the ideal function Y fo (X) in order to simplify the
construction:
qb q2,???qn),
(5)
* In certain cases, methodical errors should also be related to random ones, as,
shown below in example 1.
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20 ? CIA-RDP81 0104:11Rnn9qnnol nnil 7
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
ere q0i-thf iiinlmirsmeter:of'the el nt (gecimetric dikensions and physical cos.1
efficient., having an influence on the p_hmacterist#!_g_t_h_p m
_11,
The methodic error is equal to the dlference of eqs.(5) and
idla eqdal to their ideal
values:
(1), where the
(alculated)
The ideal characteristic of the llnear rheostat represents a proportional ratio
of electric resistance to the displacement s of the glider, represented in Fig.5a by
the broken line OA:
where K is the transfer coefficient, K = ? in ohm/mm;
wt
R is the overall resistance of the rheostat in ohms;
w is the overall number of turns;
t is the lay of the winding, in mm.
In an actual rheostat, even produced with ideal accuracy,
the output
coordinate
does not change proportionally to s, but in steps. In Fig.5a the characteristic of
* This error is known as error of turn or of resolving power of the rheostat.
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
-
the actual rheostat is represented by the stepped line, with the number of
equal
-
equal to the nuMber_of turns._ They4dth_Of.each_step is equal to tpjhe_heielt its
equal to L.
The stepped characteristic is described by th?equation of horizontal section
Fig.5 - Methodical Errors of a Linear Rheostat
a - Characteristic of the rheostat; 6 - Change of methodical error,
on sectors of the width t.
In the first section OD < s < 0: r
w
In the second section (t < s < 20: r = 2 R
In the nth section [ (n - 1)t < s < ntl :
The methodical error in any given section is ecpial to
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
-
.where (n - 1)t < a < nt; n = 1, 2, ...w.
At s (n - 1)t, the maximum error is equal to Ji.; at s = nt, the error is
equal to zero.
The graph for the change in the methodical error is shown in Fig.5b.
In view of the small magnitude of the step t, in comparison with the overall
length of the rheostat, the methodical error may be considered as random, the appear-
ance of which within the limitsArmet = 0 to Armet = a is equally probable.
, In this case the mean value (mathematical probability ) of the methodic error
is equal to
and the mean square deviation
6rni et mom' 2w'
1.R
?met-2 w ?
The mean value of the reduced methodic error, related to the overall resistance of
the rheostat, is equal to
The mean square deviation is
et
1
R Ip2u1
21/3w ?
For example, if w? = 100, the mean value of the total methodical error is equal
:to 0.005 or 0.5%, and the mean square deviation is equal to 0.003 or 0.3%.
3. Manufacturing Errors
In listing manufacturing errors, the methodical errors are considered to be
equal to zero and therefore the approximate function (5) is assumed to be ideal.
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20 ? CIA RDPRi_nina Pnno
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
(The parameters qi are considered equal to their ideal values):
Yo==f(X, 171, q2...q,1).
The characteristic of the actual element is of the type
Y=[ (X, 91+11911 (12+A q2 ??? qs),
(8)
(9)
where A qi are primary errors (deviations of the main parameters from ideal values),
appearing during manufacture of the element.
The absolute production error is equal to the difference of eqs.(9) and (8):
YpT.Y?Y0,--1(X, q2-1?q2 qn?F-Aq)-
-f (X, qi, q2...q,i).
(10)
In view of the relatively small magnitude of the increments Aqi, the absolute
production error maybe represented as a linear function of primary 'errors, if
eq.(10) is expanded into a Taylor series around the ideal values of the main param-
eters and is limited to the zero and first members of the series:
Ao
,
r A...J\aqijo ?
b=1
(n)
The relative manufacturing errors may be found on substituting eq. (II) into
eqs.(4) and (4').
In concrete calculations, the expression for relative errors is often consider-
able simplified.
We will consider a specific case which is fairly popular in practice, where the
.characteristic of the ideal element is expressed as a function of th,3 type
Y=f(X)qp, qi.
a
qnn=f(x)
1,1
Declassified in Part - Sanitized Copy Approved for Release @_50-Yr 2013/09/20 CIA-RDP81 nin4npnn9qnnoi nni .1 7
la
STAT
Declassified in Part - Sanitized Copy Approved for Release
0
t%
I .1
12 U
)1-
_
50-Yr 2013/09/20 : CIA-RDP81-01043R002300210011-7
paraiitiis qi aild7-exponents ai 'r.re jipiuintrX.1
In this case a simple expression is Obtained for the relative manufacturing
. .
error, which may be found if eq.(12) is expressed logarithmically and the full dill-
-ferential of thil-16-&-fithiCia-Ukiii;TiuWaiiNtliraliiti-
Y
Y )112r
Aq1 n 642
=a1 ? ?-t- ...2 -
41 (12
Agn Aqi
an =-- a
qn %moil 1 qi
1.1
(13)
Coniequently, the manufacturing error of the element, having characteristics of
the type (12), is equal to the sum of the products of relative primary errors in the!
exponents of the steps of the corresponding main parameters.
IXamrile 2. Let us find the manufacturing error of a linear rheostat, shown in
Fig.4.
In determining the manufacturing error of the rheostat, we will consider its
methodical error equal to zero and, furthermore, for simplifying the calculation,
assume the primary errors of the rheostat to be constant, i.e., independent of the
displacement s.
We will express the overall resistance of the rheostat by its parameters:
pL4.10-3p (D (14) w
R ? 10-3
d2
ohm - mm2
, where p is the specific resistivity of the wire in
(14)
i
L is the developed length of the wire, in mm;
q is the cross section of the wire without insulation, in mm;
d is the diameter of the wire withoui, insulation, in maq
7
di z is the diameter of the wire with insulation, in mm;
D is the diameter of the body, in mm (we will assume a round cross section of
1 the body).
?61-1177etii-uting eq.(14) into eq.(7), we obtain the ideal characteristic of the
44
Declassified in Part - Sanitized Copy Approved for Release SO -Yr 2
. C I -
STAT
Declassified in Part - Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
__ ___ _. __ ,___
rheostat, expressed by its main parametere:
4.10 ?3p(D+dh)
1 r0 -- S.
,s .., d2i
1? ?
.??? ... _ ..___________ ? ? .? ??? ... ? ... ?-????. ? -? -T?i? ,.,
I
4
a it .....] Analogous to eq. (33), we obtain the Llative manufacturing error of the rheostat
,2
i
-,
expressed by its relative primary errors.'
ti
t
1 ?
-
i
t Ail ...... Ap.1. AD+ Adiz Ad At
2? ? .
k r fpr p D-Fdiz d t
i
1 1
For example, if in a specific rheostdt the relative primary errors have the
zz ' I
Ad At
=
value ? = - 0.05; - + 0.02; -1..- = - 0?05;T + 0.03, then the overall
P D + di z U
1
,conductivity error will be equal to i
(Pk2-1) ? 0,05+0,02 ?2 (-0,05)? 0,03=0,04 or 4%.
Pr
If the primary errors are not constalit along the length of the rheostat, then
the calculation of the manufacturing erroxl becomes much more complicated, since it is
necessary to consider the law of change of the primary errors as a function of the
displacement of the glider.
In this case, the manufacturing errors of the rheostat are determined by exper-,
iment.
Fig.6 - Cylindrical Spring
Example 3. Let us find the manufacturing error of the cylindrical spring
(Fig.6).
STAT
45
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20 ? CIA-RDP81 0104:11Rnn9qnnol nnil 7
a
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
The characteristics of such a spring are determined by the formula
s ?
8D3n
P,
Gd4
where s is the displacement, in mm;
P is'the force, in kg;
D is the mean diameter of the turn, in mm;
n is the number of turns;
G is the modulus of shear, in kg/Mm2;
d is the diameter of the wire, in in.
?
