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AD/A-004 219
ANALYSIS OF THE MUTUAL INDUCTANCE PARTICLE VELOCIMETER (MIPV)
Joseph D. Renick
Air Force Weapons Laboratory
Prepared for:
Defense Nuclear Agency
November 1974
DISTRIBUTED BY:
um National Technical Information Service U. S. DEPARTMENT OF COMMERCE
IfTV ■-.:, ■■!...■■■■ ,-,■ ;,.;,, .--; , ., .,■-,-;,,. r-. -.-, ,; ,,,,„,- .^ ., . iV
AFWL-TR-74-205
This final report was prepared for the Air Force Weapons Labora- tory. Mr. Joseph D. Renick (DEX) was the Laboratory Project Officer in Charge.
When US Government Drawings used for any purpose other than procurement operation, the Gove bility nor any obligation whats ment may have formulated, furni said drawings, specifications, by implication or otherwise as or any other person or corporat permission to manufacture, use, may in any way be related there
, specification a definitely r rnment thereby oever, and the shed, or in any or other data i in any manner 1 ion or conveyin or sell any pa
to.
s, or other data are elated Government incurs no responsi- fact that the Govern- way supplied the
s not to be regarded icensing the holder g any rights or tented invention that
This technical report has been reviewed and is approved for publication.
JOSEPH D. RENICK Project Officer
FOR THE COMMANDER
F. BÜCKELBW Lt Colonel, USAF Chief, Experimental Branch
WILLIAM B. LIDDICOET Colonel, USAF Chief, Eivil Engineering
Research Division
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AFWL-TR-74-205 2. GOVT ACCESSION NO
4. TITLE fand SubtUU.)
ANALYSIS OF THE MUTUAL INDUCTANCE PARTICLE VELOCIMETER (MIPV)
7. AuTHORf«;
Joseph D. Renick
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Air Force Weapons Laboratory (DEX)
Kirtland Air Force Base, New Mexico 87117
M. CONTROLLING OFFICE NAME AND ADDRESS
Defense Nuclear Agency (SPSS) Washington, DC 20305
IT MONITORING AGENCY NAME » AODRESSf/( di«»ren( from Conlrolling Oltice)
I RPAD ivsTPiirTinv«; READ INSTRUCTIONS BEFORE COMPLEflNG FORM
3 RECIPIENT'S CATALOG NUMBER
S TYPE OF REPORT ft PERIOD COVERED
Final report; June 1973-June 1974 6. PERFORMING ORG. REPORT NUMBER
8. CONTRACT OR GRANT NUMBERf»;
10. PROGRAM ELEMENT, PROJECT, TASK AREA t WORK UNIT NUMBERS
62710H; WDNS0303; J11AASX352
12. REPORT DATE
November 1974 3. NUMBER OF PAGES
i^r/3>3 15. SECURITY CLASS, (ol Ihlt reforl)
UNCLASSIFIED
15«. DECLASSIFICATION DOWNGRADING SCHEDULE
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Approved for public release; distribution unlimited
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Raproducsd by
NATIONAL TECHNICAL INFORMATION SERVICE
US Department of Commerce SpringMd, VA. J215I
IB. SUPPLEMENTARY NOTES
PRICES SUBJECT TO CHA! M
19. KEY WORDS CContinue on reverse aide il nece&amry end Idenfffy by block nt/mber)
Mutual inductance; Soil motion measurements; Hydrodynamic flow
20. ABSTRACT CContfnuc on revere» eide ff neceeeary and Identify by bloclr number;
Three analytical areas supporting development of the mutual induct- ance particle velocimeter (MIPV) are investigated. First, the basic theory is extended to permit calculation of gage factors and determination of design criteria for complex geometry gages. A finite element model is developed to investigate the effects of divergent/off-axis flow upon the gage response. Finally, an elec- tromechanical mathematical model of the gage is developed to
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I. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE fW,
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UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PA&E(Wnn Data Knlarmit)
investigate the effects of shock-induced self-inductance and resistance changes upon gage response. The mathematical model identifies the influence of different conductor materials upon gage design, therefore a brief study is conducted to further define conductor resistance-stress and shock equilibration time characteristics. Numerical data are presented for each analyt- ical area. As a result of the numerical investigations, the following conclusions are made: (1) The gage length-to-width ratio should be greater than 5 after shock transit through the gage to ensure a constant gage factor. Generally a gage designec with a length-to-width ratio of 10 or greater will satisfy this requirement; (2) Increased gage factors without sacriiice in electrical rise time are available with a design which, instead of alternating single loops of primary and secondary turns, uses alternating multiple primary and single secondary turns; (3) The MIPV is essentially insensitive to divergent flow effects for normal gage designs where the width is less than 3 inches and the source is at a range of 1 meter or greater; (4) The gage is relatively insensitive to off-axis flow in the plane of the gage loops, the error induced being approximately equal to 1/2 (1 - cos Y) where y is the off-axis angle. However, flow off- axis in a plane normal to the loop plane results in much larger errors approximately equal to 3 (1 - cos 9) where 9 is the off- axis angle; (5) Errors caused by shock-induced self- inductance and resistance changes in the gage are effectively controlled with a large ballast inductance in the gage primary; (6) Three metals, aluminum, magnesium, and beryllium, all exhibit favor- able shock impedance, electrical resistivity, and pressure- induced AR/Ä, characteristics; each presents a relatively low neutron cross section, with beryllium being the most favorable.
L^L UNCLASSIFIED
r-'Vö
AFWL-TR-74-205
PREFACE
Appreciation is expressed to Lt Colonel Giles Willis who provided the solution methods in the electromechanical modeling of section VI and to Mr. Gerald Perry, GE-TEMPO, who assisted in the assembly of this report.
1/2
CONTENTS
I INTRODUCTION
General
Description
Background
II ' EXTENSION OF THEORY TO COMPLEX GAGE DESIGNS
Basic Theory
Extension of Theory for Complex Gage Gepmetries
Numerical Results
Summary
III OFF-AXIS, DIVERGENT FLOW RESPONSE
Development of Computer Model
Summary of Numerical Results
Summary
IV ELECTROMhCHANICAL MODELING
Previous Work
Electromechanical Model
Numerical Results
Summary
V CONCLUSIONS \ND RECOMMENDATIONS
APPENDIXES
A. Tabulated Numerical Data for MIPV Designs
B. Computer Code Listing--DIVE
C. Selection of Optimum Gage Conductor
D. Computer Code Listing--ELMEK
REFERENCES
Page
7
7
8
9
12
12
17
21
26
28
28
31
41
42
42
43
50
62
65
67
93
106
115
131
■■.,...-,.. -, - -,—«nr^-^iopf.-.,
AFWL-TR-74-205
Figure
1
2
3
4
5
6
9
10
11
12
13
14
15
16
17
18
19
20
ILLUSTRATIONS
MIPV Used in Gas Gun Testing
Mutual Inductance Particle Velocimeter Nomenclature
MIPV Undergoing Deformation Due to Shock
Cross Section of MIPV Loops, Type I Gage
Cross Sections of MIPV Loops, Type II Design
Gage Factor as a Function of Width and Number of Primary and Secondary Turns, Type I Gage
Gage Factor' as a Function of Spacing between Loops and Number of Primary and Secondary Turns, Type I Gage
Variation of Gage Factor and T\ with Gage Length-to-Width Ratio for a Typical Type I Gage Design
Comparison of Types I and II Gages
Nomenclature for Development of Divergent/ Off-Axis Flow Model of MIPV
Effects of Divergent Flow upon Gage Response
Effects of Off-Axis Flow upon Gage Response
Effects of Off-Axis Flow upon Gage Response
Effects of Off-Axis Flow upon Gage Response
Effects of Off-Axis Flow upon Gage Response
Effects of Off-Axis Flow upon Gage Response
Comparisons of DIVE Calculations with Cosine Function
Gage Undergoing Deformation for Off-Axis Shock, 6=0, Y^O
Gage Undergoing Deformation for Off-Axis Shock, Y^O, 9^0
Simplified Schematic of Power Supply and Velocity Gage with Open Secondary
Page
10
13
16
18
21
23
24
26
27
29
33
34
35
36
37
38
39
39
40
44
mmmmmmmmmmumimt .im
AFWL-TR-74-205
ILLUSTRATIONS (cont'd)
Figure Page
21 Electrical Network of Velocity Measuring System 44
22 Graph of Network Shown in Figure 21, Tree Indicated by Solid Lines, Branches by Dashed Lines 45
23 MIPV. Undergoing Shock Deformation and #
Nomenclature for Determining L(t) and L(t) 49
24 Nomenclature for Determining Gage R(t) 49
25 Glass Gage 52
26 Shock Response of AEDC Gage as Calculated by ELMEK, K » .4 ' 55
27 Shock Response of AEDC Gage as Calculated by ELMEK, K - 2 56
28 Calculated Gage Response Using ELMEK, Inductance Changes Only, AEDC Gage 57
29 Calculated Gage Response Using ELMEK, Resistance Changes Only, AEDC Gage 58
30 Calculated Gage Response Using ELMEK, Normal Resistance and Inductance Changes, AEDC Gage 60
31 Shock Response of Field Design Gage as Calculated by ELMEK, K = 0.2, L = 500 yh 63
ii
C-l Change in Resistance as a Function of Stress for Several Metals 109
C-2 Geometry for One-Dimensional Shock Response Analysis HI
C-3 Shock Equilibration of Copper in Tuff HI
C-4 Resistance Change Per-Unit-Length as a Function of Equilibration Time, t , for Several Conductor Materials, 2:00 Kbar e 114
5/6
m—f' '*'
AFWL-TR-74-205
SECTION I
INTRODUCTION
1. GENERAL
The Air Force Nuclear Weapons Effects Research Council ha« established a requirement for the experimental investigation of the radiative and kinetic energy partitioning during the initial coup- ling into the ground of a near surface nuclear burst. An under- standing of the initial energy coupling is essential to the accurate computer predictions of crater formation and ground motion used in the survivability/vulnerability assessment of Ai-r Force weapon sys- tems. Ideally, this information would be obtained through experi- ments utilizing megaton yields in near surface nuclear bursts.
However, the Limited Nuclear Test Ban Treaty of 196S prevents atmos- pheric testing of nuclear weapons; thus, all nuclear testing in this country subsequent to the treaty has been conducted underground,
As a result, the near surface burst geometry, which represents the actual threat geometry, is replaced \>y a room geometry. An addi-
tional ground shock system is generated by the room effects which overtake the shock system generated by the initial coupling and obscure information about the initial coupling. To obtain any information about the initial coupling, soil stress and motion measurements in the unobscured region very close in to the nuclear source would be required. Supposedly, this would require measure- ments in the stress region of 2 to 3 Mbar; however, any measurements
above 100 Kbar would provide useful information.
It appears to be feasible to conduct an experiment with a room configuration such that the near surface burst geometry is preserved for early times. With soil motion and stress instru-
mentation suitably placed, the effects of the shock generated by the initial coupling could then be observed. However, the ability of the scientific community to successfully conduct an experiment
of this type has been seriously questioned since the capability to make the required temperature, radiative, and ground stress and
.'
AFWL-TR-74-205
motion measurements has never been demonstrated. Until considerable
confidence is developed in the instrumentation to be fielded, a
full-scale coupling experiment is not justifiable. This level of
confidence might be achieved through evaluation of the instrumen-
tation systems in add-on experiments to scheduled underground
nuclear tests.
Instrumentation for use in an underground nuclear event would
also be suitable for use in the region close-in to a large-scale
high-explosive event since the material flow in each case is
described by a hydrodynamic model. Any calculations of cratering
and ground motion for the purpose of assessing structures surviv-
ability/vulnerability must begin with the coupling of the source
into the soil; therefore, measurements in this region would be
essential to verifying the calculated dynamic soil loading.
The Air Force Weapons Laboratory has been conducting a program
of analysis, testing, and evaluation in an effort to develop
instrumentation which can make the required soil stress and motion
measurements. Included in this program have been investigations
of the Impedance-Mismatch-High-Stress-Transducer (IMHST) (ref. 1),
Thermoelectric-Thermopile-Transducer (T3) (ref. 2), Mutual Induct-
ance Particle Velocimeter (MIPV) (ref. 3), and waveguide displace-
ment system (ref. 4). Other systems, i>uch as the crescent velocity
gage (ref. 5), DX pendulum velocity gage (ref. 6), electrolytic
stress gage (ref. 7), and the laser interferrometer system (ref. 8)
were also considered but did not appear to have potential for making
the required measurements.
The MIPV was judged to have the best potential for providing
the soil motion measurements, and a development program consisting
of analyses and testing was initiated at AFWL in late 1971. The
purpose of this report is to describe the significant analysis of
the MIPV completed by AFWL. Recent testing of the MIPV is reported
elsewhere.
2. DESCRIPTION
The MIPV consists of two sets of rectangular-shaped loops of
wire. The gage is embeded in a media (geology, grout, epoxy, etc.)
of interest and oriented so that the shock front propagates along
8
AFWL-TR-74-205
its major axis. A constant current is supplied to the primary loop,
and a voltage which is proportional to the rate of collapse of the
loops is induced in the secondary. Any number of primary and
secondary turns with any desired grouping may be utilized. Typical
gage designs have utilized alternating primary and secondary loops.
Figure 1 is a photograph of a design utilized in gas gun test-
ing at AFWL. The loops are 0.020-inch diameter aluminum wire
embedded in a strap of aluminum-oxide-loaded polymethylmethacrylate
(PMMA) which electrically insulates the gage and holds the gage
dimensions. A capacitive discharge power supply provides a constant
current pulse to the gage primary. The secondary is connected to
a 50-ohm coaxial cable which in turn is terminated in 50 ohms at
the recording instrumentation.
3. BACKGROUND
In 1963, DASA, presently called the Defense Nuclear Agency
(DNA), awarded a contract to the Engineering Physics Company
(EPCO) for development of a soil particle velocity gage for use
in underground nuclear tests (refs. 3, 9, 10, 11). The gage was
to have the capability of making measurements in the 50- to 1000-
Kbar stress region. After considering several schemes, EPCO
selected the mutual inductance principle as the most promising.
Development continued through 1965 and resulted in what appeared
to be a reliable velocity gage. The gage was fielded in RED HOT
(ref. 12) in 1966 and DISTANT PLAIN, event VI (ref. 13) in 1967
with limited success. There has been no further use of the EPCO-
designed gage since these field tests.
Stanford Research Institute (SRI) since 1971 has been conducting
a program for DNA to determine constitutive relations in rocks and
soils by in-material stress and particle velocity measurements and
analysis techniques. During the initial phase of the program, SRI
selected the MIPV as having the best potential for making the
required velocity measurements (refs. 14, 15). Excellent results
have been obtained in gas gun and explosive tests (ref. 16), where
the gage was evaluated in the stress range of from a few Kbars to
40 Kbars. Recent fielding of the MIPV in ESSEX (1973-1974) has
produced good data in the 1- to 10-Kbar stress region.
AFWL-TR-74-205
(a) Assembled gage
(b) Cross section showing 20-mil aluminum wires in aluminum-oxide-loaded acrylic
Figure 1. MIPV Used in Gas Gun Testing
10
HfUMWiwUyypTi ■y■"W7^,■""^"""'"1"^^'1'^^":,1T■,^?■""^tr,"■■7>l■■'^■ ■■■
AFWL-TR-74-205
During the AFWL development program, gages were fielded in
MIDDLE GUST IV and V (ref. 17) and in MIXED COMPANY (ref. 18) which,
respectively, were 100-ton surface-tangent, 20-ton half-buried,
and 500-ton surface-tangent TNT events. During these tests, gage
survivability well after shock passage was demonstrated. However,
gage shorting due to electrical breakdown of the gage insulation
in the 150-Kbar stress environment was observed. This dictated
further investigation to obtain insulating materials which would
maintain their properties at high-stress levels.
A series of gas gun experiments was conducted at AFWL in 1973
to evaluate the gage response in the 15- to 50-Kbar region. Load-
ing, unloading, repeatability, off-axis response, and self-
generating response were investigated with very good results.
Testing in explosive and hypervelocity gas gun experiments was
conducted during the spring of 1974. The results of these tests
are to be published in a forthcoming AFWL technical report.
Three major analytical areas were investigated to provide an
improved understanding of the gage response to the intense shock
environment. These analytical areas, which are the subject of
this report, are presented in individual sections as follows:
Extension of Theory to Complex Gage Designs
Divergent/Off-Axis Flow Response
Electromechanical Modeling
The specific requirements for investigating these problems are
presented in introductory remarks in each section.
11
mill(PifTI»l^l)[j[j^r i «rrsi^^^mMI «TITVT"
AFWL-TR-74-205
SECTION II
EXTENSION OF THEORY TO COMPLEX GAGE DESIGNS
The MIPV theory of operation developed by EPCO was applied
specifically to a single primary, multiple secondary gage design
(ref. 3). Expressions were derived for a rectangular gage geometry
for calculating the gage mutual inductance, M, and dM/da, where
"a" is gage length (figure 2). These relations cannot be applied
directly in making calculations for gages consisting of multiple
turns of primary and secondary loops such as utilized in the AFWL-
and SRI-designed gages. A requirement, therefore, existed to
extend the basic theory so that gage parameters could be calculated
for a given design prior to fabrication. This permits designing a
gage to meet the requirements of a particular test.
The basic theory developing the expressions for mutual induct-
ance and dM/da will be summarized. The theory will then be extended
to consider complex gagt designs. The results of numerical inves-
tigation of the effects of changing gage design parameters upon
sensitivity will be presented, and design parameters will be
tabulated for several designs.
1. BASIC THEORY
Consider the simple model of the velocity gage shown in
figure 2. If the magnetic flux, *, linking the secondary loop
changes, a voltage, E, will be induced in the secondary that is
equal to the negative time rate of change of the magnetic flux
such that
The magnetic flux through the surface of the gage secondary
(in the x-y plane) may be expressed in terms of the magnetic flux
density, B,
12
'!^1g5^^TvrlTI''w^^'T,,1 '""V^
(fi'lv w.f*~**^mm
AFWL-TR-74-205
Figure 2. Mutual Inductance Particle Velocimeter Nomenclature
/A"5 (2)
where dS is an element of area of the surface.
A vector, A, the magnetic vector potential, may be defined such that
(■v*X) = t (3)
Applying Stokes' theorem.
f fayX)-dS =(p%-dtc (4)
13
*:!"^^T^,fT'r-t''^',,^iT:»^«r IfWTm-»--'-
AFWL-TR-74-205
where dl is an increment of length along the secondary loop.
Equations (1) through (4) may be combined to obtain the induced
voltage in terms of the magnetic vector potential
-^ t*dt (5)
The magnetic vector potential at any point, Q, on the secondary is
V^r/ Idt
(6)
where
y is the magnetic permeability of air
I is the primary loop current
, is an element of length aloi
d is the distance from point Q to the element dJ-
dl is an element of length along the primary loop
Combining equations (5) and (6) ,
E = ^#-
•Idt ^ —£'dl d s (7)
Since by definition the mutual inductance is the flux in the
secondary per unit current in the primary,
* = MI
and
dt M - Tiff^ (8)
14
|y|l.l|Wl,Jlj.H i.|H<."t|ii'Jil'y^yfiyirey.ir->miT-<-<Ft'
AFWL-TR-74-205
which is seen to be simply a function of geometry. Equation (7)
may then be rewritten as
H..^ (9)
v.
if we require that the current be held constant, then
E = dM at (10)
Consider figure 3 where a plane shock wave propagates down the gage major axis, x, at a shock velocity, c. Assuming that the loops flow with the media at the particle velocity, u, it. is apparent that
u = - da ar
Expanding the term dM/dt
dM . dM da (It " cTa ar
The expression for the secondary induced voltage now becomes
r T dM E = I a? u (11)
recalling that M is defined by equation (8). EPCO obtained a
closed form solution to equation (8) in terms of generalized gage
dimensions (ref. 3). Their result is repeated here as equation
(12).
M = -r IT
2r -2jr2 + b2' -2 Jr2 + a2" +2 Jr2 + b2+a2
+ a In ^r2+b2' (aWr2->a2') r(aWr2 + b2+a2')
+ b In hz*a2\b*k2*h2') r(b + p + b2 + a2')
(12)
i«;
r^Jf/Tr.;»^-7fl*f?^n-.T'^»-.:.'.';;^VmVn.<- 'TWVC-T»-»«»—' T ■
1 AFWL-TR-74-205
II .^^^ ^s. .UNDISTURBED GAGE GEOMETRY
Figure 3. MIPV Undergoing Deformation Due to Shock
where the nomenclature of figure 2 applies. Differentiating with
respect to "a,"
dM Jin (^2tb2')(aWa2^r2') t 3 (^a^b^r2' - vV-t-r2')
(r(a+^a2+b2+r2')) a2+r2
(13)
Equations (12) and (13) permit calculation of gage parameters for
simple geometries consisting of a constant distance, r, between
primary and secondary loops. This would be possible for a single
primary, multiple secondary (or vice versa) design such as that
of EPCO.
There is no way known to measure the term dM/da for a piven
gage design. To establish confidence in the gage sensitivity.
16
-■ 'TffWfÄ^^^^^PWWffP
AFWL-TR-74-205
some method of experimentally determining the magnitude of dM/da
is required. EPCO noted that by assuming
M dM 3¥ (14)
an experimental estimation of dM/da could be obtained simply by
measuring the gage mutual inductance and length. Numerical inves-
tigations indicated that for a gage whose length was 10 to 15 times
greater than the width, the difference between M/a and dM/da was
less than ^S percent. Further improvements to the determination
of dM/da are obtained by applying the error term as a correction
factor. The correction factor, n, is defined as
n =
dM M 3¥ a
dM 3ä
= 1 M
a ai- ds)
and dM/da is expressed as
dM M 1 (16)
In equation (15) M and dM/da are calculated values from equations
(12) and (13) , respectively. In equation (16) M and "a" are
measured values. This method of using a calculated factor (n) to
correct measured parameters (M/a) is applicable providing the calcu-
lated and measured values of M agree very closely. As will be shown
later, this has routinely been the case.
Extension of the previous analysis is required to calculate n
for more complex geometries.
2. LXTHNSION OF TliliORY FOR COMPLEX GAGE GEOMETRIES
Consider a gage geometry consisting of m primary and n second-
ary turns. For the ith primary turn and jth secondary turn, the
mutual inductance is M- ■ and the total gage mutual inductance M,
may be expressed as
17
AFWL-TR-74-205
M -t t Mij i-1 j=l
(17)
As an example, consider the 4-primary, 3-secondary turn design
depicted in figure 4.
1 1 2 2 3 3 4
o r
r 11
+ O
1 2
+ o A
Primary Turns
1 3
Cj Secondary Turns
Figure 4. Cross Section of MIPV Loops, Type I Gage
The mutual inductance is calculated as
M=t t MiJ i-1 j=l
M +M +M +M +M +M 11 12 13 21 22 23
+ M +M +M +M +M +M 31 32 33 i»l >»2 1)3
Note that due to the same distance, r, between loops
M = M = M = M " M = M • , 11 21 22 32 33 Hi
M = M = M = M , and 12 2 3 31 *2
M = M 13 •»!
18
"" ,* ■ ' ■ -. ■■V.-K.-r'.FU-
^■Äitota*^^
AFWL-TR-74-205
The mutual inductance may then be expressed as
M » 6M + 4M + 2M 11 12 13
',
In this case, by calculating the mutual inductances M . M , and ' 0 1112
M with equation (12) utilizing the distances r , r , and r , 13 n e 1112 13
the total gage mutual inductance may be calculated. By requiring
that the loops be separated by the same distances and specifying
that there is one more primary turn than secondary turn, the generalized expression for gages with alternating primary and
secondary turns is
:
M = £ I n+m+l-2j M1;j (18) s
|
Recalling equation (15) and substituting equations (16) and (17),
n = l
m n
£ S M. . i=l ftl jj
m n M. .
i=l 3=1 1-n^
(19) ■
■l - i
where
13 = 1
M ij dH
da
and dM- /da is calculated from equation (13). Utilizing equation
(18) in equation (19),
19
fipfTZHtf.r* ■ ■■"■ 4 ■ -, '.■■7VT1—y-
AFWL-TR-74-205
n r -I V n+m+l-2j M1.
■■'-44 4— I n+m+l-Zi dM J ^
(20)
which applies to the same geometry as that associated with equation
(18). A more general expression for n would utilize equation (17)
in equation (15), resulting in
n m
n . ! - Ö_JB n m i i dM. .
U i=l j<l da
(21)
Considering the example from before (figure 4) ,
n = 1 6M + 4M + 2M
1 1 12 13 —ar m ar-
a |6 ^-Li- + 4 ^-^- + 2 13
da da da
Equations (11) and (16) may be combined to define a term, gage
factor, GF, such that
GF E _ M / 1 \ ÜT a \lTny (22)
where
M is measured mutual inductance, henries
a is measured length, meters
20
rTf*rs}~WV.T
AFWL-TR-74-205
n is calculated from equation (21)
GF has the units volts/amp • mps but is more conveniently expressed as mv/(amp • mm/psec)
Thus during a test, two measurements, secondary output voltage and primary current, must be made to obtain particle velocity.
3. NUMERICAL RESULTS
A generalized computer code (DSIN1, appendix A) calculates
gage mutual inductance, gage factor, and error term (or correction factor) for gages utilizing alternating primary and secondary turns
This will be referred to as the Type I gage. A specialized code
(DSIN2, appendix A) calculates the same parameters for the designs shown in figure 5. This will be referred tu as the Type II gage. Numerical results for a variety of designs for each type gage are tabulated in appendix A.
oo@oo
ooo@ooo
oooo@oooo 9 Loop Set A
O Loop Set B
OOGOOOOO
OOO0OOO@OOO
00®O0@OO@O0
ooooooooooo^ooo Figure 5. Cross Sections of MIPV Loops, Type II Design
a. Comparison of Calculated and Measured Mutual Inductance
Table 1 shows the results of calculations made for gages
designed for fielding in MIXED COMPANY, a 500-ton TNT event.