The manufacturing error is equal to
As AD An AG Ad
?=3? + ? ? 4 .
s. D n G
Limiting Values of Manufacturing Errors
' (15)
To find the limiting values of the absolute manufacturing errors of a group of
elements of the same type, by the "maximum-minimum" method, we will use eq. (ii), tak-
ihg into account that the partial derivatives( '_.) may be positive as well as ineg-
ative.
We will assume that, of the overall number n of partial derivatives, there are
m positive and n - m negative.
The upper limiting value of the manufacturing error is determined from the ex-
pression
A YuPrr =
r oqi
i-1
_The lower limiting value, from
wooer + ) A dy hawarr
0 91 ?
In a
OY OY
(
y ) .
qtr.ver (
+ 17 pp.r
aqi o Oqi o
1=1 1==n1-1-1
46
(16)
Declassified in Part - Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
For elements with a characteristic.of the type (12) we obtain frdm eq.(13) the
following expressions for the limiting values of relative manufacturing errors
(53-1 74rret= a,+ al
(A?Dyrrer
Y pr qi
1=1 i=-471-0
A Y\lawer
y pr -71 a (2LA Yftar+
i.1 qi
I tic n\lower
qi 1
a(&1)'
qi
i=m+i
(161)
In eqs.(16) and (161), the terms Aqi upper and Aqi lower are the upper and
lower deviations of the main parameters from the ideal values shown in drawings and
technical specifications for materials.
It must be mentioned that the calculation of the limiting manufacturing errors
of elements by the "maximum-minimum" method cannot be recommended for accurate calcu-
lations, since the laws of the theory of probability are not taken into account,
which leads to an uncorrected tightening of tolerances of prime errors. The applic-
ability of calculations, taking into account the laws of the theory of probability,
is outlined at the end of Chapter II.
4. Temperature Errors
Temperature errors appear as a result of additional increases of the main par-
ameters qi with changes of temperature.
For calculating temperature errors we will use eq.(10) but will assume that the
element is produced to ideal accuracy (ianufacturing errors equal to zero) and that
increments Agi are caused by the influende of temperature.
With an accuracy sufficient in practice it can be assumed that the increments
A qi are proportional to the change in temperature
(17,) -
,where qi is the value of the main parameter at normal temperature;
ai is the temperature coefficient of the parameter;
At is the change in temperature as compared to normal temperature.
47
Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2013/09/20 ? CIA-RDP81-01041Rnn7Inn91nn11 7
TAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
0
-.
Substituting eq.(17) into eq.(10), we obtain an expression for the -abiOlUte
'temperature error:
Yttnip f [XI qi + AO, 42 (1
c, ?
1 ? ? ? qn(1 +1,1AM ?f(X411 qi ? ? ? 'In).
(18)
Equation (18) should be used when th(i product ai A t is relatively large (for
example, if it exceeds 0.1).
For the majority of elements, the te4perature coefficients are so small that
justifiably the inequality is ai Ategl and the temperature error can be linearized
by means of eq.(11), where the value of Aqi is substituted from eq.(17):
AYte V(-aY) qictiAt.
mp dqi 0 (19)
In the specific case when an element has the characteristic of eq.(12), its rel-
ative temperature error is determined by a formula obtained by means of substituting
eq.(17) into eq.(13):
(A Y)
Y temp
Ectictiat.
(20)
The overall temperature coefficient* of such a pickup element is obtained from
eq.(20), assuming that A t = 1?C; then
a= I aim,.
i-1
(21)
Example 4. Let us find the overall temperature coefficient of .a cylindrical
spring, whose characteristic is expressed by eq.(15).
On the basis of eq.(21) and (15), the overall temperature coefficient of the
spring may be represented in the form
a aG 4aci,
* The overall temperature coefficient of a pickup element we denote as the relative
change of its output coordinate during a temperature variation of 1?C.
48
Declassified in Part- Sanitized Copy Approved for Release 50-Yr 2013/09/20 ? CIA-RDP81-01041Rnn7Inn91nn1 7
STAT
STAT
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20 : CIA-RDP81-01043R002300210011-7
t,
1 ;
.. .
__where ap, aG and ad are temperature coefficients of the parameters D, Gs and d.
If it is considered that aD and ad are equal to the relatively small coefficient
._.,
of linear expansion of the spring material and, consequently, one is equal to the
- -- - ---- ----
other, then the overall temperature coefficient of thespring will be equal to th---
e
- -- - - --
---
-1
-1
__temperature coefficient of the modulus of shear, taken with the reverse sign:
1
a aG.
1
example, for phosphorous bronze we have aG 5x10-41 from which a 5x1074.
Consequently, during a temperature variation of At = 100?C, the temperature
error is equal to
aAt=.5.10-4.100.0,05 or 5%.
5. Errors Caused by Friction
We will separate the elements with an output parameter P (see Table 1).
These elements in which the output parameter is the force P or the moment of
forces M we will, in view of these conditilons, call "forces"elements.
1
In the construction of instruments add automatic equipment, the force element is
usually mechanically connected with one or more mechanisms or other elements. Conse-
,quently, the force developed by the force ,element is able to overcome friction forces
acting not only within the given element but also in the mechanisms and other ele-
ments to which it is connected.
For mechanical systems with one degree of freedom, all forces of friction, over-
come by the force element, may be substituted by an assumed (equivalent) friction
.force, applied at some point A of the element, or by an applied friction moment,
.applied at some axis zero.
The selection of the point A or of the axis zero is generally arbitrary, but it:
is more convenient to select a point of the force element where it is connected with
other elements (in the case of translatory motion of the element), or an axis along
which it is connected with other elements (in the case of rotational motioh of the
49
Declassified in Part - Sanitized Copy Ap4.
proved for Release SO -Yr 2
. CI A-
0
Declassified in
Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
;element).
The frictional force PItp applied at the point A is determined by conditions of
. equality of the elementary work of this Rime through a small displacement ds and on
,the basis of the sum of the elementary work of the frictional forces on the actual
parts of the moving system, during corresponding displacement*
Iry , p_ 4Y1 +11 AA dak
ds
(22)
where Pi is the friction force of the liarts in translatory motion;
dyi
ds-is the translatory relationshi of point A to the ith part;
Mk are the friction moments of tqparts in rotating motion;
dak is the translatory ratio of point A to the kth part.
. ds
In an analogous manner, the friction moment applied at the axis zero is de-
termined
AA, +CV I P. dyi+IA4.
A.,tp 44.1 dp di,
(23)
dy.
where - is the translatory ratio of the axis zero to the .thpart;
dp
dak'is the translatory ratio of the axis zero to the kth part.
dyi
The sign ? in eqs.(22) and (23) means that the error due to friction is not
,single-valued since the frictional forceschange their direction during a change of
,direction of motion of the moving system.
Equations (22) and (23) give the limiting values.of the absolute error of the
..force element due to friction, expressed in units of the output coordinate (force P
, .or moment of forces M).
If eqs.(22) and (23) are divided by 4 translatory coefficient of a force ele-
ment, then we obtain limiting values of the absolute error, caused by friction, ex..
? or data, see Tikhmenev (Bib1.2)
50
STAT
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20 niA_pnDszi n n
....
Declassified in
Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
. pressed in units of the input coordinate and called the threshold of sensitivity or
the zone of rest of the system.
or
dyi
Pi --- + Mk da*
ds ds
AX1p= tP 4_
Kp \
kax )0
Alt
LIXID= P 4
' Km
dyi
M*
dy dy
OM\
kax10
(24)
(25)
A change of the input coordinate within the zone of rest does not cause move-
ment of the system, since the increase of the resultant input coordinatc. by the force
element is insufficient for overcoming the applied force (moments) of friction.
Example 5. We will find the error due to friction of the electromechanical
pressure indicator, consisting of a membrane (force element), kinematically connected
across a translatory mechanism with the rheostat slide (Fig.7).
For simplicity, we will only consider the frictional force of the slide on the
rheostat and will assume the angles of rotation of the lever to be snail, so that
(1:y b
ds a
where s is the displacement of the membrane;
y is the displacement of the slide.
According to eq.(22), the friction force applied at point A, is equal to
D
tp ? -L. -
a SI
where P81 is the friction force of the slide on the rheostat.
_
ccording to eq.(24), we determine the error of the indicator due to friction,
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20 n n
....
-
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
_
expressed in units of the pressure being nieasured:
14
-
- ? ,
P1) bPsi
? +
PtP? dP aF
dp
?dP
where F = ..is the effective area of the membrane. If, for example, a al 3mm,
b = 15mm, Pia = 0.5 gm, F = 5 c2, then th error due to friction is equal to
p= 15.0,5
ip + ? +0,5fmcm2=-- 5 trwrn wt er cot.
3-5
Errors Due to Hysteresis
Errors due to hysteresis, similarly to errors caused by friction, do not defi-
nitely depend upon the input coordinate,
coordinate corresponds to various values
the character of the !change of the input
established value.
Errors
i.e., ofe and the same value of the input
qf the output coordinate, depending upon
Ooordinate prior to the time it assumed the:
due to hysteresis are determined by means of measuring the output co-
ordinate first during the increase in in-
put coordinate and later during its de-
crease.
The graph of the characteristic of an
:element obtained in a similar manner has
the appearance of a closed loop (Fig.8),
whose axis (broken line OB) corresponds to
,the characteristic of the ideal element,
:in the case where the element does not
Fig.7 - Electromechanical Pressure
Indicator.
Declassified in Part - Sanitized Copy Approved for Release
have any errors other than those due to
hysteresis. The difference of the ordin-
52
50-Yr 2013/09/20 ? CIA RDPRi_nina Pnno
SA
ueciassified in Part - Sanitized Cop
Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
"?-? ?
0
Hates of the upper part of the loop OBC ard! of ihe-ixiai-line OB gives the upper
limiting value of the absolute error due t'o hysteresis, while_the.difference of the
lordinates of the lower part of the loop
(JAB andThi axial. line OB give8 the
F.ower limiting value for the absolute error
Idue to hysteresis.
If the pickup element also has errors
due to friction, then the experimentally
26 Yhyst
_
Fig. 8 - Hysteresis Loop '
) :AOB - Curve during rise; BCO - Curve
during decrease
X
'obtained graph of Fig.8 will express the
rum of the errors due to friction and hys-
iteresis. In order to eliminate errors due
.,;to friction, the hysteresis loops are re-
corded in practice after lightly tapping
the element or after a short-time buzzing ;(light vibration).
. 7. Errors Due to Clearance
We will consider errors due to clearance in the pickup elements, containing a
moving system with one degree of freedom, hich is secured by application of suitable
kinematic couples, i.e., guides permitting freedom of translatory motion, or sup-
ports, giving freedom of rotational motion.
In the presence of clearances in kinematic couples, the moving system may have,,
in addition to freedom of motion in the min direction, also additional small (withr
in the limits of the clearances) displacements in other directions, resulting in an '
error of the element.
The error is equal to the increase of the output coordinate, caused by addition4.
al small displacements of the moving system at a certain constant value of the input'
coordinate.
_
In the presence of clearances, the moving system in a general case may obtain ,
Declassified in Part - Sanitized Cop Approved for Release
53
50-Yr 2013/09/2n ? r-IA
STAT
4
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
-ifive additional degrees of freedom, above the one degree of freedom in the main
direction.
For
For calculating the errors due to clearances, we will construct around the ele?
ment a coordinate system ETC oriented with respect to the element in such a way,
:that the main direction of motion takes place downward (or about) the axis E (Fig.9).
The two other axes are selected in an arbitrary direction.
If the main motion of the element is translatory, then the absolute error of the
output coordinate due to clearances will be equal to
a Y Y Y Y Y
A yewAl AC + (pi cpc ( 26)
?
\al /0 \ac /0 ka?E/0 \af,n/0 ow
?c 0
where An and AZ are small translatory motions of the system along the axes n and
Acpg 0471 ,p(i) are small rotational displacements of the system about the axes 8,n
and
If the main motion of the element is rotational, then the error of the output
,coordinate is equal to
AY`tea=(?aaD0A (?aaYr,)0A1+ (---aaYt) 0"+ (?aY) A`P'+(---") AP', (27)
of, 0 &pc 0
_1
Ae is a small forward motion of the system along the axis E.
The partial derivatives in eqs.(26) arid (27) and the magnitudes of possible dis?
placements are determined from the analysis of the specific construction arrangement
,of the element.
For determining the mean and limiting values of errors due to clearances, cal?
culation methods worked out in the field of theory on the accuracy of mechanisms are,
applied.
1 Example 6. Let us find the error due to clearance, for a linear rheostat whose
Islide undergoes rotational motion (Fig.9): We will select the axis of coordinates
in such a manner that the axis 6 coincides with the axis of rotation of the
Declassified in Part - Sanitized Copy Approved for Release
54
50-Yr 2013/09/20 riA_RnoQi n4nAn
STA
Declassified in
Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
-Ifive additional degrees of freedom, above the one degree of freedom in the main
!direction.
For calculating the errors due to clearances, we will construct around the ele
.ment a coordinate system rC oriented with respect to the element in such a way,
,that the main direction of motion takes place downward (or about) the axis E (Fig.9).
The two other axes are selected in an arbitrary direction.
If the main motion of the element is translatory, then the absolute error of the
output coordinate due to clearances will be equal to
a Y
AYcZeA=(?) 0 .
1' a Y
(?) AC +-)( ATE + (4? ACPn
ac 0 a?E 0 tqn o
la
Acpc,
(26)
where An and C are small translatory motions of the system along the axes n and C;
Apg ,ATTI ,,64) are small rotational displacements of the system about the axes E,n
and C.
If the main motion of the element is rotational, then the error of the output
,coordinate is equal to
a Y aY a Y
aY /dY
AYclea=( H \ (?) Af" + (5?) "c$
o at 0 0 ipc 0
(27)
4,where AE is a small forward motion of the system along the axis E.
j
The partial derivatives in eqs.(26) ind (27) and the magnitudes of possible
.placements are determined from the analysis of the specific construction arrangement
lot* the element.
For determining the mean and limiting values of errors due to clearances, cal?
culation methods worked out in the field of theory on the accuracy of mechanisms are;
.applied.
pample 6. Let us find the error due to clearance, for a linear rheostat whose
'slide undergoes rotational motion (Fig.9). We will select the axis of Coordinates
4t/Itin.such a manner that the axis e coincides with the axis of rotation of the
Declassified in Part - Sanitized Copy Approved for Release
54
50-Yr2013/09/20-niA_pnDszinin ....
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
slide, while the axis C passes through the center of rotation 0 and the contact point
of the slide C as well as the axis n are perpendicular to the axes E and C.
Fig.9 - For Calculating the Error due to Clearance
The main motion is a rotation of the slide about the axis In the presence of
clearance between the axis of the slide and the bushing, the moving system (slide)
attains an additional five degrees of freedom and the error due to clearance in the
general case is determined according to eq.(27), as follows:
where ArGed is the change in the output resistance. However, according to Fig.9 it
Is obvious that, within the selected system of coordinates, we have
The partial derivative
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20 ? CIA-RDP81-01041Rnn2lnn91nnl L7
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release
Nor
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
2 -\,11ectric circuit transforming th -;:electromOire:fortO:,thens40*
current I;
\ ?1 -
3 ?Ilectro magnetic eys84a (frame
placement of the iiiiinter*.
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release
i.1.4;4414011
_
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
heilei:0-the link (1) (thermocouple) repro
ding_t_CL,..the tude,of T to be mea'
, -../.
, .. , Li . " ' ? . .. t
.. In instrumente;?basid on indirect me
, \ ? 4
, . , I, ?
isOnt.!dbelot'aliys react to the,
?
sical magnitude"
, the fOlari PiOkOP elsa
.cord the altitude of flight 1?ti
in inalysis?ok errors of s
r.uOural arrangssept by a pr
ishipbetiieem the quantity to be
C !
tude to. be Measured by the( ixistuntSilt4b1.tt,
?t, ) t ,
ch ;1!e funcitionally 0oiotid with the One
, , ?