Measured values of mutual inductance are compared with the
21 v
AFWL-TR-74-205
calculated values and the error between the two indicated. Induct-
ance measurements were made with a GR Type 1632-A Inductance Bridge.
The slight variation in measured values is primarily due to slight
variations in length between individual gages which results from
the fabrication technique. For gages which have a length-to-width
ratio, a/b, greater than approximately 5, the mutual inductance
essentially varies directly with length. Therefore, by considering
the individual gage length and controlling other gage dimensions, a
very accurate value of the gage factor, which, as will be shown, is
essentially independent of gage length, may be determined. The
results of table 1 are typical, i.e., gages can be constructed
routinely to within less than 1 percent design gage factor.
Table 1
COMPARISON OF MEASURED AND CALCULATED VALUES OF
MUTUAL INDUCTANCE FOR MIXED COMPANY GAGES
Gage No.
Measured mu inductance
tual (uh)
Calculated mutual inductance (yh)
Error m
1 4.411 4.356 1.2
2 4.385 4.356 0.6
3 4.398 4.356 0.9
4 4.386 4.356 0.6
b. Parametric Analysis
Figures 6 and 7 show the influence upon gage factor of
varying gage width, spacing between turns, and number of primary
and secondary turns for typical Type I gage designs. The results
are not unexpected, i.e., increasing gage width, increasing number
of turns, and decreasing loop spacing all result in increased gage
factor.
Since, as shown in figure 6, a wide range of gage factors is
available, the gage may be used over a wide range of particle velocity
inputs and still produce measurable signals. For instance, a signal
of 400 mv is produced for a O.l-mm/ysec velocity input («5 Kbar)
with a gage factor of 40 mv/(amp • mm/ysec) and 100 amperes current
in the primiry. For applications in the stress region above 100
22
••-»V»-"
AFWL-TR-74-205
SO
40
30
a e
O
U
a
20
10
Primary Secondary
I
Figure 6. Gage Factor as a Function of Width and Number of Primary and Secondary Turns, Type I Gage
23
AFWL-TR-74-205
50
40 H
P. 6
o *-» u «a u,
oo CO u
T r
30 H
20i
10H
r=.06
o-t
a = 30 in b = 2.0 in
T
Primary 2
Secondary 1
T 1 r r
3 4 5 6
2 3 4 5
TURNS
Figure 7. Gage Factor as a Function of Spacing between Loops and Number of Primary and Secondary Turns, Type I Gage
24
■pvRwn |.||IWTWW^' "^"wwufl! »mw"™*'. ^"'*"'■
fi;* ■■■:'':,'*i:- mmmmmmmtmir.^- •mmmm
AFWL-TR-74-205
Kbar, the particle velocity would generally be greater than
1 mm/ijsec, and signals above 2 volts are easily obtainable.
Note in figure 7 that loop spacing has a relatively small
influence upon gage factor. This would suggest that one would be
free to choose from a wide range of wire diameters without
seriously affecting gage factor. Past designs by AFWL have
utilized wire diameters varying from 0.020 to 0.0625 inch. Loop
spacing is generally 2 to 2.5 times the wire diameter to ensure
adequate separation to prevent mechanical contact and hence
electrical shorting between loops. Typical designs employ spacings
ranging from around 0.040 to 0.160 inch.
In many geologies at very high stress levels and for long
duration stress pulses, a gage deformation of up to 50 percent may
be experienced during the shock transit. This introduces questions
as to the influence of changing gage length upon gage factor. The
numerical results shown in figure 8 indicate that certain geomet-
rical limitations do exist, beyond which the gage factor would
undergo significant changes. For a length-to-width ratio greater
than 4 or 5, the gage factor for this design (which is typical) is
constant within 0.5 percent. Therefore, a gage of any desired
length might be utilized, as long as the length-to-width ratio
after shock transit did not decrease below 4 or 5. '
The correction factor, n» is plotted in figure 8 to clarify
its role in determining gage factor. Note that as a/b decreases,
large changes in the value of n result. This is of little consequence
since the proper application of n is in determining the initial value
of the gage factor. The results of figure 8 clearly indicate a
constant gage factor for a/b > 5 even though n is obviously changing.
c. Comparison of Type I and Type II Gages
To form a basis for a detailed comparison of the Type I and
Type II gages, many factors such as desired signal rise time, record-
ing duration, stress and particle velocity, geology, and type and
length of cables must he considered. This, however, would be a very
involved discussion and beyond the intent of this analysis. There-
fore, a simple comparison is made which illustrates how one of the
25
AFWL-TR-74-205
16
o 0) (A
£ E i
Cu B a)
O +J O rt
0) 00 rt
14 -
12 .
10
T f 1
K /— Gage Factor
MJ
s^^_^ / /
\ " ■""
^. ^---C ■ ^^^J
1 L. ' • 1 1 1 1 -^1—
60
5 6
a/b
- -40
- -30
. -20
- -10
10
n(*)
Figure 8. Variation of Gage Factor and n with Gage Length-to-Width Ratio for a Typical Type I Gage Design
designs can be used to advantage. In general, an analysis based
upon the requirements of a particular test would be required to
identify the most favorable design.
Consider figure 9 where it has been specified that loop
set A in the Type II gage is the secondary, and the definitions
set forth in figures 4 and 5 apply. Significant increases in gage
factor are available with the Type II gage without increasing the
number of secondary turns. This is important in that likewise there
will be no increase in the secondary self-inductance and thus no
increase in electrical rise time. This advantage would be partic-
ularly important in improving the rise time performance for gages
designed for the low kilobar stress region where a large gage
factor, and hence many primary and secondary turns, are required.
4. SUMMARY
Gages of any desired design may be fabricated routinely to
within 1 percent of the design gage factor. In the Type II gage
26
»■TRT-TTisTf^r»,- T't";?'n"*-'■-"•--
AFWL-TR-74-205
significant increases in gage factor are available without affect
ing gage rise time. A complete analysis would be required to
determine the optimum gage design to meet the requirements of a
particular experiment.
Considering only the requirement for a constant gage factor
duriTig gage deformation, any length gage may be chosen as long as
a/b > 5 after shock transit.
T X
40-
u a>
e 30 i
e
o +J u u.
20-
10-
O D o A
n
Type I Gages oo«o o ooo»ooo oo•oo«oo ooo«ooo«ooo oo«oo«oo«oo ooo«ooo«ooo«ooo
Type II Gages
a = 30 in b = 2.0 in r ■ .08 in
T 1
T
3
Secondary Turns, Type I Gage Secondary Turns, Loop Set A, Type II Gage
Figure 9. Comparison of Types I and II Gages
27
P|IWWA''?W"W1'"'! ''"r »Hl,,'^r-«»"^f^»<1 iWTwmrrr^^-'TTW'—- r.-f—fpT'^HP l'^WÜHT-r^.' r™ ' •■'
AFWL-TR-74-205
SECTION III
OFF-AXIS DIVERGENT FLOW RESPONSE
The theory previously presented in section II was based upon
one-dimensional flow along the major axis (x-axis) of the gage.
This environment is easily created in laboratory-type experiments
utilizing flyer plates or plane wave explosive lenses, and in fact
these types of experiments are very useful in comparing the gage
experimental and theoretical responses. However, in large-scale
TNT or nuclear events, the flow field will likely be oriented off
the gage major axis and be spherical in nature. Flow components in
the y or z directions will clearly distort the gage geometry and
consequently change the gage mutual inductance. The end result is
a response which is not totally defined by the one-dimensional,
on-axis analysis. To establish the usefulness of the mutual induc-
tance particle velocimeter as a field gage, it is necessary to
investigate the sensitivity of the ga^e to nonideal flow effects.
1. DEVELOPMENT OF COMPUTER MODEL
A simplified gage design, consisting of one primary and one
secondary turn, is chosen to reduce the computation time which
becomes prohibitive for multiple turn designs. The geometry of
the problem is shown in figure 10, where the coordinate system
origin is located at the center of the front end of the gage, and
the source is located at some selected position (x ,y ,z ) ahead of
tue gage. A spherical shock front propagates at a velocity c, and
behind the shock front the material flows outwardly at a velocity
u, along radial lines emanating from the source. Typical values
of u and c are chosen to simulate realistic field conditions. A
step particle velocity is chosen to investigate the worst case
and to simplify the calculations. The conservative relations would
not allow an actual step in particle velocity for constant stress
loading at the source of the spherical flow. However, where the
spherical flow approximates one-dimensional flow (source at a
relatively large distance from the gage), this is a good approxi-
mation.
28
,i'i—WÄ '"^P «»Ill-|JiTV<-i-n-r<-X" ..--,T-,.-.„_,
AFWL-TR-74-205
>
>-
/
o
o
i UL,
V) •H X <
«4-1
o
c
l-<
>
«4-1
O
c 0> E P. O f-i
> Q
U O
<+*
<ü Si
4J
t—i
u c 0) 6 o
0)
3
• H
29
AFWL-TR-74-205
As shown in figure 10, the gage primary and secondary are
divided into small segments. Time is stepped in increments of At,
and as the shock front propagates through the gage, the segment
positions are calculated. At selected intervals, the gage mutual
inductance is calculated, resulting in the mutual inductance time
history, M(t). By numerically differentiating M(t) with respect
to time
M. Mt^t • Mt-At
TKt
M(t) is determined. M(t) is then multiplied by the current
(assumed constant) to give the gage output voltage E(t). This
output is then compared with the one-dimensional, on-axis response
calculated by equation (11),
The mutual inductance is determined at each time increment by
first calculating the magnetic vector potential at each segment
midpoint on the secondary utilizing the finite difference form of
equation (6)
At *io-£2£ (23)
and then using the relationship
M = T Z X'At (24)
which results from equation (23) and the finite difference form of
equation (8) .
A computer code, DIVE, was developed to make the calculations.
This code is listed in appendix B.
30
■HWIIMiiii '■ * n'v .-^w-i.^ »i —r—T" -w-rw-^-mrr-
AFWL-TR-74-205
2. SUMMARY OF NUMERICAL RESULTS
Initial numerical investigation of the problem involved select-
ing time increments and number of primary and secondary segments
which would minimize the computation time while providing a
reasonable accuracy in the numerical results. M for the real flow
model is calculated by placing the source on-axis and at a great
distance so that one-dimensional flow is approximated. It is then
legitimate to compare the ideal and real flow results for the
purpose of determining computational accuracy. The result was a
gage consisting of 868 segments for each loop. The difference in
the numerically calculated mutual inductance and that determined
from equation (12) was approximately 0.008 percent. The difference
in the numerically calculated ^ and the ideal Ä was less than 0.2
percent. It is apparent that the errors due to the finite elements
are insignificant and may be neglected.
All numerical results are based upon the following gage design
and flow environment:
Gage length
Gage width
Loop spacing
Primary turns
Secondary turns
Shock velocity
Particle velocity
19,5 inches
1.5 inches
0.080 inch
1
1
4000 meters per second
2000 meters per second
The values of the shock and particle velocities are important
here only in their relative magnitudes. A worst case of c/u = 2
is selected here, where the case of very high pressure (over 500
Kbar) in geologic materials is approximated. At lower pressures
c/u » 3.
As a first analysis of the real flow effects, it is considered sufficient to investigate only the single gage geometry described
above. The expense of expanding the effort to include other geometries is not justifiable, since it is intended here only to show the general effects of the real flow upon a typical gage
design.
31
AFWL-TR-74-205
The effects of divergent flow are shown in figure 11. Here
the results are shown as a percentage error in gage response as
a function of time. The source is located on the gage centcrline
in each case. Note that for a source position as close as 1/2 meter,
the error in gage output is less than -2 percent and for 5 meters
and beyond is less than -0.2 percent. Generally, it may be con-
cluded that for anticipated field flow environments, the error
generated due to spherical flow is negligible.
Figures 12 through 16 show the off-axis response of the gage
in terms of the output of the gage, normalized by the ideal output.
Source radius was located at 5 meters in every case to observe
only the off-axis effects. The two major effects are identified
clearly in figures 12 and 13.
In figure 12 the angle off in the x-z plane, 6, is held at
zero, while the source position is varied by the angle y in the
x-y plane. The obvious effect is an increase in rise time and
slight overshoot before returning to essentially a constant level
slightly below ideal. For y t 0, a finite time interval is
required for the shock to transverse the front of the gage, and
hence the increasing rise time with increasing y is observed.
Further analysis would lead to the intuitive notion that the
output of the gage would be reduced in proportion to the angle y,
or simply, to the x component of the particle velocity u , where A
u = u cos Y
However, this is not the case as seen in figure 17, where the
normalized velocity is compared with u . It is apparent that
there must be some compensating effects to explain the discrepancy.
Observe in figure 18 the nature of the gage deformation. Since
shock arrival occurs at one side of the gage prior to the other,
the gage width is reduced somewhat behind the shock front. The
effect is to decrease the mutual inductance with time and hence
results in an additional positive output voltage in the gage
secondary. The result is that, except for the early time peak, the
gage is relatively insensitive to off-axis flow in the x-y plane.
32
vop'W'^-" '^-»imi*""i*'z"i'w""
AFWL-TR-74-205
C/5 10 CO
- 0D<3
0)
Ln
CD IS
0» 4) (U u O u c C C « 03 rt 4-> *-> 4-> I/I tn trt
O Q Q 0) <u u u
3 3 O O
U
3 o i
r o
o CVJ
o o
o 00
o (£1
u
3
o
o
to I
V) c o a w 0) & (U «50 rt U c o D. 3
O i-t li,
(U 00
>
o
(/> +J o
UJ
(1
3 oc
auooaoj ^ aoja'j
P^WWWWW 4 i IM, ■■ ' t HI'JIP M i.piWB|n»ii|T~<Tr,-v^if;~.
AFWL-TR-74-205
1.0
.8
U A O . u l-l 0) >
0) N •H
o
.4
.2
.2
o Y = 1U
G Y = 20
A Y = 30
e = 0°
-L J. 10 20
Time ^ (ysec)
30 40
Figure 12. Effects of Off-Axis Flow upon Gage Response
34
AFWL-TR-74-205
1.2
1.0
.81-
o iH «i >
0) N
•H 4
* 1
o
.21-
^><><><X(><>0 O
/DQOOOOG D-
P^AAAAA-
O e D e A e
= iol
j-
Y = 0
X
20" 30°
o
10 20
Time ^ (ysec)
-D-
A-
30 40
■/
:
■ ■,'.
■
Figure 13. Effects of Off-Axis Flow upon Gage Response
35
AFWL-TR-74-205
10 20 Time ^ (usec)
Figure 14. Effects of Off-Axis Flow upon Gage Response
36
AFWL-TR-74-205
10 20
Time ^ (ysec)
T \ -I r -
1.0 ■ Q^ -0 —o— -
.8 poo^- -D —D -
>s |AAA^^_ A A
o -o i—i I
u -
Normalized [ -
.2 - -
0 0 e Dö A e
Y
l I
= 10o = 20°
= 30°
= 20°
i
-
30 40
Figure 15. Effects of Off-Axis Flow upon Gage Response
37
AFWL-TR-74-205
10 20
Time ^ (ysec)
30 40
Figure 16. Effects of Off-Axis Flow upon Gage Response
38
p-^-.f,,,,--,■-„;, .-„^j,.^.
f;::%;^'
AFWL-TR-74-205
^ c •H O U-H o <->
rH U <ü C > 3
U< 13 0) 0) N C
•H -H iH (/) rt o EU u o 2
.4
cos Y»0 or u cos (9+e) DIVE, Y=0 DIVE, 9=0
X X X 10 20 30
Y,6 (Degrees)
40
Figure 17. Comparisons of DIVE Calculations with Cosine Function
Figure 13. Gage Undergoing Deformation for Off-Axis Shock, 6=0, y^O
39
TK;*"jr:nf7Tm>:'^,T''^''**" "?£'""»■•/■/T"- ■ ■-'
r^^VMi
AFWL-TR-74-205
In figure 13 the shock propagation is off-axis by the angle 6 in the x-z plane, while y = 0. The gage rise time is relatively unaffected, while the level is significantly influenced. It must
be pointed out that for multiple primary and secondary turns, the z dimension will be much greater, and hence, the rise time would also be increased.
Comparing the gage response with u in figure 17, an error in
gage response much greater than anticipated is observed. This is
explained by considering figure 19. Note that as the gage deforms, its orientation relative to the shock front propagation direction
changes so that the effective component of u down the deflected gage axis is not u cos 9 but rather u cos (0+e). In figure 17 cos (9+e) has been plotted, and we see that it conforms fairly well with the actual gage response, especially for 6 ^ 20 degrees. The angle e is obviously a function of the ratio of c/u, therefore, as
c/u increases, e would, for a given 6, decrease. Again, for c/u ■ 2, the worst case results.
Figure 19. Gage Undergoing Deformation for Off-Axis Shock, y=0, 6^0
40
'"(^f^yi"-Tn,,rT¥i77i?03r'""r^T'T-r" ■:—-^ ■■
AFWL-TR-74-205
PW^^ff
The intention of figures 17, 18, and 19 and the related discus-
sion is to attempt a logical explanation of the numerical results
and provide an improved intuitive feeling for the gage response.
The remaining data in figures 14, 15, and 16 show the gage response
for various combinations of 6 and y. As might be expected, it is
observed that the effects protrayed in figures 12 and 13 combine to
provide the results of figures 14, 15, and 16.
3. SUMMARY
The results of the numerical investigation of the off-axis/
divergent flow response are summarized as follows:
a. For anticipated field flow environments, the error gener-
ated due to spherical flow is negligible.
b. The gage is relatively insensitive to off-axis flow in the
x-y plane, the error induced being approximately equal to 1/2
(1 - cos y)• However, off-axis flow in the x-z plane results in
much larger errors approximately equal to 3 (1 - cos 6). An
immediate result is that if in a field test there is a plane in
which the uncertainty in shock propagation direction is a maximum,
then the preferred orientation of the gage is with the x-y plane
aligned with the plane of uncertainty.
AFWL-TR-74-205
SECTION IV
ELECTROMECHANICAL MODELING
The development in section II leading to equation (11), the
expression for the gage output voltage, was based upon the assump-
tions that the media and gage flow were ideal, i.e., one-dimensional
and aligned with the gage major axis and that no electrical param-
eters other than the gage mutual inductance were changing. The
nonideal flow effects were investigated in section III. In this
section the effects of nonideal electrical behavior are treated.
It is obvious that as the gage shortens, the primary and
secondary self-inductances will decrease. Also, the gage resist-
ance will change due to the very high stress behind the shock
from . These effects are totally ignored in the analysis of
section II and must be considered separately. The approach is to
evaluate the effects of changes in self-inductance and resistance
on gage response and, by proper circuit design, control the effects
so that equation (11) adequately describes the gage response.
1. PREVIOUS WORK
The Engineering Physics Company, during their development
program, performed a detailed analysis of the error generated in
the gage output by changes in self-inductance and resistance in
the gage primary (ref. 10). A closed form solution was obtained
for a model utilizing a voltage source as the power supply. The
solution was developed by linearizing the differential equations,
neglecting self-inductance and resistance changes in the secondary,
and making other simplifications.
An analysis was performed (ref. 16) that used a simplified
voltage source model, linearized equations, and neglected resist-
ance changes in the gage. This is an adequate model for applica-
tions in the low Kbar stress region.
Neither of these analyses were considered completely adequate
for application in the very high-pressure region because of the
42
?*wir«T"l"^T»'~7 ^•"••'."«..Mrrv'T-^rTrr'» ■■ i-
AFWL-TR-74-205
approximations used. A more detailed model was required that would consider inductance and resistance changes in the primary and secondary. Additionally, it was desirable to incorporate a capaci- tive discharge power supply in the model.
2. ELECTROMECHANICAL MODEL
A simplified schematic of the power supply and gage, assuming an open secondary, is shown in figure 20. The elements L and R are controllable parameters used to minimize the undesired effects of changes in self-inductance and resistance in the gage loops. The basic problem is to determine the proper values for L and R
for a given gage design under some specified test conditions. The circuit in figure 20 is described by a set of nonlinear differen- tial equations with time-varying coefficients which are not amenable to closed form solution by conventional methods.
a. Approach
The approach taken in obtaining a solution is to determine a set of first order differential equations in the normal form (state variable equations) that will describe the electrical model. Given initial conditions, these equations may then be solved to obtain numerical results. Due to the ease with which more complex models may be handled by this method, it is possible to utilize a detailed model of the gage that introduces as few assumptions as practical.
The electrical network that models the system is shown in
figure 21. Elements 2, 6, 12, and 13 represent the power supply; elements 1, 10, and 11 represent the primary cable capacitance, control resistance, and inductance; the gage consists of elements
5, 9, 4, and 8; and elements 7 and 3 represent the secondary line cable capacitance and termination resistance. In the network, it
is assumed that the distributed parameters may be lumped. It now
remains to determine the appropriate mathematical model that
describes the network.
b. State Variable Equations
Stern (ref. 19) describes the procedures for applying the
linear graph and electrical network theory of Seshu and Reed (ref.
43
ry™WT**r7T?*^.^T:-yc'i}tyj^Try*mFvrw'-.rKr-*'vv----''i~r ■"■■ ■**"■.,
AFWL-TR-74-205
■n^
Figure 20. Simplified Schematic of Power Supply and Velocity Gage with Open Secondary
Figure 21. Electrical Network of Velocity Measuring System
44
AFWL-TR-74-205
20) to determine the linearly independent loop and node equations,
equations (23) through (35), which mathematically describe the
network. A graph of the network is shown in figure 22. The equa-
tions are based upon the choice of the sign convention shown in
figure 21 and the tree of the graph shown in figure 22.
\
\ \ \
Figure 22. Graph of Network Shown in Figure 21, Tree Indicated by Solid Lines, Branches by Dashed Lines
e = e 1 6
e = e 2 6
e = e 3 7
e = e -e 13 6 12
e = e + e i» 7 8
e = e - e 5 6 9
i =-i - i 6 1 2
i =-i - i 7 3 t
i = i 12 13
i =-i e k
i = i 9 5
i = i 1 0 5
i = i 1 1 5
e - e 10 11
i - i 1 3 5
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
45
»^VT»7rWTrmTff*^*If ^ TTfTT-'' "rz-T.-^r-.-r .-n-
AFWL-TR-74-205
The tree branch charges, q , q , and q , and chord flux «7 n12
linkages, X and X , are chosen as state variables. By applying H 5
the basic relations
q - Ce,
X - Li,
e ■ iR,
q « i,
X ■ e,
X = L i + M i
X = L i + M i 5 5 5 •♦
apply in reducing the state variable equations to the form
to equations (27) through (31), expressions for the state variables
may be obtained resulting in
X = e + e •♦7 8
X = e - e 5 6 9
q =-i - i 6 1 2
q =-i - i 7 3'»
q = i 12 13
e - e 10 11
i - i 13 5
The basic relations are then applied throughout, except for
elements 4 and 5 where the coupling relations
1 1
(36)
(37)
(38) 1 (39) i
(40) :4 :i |
ept for f
46
AFWL-TR-74-205
i - f(x,t)
The resulting equations are
q R 7 8
C~ " IT 7
L X - MX 5 * 5
(41)
r n irq /R + R v L X - MX
>f s I
H'1 L - MX - X
\ L D/ ^ . _LL_(MX
- / D2 \ ^ L X
l» 5
(42)
_ c + c 6 1 6 2 13 \ 6 12/
ü (L.xs " M\) (43)
S = q7 or
3 7
IfLX MX 5 <♦
•)
(44)
12 - ir (— r 13X6 1
2
2 i
(45)
where
D = L L - Mz
i* s
D = LL +LL -2MM "♦5 5 ^
47
r m ^pTflW^'-'l"!'''-^» -II^I
AFWL-TR-74-205
c. Time-Varying Parameters
Note that the state variable equations are functions of the time-varying circuit elements L , L , R , and R and also L
. i» 5 « 9 <♦
and L . To proceed with a solution, expressions nust be obtained for each of these variables. An analysis of the physical model is
necessary to determine these relationships.
A step stress and particle velocity and constant shock velocity are assumed for the physical input parameters. Figure 23 depicts the gage deformation, Aa, as a result of the particle
velocity, u.
For typical gage designs, where the length is greater than
five times the width, the self-inductance is essentially propor- tional to the length, and the inductance-per-unit-length is
independent of gage length. Therefore, the self-inductance may be
expressed as
L = L0 (l + Aa/a0) , Aa < 0 (46)
where L is the initial self-inductance, and a is the initial gage length. L is calculated by extrapolation of the data of reference 21 or by measurements on actual gages.
Differentiating equation (46) with respect to time gives
1 - io f/ao]= -u |yao] ^
which is seen to be constant for a step velocity input.
The expression for resistance variations may be determined
by considering the portion of the gage behind the shock exposed to
high stress and the resistance change that is associated with that,
stress level. The situation is described in figure 24, where no
consideration is given to gage deformation. For an initial undis-
turbed gage resistance of R , length a, and width b, the resistance
per unit length is
R0/2(a + b)
48
AFWL-TR-74-205
m a •
Figure 23. MIPV Undergoing Shock Deformation and Nomenclature for Determining L(t) and L(t)
l_ \
L-Aa-J
1 1
<
u
Figure 24. Nomenclature for Determining Gage R(t)
49
AFWL-TR-74-205
The length of the shocked portion of the gage is
b + 2 xc
where xc is the shock position relative to the position of the
front of the gage prior to shock arrival. The remaining length is
b + 2 (" - *c)
By assuming that behind the shock the gage resistance is changed by a factor K, the expression for the total gage resistance is seen to be
R = Ro
2(a + b) Tb + 2(a - xc) + K(b + 2 xc)j (48)
Equations (47) and (48) apply to both the primary and
secondary loops in the gage.
Appendix C presents a study of various conductor materials
as applied in the MIPV. Included in this study is information on
the variation of resistance with stress level.
3. NUMERICAL RESULTS
Computer calculations were made to investigate this problem
on an AFWL CDC 6600 utilizing a modified Rung-Kutta-Gill/Adams-
Moulton method variable step size integration routine (ELMEK,
appendix D). First, the program was run to simulate power supply
turn-on and allow sufficient time for the transients induced in
the gage secondary to decay. The shock was numerically positioned
well away from the gage so that only the turn-on effects were
present. Then the final conditions were input as initial conditions
and runs made to observe the gage response to the shock.
The equations were found to be quite sensitive around the turn-
on initial conditions, with the automatic step size feature con-
verging to less than 10-nanosecond steps during start-up. A
constant step size of 5 nanoseconds was selected and utilized for
all subsequent calculations.
50
AFWL-TR-74-205
Because of the extremely wide range of stress and particle
velocity environments in which the MIPV may be used and the differ-
ent designs available, there are numerious test conditions which
justify numerical investigation. Examples are
• Large-scale HE tests (such as MIXED COMPANY) with field-
size gages (typically 2 inches wide by 30 to 40 inches
long, 4 primary turns and 3 secondary turns) exposed to
the 50- to 150-Kbar stress region.
• Large-scale HE tests, free-field measurements in stress
region below 5 Kbar, field-size gages (same as above but
with 10 primary and 9 secondary turns) .
• High-pressure evaluation experiments (explosively driven
flyer or gas gun) in 0.2- to 1.0-Mbar stress region with
scaled gages (0.4 inch wide by 3 inches long, 3 primary
and 2 secondary turns).
• Underground nuclear test, field-size gage in 0.2- to 2.0-
Mbar stress region.
The two most severe conditions would be those in the megabar stress
region--the high-pressure evaluation experiments and the underground
nuclear test. These two conditions will be investigated and
numerical results presented. Other test conditions may be investi-
gated as the need arises.
a. High-Pressure Evaluation Experiments
The MIPV was evaluated in May and June 1974 on the Von
Karman Gas Dynamics Facility Range G, two-stage light gas gun
(ref. 22), located at Arnold Engineering Development Center (AEDC)
near Tullahoma, Tennessee.
Two basic gage designs were evaluated at AEDC. The first
consisted of 0.020-inch aluminum wires embedded in aluminum-oxide-
loaded acrylic, as shown in figure 1. The second design, shown in
figure 25, consisted of 0.020-inch titanium wires in 0.064-inch
OD by 0.024-inch ID borosilicate glass tubing backfilled with C-7
epoxy resin. Titanium was utilized with the glass because it would
51
>. r« ' '«MNWWM
Figure 25. Glass Gage
better withstand the temperatures required for the glass forming.
Both the aluminum and titanium wire designs were investigated but
only the titanium gage will be discussed here since, as shown in
appendix C, it presents a more severe perturbation to the gage
response. The problem is further complicated by the fact that
titanium, as a transition metal, undergoes an increase in electrical
resistivity by a factor of 2 at high pressure. This condition must
also be considered.
In preparing for the tests at AEDC an extensive investiga-
tion of the nonideal electrical behavior of the gage was made.
Details of the gage design investigated which approximates the
AEDC design are
Length 6.0 inches
Width 0.4 inch
Spacing between turns 0.050 inch
52
wrT^iT,'^T-3;^^T■•'~,^•'" ■VT-::'T;,r-r^ ...,-,,-;---
AFWL-TR-74-205
ff^-mm.
Primary turns
Secondary turns
Conductor material 0.020 titanium
Insulator Horosilicate glass
This design resulted in the following electrical parameters:
R 2.017 ohms pr i
Lpri = 1.68 uh
R = 1.345 ohms sec
L =0.9 33 \ih sec
M = 0.655 uh
GF = 4.179 mv/(amp mm/ysec)
The shock environment selected was
c = 9.7 mm/ysec
u = 5.25 mm/ysec
a = 1 Mbar
K = 0.4/2.0
The investigation was accomplished in two phases. In
addition to determining suitable control parameters for this
particular gage design and shock environment, there was a require-
ment to verify the code behavior, gain an understanding of how the
nonideal electrical behavior affects the MIPV response, and
determine the most effective means of controlling the induced
errors.
In the first phase, several values of ballast inductance,
L , and ballast resistance, R , were selected to observe their 11 10
effect upon the sliock-induced errors.
In the second phase, conditions were set up to observe the
response for some extreme values of the control parameters while
restricting the code to the following calculational modes:
53
AFWL-TR-74-205
(1) Normal--inductance and resistance changes are
permitted
(2) Inductance changes only
(3) Resistance changes only
(4) No changes in inductance or resistance
Figures 26 and 27 show the effects of varying R and L
upon gage response. For K < 1, the induced errors are negative;
for K > 1, the induced errors are positive. The ballast inductance
appears to have a strong effect in controlling the magnitude of the
error while the ballast resistance has a relatively minor effect.
The implication is that the errors can be adequately controlled
solely with the ballast inductance. This result, which will be
verified, is significant in that the amount of primary resistance
has a direct effect upon the magnitude of the current in the primary
and hence dirccly affects the gage sensitivity.
Now, it is of interest to determine the manner in which
the errors are generated in the gage and verify the calculations
of ELMEK.
Consider figure 28 where only inductance changes are
permitted. Where no ballast inductance is provided, the rise time
response is affected, but no error results in the magnitude. Where
ballast inductance is included, the response is identical with the
ideal response within the normal rise time capabilities of the
system, regardless of whether or not there is ballast resistance.
In figure 29 only resistance changes are permitted. It
will suffice to discuss the results for either of the values of
the resistance change factor, K, since qualitatively the effects
of one are, in a sense, reflected about the normalized voltage
value of 1. Consider the case for K = 0.4 where the induced errors
are negative. The first observation is that■in no case is rise
time degraded. Considering the result of the previous discussion
of figure 28, it may then be concluded that the primary effect of
self - inductance changes in the gage is to degrade the rise time
of the gage output and that the ballast inductance is very effec-
tive in controlling this degradation.
54
AFWL-TR-74-205
i.o U
.8 L
<-> rH O >
(U
e »-> o
.6
.4
Nomenclature: X/X
2/300 5/300 8/300 2/100 5/100 8/100
X -L
microhenries
ohms
5
Time
10
ysec
15
Figure 26. Shock Response of AEDC Gage as Calculated by ELMEK, K - .4
55
|fflf,^:r»1tT^«fW)'F». '-TI '
im*
AFWL-TR-74-205
a»
l-H . 6 o > T3
E
O 4L
oh
Nomenclature:
8/100 5/100 2/100 8/300 5/300 2/300
10
L1 - microhenries
R - ohms
15
Time ~ ysec
Figure 2' She ock Response of AEDC Gage as Calculated by ELMEK, K = 2
56
AFWL-TR-74-2Ü5
00 a
o >
•a
6 U o
—1 1
Nomenclature:
r-
x/x
—l
2.5 - VV —L - microhenries - 11
—R - ohms 10
2.0 - -
1 .5 -
y-n/300
-
i. n -
/yr2/.^00
lY
r\~~z/0
.5 - 1 ^-o/o -
0
...... _J l •
5 10
Time ~ ysec
15
Figure 28. Calculated Gage Response Using ELMEK, Inductance Changes Only, AEDC Gage
57
AFWL-TR-74-205
2.5
2.0
<u bo (0 *-> rH O >
0) N
■H rH rt E U o z
1.5
1.0
.5
Nomenclature: X/X/X
±
Resistance Change Factor
•L -microhenries 11
J. 5
Time
10
ysec
JO/500/2 ■^2/300/2 -.-0/300/. 4 7/300/.4
2/0/ A
0/0/ A
15
Figure 29. Calculated Gage Response Using ELMEK, Resistance Changes Only, AEDC Gage
58
P«W/,?•-yy.^TT':'^r^'^"-'-'?-"TT.'.I1 rTl1^ "I■»y^ '.' ■• IT* '^"'17:". ■'"■' I' ■■ " "l T^JTiyvwyi—yryy; ■—■ ,
•UK"
AFWL-TR-74-205
The next observation is that where there is no ballast
inductance, significant errors result which are only moderately
affected by the ballast resistance. It is thus concluded that
resistance changes in the gage produce large errors, but effective
control is maintained with ballast inductance.
Now, consider figure 30 where normal resistance and induc-
tance changes are permitted. Again, as in the discussion of
figure 29, only the case where K = 0.4 is discussed. Applying the
conclusions from before, it would be expected that in every case
where there was no ballast inductance that the rise time would be
degraded and large errors would result in the magnitude of the
output. Likewise, where the ballast inductance is present, one
would expect good rise-time performance^ and small errors in magni-
tude. The results in figure 30 confirm these expected behaviors.
An additional run was made where self - inductance and
resistance changes were not permitted. This served as an additional
check on the behavior of the model. In this case the model matched
the ideal response within 0.20 to 0.05 percent.
b. Underground Nuclec.r Test
The primary difference in fielding a large gage in a field
event as compared to the scaled gage previously discussed is that
the electrical parameters are also larger. Also, for a field test,
long lines on the order of 3000 to 5000 feet would be required.
The large increase in secondary line capacitance could present a
problem, at least in the lumped parameter model, in that there
could be some undesirable oscillations due to the inductive and
capacitive components in the circuit.
The possibility of an underdamped response was investigated
by considering the secondary portion of the network in figure 21.
If a step voltage is "inserted" into the network in place of the
motion induced M and the circuit parameters are assumed to be
constant, the circuit can be modeled by the second order differ-
ential equation
a^+ B^ + yQ = ECt) df It
59
tHiwni"wirw ■ -m ?n»^f-^T»t'\
AFWL-TR-74-205
GO
r-t o >
0)
6 o z
2.5
2.0
1.5
1.0
0 -
1 1
Nomenclature: X/X/X
X
Resistance Change Factor
L - microhenries
R - ohms 10
J.
5 10
Time ~ usec
0/300/2 -'2/300/2 -O/300/.4 2/300/.4
■2/0/.4
-0/0/.4
15
Figure 30. Calculated Gage Response Using ELMEK, Normal Resistance and Inductance Changes, AEDC Gage
60
■rn^v^Hf;-^--.^'—--V'- .-IVf'■■-■■
iPiin,imniii,im -^^mmmmmmmmmmmmmmmw'-' v»1«
AFWL-TR-74-205
where a, ß, and y are time invariant coefficients, E(t) is the
forcing function, and Q is the charge on C . By lumping the long
line capacitance and resistance for typical coaxial cable (RG-213
or RG-531) and for typical gage designs with L on the order of 10
to 20 microhenries, damping ratios of 0.3 are obtained resulting in
an underdamped response. Since the purpose of this analysis is to
observe the response to self - inductance and resistance changes, a
short length (100 feet) of cable was assumed for the secondary to
avoid the underdamped response. The 5000 feet of RG-331 retained
in the primary circuit presents no problem because of its relatively
small magnitude in comparison with the other circuit components.
In this case a damping ratio of 1.9 resulted for the problem inves-
tigated.
The effects of long-line capacitance are of great impor-
tance to the proper design of field gages, and this problem must
be addressed in future analyses of the system. The approach
anticipated is st ra ighforward. First, an equivalent "IT" or "T"
network would be employed in the circuit to model the long line
and the damping ratio calculated from the coefficients of the
second order differential equation which describes the circuit.
This would provide some insight into the range of permissible
values of secondary seif-inductance which would ensure an over-
damped response. The next step would be to modify the complex
model by inserting the equivalent network and observing the code
response. This analysis is not presented here to ensure timely
reporting of the present work.
The gage design selected is as follows:
30 inches Length
Width
Spacing between turns
Primary turns
Secondary turns
Conductor material
i inches
0.140 inch
4
3
0.060 beryllium
bl
ff'yr-^ff.^^'r'W,^'iygp'*y>cffiw' T" -n-?'-
AFWL-TR-74-205
The resulting gage electrical parameters are
R^^i " 0.213 ohm pn
Lpri = 14-3 Uh
R =0.160 ohm sec
Loo,- = 9-2 wh sec
M = 19 yh
GF = 9.81 mv/(amp'mm/visec)
The shock environment is as before with K = 0.2 assumed
for beryllium.
The results are shown in figure 31 for the only case
investigated where L = 500 uh.
As shown, the error is controlled to within approximately
3 percent. It is apparent that larger values of L could be ii
employed to make the error negligible. It is worth noting that
one reason for the effective error control here is that while the
gage is considerably larger than that previously investigated, the
resistances are significantly smaller because of the larger diameter
and lower resistivity of the beryllium wire. As shown in appendix
C, consideration must be given to a conductor's shock impedance,
initial resistivity, and resistance-pressure characteristics to
determine the magnitude of AR/ü, which directly causes the resist-
ance-change-induced errors.
2. SUMMARY
The present mathematical model of the MIPV is adequate for
investigating the effects of shock-induced self-inductance and
resistance changes upon gage output.
The resulting error appears as essentially a level shift from
the ideal, accompanied by a deterioration in rise-time performance.
Both these effects are effectively controlled by employing a large
ballast inductance in the primary circuit. Resistance in the
primary external to the gage is relatively ineffective in control-
ling the errors. Therefore, the total primary resistance can be
62
PITr^'»r"T;-TT(i.V*-'l»T- ■-■ 'IV-W-WT-.';,-.»^» FWrrT'J'.1"'^ V-'
AFWL-TR-74-205
3.
o o
U
3.
<I> 6
o I
us
US p-1
w
>s Xl
0)
R) l-H 3 U
rH rt U
w as
(U M CO u
c oo
• H w
P
O
(U W c o Q. V) <u at
u o
aSBHOA P3ZTIBUUON
a>
3 oo
■H
63
7^T^.^f'"'''-l*VTV,.^i''ir7liwT7"''TTr-i'^-t'(f.w-*iiT'-
AFWL-TR-74-205
maintained at a reasonably small value to ensure maximum current
and hence maximum sensitivity with no compromise to gage accuracy.
A major improvement to the present model would be to replace
the lumped parameter model of the long lines with equivalent 'V
or "T" networks. This would represent a more realistic model of
a field installation of the system which would permit investigation
of the possibility of an underdamped response and provide a manner
in which to establish limitations upon gage secondary self-induct-
ance. This work should be accomplished prior to fielding of the
gage in a field event.
64
AFWL-TR-74-205
SECTION V
CONCLUSIONS AND RECOMMENDATIONS
These conclusions and recommendations are based upon the results
of the three principal analytical investigations of the previous
sections of this report and the conductor materials study of
appendix C. The purpose of these analyses has been to provide
improved understanding of how the MIPV responds to the shock
environment. A direct consequence of this work, however, is the
development of important analytical tools which are necessary for
the proper design of a system to meet the requirements of a given
test. Available also is the ability to study alternative designs
and the nature and extent of resulting compromises which appear in
the system performance.
1. CONCLUSIONS
As a result of the investigations of this report, the following
conclusions are made.
a. To maintain a constant gage factor during gage deformation^
the gage length-to-width ratio should be constrained such that
a/b >^ 5 after shock transit through the gage. Generally, a/b >^ 10
prior to deformation would satisfy this requirement.
b. Gage factor may be increased significantly without affect-
ing gage rise time by utilizing a Type II gage design instead of
a Type I gage. The overall system design and specific experiment
requirements must be considered to assess to overall benefits of
the Type II design.
c. The MIPV is essentially insensitive to divergent flow
effects for most anticipated experiments. Only when the source of
spherical flow is within 1 meter of conventional gages (b ^ 3
inches) will significant errors in gage output occur.
d. The gage is relatively insensitive to off-axis flow in the
x-y plane, the error induced being approximately equal to 1/2
(1 - cos y). However, off-axis flow in the x-z plane results in
65
^•l*rrr*TXwyF,irirw-nrjr.7™V' ?*mm-'*fm*IiipriWTr^* 'r-^r-'T ' v - ^ .- .^-.,-^--1
AFWL-TR-74-205
much larger errors approximately equal to 3 (1 - cos 6). An
immediate result is that if in a iield test there is a plane in
which the uncertainty in shock propagation direction is a maximum,
then the preferred orientation of the gage is with the x-y plane
aligned with the plane of uncertainty.
e. The effects of pressure-induced resistance chafes and
motion-induced self-inductance changes upon the gage response are,
respectively, to cause a level shift in the output voltago (positive
for K > 1, negative for K < 1) and degrade the rise-time perform-
ance. Both of these effects are effectively controlled with a
large ballast inductance in the primary circuit.
f. Three metals, aluminum, magnesium, and beryllium, all
exhibit favorable shock impedance, electrical resistivity, and
pressure-induced AR/Ä characteristics. Each presents a relatively
low neutron cross section, with beryllium being the most favorable.
2. RECOMMENDATIONS
The conclusions above indicate that the MIPV has the basic
characteristics of a desirable field gage; that is, it can be
designed to produce a voltage level output and, with care in
placement, small errors due to real flow can be maintained. Like-
wise, the effects of shock-induced resistance and inductance
changes upon gage response have been shown to be effectively
controlled with the ballast inductance. However, though based upon
sound theoretical approaches, there is a need to verify these
results. It is therefore recommended that these results be inves-
tigated experimentally in explosive testing. The approach would
be to deiign an experiment, make calculations utilizing the
appropriate code based upon that design, conduct the experiment,
and compare the results.
As was noted in section IV, a requirement exists to modify the
ELMEK code by providing an equivalent 'V or "T" network to properly
model the secondary long lines. This will permit designing an over-
all system which will provide a desirable overdamped response. It
is recommended that the model be modified accordingly and that
experiments be conducted to ensure that the resulting model is
suitable for actual design of field systems.
66
' i"iv'T |''^T""^ •nr v^
AFWL-TR-74-205
APPENDIX A
TABULATED NUMERICAL DATA FOR MIPV DESIGNS
i
I
The results of calculations utilizing two codes, DSIN1 for the
Type I design and DSIN2 for the Type II design, are tabulated for
several different applications of the MIPV. The codes are listed
at the end of this appendix. The different applications are
briefly discussed.
1. TYPE I GAGE DESIGN, FREE-FIELD MEASUREMENTS, LARGE HE TEST
(TABLE A-l)
This design would be suitable for free-field ground-motion
measurements in the stress region below 10 Kbar, since large gage
factors are available with the large number of primary and second-
ary turns. Results from this data are also presented in figures 6,
7, and 8.
2. TYPE I GAGE DESIGN, GAS GUN AND SMALL-SCALE EXPLOSIVE TESTS
(TABLE A-2)
Information is presented for scaled gage designs. Gage factors
are considerably reduced; however, the smaller gage designs are
more compatible with the gas gun and explosive experiments utilized
in gage development and evaluation.
3. TYPES I AND II DESIGNS, GENERAL DESIGNS (TABLES A-3 AND A-4)
Data are presented for designs where gage parameters are varied
over a wide range. These designs would be suitable for use in the
high stress region close in to an explosive or nuclear source. The
numerical results of figure 9 were obtained from this data.