_
? .
. ,
at. Of .the biiitosetrid aircraft
.if ,
the aitmosptiiric,,pr.r.,,
ar instruMehisit neqSUarY tpr.
, ? , ? ,.\ ?
sional linki eltaii11.10.41 the
functional
sured at;i: t'hr recorded
? ,STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
_gives the relationship of the atmosplieric ?weissure to the, altitude.
? .(
, An introduction Of the imoirisional link intothelittrticturiilf itrangettent,piii;aitt
ax'!ringeetents' silorin in Figs30 andp.1 are o
?
?All links in tnese 'airangements are connects&
iJ
.output
? input parameter ,Oie'f0
?
so
__
at8ii, at we'
,'.
, , ?
39, ' Fig.1.1 Barometric Altimeter.
ing met
. , _ , , _i i_t,. , , \i _ ..7 I , . , based upon .Oaspensat
? ,..-.4.3,fJ ? .
34 t -.Principa47 view;,o - zumomurai-irtange-
,_, zr,9 1 urement,:which,,,er,e wid
_ \\, . , i i i. ? .i ,.t-, - ' If9A1ti. . 36...._mentiji - Altituae tOilse =toured; It.:,.1r, 1" field! 0t tollZMT:;. al*
\ . ,, ?.? _i
?
\
_
.38_4moopheric preaSureiliotion Of tilWani..,
loops, 1;d1 ?ea.:presence'
, ?
I
40-1roid box; at.. ...h Angie. of ' deflection of the ? ` ? "'46' ?- r - -'''
--,
- '
. , ? ,L.., f.' -? al. scheme. a ,,'closed : gifei-
,
.....i , -. ?
42_ '
.," ?
, , ' `.1' ,.
pointer
, \ -
; . ? (' . -
al arrangements without parallel Unites46. Cliaracteristic,(Of MOstAisisur
14 ,
,..
ars considered.
, ', -. ?
.... Figure -.1.2 represents .a ,general open I tructural.ireapgs,uert
1
, -
, ,
strthas?ti 'consisting 44. n .liOkis ,in, series.
,.'' ' ?
., , /. , /
, . ? k , ,' ?
? \`'llif general ditPut Coordinati;I ropresiOtilits- ps
?Ilan
???';7?? ? v?Ic.???-t.-444A?ii?P.,??ty i??
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
STAT
Declassified in Part - Sanitized Copy Approved for Release
'121-41. WW1'
_ frn
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
A
?
-----4m-%4
0
4
? 6
8
10
?
?
12
14
lb
18:12. Equation of the Scale of the Instrumenti. Absolute and Relative Errors
20 The static characteristics of the measuring instrument, known as the equation
depends upon the operating principle of tho elements of thi-iiamehf: me-6444-it
coordinate In = a represents the deflecti of the reading or recording device.
Fig.12 - Generalized Ope
I - Input coordinate (magnitude to be
coordinates, Irpoit- Output coordinat
Xn.f
Structural Arrangement
easured); X1 Xn.a. - Intermediate
(deflection of the reading device).
1 tl
I 4
of the scale and indicating the ratio of the deflection of the reading device to the
__value of the quantity to be measured, may
be expressed by the formula
a=f(X). (28)
This ratio can be found by solving a
system of equations, expressing the char-
acteristics of the links in the structural scheme.
For example if in the arrangement shown in Fig.12 the number of links is n =
then the equations for the links will be
t X1=41 (X) ;
X2= f 2 (X1) ;
X3=f3,(X2);
a=f4(X3) . (29)
4,
I-
1:1:1 As a result of the solution of the s stem of equations (29) we obtain the equa-
w__tion for the scale of the instrument
t
- /4 {f 1/2 If (X)I]) = f (x).
60
(3o)
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
.??????.,
?
ti
11?
to-
14
16
Example 7.: Let Ui-find-the equan
ter, shown in Fig 10.
f the sciiiiirrlhermoelectiiFthermaaa-1
The equation for the first link (thermocouple) is,
E=ard-b71,
where E is the electromotive force;
a and b are coefficients dependent on the materials of the thermoelectrodes;
T is the temperature.
The equation of the second link (electric link) is
,j }where R is the sum of the electric resistances connected in series within the cir-
cuit of the thermocouple.
The equation for the third link (electromagnetic element) is
BSw
M
9810
where B is the magnetic induction in the air gap;
S is the area of the frame;
w is the number of turns.
The equation for the fourth link (two helical springs) is
1 121
a M,
2 bh:le
where 1 is the free length of the spring;
b and h are the width and thickness of the cross section of the spring;
e is the modulus of elasticity.
Solving this system of equations for the links according to eq.(30), we obtain
61
STAT
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20 CIA RDP81 ninanpnn9-4nnoi nt-I4
ueclassified in Part - Sanitized Copy Approved for Release
?
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Lttie equation of the scale of the instrument:
,
a
bh3E 9810 y, R ?
61 BSw (aT + bTs)
The characteristic of the scale of the instrument may be constructed by graph-
ical methods, if the characteristics of the links are given as graphs or in Tables.
Equation (30) gives the characteristic of the ideal instrument, if the output
characteristics, given in eq.(29), are taken as ideal elements
ao==fo(X). (31)
If, however, 'the characteristics of actual elements are to be considered, then
, the deflection of the reading device will be different from ideal and will be ex-
pressed by the formula
a=f (X) . (32)
The absolute error of deflection of the reading device is equal to the differ-
ence between its actual (32) and ideal (31) values, being
A a= a?ao=f (X) ?fo (X) ? (33)
? The sensitivity of the instrument is the partial derivative of deflection of
_j the specific device with respect to the quantity to be measured:
-
K
(-al)
a x 0
(34)
The subscript o indicates that the ideal equation(3l) of the scale is being
;differentiated.
,t
The sensivity of the instrument with an open structural arrangement is equal to
,ithe product of the transfer coefficients of all of its links:
\ _taxi\ fax2\ axn_ k
i\ dXn
K kd ax )0kox, )0 (axn_doai
Ks ? ? ? Ka- n = fl
1.4
Declassified in Part - Sanitized Copy Approved for Release
62
50-Yr2013/09on- rIA_Dnno.?
?-?
(35)
STAT
Declassified in Part - Sanitized Co .y Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
The absolute errors of the instrument are usually expressed in dimensions of
the quantity measured, reading
X=Xpe-X,
where Xpr is the reading of the instrument in units of the quantity being measured,
corresponding to the deflection a of the reading device;
X is the actual value of the quantity measured, causing a deflection a of the
reading device.
In order to transform the absolute error of the instrument from the dimensionm.
into the dimension X, it suffices to divide the error Aa (33) by the sensivity of
the instrument (35):
? Aa. 411at
,
K n
n Ki
I=4
*
The relative error :Of tlfe instvument is equal to
1 AX
The total relative error is equal to
x
A X
Ximax
Finding the Instrument Error from the Errors of its Elements
(37)
(38)
(39),
In general we consider the absolute instrument error to be that caused by a
'deviation of the characteristics of its elements from ideal values.
Let us assume that, as a result of calculation or of experimental investiga-
tion, the errors of all links in the structural arrangement are found, expressed in
dimensions of the output coordinate of the corresponding link
Error of the lat link=-?-(pi(X).
Declassified in Part - Sanitized Co
?
y Approved for Release
?
(40)
63
50-Yr 2013/09/20: CIA-RDP81-n1n4qpnno-v-molnr,4,,
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
0..
lirro--r of the 2nd
2-1
-
Error of the ith link
(pi (Xi_ 1).
Error of the nth link AX= 'Pm (Xn-1).
(40)
In finding the overall error of the instrument we will consider errors of A Xi
,as primary.
In the beginning, we assume that all elements of the instrument, with exception
of the ith element, are ideal.