67
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l.<»27bl£-0b 3.78112t-üb 6.a35<*<«£-0b I.a<»(»ii7£-G5 l.<»520«i£-C5
1.35b<»7E-üb 3.57Qä3c-do 6.Mla57c-lib 9.76J73c.-b6 I.35a9ttt-Ö5
1.29'»^9c-öb 3,J6799£-ab b.C6Cii4£-a& 9.17b5öt-b6 1.2b<»'«7£-ü5
2.72b00£ 7.302b2£ 1.332öl£ 2.u5*«3bt 2.o7bl3£
2. idöSli b.3b71b£ 1.2i*bl&£ 1.4ly9<«£ 2.bb2/7£
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2.35übd£ b.ld2Q7£ l.UGZit 1 .oöbdbc 2.33üai£
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2.o9J61£-06 2.7226-»^ 7.73sii£-Ub 7.d9253£ l.'»(,9äo^-b3 l.33u8a£ 2.1b9l7£-J5 2.u5lJC£ i • 332b<»£*'Ü9 2.87139£
2.7372ljc-Cb 2.57697£ 7.2b622t-bO o.357:7i 1.31592c:-35 1.2<»H16£
2 . 3 1'♦'♦ ji-0 J l.jy?5a£ 2.oC3u2£-b» 2.ö5773£
i.6y3öli-üb i,^-}im\. a.abSi«bc-yb b.Hd^bJi 1.23717_-:5 1.17llj£ l.do'»73L-b3 l>7a71u£ 2.Gllyö_-Ui t:."473J3i
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77
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78
TABU A-i. TrPE II SAGc, GENERAL DESIGNS
LENGTH (INCHES)
WIDTH (INCHES)
LOOP SPACIN;
(INCHES) PRI/SEC PATTERN
MUTUAL XNDUT (HENRIES)
GAG£ FACTOR
( ^ \ ERROP TERH
(PERCENT)
2.CO00OE*n a.ociooE+oi 2.C0000£*0l 2.000ö&E*ai Z.OOOOOEtOl 2.0000Ct*Cl 2. CO (TO Of+ 01
?.ooooiE+öa 2.0D003E+00 2.0)001E»00 2.ojoa)c*co 2.0J003E*00 2.QJ00QE«00 2.0]i)OJt*0a
6.000C0E-02 ^.0C000E-02 8.ocaoat-02 «.OC00OE-O2 s.cciooE-ae ».C0003E-02 8.00000E-02
00X00 000X000
0000X0000 coxooxoo
000X000X000 00X00X00X00
OOOXOOOXOOOXQOO
2.<.710rE-Cfc
'♦.1595'.E-06 6.I.13Q1F-C6 8.6062CE-06 l.H»0 37E-06 1.5080fiE-Of
<..6030äEO3 6.30706E+00 7.78530E*?0 1.198O7E+01 1.613)2E*n 2.13,9i»E + 01 2.83574E+31
-,i,67<»71t*0 0 •5.U0C76E»QC -5.173t<.E+00 -5.29117E*00 •U.q89S9E+00 •H.999U!:E*3C -I..68E'9E+30
2.üOOOCE*01 e.tjaooEtoi 2.00000Et01 2.000aOE*01 Z.ööflOCE+Ol 2.00000E401 2.00000E*01
2.&3003E*Or 2.03030E*00 2.GJ003E^OO 2.a')900EtOu 2.09003E+OQ 2.CJ33JE*0I3 2.00001E+0C
l.OCOOOE-01 1.00003E-01 l.OCOOOE-01 l.OOCOOE-01 i.ocoooE-ai i.ocoaoE-oi l.COOOOt-01
00X00 OOOXQOO
OOOOXPOOO 00X00X00
oocxonoxooo ooxcoxooxoo
000X000X000X000
2.27U7..P-06 3.08i,77E-Ot 3.773l7t-06 5.83<»2et-0e 7.7527ec.C6
1,02708E-C5 1.3<»29t£-CE
4.2<»781Et0C 5.776'.4E*00 7.08l75d+30 1.09362E*?! 1.<»5761E + 01 1.93072E»01 2.53220E+C1
•5.<»15<.<tE+'!P -5.12290E+00 ■<».88236E*00 •5.ni608E»00 •<..70127E*30 •i..7l766E*00 •1..39557E + Ü0
2.ÜOÜ00E+31 2.C0500£*01
T.üÜOOOE+Bl 2.ÖOOÜOE+01 2.00000E+01 2.00000E*01 2.000aOE*01
2.033O0E*00 2.03003E*03 2.CJ003c*00 2.33O03£*O0 2.03003E»00 2.03J03t+4)0 ».03O03E*O0
1.20000E-01 1.20003r:-01 1.2C0OOE-OI 1.20D0OE-31 1.20000E-01 1.20000E-01 1.200G0E-01
00X00 000X000
0000X0000 coxooxoo
000X000X000 ooxooxonxoo
000X000X000X000
2.11>llE-0b 2.6<»772E-C6 3.«.6O7eE.0f 5.3665oE-0fc 7.0b721E-06 9.36?11E-Cb 1.2H72E-05
3.95826t*30 5.3<ti*93E»CC 6.5110i»t*C0 1.0C82IE*0l 1.33186E+C1 1.76395E*ai ?.2902JE*'U
-5.18755E*Q0 •<».88C33E*00 •i..63CJ?£*00 •i.,77833£*30 ■4..<.5f.l9EO0 •«♦.«♦77'»?E*00 •'..150-.3E+30
"r.TöOOOE+Ol 2.COBOOt*01 z.oooooEm 2,duOOOE*01 Z.COOOOE^Ol 2.üOüuOE*01 2.ü0fl00£»fll
2.0Q003E+00 2.Ci;0C3E*0l5 2.00003£*CO 2.0x0OJ£+O0 2.00003E+00 ?.&3000E*00 2.oaooo£*oo
l,<»C00üt-0l 1.<»CCÜ3E-01 l.VOCOCE-Ol l.i»CCG3E-01 1.<»C000E-01 1.400C9E-01 i.<»oaooE-oi
00X00 000X000
0000X0000 00X00X00
oooxoooxoao 00X00X00X00
OOOXOOCXOOOXOOO
1.98082E-C6 2.6(»89CE-C6 3.19961E-C6 •..9758'.t-06 6.b9e3G£-C6 8.6092«.E-06 1.10«.12E-Cf
3.7li.l7£»00 i..96200E'00 6.0326SE+ÜC
1.22719E*?1 1.62537E*01 2.39112£*01
•'..983?3E*0r •i..fe6«.27E*CC ■'♦.I.0676E+00 •k.56829E*00 ■4.23772E+30 •<..26763E*00 •3,93772E+00
2.00000Et01 2.00000E*)! 2.000u0t*0l 2.0000CE*01 Z.TTTOOCE+Ol 2,0öüJ0E*3 1 2,OD0OOt»0 1
2.F3OO3E^00 2.07OO3E^O0 2.0aOC.3£*03 2.0000aE*03 2. 0l)9O3t«'C0 2.03aoaE*03 2.0a003E*00
1.6C000E-01 1.6C000E-01 1.6C000E-01 1.60Ü00E-01 1.6C000E-O1 1.6C00'»E-01 1.60000E-01
ooxoo 000X000
OCOOXOOJO 00X00X00
300X000X000 00X00X00X00
000X000X000X000
1.86'13E-06 2.'»781fcE-Pf. 2.97619E-Ct <..6«»17DE-t6 6.0152«.E-06 7.97C99E-C6 1.01381E-C5
3.5n3i»5E + 00 i..b6959e*30 5.62217E»00 8.75391t*30 1.13807E*31 1.5D757E01 1.92357£*01
•«..7973»E*00 ■<..<»6936E*00 •U.20 629£*0C •i«.379S8t*00 .<..0i.<.7SE*00 •<».0813«E»00 •3.7<,926E*00
79
TABLE ft-«» (CONT) TYPE II GAG?, GENERAL DESIGNS
LENGTH WIDTH LOOP SPACING PRI/StC MUTUAL INOUC GAGE FACTOR ERROR TERM UNoMtSI :INCHES! (INCHES) PATTERN (HENRIES)
(AMP^MM^USEQ) (PERCENT)
...comccOl 2.0300JE*00 8.COC03E-02 00X00 •♦.aoasgE-oe «».60007EtOO -2.88261E+00 ^.oo^cE+n 2.CJOOJE+0C *.0030aE-02 000X000 b.579i.9E-06 6.30257E+30 -2.7I.983E+00 -,.OD300E»G1 2.u0D0J€»a3 a.OOOOOE-02 0000X0000 8.112'»'»E-06 7.77931E*aO -2,6<»00i»E*00 >».,«0uC£*0 1 ?.03OC?£»ÖO (J.CC003E-02 00X00X00 1.25007E-05 1.19807E+01 -2.69756E*00 •♦.CCOuGE + Ol 2 03003E+C3 8.0000at-02 000X000X000 l,6798«E-05 1.61227E+01 -2.5527<»E*00 •«..OCubc+Cl 2.0000JE*CD S.OJOQOE-QZ ooxooxooxoo 2.2258(»E-05 2.13615E+01 -2.55786E+00 •♦.COOüCE+01 2.C3003E»OC e.ococa£.-02 oooxoooxoooxooo 2.9477&E-05 2.8330i»E*01 -2.'.08<»9E+0 0
•♦.«oijoce ci 2.O0OOJE+O? l.OCOOSE-Ol ooxoo <..<«3162E-06 <».2<»<t81E + 00 -2.75667E+00 *.COuOfc +0 1 2. 03000E + 00 1.O0OO0E-O1 000X000 6.01769E-06 5.77197E*00 -2.61533E*00 ■».C0«i0£*31 2.000CJE*30 l.OOOOQE-ai 0000X0000 7.36868E-06 7.07576E+00 -2.(.994TE*00 *.0uL3OE*Ol 2.0303:E*00 l.P0000£-01 00X00X00 1.13868E-05 1.09272E+01 -2.56I.90E+00 -.COOOCE*'1! 2.00001E+OC l.OCCOOE-01 000X000X000 1.51528E-05 l.<*5626E^ai -2.i»l<,<»6E«00 »♦.ccaoc;*oi 2.aü001E*0'? 1.OÜ0OOE-01 00X00X00X00 2.00727E-05 1.92892E+01 -?.«.2308E*00 •».jHOCE+n 2,C0JIJ0E*00 l.OCOCOE-Ol 000X000X000X000 2.62835E-05 2.52951E*01 -2.2710<.E*00
t.C0Jjt£^Ql ?.C3000E*00 1.200C?t-01 ooxoo «♦.12<»89E-06 3.9552$E+0a -2.e(»622E*00 «♦.C00wt':'*Cl 2.03COJE*00 1.2COOOE-Ü1 000X000 S.56141E-06 5.3i»0«»lE*00 -2.i»9829E*00 ■♦.OOi.wwP+1 1 2.oajoot*co 1.20CO0E-O1 0000X0000 6.7663<»E-06 e.sisoeE+oo -2.37850E+00 ^.COQOiEtCl ?.3J0OOE*CO 1.20003E-01 00X00X00 1.0<»B»*»E-05 1.0073<»E*0t -2.«»5071E*00 tt^üüOCt + Ol 2.03001t+03 1.2COOOE-01 000X000X000 1.38285E-05 1.33051E*')! -2.29680E*00 «♦."ajociici 2.0C3Q3E+03 1.2CCOOE-01 00X00X00X00 1.83169e-05 1.76216E+01 -2.30988E*00 ■». jüüuo^+a i 2.JJ0C3E*0C l,2C000£-0l 000X000X000X000 2.37<.25E-05 2.2875'»E*01 -2.1S6inE*00
-...aü3C':,*:i 2.0iO33E*CO l.tOOOQf-01 ooxoo 3.86660E-06 3.71117E+00 -Z,5i*7kU*n »..-OOCCEKl ».C33C3E*00 l.<.00OOE-öl 000X000 5.l7e23E-06 »♦.97751£*flO -2.39<»39E*00 *.coocci*: i ^.flJoa3t♦c,» l.<.COOOE-Ol 0000X0000 6.2621<>E-06 6.a2661E«aO -2.2717«E*0B ■,.,üOOC:*O! 2.00033E»uO i.woeoor-oi 00X00X00 9.73126f-06 9.35e07E+00 -2,35029E*00 •»««.uCj^E + O* '.C33ajEOO l.*»00OOt.-01 000X000X000 1.27279f-05 1.2258(»E*ei -2.19436E+00 ■».-oojuE + ai 2.u1l303E*Ofl 1.1.CCOOE-QI 00X00X00X00 1.68600E-OS 1.&2357E*01 -2,20981E*00 •..cou3oe*;i 2. 303'J3c*QO L't^OOOE-Ol OOOXJOOXOOOXOOO 2.16550E-05 2.Qe8'»<*E«'01 -2.05niE>00
•♦.LOOüC^ + Cl 2.CJ0B3E*C3 i.ecoace-Bi onxoo 3.6<»387E-06 3.501H.6E+00 -2.I.5775E+00 *,C3e3»£*31 '.ODBOOE^OO i.e-'oooE-oi 000X000 i..8i.880E-06 «..o65iaE + 00 -2.3008<»EtOQ -.. C 3 0 0 C E ♦ 0 1 2.C30a3E*03 i.erooof.-oi onooxoooo 5.83023E-06 5.6i618E+00 -2.176i»2E*00 (♦.LJU.uE+Cl 2. TJüü3E*03 1.6CC0lJE-0l ooxcoxoo 9.08570E-C6 e.7l»<»93E»00 -2.26060E+00
■4. ^u3 jLr «J 1 ?.QJ00J?*0j 1.6CÜOOE-01 oooxoooxoon 1.17921E-05 1.13673E+01 -2.10 36^T+00 •«,^C3Q3£*0 1 2.CJ02Qt*O- 1..600QOE-01 00*CO*ÜOXOO t.5623i»fc-05 1.50578E*01 -2.12231E*00 «.CawäCc*M ■'. C J^JE*^ l,«i0D09E-0t OOOXTOOXOOOXOOO 1.9903TE-05 1.92089E+31 -1.97D33E+00
80
, ■,'■,;■-< 'r;, V.-
TYPE TftBLc. A-«. JCONT)
II GAGc, GC'Nc.KAL DESIGNS
LENGTH (INCHES)
WIDTH (IMCH£S)
LOOP SPACING (INCHES)
PRI/SEC PATTERN
MUTUAL INHUC «HENRIES»
GAGE FACTOR
(sMFTWUsrc)
ERROR TER1 (PERCENT!
&.i.b JQCE^Ol
b.v,Q00C"t-31
2.0J0(i:£«-ÜO '. ü-lODIEt-O ?.GJDOJE+Ol
2.Q0OCJt*Dj a-cpoo^E+co
«.00CQ0E-O2 ?.ocoooL-a2 P.OGCOOt-O? 8.31CIOTE-32 a.ujoaoE-n? I.GnCOOE-32
ooxoo 000X000
0000X0000 COXOOXOO
000X000X000 00X00X00X00
000X000X000X000
7.1i»5n6E-06 9.78G95E-06 i.aoe^E-os 1.8586i»E-0? 2.4.9S8UE-C5 3,no9Pe-o5 »♦.38673E-05
'♦.5995ie*00 6.3017'»£*,)0 7.7782Dc+iiO 1.19790E+01 1.61202E+Q1 2.13581E+31 2.8325<»^+0l
■1.93163E+00 •i,8».<Oi«E+00 ■1.77173E*00 ■1.80975E+00 ■1.71i.i»9E*0 0 ■1.71798E+00 -1.62C35E+00
6.r.0JGLE + 3 1 o.CÜJ3C£+l1 6.;03JLE + 3 1
6.taC0C£tC 1 ö,C030CE+3 1 o.COGSCE+Ol
2. O)3C0E+Q j 2.CG003t*')0 2.OJOC:E*GJ P.wOO'iDEtOC 2.3)0DJLtr,: 2. UOülE + 03 a.GJOOjE+OC
1.3:000^-31 1.0C303E-01 l.OOOOCE-01 i.mcct-oi 1.03C00E-01 1.0C0C3E-01 1.SCCD3E-01
ooxoo 000X000
0000X0000 COXOOXOO
OOÜXOOOXOOO 00X00X00X00
000X000X000X000
6.687Bie-06 e.g^gecE-oo 1.09628E-06 l.&937<»E-05 2.25'.99E-C5 2.96707E-0^ 3.91,'19E-C5
'♦.2'»'«26E + 00 F.771l*E+30 7.07<»65E + ';0 l.r-^56£ + 31 l.i»i>6UlE*01 1.92859E4-31 2.52901E+01
•l.öfc8i»9E+0 0 -i.75535E+00 •1.6792,'E + 00 •l.-'2239E + 00 •1.6236i»E+0C ■i.629i»8E+30 •i.53012E+aO
o.Li.i30i+C 1 6«MOIUO£+31 b.oi;occE*r i o.t300O >C1 6«L iiUl»v£ + C 1 o. .cJOOE»; i
2. C00CJE + C3
2. cioo-tc: H.GJIJC-JE + CO
2.o:003E+33 2. C J0g3t + CC 2. 3)0yJE + 33 2. CuJC^E+CC
1.2CrO3E-01 i.?C0G0F-01 l,2C30Cc-ai 1.203000-01 i.^ruüJE-oi 1.2uC03E-01 1.2Cn03E-01
00X00 000X000
0000X0000 00X00X00
000X000X000 00X00X00X00
OOOXOoOXOOOXOOO
6.13399E-06 8.c7'«09E-C6 1.C0736E-05 1.56021E-C5 2.05'i67E-05 2.72o77E-05 3.5^617E-C5
5.95I.71E + 00 5.33958E+00 6.5C395E+00 1.3Q717E+01 1.33026E+31 1.76183E+01 2.2870iȣ + 31
■1.77?63E+00 ■1.67826E*00 •1.599S0E+00 •1.6'«73lE-*00 •1.5«»651E+00 •1.55«.6TE + 30 •1.U5511E+1C
b . G u 01/C ^ t C 1 6.C 3CJC?+: 1 6.u0GCLf-O 1 6.C 0. "'Ci+Ol ö.LOO J.C+: i b.w«C0G£'»-3l b.Li-JilC;*] 1
2. OmjE + 03 2. u j3« 3u♦G^ ?.G:OC3E+:: ?.C:POJL+CJ ?. C JGO j£ + C ] 2, GJGCJ£ + ;"'
J.MKU + C:
1.43003E-31 1.<.:CG3E-J1 i.uc-ooc-ai 1.1..CÜCE-01 1.<*CC33E-31 l.i*e;CGE-3l 1.1.C :C3E-31
onxoo 000X000
COOOXOOOü OOXOCXOO
000X000X000 ooxooxorxoo
000X000X000X000
J.75171E-Cf- 7.7üb3&E-C6 9.32332E-Ct 1^',«47E-C5 l.f'iSf'.E-G«; 2.5136SE-C5 3.2?6?et-cr
3.71062E+',0 «..97667E*00 6.325F0E+'!0
1.2255 ^+31 1,6232'»E*01 2.J879-»£+01
-1../1C50E+00 ■1.50991E+30 •1.5295i.E+3C ■1.58138E*00 •l.-»7g5lE+00 •l.i»8992F + 00 ■1.39C86E+33
b.^CCDtu^Ul &.:CJ K£+) 1 o.O 3ü0i.£-«-3 1 o.£GG0C£-Ol b.rluCi.£+31 T . . oG 3Ct *C 1 p> . L i. J 0 0 £ + C 1
?. G : ^ o: £ *r ? '. a ]Cü-)£ + C3
2 j^Zj-.tt:
2.fl33nF + Cn
,. CJOC3£*0: 2,C330H«-JU
2. C 1 K ) £ ♦ t"
1 .6GC3Cc-31 I.'.OCJOE-JI 1.6LCGJE-31 i.6c:oai.-3i 1.6CC33L-31 Lb:: 03-.-oi UbCOODE-Jl
OOXOO 000X000
ocooxrnoo COXOOXOO
oooxooQxooo 00X OCX OCX 00
000X000X000X000
p.-^iq-tE-oe 7.218I.1L-06 3.68?91£-C6 1.35276C-00 l.-'Sf.^SE-O? 2.32717E-05 2.9b573F-r,S
3.^9990E+30 '..bCi'.27E + 00 6.bl537E*3P 8.7'.327EtüO 1.136*»9Et01 1.505«» % «l + Cl 1 .92',39E+31
■1.651it5E + 00 -l.S'tS^äEfOO •1.1.6735F+00 •1.52259E+00 ■1.1.2C32E+30 ■1.U328«E*00 ■1.33U(:>8E+00
81
1 TVH^ IT Gfi^i. O^N^DSL aesiONS
LENGTH rtlHTH LOOP 5PÄC1N- o-'VUr MUTUAL INOUC \ (iNCHtS) ( I JGH-ZS» (INCHES) °ATrERN (HcNRIFS)
GAGE FACTOR ERPOR TFPM (PERCENT) (. ^ )
VMP*MM/USrC/
^.. o: c c c +":
2tC03uuE+:1 2 • £ u !J w ii £ ♦ C J
2. 5: K.)-. tc:
^.5:c53E♦',',
?. i: j o ] t ♦■ u c
2.5:ün3i*Jj ,.Li.uv,oE.+:i ?. 5JOC.
o.?r:o,U-0? OOXOO ?.7052i*L-C6 8.3: JOOL-Qi 000X000 J.^igo^-ce «.C:n0Cc.-02 0000X0000 H.6C7oCE-0f p.ccou-a? ooxooxoo ■'.o'm-.E-oe »..:CC00L-02 OOGXOOOXOOO 9.?8921E-C6 B.C:000E-3? OOXOOXOOXOO 1.2710CE-C5 ■».C :iiü]F-a2 OOO/OOOXOOOXOOO 1.696"',iE-0E 3.11.7105 + 01 - «>. 13C27E+0 0
'..90121E+?!) -7.33R2BE+00 6.8u28]£ + :0 -7.n'.^3E + 30 P.^<»75öS + :O -6.7i»379E+ir i.30i.60S*3l -6.87U90F+ao 1.7722i»5+ni -6.61126F + 0C ?.3-e91E+01 -6.51639E+00
Z.'.üCCtE »C 1 cJ tw jCi3<.^ + 5 1 2.LUJCE+!,1 B.wjOOOF+r1 2.L0C0CE+C1 c:.uOaDCE*Ol 2.i,ü JOCE*: 1
2.Liu JCC+31 2.CJJOCE+O 1 2.LijJCt+01 t.m jCE + O 1 2 . C J Ö VJ u £ ♦ 0 1 21 ii 0 ü «i w £ ♦ C1 2.0JCu0t*Cl
?. SjO^Etiil 2. SJ j^l.t1?: 2. SJOUt + SI ?.5T33?C+-J
?.':ijO(3v.E + Cn
2.5 joaH + c: 2. SOQOOEtC1;
?.5'3 3'11E + 1,'» 2.53ÜjOEtO? 2.5jüC]E+Cr ?.5303QL+J3
'.53T03E+,),3 2. 51003E + ';3 2.53303E+CC
1.CCCC3E-01 I.OUOCE-IU l.GCC30t-0l 1.C0300E-01 1.0G003t-01 i.oroo^c-oi LflOOE-Ol
1.2:303E-Ol i.2Coaot-oi 1.2C003t-01 i.^raoOE-oi 1.2CC03E-B1 1.2',3C3t-3l l.?p303E-31
00X00 000X000
CCOOXOOOO 00X00X00
OOCXOOOXOOO 00X00X00X00
OOOXJOOXOOOXOOO
00X00 000X000
0000X0000 ccxooxoo
000X000X000 0ÜX00X00X00
000X000X000X000
2.53<.13t-Cf 3.'i2ülOE-06 '..eicuE-os o.<*9-J25E-06 8.7078(}E-06 i.l5387F-0F 1.52^8?L-CE
2.3a03aE-Oe 3.l76^9E-^6 3.ft8852E-06 S.TlOgSF-O« 7.99699E-0t 1.05950E-OF 1.3872PF-05
i..6C531E+',0 e.31C'»2£+n!i 7.7bq7l<E+30 1.1996i»E+:i 1.61U62E+C1 2.13929t+01 2.83775E+01
'..3149»E + 30 5.87672E+Q0 7.21i»58t+CC 1.11351E+C1 1.48700E+01 1.96975E+01 ?.,;8923E + 01
-7.|i3E83£+0f1
-6.6881i*E+00 -6.39913E+00 ■6.5'»798E*0C -t.l6i.nt+3C -6.1755i»E+30 -5.77l.27r + 00
-6.76929E+00 -o.'.CilSi.E+OO -6.09853E+30 ■6.26377r+0C •5.86i»7i»E+00 ■5.88299E+90 ■5.'»71?7E*00
2.i.0 JÜCE+L 1
2.C0JOCE+i 1 2.LjjObi+Ol 2.CCJgO£+31 i:.wüOCCE+31 2.30300£+:3 2.UL0ÜCE + C 1
'.5JQ03E<-':: 2.iTSOJc+C 3 2.53 3 31t+0 3 2.53J03E+CC ?. 5 3 0.3E + 3'1
?.i0300E+0C 2.EJÜ-3E+0:
Lucroct-oi l.btaDOE-Ol 1.UC300E-31 i.'.^oaE-oi 1 .I4QCG3E-01 i.^coooe-oi LU^JOOE-ai
00X00 000X000
0000X0000 00X00X30
000X000X000 00X00X00X00
000X000X000X000
2.2025fF-06 2.97181E-0f- 3.6187CE-0fe ■5.6C68»E-Cf, 7,'.C'»28E-06 9.80911E-C6 1.27363E-05.
'».36997E+00 E.51132E+00 6.73095E+30 1.0«»11JE + 31 i.380Z}E*i31 1.82808E+31 2.38306E+31
•6.52957E+00 ■6.1«»?S5E+9C •5.83C90E+00 ■6,011!»2E + 00 ■5.60Ü88E+00 •5.6259f.E+00 ■5.206q6E+00
2.oüi.3CE + 3 1 2.CÜ0DÜE+?! 2 iljuG30£+0 1 2.Cü j()fic*31 2.0000C£+ei 2.20O3CE+01 2.CÜ33CE+01
2.5JO0JE+ü3
?. 53JJ0E + C'! 3.5j033i. + Crj 2.530C3E+03 ?. 53a0j£ + 00 2.53003L+0T 2. 5ü33uL«-00
1.60303^31 l.<i)L303t-01 1.6CüOa£-01 1.6C3D3t-31 1.6C000E-01 l.e^OOOE-Ol l.eOC33E-Ol
00X00 000X000
0000X0000 00X00X00
000X000X000 00X00X00X00
000X000X000X000
2.Ü8366E-06 2.795<59E-06 3.38715E-06 5.2O&29F-06 6.89851E-06 9.1393<»E-06 1.1775'*E-05
3.85821E+00 E..19607E + 00 6.31U6SE+00 9.78873EiOO 1.28883E+31 1.70697E+31 2.2082aE+01
•6.31058t*00 ■5,91312£+00 ■5.58923E+00 •5.78?95E*00 ■5,36i.52E+00 ■5.39635E+00 •<..9723<»E+00
82
TABU «-<* (CONT) TYPE I I GAGE, GENERAL DESIGNS
LENGTH WIDTH LOOP SPACING PRI/SEC MUTUAL INDUC GAGE FACTOR ERROR TERN (INCHES) (INCHES) (INCHES) PATTERN (HENRIES) / Nf \ (PERCENT)
*.OOQOOc»ffI 2.53 001EVC 0 8.00000F -02 00X00 5 . 22395E - 0 6 U . 9 5 6 5 3 E O O - 3 . ? 3 5 5 0 E * 0 0 <».oooaoE»pi 2. 5 0 0 0 3 E + 0 0 8 . » P 0 0 0 t - 0 2 000X000 7 . 19375E -C6 6 . 8 3 5 7 S E + 0 0 - 3 . 5 7 9 3 2 E * 0 0 <».00000E*91 2 . 5 0 0 0 J £ » 0 0 8 . 0 C 0 0 9 E - 0 2 0 0 0 0 X 0 0 0 0 8 . 92095F - 0 6 8 . 4 8 7 7 1 E + 0 0 - 3 . M»996E*00 * . O O u O O t O l 2. 53000E+00 S.OOOOOE - 0 2 00X00X00 1 . 37162E - 0 5 1 • 30<»20E* 01 - 3 . 51285E*0 0 4 . 0 0 0 0 0 E + 0 1 2 . 5J003E+GC S.OOCOOE - 0 2 000X000X000 1 . 85S51E - 0 5 1 .7701<»E*31 - 3 . 33671E+Q0 "».C06JCE»C 1 2 . 5 0 0 0 0 E » 0 9 S.OOOOOE - 0 2 00X00X00X00 2 . 4633CE -C5 2 <3<t611£ + 01 - 3 . 3 « 1 6 2 E * 0 0
~57 OOOOOE+Ol 2 . S0003E»00 S.OOOOOE - 0 2 000X000X000X000 3 . 2 9 M 2 E - 0 5 3 • 142 98E+ CI - 3 . 158OQE+00
<».JOOOOE»CI 2 . 5aaoiE*oo 1 . 0 C 0 0 0 E - 0 1 00X00 k. S«»201E - 0 6 k . 6 C 0 6 i £ » 30 - 3 . 5 8 9 2 5 E * 0 0 4 . 0 0 0 0 C C + 0 1 2 . 5000JE+00 l.OOOOQE - 0 1 000X000 6 . 623<»2E - 0 6 6 <303<»lE + 30 - 3 . <»2197E*00 •*. 00000E + 01 2. 50 000E+03 1 . 0 0 0 0 0 E - 0 1 0 0 0 0 X 0 0 0 0 3 . 16<»<»8E - f t 7 • 785<»J£* 00 - 3 . 2 8 3 5 2 E * 0 0 » . 0 0 0 J 0 E * 0 1 2 . 53 003E+03 l.OOOOOE -ai 00X00X00 1 . 25826E - 0 5 1 • 1982<»E* 01 - 3 . 35685E *0 0 <».bTffOffE*Tl 2. 5 u 0 0 3 c * 0 0 l.OOOOOE - 0 1 000X000X000 1 . 69031E - 0 5 1 . 6 1 2 5 2 E * 01 - 3 . 17330E+00 • .0000CE+G1 2. 5JOOJE+OC l.OOOOOE - 0 1 00X00X00X00 2 . 23968E - 0 5 2 . 136<»SE«-01 - 3 . 17929E»00 -.uiiOOCE+Ol 2. so OOOE +oa l.OOOOOE - 0 1 000X000X000X000 2 . 9.6<»9*E - 0 5 2 • 83355E + 01 - 2 . 99C25E»00
<•. OfiO OOE + O 1 2. 53 003E *C 0 1 . 2 0 003 £ - 0 1 00X00 53ii 82E - 0 6 k • 310 30 £ • 0 0 - 3 . "»6 G6«»E +3 0 <*.00au6E«01 2 . 500Q3E+0Q 1 . 2 C 0 0 3 E - 0 1 oocxooo 6 . 15950E - 0 6 5 , 8 6 9 7 1 £ • 0 0 - 3 . 28U39E+00 <*.C0000E + 0 1 2. 50 003E+0 3 1 . 2 C 0 0 0 E - 0 1 0000X0000 7 . 55036E - 0 6 7 . 2052<»£+ 00 - 3 . 13969E+00 *.GOOOaE»3l 2. 53003E>00 1 . 2 0 0 0 3 E - 0 1 coxonxoo 1 . 16b2*E - 0 5 1 . 1 1 2 1 1 E + 0 1 - 3 . 2198TE+00 < * . £ 0 0 0 0 c * 0 l 2 . 53Q03E»00 1 . 2 C 0 0 0 E - 0 1 •300X000X000 1 . 55<*38E -C 5 1 .*8<»90Ef01 - 3 . 0 30 S8E+3 0 %.OOOOOE*OI 2 . 5 3 0 0 3 t * 0 3 1 . 2 0 C 0 0 E - 0 1 00X00X00X00 2 . 05913E -C5 1 . 9 6 6 9 0 E + 3 1 - 3 . 3'»'j6'. ,r»00 <•. jooooE+ai 2 . 500Q3E»0 0 1 . 2 C 0 0 0 E - 0 1 OOOXOOGXOOOXOOO 2 . 70117E -C5 2 . 58500 £ • 3 1 - 2 . 8l»831E+00
<».oO «0G£»0 1 2. 5uOOOEVCO i.*»onoo: - 0 1 00X00 u. 26851c. - 0 6 (» . 0 6 5 3 0 E + 3 0 - 3 . 3<»521 E»3 0 -..CQuUCE+01 2 . 53 OOOE+OO 1 • i»C0 0 0 E - 0 1 000X000 5 . 76919E - 0 6 5 .5m»31E + 00 - 3 . 16166E+00 '•.uOOOOE+O 1 2. 50003t«-00 1.4GQ03E - 0 1 coooxoooo 7 . 03«,86E - 0 6 6 • 72161E+ 00 - 3 . 01219E+0 0 -..aooocE + oi 2 . 530G3E*C0 l.OOOOOE - 0 1 00X00X00 1 . Q8911E -G 5 1 • 0 3973E+ 01 - 3 . 3 9963E»0 0 4.0«)aaOE + Ol 2, 5 3 0 0 3 E » 0 3 l.OOOOOE - 0 1 000X000X000 1 . *4087E - 0 5 1 • 3 7813E* 01 - 2 . 90632E *0 0
• j uO G OE + 3 1 2 . 5 0 0 0 J E + 0 3 1 .4G003E - 0 1 00X00X00X00 1 . 90863£ - 0 5 1 . 8 2528E+11 - 2 . 91956E +30 '••uOGOOE + O 1 2 ( saoaoe.+co l.OOOOOE - 0 1 000X000X000X000 2 . 48283E - 0 5 2 • 37886E* 01 - 2 . 72556E+00
<*« uO 00 Q£>0 1 2. 53003E+0C 1 . 6 0 0 0 3 E - 3 1 ooxoo < • . QI.205E - 0 6 3 . 9535"»~ + 0 0 - 3 . 2<»015"*0 0 <,.003 JOE*'' 1 ?. 5 i 0 T 3 £ + 0 3 1 . 6 0 0 0 3 P -0' . 300X000 5 . '.32 9 ? t - 0 6 5 •18906E+C3 - 3 . 35C62£*90 <• • J0C u uE »C 1 2 . 5300J£-t-00 1 . o 0 GOO £ - 3 1 COOOXOOOO 6 . 5918 Si; -C6 f. .3!i53<»E*0C - 2 . 89753E +-0 0 '•.CO JUOE + Ci 2. 5Goo3E*oo 1.60C03E •31 00X00X00 1 . J2282F -C5 9 . 77<»72E* CO - 2 . 9915SE *3 3 V.uJDOaE+31 2 . 5 0 03 3£ +G 3 1 , 6 0 0 0 3 F. - 3 1 000X000X000 i a 3<»38 7E - 0 5 1 • Z 867-.E + 31 - 2 . 795i»0£+00 <• «G000 0£*3 1 2. 5 3 3 0 3 £ * 0 0 1.6caonE - 0 1 00X00X00X00 1 . 78013E -C 5 X . 7 j l » l 7 £ + n l - 2 . 81236E+30 ••.G0000E + 0 1 2 . 5u003£+G7 1 . 6 0 0 0 0 E - 0 1 000X000X000X000 2 . 29789E -C5 2 .20<»Q1S«-01 - 2 . 617<»G E »00
33
ffmm MV-iWWiV'frrrmn. T:/H(T>,",',-;»T- •W ™ ■ r.-^.jr^n^Tfir ■,
TA^LF Ä-U (CONT» TYPE 11 GAGE, GENERAL DESIGNS
LENGTH (INGMtS)
WIDTH (INCHES»
LOOP SPACIN3 (INCHiS)
PRI/^EC "ATTtON
MUTUAL INDUC (HENRTfS)
GAGE FACTOR
(rr-P^MM/USFQ/
SRROR TERM (PERCENT)
6.COCOüE+01 6.C00C0E+01
6.C000C^+0l 6.l,ü000E*nl 6.0DaOQE+Ql 6.uü3üOE + !J 1
?.500JJE*C3
?.5i001E+00
?.5303TE*Ü0 a.saoojE + o1? ^.soaoaE^oo 2.5jünjE+00
W.003DQE-02 8.00003E-02 «.00600E-02 8.0C000E-02 8,CaC00E-ü2 ?.CC000E-02 9.CCCO0E-I12
OOXOO 000X000
0000X0000 00X00X00
000X000X000 00X00X00X00
000X000X000X000
7.7i»16CE-06 I'.gSSeeEOO -2.50U78E*00 1.06659E-r5 f .83i,i»^£+00 -2.<«0190E*00 1.3232?t-C5 e.'tesge'i + OO -2.31620E+00 2.C3'»07E-Cr 1.3C39<»E + 01 -2.35e33E+00 2.75762E-C5 1.76975E + ')1 -2.2'439i»E + 00 3.65'»96E-C5 E.a^SSgE + Ol -2.21.599E+00 '».»»9Q52E-C5 3.i<»220E4Cl -2.12599E-fOC
6.C000CEV1 1 o.COOOCE+3 1
e.oüCJuE+pi 6.B0JCCE+? 1 ö.OOOQüE+O 1 o.OOOOCE+C 1
2.5000aE+CO 2.5JJOJE+C0 2.5üOO'3E+0!) 2.5003JE+Q? ?. 5Jaao=.+oo 2.50000c*CC 2.5J0QJc*ÜJ
1.0rOOQ£-01 1.0C003E-01 l.CQCOOE-01 l.OCOOOE-01 1.0C303E-31 l.ncOOSE-Ol
00X00 000X000
0000X0000 00X00X00
000X000X000 00X00x00X00
000X000X000X000
7.17887E-C6 9.82516E-0e LZlie^E-Of 1.86689E-05 2.50935E-C5 3.32',85E-0? '..(♦0418E-0&
«..sggyeE+ao 6.30211E+T0 T./TijroE + O!! i.ig'AgsE+ni 1.61213E*(!1 2.13596E+01 2.8327TE+01
•2.<.0837F+aO ■2.2983QE+00 ■2.2073i»E+00 •2.25?39E*00 •2.13521E+00 -2.13953E*00 •2.01608E+80
6.00000t*01 6.COÖ«uF*01 6.0ÜQ00E+T1 6.-OQuLE+01 6. L ü Q J C £ + ^ 1 b.-OOJuF+Jl 6.CC3 JliE + 01
2.5ja03£+00 ?.53flOJt*(iO '.SOOCOEtCO 2.53003E + C(! ?.5J00JE+CJ
2.5J03JE+C]
2.5J0a3E + u(!
1.2n003£-01 1.20000E-01 1.2CC0JE-01 1.20CC3c-01 1.2C000E-Q1 1.20000F-01 1.2CüüO£-01
00X00 OOOXJOO
0000X0000 ocxooxoo
000X000X000 ooxcnxooxoo
000X000X000X000
6.72019E-06 9,ll»091E-06 1.12iniE-05 l.r31l5E-0E 2.30Ö59E-0E "^.C58^3F-'!,; (♦.CIMIE-OS
'.t339'»4E + 00 5.868«.lE + 00 7.2035ÜE+00 1.11185E+T1
1.96D«»2E+31 2.581.22F+C1
•2.32366E+00 •2.20781E+00 ■2.1129'3E+00 •?.16581E*00 •2.0'«2UFfOO ■2.0<.87qE+?0 •1.92360F+00
6.C0000E4-P1 &.00uuuc*3 1 6. t 0 J J ü E «-01 b.üOuOCc+C 1 6.ujo00£+r 1 6.C00J(J£*31 b.OOuQuE+Ol
2. B^OOOL + OT
2.i>]0QnE + 3p
2.5flu03EtQQ 2.53033^0" 2.?Q003c+CJ 2.530Q3£*Jü
'.5Jil03E*0j
l.'tO0Q3E-01 1.4COO0F-Ü1 1.UCCP3E-01 l.ttPCCOE-ai l.itCflQOF-ll i.uccroE-oi
00X00 000X000
0000X0000 00X00X00
000X000X000 ooxooxonxoo
nooxoooxoooxooo
b.^33<«2E-(]6 8.C6498E-06 l.C<»l»ä9E-C6 1.61721E-C5 2.1'»08c:£-ü5 2.83571E-05 3.^9103F-C=
'♦.0bi»4JE + 0(3 5.5C301E+00 fc.71988E+00 1.039i«7E+0l 1.3777'4E*01 1.8?i»76E+01 2.37809E+01
•2.2<.769E+00 •2.12718E+00 •2.02929EO0 ■2.08e96E+00 •1.96C7';c+00 ■1.96973F>aO •1.8'»3^7t+00
6.CQuJ3E+31 b.caoaoc+31 b.QCOOÜE*'1 1 u.üflJOlHO 1 6.u03QCE + C 1 b.CüOflCc*0 1 b.cooauF:+oi
2.5J3J3E+01: 2.5)303£*C3 2.5Jü00£+33
?.53033EtC3 2.53003E+GC 2.5J303E+CC
1.6C000E-01 l.f.p:C0E-01 i.eccoat-oi l.bOOCOL-01 1.6CQC3t-31 1.6C003E-01 1.6P0Q0E-31
00X00 000X000
ocooxrooo 00X00X00
000X000X000 00X00X00X00
300X000X000X000
5.999^8E-Ce 8.Ü6657E-06 g.rqifU'.E-Ofc 1.5193CF-P5 1.997i»lE-0F 2.b<«669F-05 3.U1729E-0?
3.85267E*"Q 5.i877bE+C3 e.3a361E*-00 9.77212E+00 1.28635E+01 1.70365E+01 2.23323E+01
•2.17P&0E+00 ■2.05i»32E+Q0 •1.95<.?3E + 00 ■2.01628E+00 •l.«8fli»2E+00 •1.8997PE+05 ■1.773S5E*00
84
MBBWIQBPiWBWBi^lP Mfff.';. • .'■■■:l,i;.-r;\- i:- ■ '■.. : .-■•■ y *■.:■'■;•..., ..■■t.
TÄRLE A-l» (CONT) TYPE II GAG?, GENERAL tlESIGNS
LENGTH (INCHES)
WIDTH ( INCH-IS)
LOOP SPAC1N3 (INCHES)
"KI/SEC PATTERN
MUTUAL INDUC (HENRIES»
GAGE FACTOR
\AMP»MM/USEC/
ERROR TERM CPERCENT)
2f.»0 JC£ + 3 1 i.COuQ^E O 1
2.tCüGJ£+Ci 2.CaüJb£>Ql 2.CJüCCE*01 2.ÜC000E+01
3. ooaiot + oo T. OJ030FtO: i. JJOOJE*1»
3. OJOO JC«'3';
3.C0QO3E*-O0 3.OJ0J1t+u0
8.0C:CDE-T? 8.0CÜ00E-32 8. JuOOOE-0' «.0r003t-02 8.ÜC0OOE-02 e.0CC&JE-3? a.orcooE-02
ooxoo 000X000
0000X0000 00X00X00
000X000X000 ooxooxooxoo
000X000X000X000
2.qi033E-06 i*.01961E-06
7,67831E-06 1.0(H»67E-05 1.38500E-05 1.86159E-05
5.2550<»E*00 7.28275E+00 9.a8227E+00 1,39331E+01 1.90307E*01 2.S2305E*01 3.<»0 55<»E«-01
•9.01903£»00 •6.6<ta78E^00 ■8.33710EtOO -e.l>8089E>30 •8,05890E*00 ■8.05931E+00 ■7.60559£*00
2.0030uE't'01 2.i,üw30E*r 1 2.00JJCE+C1 2.C,0000£»01 2.üOUOCE*0 1 2.C0600E + Ü 1 2.COO00£*01
3.OJOOJE^OO 3. üSUOIcKül 3. coomi + ca 3. OlOOJi+Ol i. 0u0u3tt00 3.03030E+03 3. 03003E* j?
1.QCOJOE-J1
l.CCOOOE-01 1.300CPE-01 1.0C0OOt-31 l.OOOOOE-01 1.0C00uE-01 1.0C300E-31
00X00 000X000
0000X0000 ooxooxoo
000X000X000 00X00X00X00
000X000X000X000
2.70i»50E-06 3.712t,8E-06 '♦.59129E-C6 7,06790E-06 g.s^sabE-oe 1.26'»5'*E-06 i.sa^ooE-os
<>.89880E*00 6.7<*9<»tE«aO 8.37293E+00 1.2S700E<'01 1.7«»«»55E*01 2.31206E*01 3.09269E*01
•8.67598£*00 •8.27678E*00 •7.9<»265E + 00 ■«.105i»0£+00 •7.65685E+00 •7.663<»6£*00 •7.18727e+00
^.OOCOC^tCl 2.t03aCE+31 2.u200C£+01 2.UuOCr*0 1 2.!;0OCCc + 01 2tJOuuOu+Cl 2..03CC£+]1
3. C300 J£ + 3^ 1. CbOCjc + C] 3.3030JE+C3 3.ai3QlE*JQ 3. ü3aJH>0Q 3,0 jQCJt «-Cü 3.0 3D0 3E + G''
1.2CJ0ÜE-01 l.?Cu03£-ül 1.2C00aP-0l 1.2C0ü0E-0l 1.20Ö33E-01 1.2CuCaE-01 1.2r!303E-01
00X00 000X000
0000X0000 00X00X00
000X000X000 00X00X00X00
000X000X000X000
2.53689E-06 3.<»6279t-06 <».26083E-06 b.57263E-06 a.soa^^E-oe i.l672?E-05 1.5i.l35E-C5
«..6080i»E + 00 6.31<»53£+00 7.79525E*00 1.200<»6E*01 1.61585E+01 2.1<»0 92E*01 2.8'»020E + 01
■8.37321E+00 ■7.9i»967E*00 •7,5970<»E+30 •7.77768E+0fl •7.30»"' '♦OO •7.3:^ . jt+OO ■6.82892EiOO
2.oacocc*c i 2.Cü00üt*3 1 2.00^:»,^+: i 2 .wCOuCc + o 1 2.CÜO0CE+C1 2.CQJ00'+01 2.L0JQ0E+C1
3.li3303E+03 3. OJOQJt^uO 3. O:OOJE + O" ■'. O^OCTE+OJ 3. c joiaz^o11
3.C303JE+C0 3. C00:3E + O3
i.uccoaE-oi l.LSOOOE-Ol 1.1.0000E-Q1 l.<»QOaJE-01 l.ttOOOOE-Ol i.i»caa3E-oi l,i»PCC3E-0l
00X00 000X000
0000X0000 00X00X00
000X000X000 ooxooxooxoo
000X000X000X000
2.39569E-06 3.25278E-06 3.9a3<»H-06 ötlS^OlE-Ofe 8,19612F-06 1.18593t-0E l.'»229'«£-05
<*.3625<>E*Q0 5.9'»77<»E*00 7.30872E+00 1.1276QE*01 1.50786E+ai 1.99745E+01 2.62975£*01
•8.10 0<»(»E^0Ö ■7.65612E+00 •7.28817E»00 •7.<»8560E+00 ■6.99965E+00 ■7.01958E*00 •6,5l<»76£*00
2.cücatE+r i 2.UUJüO£>C1
2.1JOüOC? + '?1
2«C30UOE+J 1 2.C000CS + 3 1 2.Qüü0uE+,7 1 2.üJ0';Gi£«-ai
3,&30a3£+30 3.00ÜQJE+CO 3.jJOOJE+OO 3.aaaoit+oo 3. O30O3Et3: 3. ooon€ + oo ■».Ü J013E + 0^
i.e.coaat-ai LecooaE-Ti i.e^ooac-oi 1.6P000E-01 1.6L0B3E-31 l,6C000t-Ql i.scaoaE-oi
00X00 000X000
0000X0000 00X00X00
000X000X000 00X00X00X00
000X000X000X000
2.27383E-06 3.Ü7187E-06 3.7i,i.96E-0 6 5.79983E-0f ■'.67?0<»E-06 l.ül6<»3E-05 1.32235E-05
«..15020E*aO 5.63092E+00 6.88917E+00 i.a6«.«lE + ai 1.<»1512E*01 1.87U36E+01 2.'»5029E*01
•7.85115E+00 •7.38888E+00 •7.0081'♦£♦00 ■7.22139E*00 ■6.72211EtOO -6.7<»875E*00 •6.23<»50E*00
85
TABLE A-l» (CONT) TYPE II GAGE, GENERAL DESIGNS
LENGTH WIDTH LOOP SPACING P R I / S E C MUTUAL INOUC GAGE FACTOR ERROR TERM ( I N C H t S J ( INCHES> ( INCHES 1 PATTERN ( H E N R I E S ! ( W \ (PERCENT!
VAMP^NM/USEty
H . C O O O C E + C I *. ooaait+Gc 8•GO 300 E - 3 2 0 0 X 0 0 5 . 5 7 7 6 2 E -oe 5 .2< .8 32E»9C -<». 600 75E»0 0 <».roooGcOi 3 . 3JGG3£*C3 " . 9 C 0 0 9 E - 0 2 0 00X000 7 . 7 1 5 8 3 E - 0 6 7 . 2 7 2 6 6 E * 30 - U . ( » 2 2 9 9 E * 0 0 • » . u u 0 0 G E * j 1 3. 3 3 09 3E• 03 » . 0 0 : 0 3 E - 0 2 0 0 0 0 X 0 0 0 0 9 . 6 0 7 7 0 E - 0 6 9 . 0 6 8 8 2 E + 0 0 - 4 . 2 7 3 7 2 E + 0 0 <•. oc ooos+; 1 3 . 3 3 03 3 E 0 3 6 . 0 C 0 0 3 E - 0 ? ooxooxoo l .<»7i»95E - 0 5 1 . 3 9 1 3 0 E + 0 1 ••<». 3<»3?8E»0 0 • * . C J U J J £ » C 1 3 . U C J C J C + G G 6.CCCG0E - 0 2 oooxoooxooo 2 . 0 1 0 - . 1 E - C 5 1 .9C00 5E+G1 - 4 . 1 < . 2 0 3E + 0C <..»:03CCE»0 1 3 . 3 3 OG 3E + 0 3 H.OOOOOt - 0 2 ooxcoxooxoo 2 . 6 6 5 3 5 E - 0 5 2 . 5 1 9 3 1 E + 3 1 ••<«.1<»287E>0C <» . OG 3 u l c »G 1 3GuGCE*G3 P . 3 C 0 3 3 E - 0 2 oooxoocxoooxooo 3 . 5 8 9 5 6 E - 0 5 3 . 399f»9E* 0 1 - 3 . 9 2 816E+30
* • G 0 u u OE »C1 3. 3 0 3 0 j £ » 3 3 1 . 3 G 0 0 3 E - 0 1 ooxoo 5 . 1 9 0 8 1 E - 0 6 I . . 8 9 2 0 7 E + 09 - 4 . U 3 5 6 3 E + 0 0 G 3u 5C E + 3 1 3 . OCOCJE+OC 1 .G0G33E - J 1 0 0 0 X 0 0 0 7 . 1 3 7 7 7 E -C 6 6 . ' 3 9 J 2 E * 00 - <». 2W439E + 0 0
. . . C 0 0 0 G E + 5 1 3 . C 1 3 3 C t * C C 1 . 0 0 3 0 3 c - 0 1 0 0 0 0 X 0 0 0 0 8 . S k O U E -C 6 s.ssgnE + ̂ o -<» .0d«.97E+90 •» . GO .,2 OE *G 1 3 . j 3 3dGE *C 0 1 . 0 3 3 G 0 E - 0 1 0 0 X 0 0 X 0 0 1 • 3 5 9 9 J E - 0 5 1 . 2 8 I . 9 9 E M 1 - < » . l b 3 7 3 E * 0 0 * . J C 3 G E +0 1 3 . GuQ£3£+Q3 1 .CCC30E - 0 1 0 0 0 X 0 0 0 X 3 0 0 1 . 8 3 9 2 9 E - 0 5 1 . 7 4 1 5 2 E + H 1 - 3 . 9 5 0 8 S E + 00 4 i t ilCCCE *0 1 3 . 0 j 0 0 3 c * 0 0 l . O f G O C L - 0 1 0 0 X 0 0 X 0 0 X 0 0 2 . 4 3 7 7 0 E - 0 5 2 . 3 0 8 0 3 E 4 9 1 - 3 . 9 5 i . 9 0 E * 0 0 - » . C 3 G 0 l t * 0 i 3 . 0 3 uG3£*G G 1 . 0 G 0 0 3 E - 0 1 0 0 0 X 3 0 0 X 0 0 0 X 0 0 0 3 . 2 5 394E - 0 5 3.3866<»£ + 31 - 3 . 7 3 1 1 9 E + 3 0
* . .G3udG£+0 1 3. U J 3G 3E•O 3 1 . 2 C C 9 3 £ - 3 1 ooxoo <•. 8755CE -C 6 1 . 6 G 1 3 2 E + 0 0 -km29C18E* 00 H . O C C J 0 E + 1 1 3 . 0 3 9 0 J £ * 0 3 l.aooanE - 0 1 0 0 0 X 0 0 0 6 . 6 6 7 1 5 E - 0 6 6 . 3P<» «•<»£• OC -< • . 3 8787E + 0 0
. C COG + 0 1 3 . G J Q O J E + GO 1 . ? C 3 3 3 E - 0 1 0 0 0 0 X 0 0 0 0 9 . 2 1 6 2 6 c . - C 6 7 . 7 9 1 8 1 E * 0 0 - 3 . 9 2 0 2 8 E + 0 0 i».G00 0 u t * 0 l 3. G J 3 3 3 E * 0 : 1 . 2 C C 0 9 E - 3 1 0 0 X 0 0 X 0 0 1 . 2 6 6 4 1 E - 0 5 1 . 1 9 8 M » E * 0 1 -<». 00 758E »00 - . . . • 33 0GE + 0 1 3 . 0 J 003E + G 3 l . ' G G O O E - 0 1 0 0 0 X 0 0 0 X 0 0 0 1 . 7 0 0 6 7 E - 0 5 1 . 6 1 2 8 3 E + 3 1 - 3 . 7 8 5 9 8 E + 0 0 •».COGu'3"+2 1 3 . 0 j 30 G£ + C J 1 . 2 3 2 3 3 E - 0 1 0 0 X 0 0 X 0 0 X 0 0 2 . 2 5 3 i . wE - 0 5 2 • 1 3 6 8 9 £ » 31 - 3 . 7 9 3 3 9 E + 0 0 • . .CG303E + G 1 3. C 0 0 0 J E • C j l.arcooc - 0 1 0 0 0 X 0 0 0 X 0 0 0 X 0 0 0 2 . 9 8 2 1 3 E - O f 2.33<»17£ + « 1 - 3 . 5 6 3 7 5 E + 0 0
•». k GO 0C£ +01 3 . j w G 3 1 £ * C ? 1.UC 30 3 £ - 0 1 00X00 • . . 6 0 959E - 0 6 4 . 3 5 5 8 1 E + 3 0 - i » , 15942E + 3 0 <• # u 3 w u C ^ +G 2 3. j 2 Q 3 3s. • C 3 1 . 4 C ? C ? E - 0 1 0 0 0 X 0 0 0 o . 2 7 C 8 2 E -C 6 5 . 9 3 7 6 6 E + 0 0 - 3 . 9 < » 7 8 1 £ » 0 0 <», COOG CI+G 1 3 . g J J G J E + L * l . f »CC03E - 0 1 0 0 0 0 X 0 0 0 0 7.C- 9171E -C 6 7 . 2 9 5 2 i £ * 00 - 3 . 7 7 1 6 5 E + 0 0 -».»CJGCE*G1 3 . 0 J O C G E + C J 1 • <»0 CG j E - 3 1 0 0 X 0 0 X 0 0 1 . 1 8 7 8 4 E - 0 5 1 . 1 2 5 5 9 E • 3 1 - 3 . 8 6 8 9 3 E + 0 0 >.. 'J GOGGE * J * Cu 331E*G3 1 . u 3 0 0 0 L - 3 1 3 0 0 X 0 0 0 * 0 0 0 1 . 5 8-.5BF - C 5 1.5C"»8<»E* G1 -3 .6< .C»2E»0C •» .C jGCGE*01 3 . G J G C ? £ » 2 * i.wcao?E - 0 1 OCXOOXOOXOO 2 . 0 9 9 2 8 L - 0 5 1 . 9 9 3 4 3 E + 21 - 3 . 6 5 1 5 9 £ * 3 0 u . C C O G j £ * C i 3 . ojOJJc»rc 1 . 4 C 3 G 3 E - 0 1 0 0 0 X 0 0 0 X 0 0 0 X 3 0 0 2 . 7 5 6 8 1 E - 0 5 2 . 6 2 3 7 2 £ + 3 1 - 3 . i » 1 8 1 1 E » 3 0
• » . K J G G C E * 1 1 i . CICO 3£•G3 1 . 6 0 C 3 C E - 3 1 00X00 <•. 3 7 9 " & t - 0 6 •..1<.3<»8E + 0C - i » . 0 < . 0 1 5 e + 0 C •».».GCCCC*01 i . G30G3E+GC - 3 1 0 0 0 X 0 0 0 5 . 9 2 9 9 6 E -OC 5 . 6 2 G 8 4 E + " 0 - 3 . 3 2 C 6 8 E + 3 0 • » . C G « - j C £ * 2 i 3 . C 3 j j 3 E » C i i .oOOOCE - 0 1 0 0 0 0 X 0 0 0 0 7, 2<»0 1?E - C o 6 .87571.E+CG - 3 . 6 > » l 2 3 f + 30 • . . W G J O C E + H 3. o:CCJE*CC 1.6CG9CE - 3 1 0 0 X 0 0 X 0 0 1 . 1 2 0 2 2 E - C ? 1 . 3 o 2 7 9 E + C l - 3 . ' 4 3 9 8 E +0 3 •••v c c«u — ^ <* l 3. : : 3 : J E * G - 1 . 6 - 3 3 3 E 0 0 0 X 0 0 0 X 3 0 0 1 • 4850 6E -C 5 1 . i . l 2 1 3 " * 0 1 - 3 . 5 1 0 7 T = > 0 0
* C 1 3. G - u C 3 E + C j 1 . 6 f 0 0 3 C - 3 1 0 0 X 0 0 X 0 0 X 0 0 1 . 9 6 7 2 S E -G 5 1 . 9 7 3 3 ' . < > 3 1 - 3 . 5 2 5 1 1 £ + 3 0 •«« 3. 3 J O G J E + ' C i . 6 r o r i E - 3 1 0 0 0 X 0 0 0 X 0 0 0 X 0 0 0 2 . & 6 S 3 6 E - C b 2 • 2 7£ + 01 - 3 . 2 8 9 2 3 E + 3 0
86
"I "■
RMMMMMMM.
mmmvmemmm .■■I.IH'P»I|»II»I.IH!IIU,II IIIWM'JI"!