The partial error of deflection of the reading device, resulting from the error.
la the ith element may, with an accuracy sufficient for practical purposes, be con-
sidered as a linear function of the primary error AXi:
A2(i)
e )
axi '
(41)
The index i of the error Ace indicates, that this error is partial, caused by
the influence by the ith element.
The linear approximation is completely proper in the given case, since all
partial errors of the instrument lie within the limits of the field of tolerances,
whose dimensions are very small relative to the length of the scale of the instru-
ment.
Ba
(
The partial derivative --- may be represented as the product of transfer co-
i/o
efficients of links from (i + 1) to n:
(a2) =(axii(ax,+,
dx, 0 ,ax, 0 ax,,,),
? ?
(aXn_i\ ( )
kaxn_2/0\axn_1/0
K 116+2 ? ? ? Ka ? 1 Kn= n K3.
s=1-1-1
(42)
STAT
64
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
.V ? - - ? "e
????????Irkri
,
the overaIl-err6r of deflection ot th reaIig Ece, during simiatineous ac-
:tion of errors of all links, is equal to the sum of the partial errors
tla= 4041) =-- (eX, 11 Ks).
(43)? ---
In order to find the absolute error of the instrument, expressed in units of
the quantity measured, the error of deflection (43) should be divided by the sensi-
tivity of the instrument; according to eq.(37) we obtain
n n n
1 (AXI El K. )
4.17 1.4 s=--14-1 I, ??,
_ .
K n 1
i.i. fl K.
1.1 s=1
(44)
The relative error of the instrument is computed by substituting eq.(44) into
eq.(38), and the total relative error by substituting eq.(44) into eq.(39).
For specific calculations,, the expressions for the relative error of the instru-
ment are often considerably simplified.
We will consider the simplest specific case, where the instrument consists of
linear links, whose characteristics are expressed by the equation
= X;
X2 = Ka ;
X3 = X2 ;
?
=-- ?1;
X, =-- KnXn?i?
1
(45)
Using eqs.(45), any intermediate coordinate may be expressed in terms of X:
== KiX;
X2 = K2 X1 KiK,X;
X= K3 Xg = K3 X
. ;
65
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
?
?
, from which
X KiKiKa
. ?KIX Xill Ks,
s.1
n?
s=4 sx ?
(46)
Substituting eq.(46) into eq.(44) and dividing both parts of the equation by X
we obtain:
AIX=x14)(1
ALI Xi
(47)
From eq.(47) it follows that the relative error of the instrument, consisting
of linear links, is equal to the sum of the relative errors of all links.
Analogously, it may be shown that the total relative error of the instrument,
consisting of linear links, is equal to the sum of the total relative errors of all
links:
LX AXI
X max (X1) max
1=n
(48)
Example 8, Let us find the overall relative error of a thermoelectric (Fig.10),
where for simplification we will assume the characteristic of the thermocouple to
be linear, i.e.,
E=a71,
In this case, the overall.relative error of the instrument may be expressed by
the relative errors of its links by means of eq.(47), as follows:
dE I MAn
7' ?Egli M "I- a
66
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
STAt
/
1?9,
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
AE
=where ? is the relative error of the thermocouple;
a
is the relative error of the electric circuit;
is the relative error of the electromagnetic element;
is the relative error of the springs.
?
3. Conditions of Physical Interchangeability of Elements of a Measuring Instrument
The physical interchangeability of pickup elements, as well as the dimensional
interchangeability of parts and subassemblies, may be obtained by one of five meth-
ods, accepted in the construction of machinery (Bib1.4):
Method of absolute interchangeability;
Method of incomplete (partial) interchangeability;
Method of selection or of selective assembly;
Method of fitting;
Method of regulation.
The conditions of absolute physical interchangeability are determined by means
of eq.(44) if, in the latter, concrete values of errors of the elements are substi-
tuted by their limiting values (calculations of "maximum-minimum"), taking into
account the signs of the transfer coefficient.
Let us assume that, of the overall number n of links, there are in links for
which the product IlKs is positive and (n-m) links, for which the product HI( is
s=1 s=1
negative.
Then the upper limiting error of the instrument is equal to
APPer
== ?
Ammd
n As i=m+1 n Ks
s=i
? (49). .
STAT
67
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
31-
AE
'where --- is the relative error of the thermocouple;
?
AI
is the relative error of the electric circuit;
AM
is the relative error of the electromagnetic element;
Aa
is the relative error of the springs.
a
_
3. Conditions of Physical Interchangeability of Elements of a Measuring Instrument
The physical interchangeability of pickup elements, as well as the dimensional
interchangeability of parts and subassemblies, may be obtained by one of five meth-
ods, accepted in the construction of machinery (bib1.4):
Method of absolute interchangeability;
Method of incomplete (partial) interchangeability;
Method of selection or of selective assembly;
Method of fitting;
Method of regulation.
The conditions of absolute physical interchangeability are determined by means
of eq. (44) if, in the latter, concrete values of errors of the elements are substi-
tuted by their limiting values (calculations of "maximum-minimum"), taking into
account the signs of the transfer coefficient.
Let us assume that, of the overall number n of links, there are in links for
which the product flK is positive and (n-m) links, for which the product nK is
a=1
negative.
s=1
Then the upper limiting error of the instrument is equal to-
m 12
141,fr X lower
ima Pr L1X
AX
AMMd k .(4.9)
" ? S M n K
.s
67
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
and the lower limiting error of the instrument is equal to
111
viewer ).1 ,lower
I
fl Kc
s,1
xiupper
n K
i
s=1
(50)
where AXiupper and AX. are
are the upper and lower values of the errors of the ele-
ments, permitted by the technical specifications for a given element.
If the instrument consists of linear links, eq.(47) will yield the following
expression for the limiting relative errors of the instrument:
(A_Xr ?pper_ ( upper ,y!) lower
, v.
Ammo aims \
-1
m n
/AA' \ lower. I ( .A.Xi) tower+ Is-,1 f_62(L) "Mr.
X ) 1, xi ) 4..11 k x1)
, I i=m+i
(51)
(52)
Analagous expressions are obtained for the limiting values of total relative
errors, if eq.(48) is used.
The method of absolute interchangeability, the basis for which is the "maximum-
minimize calculation, is principally and practically without justification in most
cases, if it is taken into account that primary errors are mainly random quantities
and the probability of addition of their extreme values, especially in multilink
chains is quite negligible.
N. A. Borodachev cites, for example, the following data for chains with ten
primary errors, uniform in magnitude. The probability of obtaining a resultant err-
or within the limits of 0.9 to 1.0 of its value, computed as a maximum, is as fol-
lows:
a) With distribution of primary errors according to the law of equal probab-
ility (which is rarely encountered in practice) it is less than one case
out of a million.
of primary errors according to the normal law (close to
68
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
practical cases) it is less than one case out of 1015 cases.
The method of partial interchangeability, where the overall error of the instru-
ment is computed in accordance with the rules of the theory of probability, seems to
be more justified, as accepted for measuring and kinematic linkages.
The applicability of this method of computing is primarily derived from an ex-
planation of the laws of distribution of primary errors by means of statistical
treatment of the results of control of a large number of elements and, secondly,
from the application of the following rules for the addition of errors (Bib1.2):
a) Systematic errors and quantities, characterizing centers of grouping
of deviations of random errors, are added algebraically;
b) Quantities, characterizing the scatter of deviations of independent
random errors, are added quadratically.
The possibility of appearance of any possible value of a partial error of the
instrument, independently of the value of the other errors, is referred to as the
independence of errors.
The formula for computing the tolerances of the overall error of the instrument,
derived on the basis of the above rules, is of the type
AX1. = .X0
where A.X0 is the systematic part of the overall instrument error, equal to
= (---) (A01+ai9;
axi 0
i
(53)
(54)
Here, 6X is the practical limiting random part of the overall error, 'equal to
n - _
- ?
6A,== ax ax do ( \ 2 6.12 = /
.4mJ1.