T4iiL[ r.-u (rcNT) TYP? II &aGr, GPNtCAL n£ ;IGN
C
ut (LiTH .v I L r M
(IHCH^S) "Kl/SEC Pfi TTEHN
MUTUAL IKDUC (HENRIES)
oar,;: FACTOR
^)
ER"Oh TERM {PERCENT)
■J«.JO-J:
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"V'T ""TT.""!^*--!
■■■■■■■MBWWWMI^WtlWWWWWMWWaWWWi 1 !
P*0Gi<4H üSI^KINPJr.OjrPUI.PJi^H.rrt3
airttHSlÜN 4Q45J) •b«R(3u) ÜIM-.N5iON -Ul(15).-O2U3i«Hüm5)«Tir(a)tT<U(0) PI = J.I««15^26 A,1U=.0üüJäl25&b9
RtAO Hi (H31 (i) il 31 FORMAT (AotAbfA)»,
kiAO itf» (Ha2 (I) , l = l,o) 3^ FJKKAT (Ac,Uo,Aö,A>,M?,Ai,A2,A^>
KiAO J3» (H03 (I),1 = 1,2) 33 FORMAT (A^,A7)
1 CONTINUE. k-AO •», (TAT(I),I = 1,)) kcAO •«, (Tin i)fl*l,3»
■♦ FORMAT UALJ) K-AO =>,Y,X,_R,J< RwAO 6,ITfNMAX.iCR
5 FORMAT U(F9.5,1X)) o FORMAT (3(12,IX))
IF(IT-l)l.,lü,i9 lu CONTI^ÜL
PRINT 2Ü 2u FORMAT (1M1)
PRINT i»*,(TAT(I) ,I = 1,J) PRINT '♦'♦.(TIKI) ,I=li»)
*♦>♦ FORMAT (ÖAlu) PRINT 22 PRINT 22
22 "FORMAT '(* •) PRINT il,(-401(1)
7=IN3JI,TAPi9=0UTPJT)
91
92
93
1=1,12) FORMAT (<«XfA6,aX,A5,iiX,A^,lXiA7,2X,A3,3X,A3,2X,Ab,lX,A5,2X,A^,lX,A
16t2X,A3,lX,A4) PRINT a2,(H32(I),1=1,9) F0RMA^J3K,Ad,3X,Aö,?X,Aä,3X,4i,lX,A5,2X,A9,oX,A2,7X,A9) FäINT 93,1 rfÖ30) .1 = 1.2) FORMAT (6bX,A^,Ä7) PRINT 22 X=X»C(/ Y=Y»C\/ R=R»Cj/ ÖRsDR^üV " IT - CXIT T£ST NNAX - MAX NUMBER OF PRIMARY TURNS TO tic iOMSIU-RtiO liR - MAX NJMBiR OF TlNcS OR IS ADOiO TO R INTF - NUM3 TAT - TABL^ TIT GAF
ft OF PRIMARY/StiOMJARY _ HEADING
TABLE HtADINS GAGIL FACTOR U HlLLlVOuTS P_R
INTERFACES
4MP»l1M/hICR0SE£UN0
SET R DO 100 1=1.ICR PRINT 22 IFJ 1-1)16.15.17
17 R=R ♦ OR lb CONTlNUt;
SET S AND iALCULATE aa ANO E^R rOR. EACH M
88
m.vwpHijißf^Typiwirvt. '■/ r.f% r- Ji^*'".""f«Ti,r'3l i"rfv» vf
■wm
flM-NHÄA-l üü eü M-l.il
M=saRr(K»A ♦ y»Y»
MM=sa'<T(x»x ♦ Y»r ♦ J'S)
1 y»AuJ^(BMlf*C)/(S»(V*A4))) ) PJli=(AMÜ/PI» »(ÜLO^C'CXfÖ)/(i'tX^a.*) )» ♦ X»lAA-Bi/l3»J)» QPL = lia(H)/X iRR(H» = (pja-uPL)/p.n
i CONTINUE.
ia j
7i
2ij
94
Uü 2QJ N=2tNMAX J'H-l H?-U Ni=N-l Ji=N-l INTF=2»(N-1) UX=u. QY = 3. L0wF=INTF ÜO 3JI] J = 1»JÜ UX = ax ♦ CJ£F»au(J) (1Y = QY ♦ GüiF»aQ( JM(l.-cRR{ J)) u0^F=i0iF-2. C0NII>JUi Q=UX t<=l. - Qx/^Y oUF=lOüüG-o.»Q/(X»(l.-cR))
X = X/L;»/
Y = r/Gy/ R=R/Gi/ PRINT 71,X.V^jNP.MStiiGAF.i^ FORMAT {lX,3(£12.5,lX»,2X.I2,i»X,I2,2X,i(£l2.5flxn X=X»^»/ y=Y*i;(/ R=R»C\/ CONTlNJi C0NTINU£ GO TO 1 CONTI^O- GALL EXIT cNO
89
P*OG*AH OSINZdNPUr«OUTPUTfPUMuHtTAP£rsiNPJTiTAPrLft'OUTPJT) DIMENSION (ja(20»,£RRC2a» DIMENSION H01(15)«H02U5)«H03(15) DIMENSION NPATTdO» .P*TT (101 »TERNNC ta> »NMAXXili) DIMENSION TIT(8)tTAT(8> PI«3.1<*15926 ANÜs.00aCul25&69 C«**.a25<i READ Jl, CH01CII f Utt LI)
31 FORHAT (Ab.AS(Ai»fA7,Ar.A6,A5,Ai»,Ab.A5,A<*) R£A0^32,JHD2 (IJ ,1*1,7»
32 FÖRMAT ( AÖ, At», Ä8 ,Ä7', A9, A2, A9) READ 33,(H03(I),1*1,21
33 FORMAT (A<»,A7) READ 3<*, (NPATT(I),I»li7)
3<» FQRHAT (8IIÜI READ 35,^PATT(I) ,1 = 1,71
35 FORMAT (7Ä9) READ 36,(r£kNN(I),I=l,7)
36 FORMAT (7A7) RcAO 37,(NMAXX(I),I*1,7)
37 FORMAT (712» READ 6,10^
6 FORMAT (12» i CONTINUE
READ '«, (T^ril) ,I*l,d) READ <*,(Tir(I) ,I*l,d)
i» FORMAT (bAlü) R-AO 5,Y,X,k,[)R
5 FORMAT (<»{F9,5,lX») RtAO 7,IT
7 FORMAT (12) IF(IT-1»1^,10,9^
lg CONTINUE PRINT 20
2ü FORMAT (1M1> PRINT <*. (TAT (I) , 1 = 1.0) P<INT <♦, (TIT (I» ,I = l,(J) PRINT 22 PRINT 22
22 FORMAT (♦ »» PRINT il , (ROKI) ,1 = 1,11)
91 FORMAT (<,x,A6,oXt4S'»X,Ai,,lX,A7,6x,A7,5X,Ao,lX,A5,2XfA'4(lX,A6,2X,A
PRINT 92,(rt02(I»,1=1,7) 92 FORMAT ( 3X,Mo,9X,A&, 5X , AÖ, 4X,A7,7X,A9,oX,A2,7X. A?)
PRINT 93,(H03(I) ,1 = 1.2) 93 FURMAI (71X,A*,A7)
PRINT 22 PRINT 22 X = X»C>/ Y=V»CJ R=R»C; üR=OR*:V IT - EXIT T-ST NMAX - NU.i3£K OF LOOP INTERFILES ä£TXE=.N z*l<iHz PXl AND 5E3 T0RN5 IiiR - MAX NJMBER JF TlMoS DR 13 AUQ^J TO R NPAT - Pki/Sci PATTERN
90
] PAT.Td^N - PKl/Sdi PATTiRN TAT - FABLE H£ADINS TIT - FABLE HcAOI^; UAF - liAUr. FACTOR IN HILLIVOLFS P.R AMP»MM/HICi*OS£SOMO
btT R 00 IJi] 1 = 1,IC« IF(I-l)löilD,l/
l^ R=R ♦ JR lo CONTINUE
00 2Gu J=i,7 NPAT=SPATI(J) PAT=PATr(J) TERN=Fc:RNN(J> NHAX = N/1AXX( J) ScT S AND :AL^JLAIE Ui AND tR< FOK EACH M
Nal.NiAX
m*
'■■■
if
00 oü P = rt S=P»R A = saRTU*X ♦ Q=SQRT(X»X ♦ C=SQRT(r»V f AA = S(iRT(X»X QO(M)=(AMU/:>
i r*AL0&(6*(ir PDU=(AMU/PI) QPL = CU(MWX tRR(M»=(Paa- CQNTiNuE IF(212-NPAT) IF(31J-NPAT) IFUl^-NPAT) 1F(21212-NPA IF(3lili-NPA iF(2121212-^ IF(31il313-M U=2.MQ(m)
liO TO ?u Q=2.MUCm) [.R=l.-(Q/2.)
D) GO TO 7Ü
S-» U = 2.*(iQ(l) £t=l.-(Q/2.)
1)+QQ(-»)/ (1.- ijü T ■ j 7 Ü
5o Q=t.»(QQtl) tR=i.-ra/ (^.»
&Q TO 7Ü 53 ^-♦.♦uau)
LR=1.-U/ (<♦.♦ 1)*2.#(UQ(3l/
GO TO 7u
du
51 53
57 3=« bl 50
5^
S»S) 3»S»
♦ Y»Y f S»S) :)»(2.»S-2.»C-2.»3*2.»AA»X»AL0G(C»(X*8)/(S»(X»AA)>) ♦ ♦C)/(S»{Y*AA))) ) MALOJCMX+BI/CSMX + AA) >» ♦ X*(AA-di/(B»a))
UPD/PDJ
51,5J ,51 pi,52,53 55,5^,55 T» 5 7 , 5 b, 5 7 1)59,50,5^ PAT)bl,bü,ol PAr)62,b2,62 ♦ iQ(2») /UQ(imi.-ERR(in *CU(2)/ (l.-tRR(2)) )
«■ 3Q(2» ♦ U(l(3) ) /(aQ(l)Ml.-£RR(l))»QU(2)/(l.-cRk(2))*QQ(3)/(i.-t.RR(3)
♦• aQ(2) t aa(3) •■ aac»)) /(JQ{l)/(J.-cRR(l»)fQa(2)/(l.-ERR(2))*Qa(3)/(l.-iRR(3) LR^C»)))
♦ QQ<2)> *■ I .♦(Qai*) * Qat5) ) (Ua(t)^(l.-cRR(in*aQ(2»/(l.-£RR(2n) ♦g.^tUQC'fJ/Cl.- J/tl .-iR-<(3) > n
♦ aQ(2) ♦ ii(i)) ♦• 2.»(Q(i(5) ♦ aatb) ♦ ua{7n (aU(l>'tl.-£RR(in*QQ(2) /(l.-ERR{2))*QQ(3)/(l.-£RR(3)) (l.-£RR(5))taa(ö)/(i.-iRR(b»)♦Ua{7)/(l.-ERR(7) )))
♦ 'aQ(2)) «• «♦.♦(oac») * ciut5)) ♦ 2.»((10(7) ♦ aii(dH
91
r
£«-l.-Q/(3.»{Q0(l)/(l.-cRi<(lH*aQ(^»/(l.-tKT(2)) )♦■».» (UQ( «)/(1.- l£RR((»n»Qa<5l/(l.-£RR(5) ) > tZ.» (QU< 7» / (1.-£K-i( ^) * iQuls) / (I.-cR<( i H 1))
GO TO fO 62 Qs6.*(QQ(l> ♦ QQC2I ♦ QQiJH ♦ «.»(JQCi?) f QUCi) ♦ CU(m ♦
i 2.»(aQ(9» ♦■ QQ(iu) ♦ QQ(il)) £R=l.-Q/(o.»(a^(ll' (l.-ERR(l)»»Qa(2)/(l,-r.«R(2)) ♦QÜUä) Ml.-tRR( 3) )
l)f<».*(QQ(3)/(l.-£R<(5MfQQ(6)/(l.-^RR(b) i*(iH7i/ (1.-iRk( 7)) ) « 22.*(Qll(9)/(l.-£RR(9n»QQ(ld)/(l.-£'tR(läMtaa(ll)/(l.-£RR(lin))
Ti CONFINUc GAF*l00 0ßJö.»a/(X»(l.-£Rn ERsläfl.'ER XsX/Crf
RsR/C«/ PRINT 71,X,rfR,PAT,Tc^NtQ,GAF, = ^
/i F0RH4T (lXf3(£12.JtlX),lXtA8tA7,lX,j(£12.5,lX)) X=X»C^ YsV»C R=R»Crf
200 CONIINUL PRINT 22
103 uONTINUd GO TO 1
99 CONTINUE. CALL EXIT ENO
92
»WWW^iW^^WWWWWUMIIllJ ■* ■
AFWL-TR-74-205
APPENDIX B
COMPUTER CODE LISTING--D 1VI:
9 3
i.piiwnviiMPPii ***** mil i " *w«V' ■ T-T—■
C c c c
c c c Ml
c c c c c c a c c c c c
PROGRAH 01VE(INPUT,OUTPUT,PUNCH,TA?E7«INPUT,TAPE«»OUTPUT) COHHON kXdZlS) .YV(i20ü) ,0(630),ER(6aO) ,SR(600) COHrtON XC0(12Ja),YCO(l2aC>,X(1200),Y(1230)•ALFA(120 0),SEGRC(1200) COHHON AY(1203) •AX(120 0),E (600)«OHITA(600) COHHON XC(12J0) ,VC(l2Qj),OSX(1200),OSY(1200) COHHON J,CüR,PItAHU,O,U,CV,C,0T,XS,YS,ZS,R,RArj COHHON N1,N2,N3,N<»,N5,JHAX,NSHK,HSHK,T(60J) COHHON XXCO(12JO),YYCO(120J),ZZCO(1230) ,ZCO (120G) ,Z (12 09), 7Z (1200) COHHON OSZdZOO) ,OSXX(1200),OSYY (120Q),OSZZ(12C0),ZC(1230) COHHON XXC(12QÜ),YYC(1201) .ZZC(1203),AZ(120 0) COHHON ALFAA(12oJ),SEGAA(12Q0),BETAA(1200),BETA(12C0)
THIS PROGRAH COHPUTES THE TOTAL RESPONSE OF THE HUTUAL INDUCTANCE PARTICLE VdLOCXMETER IN A DIVERGENT, OFF-AXIS FLOW FIELD. THE IOEAL RESPONSE OF THE GAGE AS DEVELOPED THEORETICALLY IN THE OASA 1*31 REPORTS CORRESPONDS TO ON-AXIS, ONE-OIHENSIONAL FLOW. THE IDEAL FLOW RESPONSE IS CALCULATED AND COMPARED WITH THE TOTAL GAGE RESPONSE AND PRESENTED AS A PERCENTAGE ERROR. INCLUDED IN THIS ERROR IS ALSO THE EFFECT OF THE CHANGE IN GAGE LENGTH UPON THE GAGE SENSITIVITY. ^
THE SHOCK SOURCE IS LOCATED AT THE POSITION (-XS,-YS,-ZS) AHEAD OF THE GAGI CENTER. THE SHOCKFRONT PROPAGATES SPHERICALLY FROH THIS SOURCE.
THE UNOISTUR^D GAGE IS DIVIDED INTO N5 SEGHENTS. TIHE IS STEPPED IN DT TIHE INCREMENTS AND THE SHOCKFRONT POSITION CALCULATED. EACH GAGE SEGHENT POSITION IS CALCULATED AT EACH TIHE STEP. PERIODICALLY THE TOTAL GAGE MUTUAL INDUCTANCE IS CALCULATED. THE GAGE RESPONSE IS THEN CALCULATED AS THE NEGATIVE TIHi 0£RIVATIVE OF THE CURRENT TIHES THE HUTUAL INDUCTANCE.
C C c c
C c c c c c c c c c c c c c c c c
c c
C IS SHOCK VaOCITY U IS PARTICLE VELOCITY
XCO, YCO, Z30, XXGO, YYCO, AND ZZCO ARE INITIAL SEGHENT INTERSEC POSITIONS
GAMHA IS ANGLE OFF OF SOURCE IN THE X,Y PLANE RELATIVE TO THE GAGE CENTERLINE
THETA IS ANGLE OFF OF SOURCE IN THL X,Z PLANE RELATIVE TO THE GAGE CENTERLINE
THZ ANGLES ALFA AND BETA DEFINE THE SHOCK PROPAGATION DIRECTION IN THE X.Y.Z LOOP
THE ANGLES ALFAA AND BETAA DEFINE THE SHOCK PROPAGATION DIRECTION IN THi XX,YY,ZZ LOOP
SUBSCRIPT I IDENTIFIES SEGHENT IN SECONDARY LOO? SUBSCRIPT < IDINTIFIES SEGHENT IM PRIMARY LOO0
SUBSCRIPT J IDENTIFIES TIME INCR'iHENT PDMX IS PARTIAL DERIVATIVE OF ThE HUTUAL INDUCTANCE WRT X AS DERIVED iN DASA 1-31 REPORTS QD IS HUTUAL INDUCTANCE DERIVED IN OASA 1-31 REPORTS
94
rfWJJ'f^w^'fU^i^ T ' r j—Wsl^TOJCTI^r» 1wy»i»iTO?PTT'1-,.F»W'TOW7^' ''-r^'W'w^»rf7;y^w!TT»rii«w^ ™^«iFyi".' '^rrir- F^lfl^H-
ONITI IS DERIVATIVE OF MUTUAL INDUCTANCE WRT TIHE - IDEAL FLOW
_C_ _C_ C
_C_ c
c _c_ ^_ c_ c
DMITA IS DERIVATIVE OF NUTUAL INDUCTANCE HRT TIME FOR ACTUAL DIVERGENT/OFF-AXIS FLOM
ER IS PER3ENTAGE ERROR IN_G*GE RESPONSE DUE TO NON-IDEAL FLOM OSO IS INITIAL SEGMENT LEMCTH" Yjj IS INITIAL GAGE WIDTH, HETERS XL IS INITIAL GAGE LENGTH, METERS VNSEG IS NUMBER OF SEGMENTS ACROSS GAGE WIDTH XNSEG IS NUMBER OF SEGMENTS ALONG GAGE LENGTH YNSEG MUST BE AN OOP WHOLE NUMBER RATIO MUST BE_ AJt OOOlWHOLE NUMBER, TYPICALLY BETWEEN T N1,N2,N3,NW IDENTIFY CORNERS N5 IS NUMBER OF GAGE SEGMENTS
AND 15
T
c_ c
JJ AND INTl DETERMINE INITIAL VALUE AND INTERVAL IN J FOR CALCULATING MVP
1 CONTINUE READ ♦JLtiEXlT
kk FORMAT (III
C IF JEXIT « l._REAO DATA IF JEXIT = 0, CALL EXIT
IFUEXIT - ll8««,s»5,%5 »5 CONTINUE __
READ 17,JDIA6 JLlJrQRMAT_JJjX ,
C IF JOIAG a 1, PRINT OUT DIAGNOSTICS C IF JUIAG g Q, OHMlf DIAGNOSTICS
PRINT 9 9 FORMAT llHlj
READ <»0,RS kü FORMAT CtFlO.O)
READ Vl,YW,D,C,U,OT,GAHMA,THZTA,RATIO kl FORMAT (8F13.0)
READ <»2,YMSEG,JMAX,JJ.INT1 k2 FORMAT <F10_. Ob7i5)
READ ^3,QATE »3 FORMAT 1A_9J
INITIALIZE
CV*l»/_.025'* ANU=.300001256bk PI=3.1415926 CUR=
■-57, RAO« 295779 DTs=oT»iaaoaao.
95
tuuMMMy^üidUM ■^.A-■,.,-...■„., "■-■-'ift '•■"■iliiifriitftiiiii in " •'■miiimf-^iiiBiiiit--'-'-—imi-i
HW imppHiiippi, wmnw^" mirm pppnpnip m wanftfiw ^' '^r*™™imimmmwmmmmp*^w™rT«m<Frin '.*■"'! ■Hn"V'lT"W?*f" ^/T^'
C
JGRI1F = JJ
PRINT i»90 »♦9L FORMAT (• Ocl/ELOPMENT OF HUTüÄL INOUCTAMC: PARTICLE VTLOCT^TEP ♦)
PRINT 53 PRINT <m
♦ 91 F^RHAT (♦ INVESTIGATION OF OFF-AXIS/lDlVERGENT FLOW RtS?0N?? •) >RINT 53 PRINT i>92
492 FORMAT <• PROGRAM OIVE •) PRINT 53 PRINT i*3Z
-►93 FORMAT (♦ AFML/OIX ♦) PRINT 494,0ATE
^9- FORMAT (15H DATE OF RUN - ,A9) PRINT 53 PRINT 53 PRINT 53
♦" • V # '»♦»♦♦»♦»»♦♦»♦•»♦»♦♦#»♦• »»♦♦»»
XL=YH»RÄTIO OSOsYM/YNSEG XNSEGMXL/YMJ'YNSEG NY=VNSEG NX-XNScG N1=CNY ♦ l»/2 N2=M1 ♦ NX N3=N2 ♦ NY NV« N3 > NX N5 = N<» ♦ CNY - t)/2 GANMA = GA"iMA/RÄO THETAsTHETA/RAO OENOM = l ♦ (TAM(THETAn»»2 ♦ (TAN (GAMMA) ) ♦•2 XS=SQRT(CRS*»2)/QENOM) YS=SQRT((RS»»2)*((TAN(GAMMA»)»»2)/OENOM) ZS=SQRT((RS*»2)M(TAN(THETAn»»2)/OENOM) GAMMA=GAiMA*RAO TH£TA=THcTA^RAO
XS=-XS YS=-YS zs=-zs__ PRINT 53 3
600 FORMAT <• GAGE GEOMETRY •) PRINT 53 PRINT 501,XL
5(11 FORMAT (5X,1-»H GAGE LENGTH =,F5.1,7H INCHES) PRINT 5 3 2tYH
502" FORMAT (5X,13H GAGE WIDTH =,F-».1,7H INCHES) PRINT 5(13,0
503 FORMAT (5X,15H LOOP SPACING =,F'».2»7'H INCHES) PRINT 51'*.N5
5lt FORMAT i5X,2lH NUMBER OF SEGMENTS =,15) PRINT 53 PRINT 53 PRINT 53%
50H FORMAT (• SHOCK SOURCE POSITION ♦)
96
mmUtoaaamtM rnuintHtUatäämUämvuaäu^ii.^,.,.^.^ '!■ mr I I
:
i Co
^7
31..
512
51J
19
2L
21
22
do
FORrtAT PRINT FORMAT PRINT FORMAT PRINT FORMAT PRINT FORMAT PRINT FORMAT PRINT PRINT PRINT FORMAT PRINT PRINT FORMAT PRINT FORMAT PRINT PRINT PRINT FORMAT PRINT PRINT GAMMAs TH:TA= XL=XL/ YM=YM/ Ü = Ü/C\/ OS0=0 5 xs=-xs ys=-YS zs=-zs 00 19 XCO(I) YG0I1» NA = N1 no 2: IA=I ♦ YCOCIA NA=N1 00 21 YCO(I) XC0(NA N9=N2 00 22 IA^I + XCOJIA DO 23 IP=N5 XCOdP YCOdP NCsN3
(5X,3^ RS =,F9.'.,''H MtTERS) 5-5.XS
(3X,5H XS =,F9.<*,7H M£TERS) 5J6,YS
(5X,5H YS =,F9.>*t7H M£T£RS) 537,ZS
(iX,5H ZS =,F9.^,7H METERS» 503,GA.HA
(5X,8H GAMMA =,F5.1,3H JtGREES) 5Ü9,THITA
(3X,8H THETA =,F5.1,8H DEGREES) 53 53 51J !• SHOCK PARAMETERS »I
53 511,C
(5X,17H SHOCK VELOCITY =,F6\0,11H METERS/SEC) 512,0
(5X,2jH P4RTICLE VELOCITY =,F&.ltllH HETERS/SEC) 53 53 513,OTS
(2X,17H TIME INCREMENT =,F5.2,9H MICROSEC) 53 53 GAMMA/RAO THITA/RAO :v CV
O/CV
1=1.Nl = j •
=0S0/2. - 1 1=1,NÄ
1 )=YCO(I) + OSO * 1 I=NA,N2 = Y30(N1) )=OSü - 1 I = NA,NT
1 )=<CO(I) ♦ OSO 1=1,N3 ♦ 1 - I )=XCO(I) )=-YCO(I) - 1
I
97
ti^^mmtutt JMUlMmimliiiii nn '■ ira^ '-'-"■■■"■■' ■■' .-,..,..■■ ..-. r,- ■ ■■■ij
HRjnpqiPiqpPKpR^^ -ww." itl'»wwiiW«.w^(i|.Hiwwi*Wt;""«Wfl>ra"»«i"!«lWPP«mi< ,i)iigji«iwijii •'
00 18 IsNZ.SC IA=I ♦ i XC0(IA> = )(C0(N2) YC0(I«)»YCO(I> - 0S0
10 CONTINUE 00 25 1*1.N5 XXC0(I>=XCO(I) YYC0(I)»yC0(I) ZC0(I)=O/2. zzcom=-o/2. ALFA(I)=ATANl(VCO(I) ♦ YS)/(SORT(«XCO(I) ♦ XS)»»2 ♦ IZCOfl) ♦ ZS)»
i»2n» ALFAA(I)=ATAMt(YYCO(I» ♦ YS) /(SORT((XXCO (I) ♦ XS)*»2 ♦ (ZZCO(I) ♦
1ZS»»»2))) BcTA(I>=ATAM{(ZS ♦ ZCO(I))/(XS ♦ XCO(I>?> a£TAÄ(I)=ATAN((ZS *■ ZZCO(I))/(XS + XXCO(I))) S£GRC(II=SQRT((XS ♦ XC0(I))»»2 ♦ (YS ♦ YC0(I))*»2 ♦ (ZS ♦ ZCO(I))»
1*2» S£GAA(I)=SQRTllXS ♦ XXC0{in»"2 + (YS ♦ YYC0(I))»»2 ♦ (ZS + ZZCOd
in**2) 2i CONTINUE
IF(YS ♦ YCOJN-») »89, 89t90 89 CONTINUE
S:?(1) = SQRT(XS»*2 ♦ ZS»*2) - .032 GO TO 91
9. CONTINUE SR{1)=SQRTUYS + YC0(N<»))»»2 ♦ *S**Z ♦ ZS»»2) - .002
91 CONTINUE T(l»=3.
c 00 2ÜJ J=1.JHAX JA=J+1 JC=J-l T(JA)=T(J) ♦ OTS S^(JA)=SR(J)*3*ÜT NQ=N3 ♦ 2 IF(SR(J) - SEGRC{NQnilJ,110,999
ii: CONTINUE Q(J)=3.
i
CALL COPOS IF(J - jj) ii.u.ia
1. CONTINUE
c JJ=JJ t INT1
CALL COM\/P
PRINT 9 PRINT 239. TU)
239 FORMAT (IX.^H T =,F6.1.9H MICROSEC) PRINT 2-»J.NSH<
Zk^ F3RHAT (IX.7H NSHK =,!*) PRINT 2'»1,.,1SH<
24i FORMAT (IX,7H MSMK =,I-t) PRINT 53 PRINT 53
98
bttUtüt „..^aim^.^,-. ^^„^ (HI irnrmiilihllr ' - -'—-■
gjjpppiiwppi m$mmm W"" WWPPWipwiBlMWHIWiWPIIIWPlIHgWHWWIWPIW! »VWBIIIIIIBlUMl.millili, i"
231 PRINT 231 FORMAT" ("♦'
1 XXC PRINT 53
XC m
YC ZZC •»
ZC
c c c
XC, YC, ZC, XXC, YYC, AMD ZZC PRINTOUT IN INCHES
67
39
8b
3- 3^
00 307 1=1 X3(I)=XC(I YC(I)=YC{I ZCm=ZC(I XXC(I)=XXC YY3<I)=YY3 ZZC(I)=ZZC 00 67 1=1, PRINT 36,1 FORMAT C1X IF(SR(J) - CONTI.^Ut N;=Ni+l 00 88 I=^£ PRINT 86,1 NF=NSHK-? NG=NSHK*5 00 i* I=NF PRINT 86,1 CONTINUE MH=MSHK - NlslSHK ♦ 30*31 I = NH
86,1 I=M 86,1 I = N^ So,I
31 P^INT 10 32
32 F^INT 00 3 3
33 P^INT IF(JQIAG)5
;99 CONTINUE PRINT 9 PRINT 6QJ
6C. FORMAT (♦ 1 lETAACH oRINT 33 oo eai i=i ALFA(I)=AL ALFAA(I)=A ^£TA(I)=^c a£TAA(I)=3 P^INT 5C2,
iic F0R1AT (IX 4LFA(II=AL ALFAA(I)-^ BITAdlslä liTAA(I»=l
ojl CONTINUE ^RINT 9 ORINT ^9
>.i FOrMAT (• PRINT 5 3
,N5 )*Z\I )*C>I ) »cv (I)»CV (I)»CV (I)^CV Nl ,XC(I),YC(I),ZC(I),XXC(I),YYC(I),ZZC(I) ,I^,lX,7(£15.8,lxn
S£GRC<1)»38,39,39
,NSHK,5 ,XC(I),YC<I),ZC(I),XXC(I),YYC(I»,ZZC(I)
, NG ,XCtI),YC(I),ZC(I),XXC{I),YYC(I) ,Z7C(I)
,NI ,XC(I),YC(I),ZC(II,XXC(I), YYC(I» ,ZZC(I) , N>t5 ,XC(I) ,YC(I).ZC(I),XXC(I),YYC(I),zz:{i) ,N5 ,XC(I> ,YCtI),ZC(I),XXC{I),YYC(I) ,ZZC(I) 98,593,599
I ALFAOEG) G) SdGRC(1ET:RS»
ALFAA(OEG) SEGAA(^ETERS) •)
9ETA (OEG)
,N5 FA(I) LFAA( TA(I) ETAAt I.ALF ,15,2 FA(I) • TAAC TAd) .r4A(
♦RAO i)»Ra •RAO I)*RA A(I), X,6(£ /RAO I>/R4 /RAO I)/RA
a ALFAAd) ,9£TA(I) ,9ETAA(I),SEGRC(T) ,SEGAA(I) 1H.7,2X»)
I
.1
PRINTOUT FROM COMVF •)
99
-■ - ■ ■'-' ■' ■'■ • ■ ' ■ -- - iiiuUtiuk iiii«inmiiMiir<«iiiiiiiiiiMii i
. wmm IP1 'WW'WWillWWWII^ppillllipilipii wimmmmmm 'Jiui.Wi^iu.iiipi.mnin
53 F3R1AT (♦ ♦) PRINT 239,T(J) PRINT 53 PRINT 3 3 PRINT 55
5^ FORMAT (♦ I AX PRINT t3 DO 5J 1=1,N5
5. PRINT 5l,I«ÄXm ,AY(T) ,AZII) 51 FORMAT (lX,I-»,lX,7(£l<,.7tlX>)
AY ÄZ ♦)
•.9h CONTINUE
00 96 1=1,N5 9-. o(j)=o<j) *• Axm'nsjni) + äY(I)»OSY(I) * AZ(i)*nsz(i) 11 CONTINUE
2: CONTINUE
999 CONTINUE JJ=INT1 ♦ 1 00 3CJ j=ja, JCINTl JA=J - INTl J3=J ♦ INTl TINT1=INT1 ÖTT=0T»2.*TINT1 E(J)=-{Q(J91 - Q(JA))/0TT 1MITA(J)=£(J)/CUR AA=SORT(XL*XL + YW'YM) 3^=SQRT(XL*XL ♦• 0*Q) CC=SQRT{YM»YH ♦ 1*3) OJ=SQRT(XL*XL ♦ YH»¥W + D»01 FÜ«X=(AMU/PI)»(AL'DG(CC»(X1. ♦ 33)/(D»(XL ♦ DO))) + XL'COO - 99»/(gq
1»31)) OMITI=U»?ÜMX ':R( J) = (D:'1ITA( J)- 3MITI)»1CJ./0HITT
3C' CONTINUf QQ=(A»1U/DI)M2.»l-2.*CC-2.*33*2.*0'DfXLML0G(CCMX«,. + B3) /(0»(XL*DO) ) l)*YH*AL0Gte8»(YM+CC)/(O"(YH+OT) ) )> JRCAL=1J^.♦(1(1) - QQ)/QÜ PRINT 9 PRINT 788,QQ
76a FORMAT (1X,32H THEORETICAL MUTUAL INOUCTANCE =, IX, El^». 7, IX, H HENR 1HS) FRTNT 53 PRINT 736,£RCAL
78': FORhAT (1X,31H £RR0R DUE TO FINITE ZLEMINTS =, IX ,E1^. ^ , IX, 8H PERCE INT! TRINT 63 PRINT 53 PRINT 793
79c FORMAT (• TIM£ PRINT 791
791 FORMAT (♦ (MIGROS£C) PRINT 33 00 792 J=JGRMP,JrtAX,INTl
792 PRINT 891,TU) ,Q(J) PRINT 53
MUTUAL INO
(HENRIES)
♦)
•)
100
■MMMMM
o nnwpillllppilVPi
TIME JM/JT IDEAL
(MICROSEC) (H^MRIES/SEC)
•1UTUAL INO NORMALIZEH
(HENRIES) VELOCITY
•)
PRINT 53 PRINT 53 PRINT 53 ORINT 53 PRINT 831
ftd: FORMAT (• 1CTUAL PRINT 881
881 FORMAT (♦ IS/StC) (H^MRIES/SEC) VELOCITY ♦! PRINT 53 PRINT 53 00 89Ü J=J0,X.INT1 VNORM=OMITA(J)/OMITI PRINT 891.T(J),Q(J),ER(J),0MIT4U),0MITI,VN0RM FORMAT (IX,6(£1^.7,2X>) CONTINUE PRINT 9 GAMMA=iAM1A»R*0 TH£TA=THETA»RAO YW=YM»CV XL=XL»CV 0=0*CV GO TO 1
88ö CONTINUE CALL EXIT ENO
ERROR
(PFPCENT)
OM/OT A
(HENRIE
69w i9l 996
101
mmäk UMk ■■--"■■'■-■■'■ ■ - »I I'i ii Oil'■^■"'■"-•—"■"-
mm mmimm^vf ". •> IWWpWPWBpwpt rnrnm/r ■-»nrw- F'*~jrnt*rp*?> »WWy,l«i,lit,)HWH>H»ini.iJi>li,.i,li,<lw,,w„ ■r^rT^rwMwr,Tr^,7Wff ' ^ TfT'ffTpjM|t»'^»rfl(J^i7Tw
3EGRC(12CC)
SU8R0JTINE CO?OS COMMOM XX(12Jj),YY(l2u3),Q(6ai),£R(6;0)»SR<6i3) COMMON Xw0(l2;w;) ,YCO(123J),X(120C),<(1200) .ALFAdZüü). COMON 4Y(12:3) , 4X(12Ca> ,i(60:)«OMITft(61J) COMMON XC(12J:) ,YC(120'Jl ,33X1120:) ,OSY11200) COMMON J,CU^,oi,AMU,0,U,C\/,C,OTfXS,YS,ZS,R,RA'D COMMOM Ni,N2.N3,N4,N5,JMAXtNSHK,MSHK,T(63:) COMMON XX:O(l2Jü),YYCO(120Q),ZZCO{12J0» »ZCO (12CC)tZ(1233)♦ZZ(1200) COMMON 0SZC12JÜ)«OSXX(12C3),OSYY(123:) .OSZZ(12CC).ZC(l?0C) COMMON XXC(123 J) .YYC(12a•:) ,ZZC(12QC) ,AZ(12J0) COMMON ALFAA{l2wl) ,SZGAA(12CJ) iB£TAA(1230> ,BETA(12C0>
C c c c c c c c
NSHK INDICATES FIRST INTERSECTION POSITIO." IN UPPER HALF MHICM IS UNSHOCKtO
MSH< INDICATES FIRST INTERSECTION POSITION IN LOMER HALF WHICH IS UNSHOCKED
C c
MX=3 MY = Ü MSHK=N--2J
HIGH 3IO£ CALCULATIONS OF XC, YC, AND ZC
61
8v 81 67 55
C C c
NA=N2 - 1 00 S>5 1 = 1,NA IFCSRCJ» - S£5RC(I))6G,60,61 CONTINUE SEGRCII)=SEGRCII) ♦ U»0T XC(I)=SEGRC(I)»COS(ALFA(I))»COS(BtTA(I)) - XS YC(I)=S£GRC(I>»SIN(ALFA(I)) - YS ZC(I)=ScGRC(I)»COS(ALFA(I))*SIN(OETA(I)) -ZS GO TO 67 YCJXlsYCOd) xc(n*x"cö("i> ZCII)=ZC0(I) MX=MX ••• 1 IFIMX - udu.sj.ai NSI K=I C0^TINUE _ CONTINUE CONTINUE XC(^2)=XC0(N2) YC(N2)=YC0(N2) ZC(N2)=ZC0(N2>
HIGH "STOE CALCliLAtlONS OF XXC, YYC, AND ZZC
= 1»NA • SEGAAdM 160.160,161
NA=N2-1 00 155 1= IF(SR(J)
161_ CONTINUE _ 'SE'GAÄ(n=SEGAA(i") > U*OT XXCII)=SE6AA(I»»C0S(ALFAA(I))»C0S(BETAA(I)) YYC(I)=S£GAA(II»SIN(ALFAA(I)> - YS
XS
102
tmM I HIM—^M— _J
^P^^PWPWI^w r- ,11. ii)i|i. JUIIIP|II^»W»^PWW^.'.MIII .i^unM|ii>,i,,iW,iiiiiiLfiiiiiiiii>i i -w/ IXUiiunaiWKiiiiiu
16.
is; 15:
c c
71
9.
/; 7,
c c
ZZCm=SE_GAMI)»C0S(ALFAA(imSIN(3ETAA{in - ZS 'GO T0"l67' Yrcm=YYco(i) XXC(I)=XXCO(I) ZZC(I)=ZZGO(;) CONTINUE CONTINUE XXC(N2)=XXCO(N2) YYC(N2)=YrCO(N2) ZZC(N2)=ZZCO(N2»
LOH A^O ^IGHT SIOE CALCULATIONS OF XC, YS, ANO ZC
N3=N2 *~ l~ 00 7u I=NB,N5 IF(SR(JI - SZG<CII))7l,71,72 x:(I>=X30(I) VC(I>=YCO(I) ZCII)=ZC0(I) GO TO 77 CONTI^Uc S£GRC(I)=S£G(<C(I) ♦ U'OT XC«I)=SEGRC{I)»C0S(ALFA(I))»C0S(8ETA(I)) - XS YC(I)=S£GRC(I)»SIN<ALFA(I)) - YS ZC<I>=StG^C(I)*C0S(ALFMI))»SIN(9ETA(I) ) -Z'J f1Y = MY ♦ 1 IFIMY - 1)93,9J,91 «SHK=I-1 CONTINUE CONTINUE CONTINUE
LOW 4^0 RIGHT SHE CALCULATIONS OF XXC, YYC» »NO ZZC
17;
17; 17.
c C
Na=N2 - 1 00 17J I=N3,N5 IF(SR(J) - SZGAAd) ) 171, 171,172 XXC(I)=XX:0(I) YVC(I)=YYC0(I) ZZC(I)=ZZCO(I) GO TO 17^ CONTINUE 5£GAA(I)=S-:GAA(I) + U»OT XXC(I>=S£GAAII) •C0S(ALFAA(in»C0S(8£TAA(I) ) YYCfI>=SiGAA(I)»SIM(ALFAA(I)) - YS ZZ3{I)=S£GAA(n»:0S(ALFAA(I))»SIN(Q£TAA(I) ) CONTINUE CONTINUE
7S
CALCULATIONS OF X, ftNO Z
NC=N3 - 1 03 57 1=1,MC ia=i ♦ 1 X(IA)={X:(IA» Z(IA)=(ZC(IA)
xccin/2. ZC(I))/2.
VCIA» = lYCtIA) t YCtin/2. X(1)=(XC(1) ♦ XC(N5))/2.
103
L_ H^MMMyMMItt i mnjnajj^aMmiiMa
mm*** *m WÜWPBT^^ n. mnipiiiiwwwimwwp in mm im HHIIIIJHPIIJWWII
■MMM WW(P>I^M.HMIL.IIW
c c
c c c
15",
C c c
15.
Y(l)«(VCCll ♦ Z(1)=(ZC(1) t
TC(N5)»/2, ZC(N5))/2.
CALCULATIONS OF OSX, OSY, AND OSZ
3SX(1)=XC(1) - 0SY(1)=YC(1J - 'DSZ(i>sZC(l) - 00 50 1=1.NC U=I ♦ 1 0SX(IA)sXC(IA) 0SZ(IA)sZC(IA) 0SY(IA)=Y3CIA)
XCIN5» YC(N5I ZCIN5)
- xcm - ZCd» - YCCII
CALCULATIONS OF XX« YY« AND ZZ
NC=N5 - 1 00 157 1=1,MC IA=I ♦ 1 XX(IA,«<XXC(IA» YY{IA1=(YYC(IA» ZZ(IA)=(ZZC(IA) XX(1)=(XXC(1) ♦ YY(1)=CYYC(1) ♦ ZZ(l)»(ZZC(l) ♦
* xxccin/2. ♦ YYC(I))/2. ♦ ZZC(I))/2. XXC(N5>)/2. YYC(N5n/2. ZZC(N9))/2.
CALCULATIONS OF OSXX. 0SYY, AND OSZZ
0SXXII>=XXC(1) - 0SYY(1)=YYC{1» - 0SZZ(1)=ZZC(1) - 09 150 1=1,NC IA=I ♦ 1 QSXX(IA)=XXC(IA) OSYY{IA)=VVC(IA) OSZZ(IA»=ZZC(IA) RETURN END
XXC(N5) YYC'N5> ZZCCM5J
XXC(I) YYCd» ZZC(I)
10^
L MMM,
^•S»!*»*»,»,«««^.», mw. ■ mi ^ii in i. i ■ a
SUBROUTINE COHVP COHHOM XmZJjl »YYIlZaai »QC630>f£Ri630>»SRC600) COMMON ICCO(i2}j)fYCOfi20J) «XCIZQC) f Y (1200) . ALFA f 1230) .S^GRCC 1200» COMMON AY(12JJ)VAX(12C0)«£(6aG)tOHlTA(603) COMMON XC(l2üJ)vYC(12Qa)«OSX(1200)tO5;Y(12C3) COMMON J,CUR*PI«AMU*O«U«Ctf«C«0TvXStYS«ZS,R(RAO COMMON Nl,N2tN3tN<»tN5«JMAX,NSHK«MSHKtT(6j:) COMMON XXCO(12JG)f VYCO(12C 3). ZZCOt 1230) fZCO(12CG) •.2(1210 )t 72 (1209) COMMON OSZ(123J)*OSXX(12CJ)tQSYY(120])tOSZZ(120C)tZC(1200) COMMON XXC(123j)«YVC(123C)tZZC(1203)«AZ(1230) COMMON ALFAA(123J)«S^GAA(123i>)«e£TAA(1203) vBbTA(1200)
C c c
THIS SUdROUTINi CALCULATES THE MA&NcTIC VECTOR POTENTIAL AT EACH ScGMZNT IN TH£ X,V,Z LOO?
AX IS X C01PN£^T OF MVP AY IS Y COMPONENT OF MVP AZ IS Z COMPONENT OF MV» A IS MAGNETIC VECTOR POTENTIAL (MVP)
9,
FATs(AMU*CUR)/(^.»PI) OO 90 1*1.N5 AX(I)»Ü. AY(I)=G. A7(I)=J. OO 91 K=l,N5 '»s3QRT((X(I) - XX(K»)*»2 ♦ (Yd) - YY(<))»»2 ♦ (Z(I) AXX=FAT*OSXX(<)/R AYY=FAT»3SYY(<)/R AZZ=FAT»OSZZ(<)/R 4X(I)3AX(I) t AXX 4Y(I)=AY(I) ♦ AYY AZ(I)*AZ(I) ♦ AZZ CONTHUt CONTINUE RiTUSN END
ZZ(K))»»2)
105
■■-'-»^■Mtfrp J
wwim ' i m.im ^ i i—«mmiju.,!! , .„.i ■f'^'WFTrTWrTT7"li!,',!J,«,7'-A-'WWT--- .oR^^^r^^p^j-^,,^...
AFWL-TR-74-2Ü5
AFPENÜIX C
SLLLCTION OF OPTIMUM GAGH ^UINI/JCTOR
The analysis in section IV identified the manner in which the
shock-induced resistance changes cause errors in the gage output.
As indicated in section IV, these error? are controller! with a
large ballast inductance in the primary circuit. It was also
evident that these errors were influenced by the ballast resist-
ance, though not to the same extent as the ballast inductance.
Since there are many metals which can be used as conductors in the
M1PV and each has its own resistivity, resistance-pressure, and
shod; impedance characteristics, it is of interest to compare these
characteristics. This information is required for gage design and
must be considered along with other factors such as radiation cross
section and the practical aspects of availability, cost (beryllium
cost- approximately $2000 per pound), and compatabi 1 ity with
fabrication techniques in selecting the optimum conductor
material.
1 Till-; IDEAL CONDUCTOR
The characteristics of the ideal conductor are identified by
considering the total gage environment. The gage must rapidly
equilibrate with the shock-loaded geology in which it is embedded,
and it will be exposed to extreme pressures and an intense radia-
tion environment. The ideal conductor would, therefore, have a low
resistivity and exhibit little or no change in resistance with
change in pressure, have a shock liugoniot which is identical to
that of the media in which it is embedded, present a low neutron
cross section, and present no insurmountable fabrication problems
in its use. Several metals and their alloys have some of these
characteristics, but none exhibit all.
For the moment, if the radiation cross-section characteristics
are ignored, and it is assumed that no fabrication problems will be
encountered, the criteria for selecting the most favorable gage
106
»MdMMHHiHM ^-.ij.-...^«..^^,—nm....... t....i...i;|rfni)jjn
■ •I|IHM:.*PHII
yWHwtnmimu*,»
Püü pppi nin.j.n!iiiiiii)i>ifnnij,iinniii» ii i ii>m.i,iiiiinn)i.ii.
AFWL-Tk-74-205
conductor material is reduced to that of determining which material
would result in the minimum resistance change while meeting equi-
libration time requirements. The materials under consideration may
be ordered in this respect and the radiation and fabrication
characteristics considered separately.
2. CANDIDATE CONDUCTORS
The candidate conductor materials and their important charac-
teristics are listed in table C-l. For the analysis, a shock
pressure of 200 Khar will be assumed. It is further assumed that
the results obtained at 200 Khar would not differ in terms of which
conductor material was most favorable from results obtained at
higher pressures. Three columns in table C-l require comments.
The resistivity data assumes a high purity metal. Alloying
and impurities will increase the resistivity in varying amounts.
For instance, several magnesium alloys have a resistivity of
12.5 \i9. • cm, a 300 percent increase over the pure metal, while
both aluminum and copper alloy systems exhibit increases in
resistivity of only a few percent over that of the pure metals.
If a pure metal is not used, the particular alloy should be
considered. For this analysis, a pure metal is assumed.
The column headed "(R/R ) 200 Khar" indicates the resistance
of the material at 200 Kbar pressure relative to the resistance
at zero pressure. The data is based upon the extrapolation of
Bridgeman's static data (ref. 27>) ihown in figure C-l. Since
Bridgeman's data above 30,000 kg/cm2 is itself corrected, it must
be recognized that some uncertainty exists in the table C-l data.
However, the relative magnitudes in R/Rn between materials are
preserved, and the results are probably affected insignificantly
by the uncertainties. Nevertheless, until high-pressure dynamic
resistance-pressure data become available, the R/R0 data must be
considered, at best, an estimate for pressures to 1 Mbar.
The column headed "(p c) 200 Kbar" is the slope of the shock
loading path in the stress-particle velocity plane at 200 Kbar.
The intent of this column is simply to indicate the relative magni-
tudes of the shock impedances.
107
BMBiliiMMBl tttrt -JA^ai**.*....^***^ .....^ ■^.L^.^-^..-,.' -■'-„ihy ■
1 jpppnPTCMpinpp
AFWL-TR-74-205
^ffTTirv« t> ^rvi^\i7^^^'giwwTT,"'tW;irT"!ni,i^^ '^^^^▼^r^rf^w^ 1
1 e
o
c o
• H ^J i-H
• H re W 4->
C a> re B M H
u
re
re
o o
CO
< i—i
OS Ui s-
o H U
a o u UJ H < Ö
a is < u
o
u H ^ CO
CJ X3 O Ui
>x ^-s
•M m
■ l-l g l/l u C\ (Ü w o ^.—'
^^ re o x
o r- •-D LO r~- 00 IS) o i-H .-* rj «*
o O LO r- o '* LO 00 i-H to h~ 00 (-■ LO en
r-t i-H n «tj- 00
LO 00 1-0 O r^ r- Ol r~- LO 00
« r- \D 00 t~-
IX
H u
< u
>> •»-■ /—\ •H e > u
•H ■P • [/)
• H a (^ a. 0) —/
OJ
K) vO i/) r*- ^r O '-> o vO
^r LO (^i >—t
>—t 3 3 e s re •H •H 3 3
•H IA i-H c •H !H
^ 01 r-H • H c a> 01 ß r-^ e re a 4-> oo u 3 p cu re re <D i-H ■H o S S 03 < H u
108
HMMfeaMH i^MMMttMIIII ■Mm ^ ■•■ - ■
AFWL-TR-74-205
300
200
CO
100 -
i r Aluminum
Copper^. Magnesium ^\ \
1 ■Beryllium
■Titanium*
Extrapolated Data
* Transition Not Considered
_L J_ .7
R/R .9 1.0
Figure C-l. Change in Resistance as a Function of Stress for Several Metals (Bridgeman, ref. 23)
109
■MMMMMI - —-'- I ~-^^.
**— Wmmmm mmmm 1*m«„mm,,« - ■ r^-TTIW»
AFWL-TR-74-205
3. COMPARISON OF CANDIDATL CONDUCTORS
The conductor material characteristics listed in table C-l
interact in a straightforward manner to influence the change in
conductor resistance per unit length, AR/9,. The problem begins
with a specified equilibration time, t , and concludes with a
calculated AR/2,.
A one-dimensional analysis of the conductor shock response- is
performed to determine the minimum cross-sectional area that is
required for circular conductors to meet the equilibration time
requirements. The geometry is shown in figure C-2. In general,
the conductor will be stiffer than most geologies; therefore, for
this analysis it is not necessary to consider many geologies, only
a representative one. The chosen Ilugoniot approximates a saturated
tuff with a density of 1.96. The Ilugoniot in the stress-part ic le
velocity plane ii shown in figure C-3. The states in the conductor
during equilibration with the geology arc determined by the inter-
actions at the conductor-geology boundaiics. fin reality, an
insulating material electrically isolates the condutor from the
geology. Here it is assumed that the insulation and geology
Hugoniots are identical to simplify the one-dimensional analysis.!
In figure C-3 the equilibraLion of copper is shown. Due to the
large difference in Hugon^ot between copper and the geology, a
large number of transits are required, seven in this case, to reach
a conductor particle velocity of 0.99 times that of the geology.
A lesser number of transits is required for the other conductor
materials; five for titanium, three for aluminum, three for
beryllium, and two for magnesium.