69
(55)
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20: CIA-RDP81-01043R007mn91 nni 1_7
STAT
;
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
In eqs.(54) and (55)
( cic)((vi )
is the transfer coefficient from the input coordinate of the instrument X
th.
to the output coordinate of the I link;
Aoi is the coordinate of the center of the field of tolerances of the ith prim-
ary error (if this error is random) or its algebraic value (if the ith error is sys-
tematic),
a1 is the coefficient of relative asymmetry of the ith error;
61 is one-half of the absolute magnitude of the field of tolerances of the ith
?
random error (if the therror is systematic, then 45.1 = 0);
Xi is the relative mean quadratic deviation of the law of distribution of the
a.
ith error(X. whereof is the mean quadratic deviation of the ith error);
6.
X is the relative mean quadratic deviation of the overall instrument error;
a
where 0 is the mean quadratic deviation of the instrument error;
X.
" '1 is the coefficient of relative scatter of the primary error.
X
If the fields of tolerances of all primary errors are given and the law of
their distribution is known, then all coefficients of the right side of eq.(55) are
determined) with the exception of the coefficient X which may be made variable.
An increase in X permits a decrease of the tolerance of the instrument error,
but in this case the so-called percentage of risk, i.e., the probable percentage of
instruments with errors exceeding the limits of established tolerances, is increased.
In determining the percentage of risk, it has been accepted to use the quantity
Since, in most cases of practical problems, the distribution of the overall
error may be considered subject to the normal law, the percentage of risk is deter-
Declassified in Part - Sanitized Copy Approved for Release
70
50 -Yr 2013/
. C -
_ nneNn
STAT,
Declassified in Part - Sanitized Copy Approved for Release ?50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
mined according to the formula
P5i) ?0(01 = [
2 e ? 2- dti, (56)
2it
where (I) (t) is a Laplace function.
The computed values of the integral 4)(t) are given in Tables used in courses
on the theory of probability and mathematical statistics.
Table 2 gives certain values of the integral 4) (t) and the percentage of risk,
as a function of the parameter t.
Table 2
rP(1)
p go
0(t)
P %
1,00 0,6827
32
3,00
0,9973
0,27
1,65
0,9011
10
3,29
0,9988
0,12
2,00
0,9545
'4,5
3,89
0,999
0,01
2,57
0,9901
1,0
In practice the triple mean-square deviation is considerea to be the limit of
the normal curve of distribution, i.e., 6X = 3c, which corresponds to the percent-
age of risk F= 0.27% (for t = 3).
Applying this condition, Tables yielding the values for the coefficients ai and
ki for the most frequently encountered laws of distribution of primary errors in
manufacturing, have been prepared.
After determining the coefficient ki and computing the error 6X, the upper and
lower limits of the overall instrument error can be found from eqs.(53), (54), and
(55).
11
c)X
-I XIII'?" = Xo ax= - (A.+ ctiai) +
cai 0
71
(57)
STAT
Declassified in Part - Sanitized Copy Approved for Release ?50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Y"
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
11
()X b282
cm,
1=1
11
tarcr= Ax0?ax.)-1 Poi+ ctiai) ax
?
axi 0
b=1
_if(?X7a k61.
1=1
(57)
ax
axi0
Fox' an open undeveloped structural arrangement, the transfer coefficient()
is inversely proportional to the product of the transfer coefficients of the links
from 1 to i, i.e?
'Ox'
ax,1 1
I
1
ax, ax2 ax, K1K2.?? Kt n Ks
X diVi? ? ? ax s-1
(58)
Substituting eq.(58) into eq.(57), we obtain the final expression for the tol-
erance of the overall instrument error, constructed according to an open structural
arrangement,
11761
n n
Ani+aibi
i + 11
n Ks
fl K5
i=4 5?1 i =1 s_____,
A
7-- n 14bl x rwer_, , 1 Ant ? a it i
i V I
i? I s=1 1/ . 1=1 s=1
U Ks 11 Ks
(59)
Equations (59) are also conditions of partial interchangeability of the elements
of the measuring instrument.
The method of selective assembly, widely used in instrument making, and espec-
ially the method of regulation require special consideration. It can only be indi-
cated that these are also based on assumptions which have been accepted for analo-
gous methods in the theory of measuring and kinematic linkages.
72
;
STATI
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-fl1n4npnn9-4nnoinn44
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
BIBLIOGRAPHY
1. Balashkin,B.S. - Technology of Machine Building. Mashgiz (1949)
2. BorodachevIN.A. - Analysis of the Quality and Accuracy of Manufacturing.
Mashgiz (1946)
3. Braslavskiy,D.A., Logunov,S.S., and Pelpor,D.S. - Calculation and Construction
of Aircraft Instruments. Oborongiz (1954)
4. Bruyevich,N.G. - Accuracy of Mechanisms. Tekhteoretizdat (1946)
5. Bykhovskiy,M.L. - Accuracy of Electrical Computing Linkages. Izvestiya AN SSSR,
OTN, No.8 (1948)
6. Gavrilov,A.N. - Technology of Aircraft Instrument manufacture. Oborongiz (1951)'
7. Kutay,A.K. - Interchangeability, Tolerances, and Adjustments in Instrument
Making. Mashgiz (1947)
8. MalikovIM.F. - Fundamentals of Meteorology 1 Committee on Measures and Measuring
Instruments (1949)
9. Tikhmenev,S.S. - Theory of Aircraft Instruments. 1/VIA imeni Zhukovskiy (1940)
10. YakhinIA.B. - Technology of Precision Instrument Manufacture. Oborongiz (1940)
STA
73
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20 ? CIA-RDP81-01041pnn7'Inn91nn11 7
Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
???
ABBREVIATION OF THE EQUATION OF MOTION OF A RAPIDLY
ROTATING GYROSCOPE ON GIMBALS AND THE INFLUENCE OF STATIC
IMBALANCE OF THE GYROMOTOR ON THE BEHAVIOR OF THE GYROSCOPE
by
Cand. Tech. Sci., Docent G. A. Slomyanskiy
We will consider two of the most popular cases of arrangement of the symmetri-
cal gyroscope in a gimbal suspension (Figs.la and lb).
Fig.1 - Arrangement of a Gyroscope on Gimbals
a - Vertical axis of the outer frame of the gyroscope, 6 - Horizontal axis of the
outer frame of the gyroscope..
The system Ent', is stationary, the axis xsylz rigidly connected with the inner
frame, the angle a is the angle of rotation of the outer frame, the angle 0 is the
angle of rotation of the inner frame, the angles a and 0 are indicated in a posi-
tive direction.
If the inertia of the frames of the suspension is not taken into account, then
74
STA
-
STA
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20 ?
nnnonnn nnw
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
nf avroscope and causing nutation;
the equations of motion of the axis of the gyroscope for both cases of arrangement
of the gyroscope in a universal suspension are of the following type (Hib1.1):
A (0.+ ci2 sin p cos p) ? Hci cos p = (1)
Where Mx,, My,land Mz are the projections, upon the axes xl, yl, and z, of the mo-
ment M of the internal forces applied to the gyroscope. Then, equating Mz = 0, we
will have H = const according to the third equation of the system (1). In this case,
the motion of the axis of the gyroscope is fully characterized by the two first equa-
tions of the system (1). Only moments (or their projections), directed along the
axes of rotation of the frames of the suspension, i.e., along the axes xl and for
the gyroscope shown in Fig.la and along the axes xl and n for the gyroscope shown in
Fig.lb, can show any influence upon the behavior of the axis of the gyroscope in
universal suspension. We will transform the first two equations of the system (1)
in such a fashion that they will include only moments directed,along the axes of
rotation of the inner and outer frames of the universal suspension. These moments
we correspondingly designate by Min and Mont. Since Mdn = Mx, then it is only nec-
essary to transform the first of the equations in the system (1). Multiplying this
equation by cos 0 and taking into account that My, cos O = Mout (in the case of
Fig.la, the moment Mout = it and in the case of Fig.lb the moment Mout = M ) we
will rewrite the first two equations of the system (1) as follows:
A (cos p ?2d0 sin p) cos (3+ H cos
A (0 ? cil sin p cos p)?tici cos p = ? M
(2)
Abreviation of the Equation of Motion of a Rapidly Rotating Gyroscope in Gimbals
We will assume that the gyroscope rotates rapidly. In order to obtain abbre-
viated equations, characterizing the basic precessional motion of the gyroscope in
gimbals, the terms in the equations of the system (2) containing A, depending upon
75
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7 -
STAT
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
_
the equatorial component of the kinetic moment of the gyroscope and causing nutation;
will have to be discarded. Then we obtain
HO cos = m.?t,i
Hci cos = Min ? 1
(3)
Equations (3) are applicable to any given values of the angle p 1 satisfying
It It
the inequality _ < 13 < They differ in this respect from the usual abbreviated
2 2
equations
=M,
applicable only for small values of the angle 0.
Law of Precession of the Rapidly Rotating Gyroscope in Gimbals
We will designate the angle between the frames of the universal suspension by Iv.
Figure 1 indicates that IV= ?2- - 13.
troducing the angle w we obtain
?
Solving eqs.(3) with respect to a and p and in-
Min
HsintIJ
Mout
1-1 sin 4)
These expressions determine the rate of precession & and 13 of the rapidly rota-
ting gyroscope about the axes of rotation of the outer and inner frames of the sus-
pension, and are applicable to. any given values of the angle w, except for w = 0,
permitting the formulation of the law of precession of a rapidly rotating gyroscope
in gimbals in the following manner: The internal moment M, acting upon the gyroscope
about the axis of rotation of any of the frames of the universal suspension, causes
the axis of the gyroscope to rotate (precess) about the axis of rotation of the other
frame with a speed
(2
11 sin 4, ?
(4)
STAT
f.4
76
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20 ? CIA-RDP81-01043R0o2lon71nn1 1_7
?r?
Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2013/09/20 : CIA-RDP81-01043R002300210011-7
"mm=3=4
!During this rotation, the vector H tends to coincide with the vector M. In this
_
fashion, the same moments acting about the axes of rotation of the frames of the
universal suspension increase the rate of precession as the angle between the frames
?
deviates from a right angle. The smallest values of the rates of precession a and 0
are obtained when the frames of the universal suspension are mutually perpendicu-
n
lar = 0,w = ___). From this follows a conclusion important for the technical
2
utilization of the gyroscope, namely that at mutually perpendicular frames of the
universal suspension, a rapidly rotating gyroscope possesses maximum stability with
respect to upsetting moments applied to it.. The more the angle between the frames
of the universal suspension differs from a right angle, the less stable will the
gyroscope be. Therefore, in cases when it is absolutely necessary that moments of
friction on the axis of the universal suspension, moments caused by insufficient
balances of the frames of the suspension, and also moments of transfer forces of
inertia, have as little an influence as possible on the location of the axis of the .
gyroscope, then special devices for automatic suspension of the frames of the uni-
versal suspension in a mutually perpendicular position have to be considered.
Equation (4) is also applicable in cases, when the freedom of rotation of the
gyroscope about its dead point is secured not by means of a universal suspension,
but in some other fashion; as an example, the following may be. mentioned: gyrostat;
a gyroscope suspended by one end from a string, and others. In that case, the
angle w is the angle between H and 0, if H,M-= const; if, under the action of
the moment MI the angle between H and M changes, then w = HI M.
As an example of application of the laws of precession formulated by us, we
will consider how a displacement of the center of gravity of the gyromotor (the in-
ternal frame with the rotor of the gyroscope) in the direction of the axes yl and z,
at various values of the angle w between the frames of the universal suspension,
will influence the gyroscope shown in Fig.la. will assume that o, yll and z are
the coordinates of the center of gravity of the gyomotor and that the weight of the
77
F.11
STAT
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20: CIA-RDP81-01043R00230071nn1 1-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
z
gyromotor is equal to P. Then, at an angle between the frames equal to W < 1 the
2
moment
P (z sin e? ?y' cos 4)),
will act on the gyroscope, which in accordance with the law of precession (4) will
cause a rotation of the gyroscope about the axis of the outer frame with a speed of
P? ('r ctg (I)).
ir; _ _ _
(2.a.
H sin qi
This expression indicates that a displacement of the center of gravity of the gyro-
motor along the axis z causes a precession whose speed does not depend on the
angle W between the frames. The rate of precession, caused by a displacement of the
center of gravity along the axis y', is proportional to cot W.
Integration of Abbreviated Equations of Motion of the Gyroscope
The possibility of integration of eqs.(3) depends on the character of the mo-
ments Min and Mout. We will consider certain important cases where integration of
these equations is possible. We will assume that the moment Mout is a function only
of the angle 0 (or is constant), i.e., that Mout = Mout (0). Then, assuming, that,
during the instant of initial motion o = 00 at t = 0, we obtain the general solution
of the first equations of the system (3) of the following type:
t = H
cos 'OP
M0 o () ?
(5)
It is obvious that the possibility of expressing the given integral in terms of
elementary functions is determined only by the type of the function Mout (0). The
relation (5) permits a determination of the time of transition of the gyroscope
from one position to another. For integration of the second equation in the sys-
tem (3), We will initially assume that the moment Min depends only on the angle 13
78
-4
STAT
Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2013/09/20 ? CIA-RDP81-01041Rnn7Inn9inn11 7
Declassified in Part - Sanitized Copy Approved for Release
50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
(or is constant), i.e., Min = Min (p) and, therefore, that this moment is a function
of the angle a only (or is constant), i.e., Min = Min (a). In each of these cases,
the general solution of the equation under consideration can be expressed in quad-
ratics and, under the conditions that a = ao at t = 0, is of the type
= ao
I M (5) dt ( for M=M (p));
(6)
H cog 1
and
H
da di
(a) Y cosn p
( for Min = Min (0). (7)
'Wi
In a number of practically important cases, the integrals appearing in eqs.(5),
(6), and (7) may be expressed in terms of elementary functions.
Dividing the equations of the system (3), we obtain the differential equations
of the trajectory of the top of the gyroscope along the surface of a unit sphere:
M out
da M in
(8)
From eq.(8) it follows that the type of trajectory, described by the top of the
gyroscope, depends only on the character of the moments acting on the gyroscope
about the axis of rotation of the inner and outer frames of the universal suspension.
Mout
The entire fixed value of the ratio --- corresponds to the fixed value of the ratio -
Min
, independently of the angle between the frames of the universal suspension.
da
In a number of technical gyroscopes the moments Mout and Min are functions of
the angle j3 only (or are constant),. i.e., Mout = Mout (j3) and Min = Min (0). In
this case, the general solution of eq.(8), under the initial conditions a = ao and
P po at t = 0, is of the type
7.0 + A4'n (3)
M 000) sj.
(9)
STAT
79
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R0o2mm1nn1 1_7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
In cases when the moment Mout depends only on the angle p (or is constant)
i.e.; when Mout = Mout (0), and the moment Min is a function of the angle a only (or
is constant), i.e., when Min = Min (a), the general solution of eq.(), under the
same initial conditions, is of the type:
M1,() . mout
(10)
In a number of practically important cases, the integrals in eqs.(9) and (10)
may be expressed in terms of elementary functions.
Min
' We will note that, at = k = const,
mout
scope along the surface of a unit sphere is
the equation
the trajectory of the top of the gyro-
determined, as follows from eq.(9), by
a=ao-f-k(13--pc)?
Mout
If ;.7.--=q cos 0 where (q = const), the top of the gyroscope will describe a loxo-
min
drome on the surface of a unit sphere.
The relationships considered permit an investigation of the behavior of techni-
cal gyroscopes for any values of the angles a and 13 except where 0 = ? ?2 '
Influence of Static Imbalance of the Gyromotor on the Behavior of the Gyroscope
We will consider a gyroscope, shown in Fig.la, and assume that the axis of ro-
tation of the supporting frame is vertical.
We will assume that the center of gravity of the gyromotor does not coincide
with the center of rotation of the inner frame (axis xt) and
point C with the polar coordinates 1 and T (Fig.2). Here, 1
the axis xt to the point C, w is the angle between the plane
is located at the
is the distance from
xtz and the plane pass-
ing through the axis xt and the point C; the angle w is considered positive in the
direction of positive yl. The weight of the gyromotor will be denoted by P. The
80
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
.14
STAT
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
161.4?111:TV.Tro;''r?,4,77.7`...-,-7,24:7#1,7=1?7:'?
following moment with respect to the axis of rotation of the inner frame will then I
act on the gyroscope:
M 1,41 .1P cos (p +
The moment will be considered positive, if it tends to rotate the gyromotor
counterclockwise, looking at it from the
positive end of the axis xl (Figs.la and 2).
Aside from this, friction moments will
act on the gyroscope in the bearings of the
()P
frames of the universal suspension and will
--
P
be constant in magnitude and opposite in
direction to the angular velocities a, 13
Fig.2 - Schematic of the arrangement
and polar coordinates 1, T of the
center of gravity C of the gyromotor.
Since, at positive a and 0 1 the vectors of
these angular velocities are erected along
the axes (+ ) and (-xl), the friction
moments along the axes xt and r, are equal
to
MT. 41 sign And MT. out sign 6,
where MT.in and T.out are the absolute values of the friction moments about the
axes xl and t .
Thus, in the case under consideration, the moments applied to the gyroscope
with respect to the axis of rotation of the inner and outer frames of the universal
suspension, are equal to
IP cos (p + co + Al r. in sign f.3'
M0=0L4t sign &
(n)
In order to set the gyroscope in motion (on a rigid support) it is necessary
113,C.
Declassified in Part - Sanitized Cop Approved for Release
?
81
50-Yr
2013/09/20: CIA-RDP81-01n4n;mn9qnno1nni 4 7
STAT
K
0'
Declassified in Part- Sanitized Cop Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
74V
following moment with respect to the axis of rotation of the inner frame will then I
act on the gyroscope:
114 pi? 1 P cos (p + to.
The moment will be considered positive, if it tends to rotate the gyromotor
counterclockwise, looking at it from the
positive end of the axis xl (Figs.la and 2).
Aside from this, friction moments will
act on the gyroscope in the bearings of the
frames of the universal suspension and wil)
p
be constant in magnitude and opposite in
direction to the angular velocities a, p.
Since, at positive a and 0 1 the vectors of
these angular velocities are erected along,
the axes (+ ) and (-xl), the friction
moments along the axes xl and are equal
OC-
Fig.2 - Schematic of the arrangement
and polar coordinates 1, w of the
center of gravity C of the gyromotor.
to
MT in sign 1.3 and ? gr.out sign Ct,
where MT.in and MT out are the absolute values of the friction moments about the
axes xi and r. .
Thus, in the case under consideration, the moments applied to the gyroscope
with respect to the axis of rotation of the inner and outer frames of the universal
suspension, are equal to
114 . 1P cos (p mt in sign 0,1
Mout= ?MT.,c141. sign a.
In order to set the gyroscope in motion (on a rigid support) it is necessary
81
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01(14Rnn9qnnoinni 4 7
A
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
to have the absolute value of the moment Mpin larger than the friction moment MT.in,
i.e., it should satisfy the inequation
IP I cos ([30 + > MT. ill
where 00 is the initial value of the angle 0.
If the unbalance LP is equal to or less than the friction moment MT.in motion
of the gyroscope is impossible for any values of the angle 00. If, however,
1P > MT.in then the gyroscope will only start moving at angles 00 satisfying the in-
equation (12).
Substituting in expression (12) an equal sign for the sign > we obtain an equa-
tion for the limits of the region of rest. The graph in Fig.3 has been constructed
according to this equation. If ----- and (p T) are such that the point corres-
MT.in 0
ponding to them in Fig.3 lies in the hatched region, which is the region of rest,
then the motion of the gyroscope cannot start since the inequation (12) is not sat-
isfied. On the other hand, the graph in Fig.3 indicates that, at /13 > MT.in, there
always exist such values of Po at which the inequation (12) is satisfied, and also
values at which it is not satisfied. Therefore, the balancing of the gyromotor and
the checking of its quality must be done at several values of the angle Pe
(12)
Furthermore, the graph in Fig.3, shows that it is impermissible to limit the
balancing of the gyromotor to only two values of the angle Po, 900 apart, since such
a method of balancing may prevent an unbalance from becoming apparent. This is pos-
sible, if /P and ((3o + T) are such that the point corresponding to them in the
BTITI
graph of Fig.3 falls into the crosshatched region. For example, ift, IP = 1.2 and
MT. in
the balancing is in one case done at 00 +q) = 40 and in the other case at p. + T +
900 = 1300 or at 0 + T- 900 = - 50, then an unbalance will not become apparent,
0
since the corresponding points Al, A2, and A2, (Fig.3) are located in the region of I;
F.1
rest, whereas the point Al is in the crosshatched region. 0
R
Considering further, that the inequation (12) has been satisfied, we obtain,
82
STAT
Declassified in Part- Sanitized Copy Approved for Release ?50-Yr2013/09/20:CIA-RDP81-0104riRnn9qnn9lnni1 7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
on the basis of eqs.(3) and (11), the following expressions for the rates of preces-
sion 0 and a :
----- sign et
H cos p '
IP
cos(+cp) + sign (3
H cos p
MT. in
H cos 0
(13)
If in the expression for a, the member containing MT.in is omitted and if it
is assumed that cp = 0 or it i.e., if it is assumed that the center of gravity of
the gyromotor lies in the plane xlz, then we obtain the result that the velocity of
precession a, caused by the unbalance of IF, is equal to ? LE = const and, conse-
quently, does not depend on the angle 0 or on the angle between the frames of the
-180 -135 -90 -45
0 45 90 135 180 4o.130
Fig.3 - Regions of Rest of the Gyromotor
a) Region of rest
universal suspension. In all other cases of location of the center of gravity C
(see Fig.2), and also at MT.in II 0, this velocity is a function of the angle 0 .
The velocity 0, as may be seen from eq.(13), does not depend on the magnitude of
83
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
. the unbalance IP. The absolute value of the velocity 0 increases as a function of
the increase- of 431 from 0 to ---.
2 v4e should mention that 13 is independent of IP
only in the case where the axis of rotation of the inner frame is vertical, which
was also assumed in the beginning of this Section.
'Substituting eq.(11) for M in eq.(5) and integrating, we obtain
t== ?sign& -// 4
T. 01.14.
(sin 13 ? sin rtio).
From this we obtain the law of change of the angle 13 with time
(14)
MT. out
!1=--- arc sin ( sin (30? sign a
(15)
albstitArtinginegeNth"Olnent.by its value from eq.(11), we obtain
Min
H
M, sitzn 0+ IP cos (3+ cf)
dt.
coq
Eliminating, by means of eq.(15), the angle P and applying the substitution:
we obtain
a = Gt
0
sin po ? sign a MT. Out t =__ x;
sin p, x0; ? sign a dt= dX ,
sign a
M1.. ou.4
{MT. in sign 0 $
dX
xt,
X
X d X
?1P sin c .
' 311?X
/P cos cp ?
x?
Using the integrals which are part of the last expression, we have
a =-- at)
sign ci
MT. otAt
IMT.
sign 1.3 arc sin X+
+IP cos yX + IP sin p V1 ? X2ry:
84
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/09/20: CIA-RDP81-01043R007mn91nn1 1_7
;`,4
STAT
Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2013/09/20: CIA-RDP81-01043R002300210011-7
?
?
?
The first of the equations in the system (13) indicates that, at H > 0 and -
- MT in (which in turn is possible at 0 < (I) < n), the frames of
the universal suspension will, in time, coincide and the gyroscope will lose one
degree of freedom.
At/ P sin T < MTin
(if n. <