Having defined the number of transits required for equilibra-
tion, the conductor thickness, or diameter, may be determined if
the time required for equilibration, t , is defined. Several cases
will be considered to observe the variation of LR/I with t . The
conductor diameter may be determined from the expression for the
equilibration time
t = d e \ i 2 n'
110
•MM^MMMMMMMMI „MMMIMM
—"■'■■■ — ■ i ! l
m^^^mm mmmm- 1ijHI)ij,i;HiH«iiinigiBliiii|iiui,iiliii;»lifiTW|ii.mii|i mntwnm^ I " |
AFWL-TR-74-205
Shock
C u ^ l" w • • t »
_L . / ^
Conductor
^ <• " 6 .* ' o .c ^ o ro^. . .^ Geology «f <> < *
C b ft ♦ _ u
Figure C-2. Geometry for One-Dimensional Shock Response Analysis
500
400
« 300
in m
u is; 200
100
Copper Hugoniot
Copper Reflection^ Hugoniot
^-Saturated Tuff Hugoniot
= 1.96
Assumed Tuff Reflection Hugoniot
_L
12 3 4
Particle Velocity, mm/ysec
Figure C-3. Shock Equilibration of Copper in Tuff
111
taM - - ■ ■ ■- - ■--■-' UM - I l^———1——m I
BBMWMBMMHMWIilWWI wnts' ■ ■ ■vOT/i^.iwxf.iwiv'»' »''«•* wmymtmHmmm.
AFWL-TR-74-205
where
d is the conductor thickness, mm
n is the number of shock transits required for equilibration
c is the shock velocity for the first, second, ... nth shock transit
The cross-sectional area is determined from the diameter. The
resistivity and area provide the resistance-p^-unit-length, and
from R/R0 data of table C-l, AR/Ä, may be determined. The results
are summarized in table C-2 and presented in figure C-4 on a semi-
log plot.
In figure C-4 magnesium, beryllium, and aluminum all indicate
values of AR/fc an order of magnitude less than copper and two
orders of magnitude less than titanium. They also aie more favor-
able with respect to radiation cross section with beryllium being
the obvious choice because of its low atomic number.
4. SUMMARY
The results of this study identify the relative merits of
several conductors and provides insight into how they would respond
in the problem of section IV.
Magnesium, beryllium, and aluminum are the most favorable of
the materials considered. Aluminum would be the preferred choice
in the high explosive environment due to its availability and ease
of use. However, at low pressures copper has been shown (ref. 16)
to exhibit better survivability characteristics. In the nucleai
environment beryllium would be preferable due to its low thermal
neutron cross section (n.0085-Be, Ü.22-A1, 0.063-Mg) (Ref. 24).
Also, beryllium's relative high melting temperature of 12780C is
more compatible with the softening temperature of glass systems
under consideration for use as insulators.
112
r"1 ' I" H'li I I II II II KK^B*
I 'I. 'fWfipiipWWiWWIiViiWPmWrW11'1 »imw'vtmvwmn.v WP
AFWL-TR-74-205
Table C-2
SUMMARY OF CONDUCTOR RESPONSES
te d R/H ARM
Mtl Transits (ijsec) (in) (mil/in) (mfi/in)
Mg 2 0.5 0.067 0.4980 0.1618
1.0 0.134 0.1250 0.0406
2.0 0.269 0.0310 0.0101
5.0 0.672 0.0049 0.0016
Be 3 0.5 0.061 0.7950 0.1606 D V
1.0 0.122 0.1990 0.0402
2.0 0.244 0.0500 0.0101
5.0 0.611 0.0079 0.0016
Al 3 0.5 0.045 0.6560 0.2145 r\ x
1.0 0.089 0.1680 0.0549
2.0 0.178 0.0420 0.0137
5.0 0.446 0.0067 0.0022
Ti 5 0.5
1.0
0.023
0.045
39.8
10.4
5.970
1.560
2.0 0.091 2.54 0.381
5.0 0.228 0.405 0.0607
Cu 7 0.5
1.0
0.013
0.026
4.96
1.24
0.1056
0.2641
2.0 0.052 0.310 0.0660
5.0 0.130 0.0496 0.0105
113
m
AFWL-TR-74-205
10.Or
es E
001
te> usec
Figure C-4. Resistance Change Per-Unit-Length as a Function of Equilibration Time, t0, for Several Conductor Materials, 200 Kbar
114
IHMilMr»li 111 mi id-aim lOMmi
AFWL-TR-74-Z05
APPENDIX I)
COMPUTER CÜDF. L 1 STING --liLMIiK
^ :;:.
I
i
us
- • - - - - ■ - ■ - — — , mi jMiaiaa——M
PROGRAH ELHEK(INPUT*OUTPUTtPUNCH,TAP£7sIKPUTfTAPF.6sOUTPUT) C C DIFFERENTIAL EQUATIONS INTEGRATION ROUTINE -C C RUNGE-KUTTA-GILL / AOANS-HOULTON METHOD C VARIABLE STEP-SIZE INTEGRATION ROUTINE C C H SIHUTANEOUS EQUATIONS C
C HAIN PROGRAH OF THE SU3R3UTINE /OR DRIVER C c
COHHON y(2(»), Yl(2(»); Y2(2l»)t Y3C2%), YH(2(»), Z(2<*), Zl(2<») COHHON Z2(2<i)* Z3(2«il, ZkiZk), YI(2^)« E(2«»l, XI, XC, X, XI, X2 CflMMOILJllt X<», DXI, OX, DXH, EHIN, tMAX, KONT, LC, HP, HC, Hl, H2
COHHON H3, HI», N, HT COHHON CPO,RSatR90,XQ,YQ,R,AK, AH ,AL<»0, ALSO ,CP, ALII,NZ.OHOX.V COHHON U,CAP1,R2,R3,CAPD,CAP7,R10,CAP12,R13,NHAX COHHON UO,CO,CURRI,NCI,ZAP,RSEC COHHON RHKl(6)fRHK2(8)
_a__ C READ INITIAL CONDITIONS AND CONTROL DATA 1 READ lOa, N, XI, XC, OXI, DXH, EHIN, £HAX, HP, HC, HT, NZ 100 FORHAT (I3,6E10.3,312,13)
IFIN-990)3,3,50 : IF N GREATER THAN 990, CALL EXIT
3 READ iOi.{Yl(Ii,I*l,NI 101 FORHAT (5E1<».7)
READ 30ü,C0,UQ,AK,CP0 300 FORHAT UFIO.O)
READ 310,NHAX,XQ,YQ,R 310 FORHAT (12,8X,3F10.0) REAJl_32fli£ API,RZiRS^L^O.ALSO,CAP6,CAP?
READ 320,RaO,R9G,CAP12,R13,RSEC READ 320,R13,AL11,XPRT
. 320 FORHAT (8^10.3) READ 330,COND
330 FORHAT (A10) BEAU.-ikiUiRHKlCI) .1 = 1.»)
READ 3<»d,(RHK2(I),Isl,S) 3<ia FORHAT (6A10)
READ «tO»ZAP kkO FORHAT (F5.0)
C NCi*fl-
co«ca/iaoQ. UQ = U0/1000.
c C CPU INITIAL SHOCK POSITION, HETERS (-10 HcTERS FOR TURN-ON) C CP SHOCK POSITION C Ctca SHOCK VELOCITY, icTERS/SECONO C U,U0 PARTICLE VELOCITY, HETERS/SECOND C AK RESISTANCE CHANGE FACTOR C C NHAX NUHBER OF PRIHARY TURNS C XQ INITIAL GAGE LENGTH, INCHES C YO INITIAL GAGE WIDTH, INCHES
116
c c c c c c c c c
c c c c
c c
R UP V
Cl ?2
Amo AL50 C6 C7 R89 RSuC «43 Rlü Ai.ll Cl^ R13
N XI XL OX NZ XPiiT
SPACING BETWEEN LOOPS» CAGE OEFORMATIOM DEFORMED GAG£ LENGTH
INCHES
PRIMARY LINE CAPACITANCE (CAP1) POHER SUPPLY SHÜHT RESISTANCE SECONDARY LINE TERMINATION, 50 OHMS INITIAL SECONOARY SELF INDUCTANCE INITIAL PRIMARY SELF INDUCTANCE POHER SUPPLY CAPACITANCE (CAP6) SECONOARY LINE CAPACITANCE (CAP7) INITIAL RtSISTAN^t, GAGE SECONOARY SECONOARY LINE RESISTANCE, LUMPED WITH R8 INITIAL RESISTANCE« GAGE PRIMARY PRIMARY LINE RESISTANCE BALLAST INDUCTANCE IN PRIMARY POHER SUPPLY CONTROL CAPACITANCE (CAP12) POHER SUPPLY CONTROL RtSISTANCE
NUMBER OF STATE EQUATIONS INITIAL TIME CUTOFF TIME TIME INCREMENT NUMBER OF V VALU£S TO BE PRINTED OUT PRINTOUT CONTROL, TIME INTERVAL
C c c w
c c c
c c
501
395
W96
50 a
1*00
(»01
*»02
MP=0 PKINT COMPLETE OUTPUT MP=1 PRINT INITIAL AND FINAL RESULTS ONLY MC=0 AUTOMATIC STEP SIZE CONTROL MC=1 NO STEP SIZE CONTROL MT=0 RKG PLUS A-M METHOD MT=1 RKG METHOD ONLY
iUKRI INITIAL VALUi OF CURRENT NCI COUNTER IN OERV, CURRI CALCULATION ZAP DETERMINES CALCULATIONAL MODE
ZAP= -1. INDUCTANCE CHANGES ONLY ZAP= 0. NORMAL ZAP' ♦!. RESISTANCE CHANGES ONLY
NPRI«NMAX NSJ.C=NPRI - 1 DO 777 LU=1,2 PRINT 501 FORMAT (1H1) PRiNT 395 FORMAT <♦ PRINT <»96 FORMAT (♦ AFML/DEX •) PRINT 5Ü0 FORMAT (♦ *) PRINT i»J0 FORMAT (• GAUE DESIGN M PRINT <»01,XQ FORMAT (5X,9H LENGTH s,F7.3,lX,7H INCHES) PRiNT <»02,YQ FORMAT (i>X,«H HIOTH =,F7.3,lXf7H INCHES) PRINT <»a3,R
MUTUAL INDUCTANCE PARTICLE VtLOClMETER DEVELOPMENT •)
117
i U ii Mi -"•-'
(»03 FORrtAT (5X.15H LOOP SPACINv« *7TW7TTTT77* INLHb^) PkINT i*ik,HPRl
«»01* l-O^MAT (6XtIliiX,iUH PRIMARY VURNS) PRI.IT HOS.NSEC
«♦05 FORMAT (6X« I1V1X,1SH SECONDARY TURNS) PKINT HJOtCONO
W30 FORMAT (5X^1H CONOUCT3R HATtRIAL -,A10) PRINT 500 »RiNT ktb
•♦06 FORMAT (• SHOCK PARAMETERS ♦) PRINT ^07,CO
i»07 FO-iMAT (sXfl7H SHOCK VELOCITY = ,F6.3,1X ,15H MM/MlCROSECONO> PRINT <»0 8,U0
UL6 FOKMAT (5X,2bH PARTICLE VELOCITY = ,F6 .3,lXf 15H MM/I1ICROSECONDI PRINT <»09,AK
1,09 FORMAT (5X.27H RESISTANCE CHANGE FACTOR -tFV.2) PRINT «»iU.CPO
«.10 FOKMAT (5X,25H INITIAL SHOCK POSITION =,F9.5.1K,7H METERS) PRINT 5üü PRINT kil
Wil FORMAT (• ELECTPICAL PARAMETERS ♦) PRINT m2,CAPl
»♦12 FORMAT (5Xt*H Cl = ,l X ,£14,7 ) PRINT (»13,R2
♦ 13 FORMAT (5X,*H R2= , I X ,Eli».7 ) PRINT <*U.R3
hlh FORMAT (5X»(#H R3=f 1X,E1>».7) PRINT i»15«AL<t3
♦ 15 FORMAT (5X,6H AL«»0= . IX , ElV. 7 ) PRINT ,16tAL50
♦ 16 FORMAT (5X,6H AL50= . IX.El-^. 7) PRINT (♦17,CAP6
1.17 FORMAT (5X,i,H C6=, IX , Ell*.7» PRINT *18,CAP7
Wlö FORMAT (SX.WH C7= , IX ♦£!■♦ .7) PRINT (»l^.RSO
♦19 FORMAT (5X,5H ReC=,1X»L1W.7» PRINT «♦J2,RS£C
«♦32 FORMAT (5X,6H RSEC= «IX, Ell». 7) PRINT <t2C,R90
«♦2J FORMAT (5X,5H R90= , 1 X,£l«».7 ) PRINT ^21,CAP12
«.21 FORMAT (5X,5H C12=, lX,ili».7 J PRINT «♦22,R13
•.22 FuRMAT (5X,5H R13=,. ,.li-.7) PRINT 5J0 PRINT «»23
♦23 FORMAT (• TIME PARAMETERS •) PRINT *»2^,XI
W2-* FORMAT {5X,13H START TIME. = ,E1«». 7,IX ,dH SECONDS) PRINT ■♦25,XC
'♦25 rORMAT (5X,lnH CUTOFF TIME s.ElW.? ,1X ,aH SECONDS) PRINT i»26,0Xl
kZb FORMAT t5X,llH TIMcSTEP =,E1H.7,IX,«H SECONDS) PRINT 50C PRINT i»27
U27 FORMAT (* VARIABLES •) PRINT I*?* ,R1C
118
•MMiMaaaMa Ma^MB J
«128 FORHAT <5X,5H RIO», PRINT %29«AL11
lX,Elb.7)
i»29 rORHAT (ÖX«6H ALU« »1X.E1<*. 7 1 PRINT 500 PRINT 500 PRINT 540 PRINT 5Ü0 PRINT <*31f(RHKl(Il, 1=1,8> PRINT «♦31,(RHK2(I»i I>1,8)
hii FORHAT (1X(8A10) 777 CONTINUE
XQsXQ*.025<» '
YQ»YQ».025<» R»R».025<» UO=UO*1000. C0=C0»ia60. IRK»N ♦ 1 DO 17 I«IRK,NZ ZIII«J.
17 Ecij*a. C INITIALIZE FOR STA1T-ÜP 2 KONT » 1
HI a 2 H2 » a M3 « 0 H% « 0 LC s d
C c
£F » i.OEll
CHECK ALL INPUT OPTIONS IF (OXH) 11*10,11
19 OXH s lO.Q • oxi 11 IF (EHAX) 13,12,13 ik EMAX - 10.0E-06 13 IF (EHINI 15,1%,15 Ik EHIN * EHAX / 11.0.9 15 X = XI
DO 16 1 * 1, N Yd» = YI(I)
IB CONTINUE OX » OXI XXX«XI ♦ XPRT - OX
C OBTAIN THt OERIVITIVE OF FUNCTION AT INITIAL CONDITIONS CALL OERV
C PROVIDE OUTPUT OPTION ON INITIAL CONDITIONS CALL PRNT (1)
C
c C HQtfE DATA IN ARRAY PRIOR TO NEXT POINT 23 CALL HOVE
C C TEST CONDITION OF SYSTEM 2 RKG ON START-UP (4 POINTS) AND AFTER CHANG:. IN INTERVAL C A-H ON ALL OTHER
119
IF IHT) 22,21,22 21 KONT s KONT ♦ 1
IF UONT • ki 22,22,23 C : PERFORH RKG INTEGRAFION TO PICK UP POINT N+l FROH N
22 CALL RKGI 00 2<» I s 1, N Ed) • 0.0
21» CONTINUE GO TO 25
C L PERFORM A-M INTEGRATION FOR POINT NU USING N THRU N-3
23 CALL AOMI C CHECK FOR START UP ERR3R
IF (Z2(l) - tF) 25,25,(»0
C
3 PROVIDE OUTPUT 0PTI3N FOR PRINT N 25 CONTINUE
IFIX-XXX)77,88,88 88 CALL PRNT(2)
XXXsXXX ♦ XPRT 77 :ONTINU£
C C STEP INDEPENDENT VARIABLE X AND INTERROGATE CUT-OFF
CALL iTRO (HCO) IF (HCOI 30,20,30
C OUTPUT FINAL VALUES FROH INTEGRATION ANÜ REINITIALIZE 3i CALL PRMT (3)
GO TO 1 WD PRINT 2^0
PRINT 2J1, (Ed), I = 1, N) PRINT 2d2 DXI s OXI / 2.0 GO TO 2
C 50 PR*NT 203
GO TO 51
200 FORMAT (35H INITIAL INCREMENT UNSATISFACT0RY/1ÜX15HERR0R RANGE M IAS)
231 FORMAT (d£l5.n 232 FORMAT (3X21HHALVING THE INCREMENT/) dD3 FORMAT (1«»HA1A*ENO OIF^EQ)
51 CALL EXIT ENO
C
120
I ! H'lfc^'1 --■■■■--■■ ■■■■ ■■ ■ J
mm Mi i ujiniiwpiiqppppippppii mmm 11 "'■'"'■^'■^ Jim
t,l^*^^^mmm*m»m»-v^.>ivmnmnm<^' IMIWMIMWMRItrMMMMwmHnnNM irMIIHmMMHBMMMHBi ^ ^ ' ' M'*,'«WVi-^V(w-MWWtrW^ ■*w»w^^,>t«»»'-*W(ilÄ irrWWfft^wTtw iWWWBIWIWgWBMWWWWIW ■■
c c c c c c c C c c c c c
c c c c c c c c c c
SUBROUTINE 0E«V SUMMON Y(2<*), Yl(2l*), V2(2<*)* V3(2<»)t Yi*(2V» t Z(2<»». Z1C2'») COMMON Z2(2W)t Z3t2^1, Z<*(2^), VI(2<*), E(2<*)« XI, XC, X, XI, X2 COMMON X3, XI», 0X1, OX, DXM, EMIN, EMAX, KONT, LC, MP, MC, Ml, M2 COMMON M3, HU, N, MT COMMON CPü,P.80,R90,XQ,irq,K,AK,AM,AL<»0,AL50,CP,ALll,NZ,OHOX,V COMMON U,CAP1,R2,R3,CAP6,CAP7,R1(.,CAP12,R13,NMAX COMMON UO,Ca,CURRI,NCI,ZAP,RSEC COMMON RMKi(6),RMK2(6)
PARAMETER LIST
y(i) - LAMOA k r(2) - LAMDA 5 Y(3» - Q6 YC«,) - U7 Y(5) - Q12 Y(6) - 15 Y(7» - AM Y(8) - Lk Y(9> - L5 YdOl - Rö ^ Y(ll) - R9 Y(12) - CP Y(l3) - UP Y(l^) - 15 DOT Y(i5» - £3 UBSOLUTt VALUE) Y(16» - t3 IDEAL (ABSOLUTE VALUE), INCLUDES SECONDARY ATTENUATION AND IS CALCULATED UTILIZING THE CALCULATED VALUE 0.f CURRENT, Y(6», AT EACH IIMt STEP Y(17) - E3 IDEAL (ABSOLUTE VALUE), INCLUDES SECONDARY ATTENUATION AND IS CALCULATED UTILIZING THE INITIAL VALUE OF CURRENT, CURRI, PRIOR TO SHOCK ARRIVAL Ydd) - NORMALIZED £3, E3/E3 IDEAL, BASED ON Y(6) Y(19) - NORMALIZED E3, E3/E3 IDEAL, BASED ON CURRI Y(20) - AMOUNT OF CURRENT CHANGE DURING SHOCK TRANSIT
C = C0 C1=CAP1 C6=CAP6 C7=CAP7 C12=CAP12 CHECK ON SHOCK CP=CPa ♦ C*(X • IF(CP)22,33,33
22 :ONTINU£ v=xu CALL MUTI DMOX0=OMOX U = ö. DMOT=ö. ALtsÄL^O AL3=AL5a ^LOT^=0. L)LOT5 = 0. R8=Rö3 + RSEC ^9=R9 3 UP=0.
POSITION XI)
121
tmm tUtämum - afahiBXifäiim J
»1.11.1'WIMWHWUIII"!'1!. ' -rf^WK-W (■■Il"t 'Jfm 'VT^mrv mil,, w < ■ 1,11, fff, , .^
GO TO 99 33 CONTINUE
UsJO 0T=CP/C up=u»or V=XQ - UP
CALL NUTI aMOT=-OMDX*U OLOX4sAL^0/xa DL0X5=AL5a/XQ AL4 = AmJ - DL1}X1»*UP AL5=AL50 - DL0X5»UP OLOT^s-OLOX^^U 0L0T&=-0L0X5»U R8aCR60/(2.»(Xü ♦ Y(1»))»(YÜ R9= <R90/(2.»(xa ♦ yamMYCi
c c
♦ 2.*<X(i - CP» ♦ AK»(Y(J ♦ 2.»CPn*RSEC ♦ 2t*(XQ - CP) ♦ AK»(YQ ♦ 2.»CPn
CHECK CALCULATION'^ IF (ZAPl4'*f 99,66
hi* CONTINUE
MODE
INDUCTANCES CHANGES ONLY
66
R8sR60 ♦ RS£C R9=R90 GO TO 99 CONTINUE
♦ R10»/DET)»ULVY(2I- - Y(l»»DMOT) - «(ALU*
\ I
RESISTANCE CHANGES ONLY
ALt^AL^O AL5=AL5a DLDT«f=0. DLDTSsQ.
99 CONTINUE OETsAL^ALS - AM»AM 0ETDT=AL*»#0L0T5 + AL5»DLOTi» - 2.*AH»DH0T Zll)*Ui*i/C7 -(R8/DtT)»(AL5*YCl) - AH*YC2)I ZC2)s{0ET/<0ET ♦ ALll»ftL«»)»»(Y(3)/C6 - ((R9
lAM»y(l)> - (AL11/DET)*CY(2I»ÜLDT<* - AM»Z(1) 20tTOT)/(0ET»0ET)»*{AM»Yll) - AL«»»Y(2))) Z(3)=- (C6/(C6 ♦ Cl»)»(Y(3)/(R2»C6»+ (1./R13)»CYC3I/C6 - Y(S)/C12)
1 + (l./0tT)»(ALi»*Y(2) - AM»Y{l>n ZC^Is - Y(«»)/(R3»C7) - {AL5*YC1) - AM»Y(2>)/DET Z(5) = (1./R13)»(Y(3)/C6 - M5)/C12) Y{6)=(AL*»*Y(2» - AM»Y{l)»/DET IF(NCI)30,50*51
50 .URRI=Y(6) 51 :ONTINUE
NCI=NCI ♦ 1 Y m=AM Y(SI=AL<* Y{9»=ALö Y(10»=RB Y(11)=R9 Y(12)=CP Y(13)=UP
122
jteumn^, -HMMIMMM«^
wmmm «!
■PPWR IIPMIiWIWIIllHII ^ 'l.|lww^pwwlWl.|^,llw^l■-"IW'■ w^nwn^mmupwipwi^Tw;. »m ■ wupwmnp»
r(iW»=(l./DET)»(AL«4Z(2) ♦ Y(2»•DL0T^ - AM»Z(l) 1 (üeTQT/(DtT»OtT))»(AM»y(l» -AL<«»V(2)) YC15)=YU)/C7 YU5> =A8S(r(15> ) Y(lbl=Y(6»*DM0X»U»(R3/(i<3 ♦ RdC ♦ ÄStO) Y(iöi-ABS(Y(X6t » iF(Y(i6)»loitiae.iai
101 CONTINUE Y(17)=CUR9I»0M0X3»Ü»(R3/(R3 + R60 ♦ RStC)) Y(i/)=ABS(Y(17» ) Y(ia»=r (15)/Y(16) Y(19)=Y(15)/Y<17) GO TO UJ
102 ;0NTINUc Y(16)=0. Y(17)=0. Y(i6)=0. Y(19)=0.
10 3 CONTINUE Y(20)=r(6»/CURRl n£TURN END
Y(1»»ÜMDT) ♦
aOMHON COMMON COMMON COMMON COMMON COMMON COiMO.'l
HZki, ZHZk) Xlf XC, X, XI« X2
LC, MP, MC, Ml, M2
*iQ
303
SUBROUTINE MUTI COMMON Y (2^) , Yl(2'i), Y2(24), Y3C2**), iHlZki,
Z2(24), Z3(2%», ZU(2<»), YI(2<,), HZk) , X3, XI«, OXI, bX, OXM, EMIN, EHAX, KONT, M3, Ml», N, MT CP3,RoO,R90,XQ,YQ,R,AK,AM,ML>«0,AL£J,CP«ALll,NZ«OMOX,V U,CAPl,R2,R3,CA'»6,CAP7,R10,CAF12,Mi,NMAX Uu^T^URRI.NCI^AP.RSEC RMKKö) ,RMK2(6»
DIMENSION QU(5&) ,ERR(51) PI=3.mi5926 flMU = .30 JUJ1<;»6P'9 Z=YU MM=NMAX-1 DO 80 M=1,MM P=2»H-l S = P*R A=SQRT(V*\/ ♦ Z*Z) 3=3URT(V»V ♦ S*S) C=JQRT(Z*Z <■ S*S) AA=S^RT(V»V + Z»Z + S»S) QQ(M)=<AMJ/Pl)»(2.»S-2.»C-2.»B + 2.»AA*\/»ALOÜ(C»<V + B>/<S»(V+AA)»)
1 Z'ALOG(B»(Z>C)/(S»(Z+A4)»») PiU=(.'■1,J/PI»MALOG(CM\/ + B»/(S* (V+AA) )) ♦ V* ( A A-B) / (B*B) ) JPL=Qü. ■*> /V £KR(M)=(POQ-QPL)/P0Q CONTINUE JC = NMAX-1 INTF=2»(NMAX-1) QX=0, aY=a« CÜEF=INTF 00 J03 J=i.JC uX = ax * COt.F»QU(J) QY=QV ♦ C0EF*QQ(J)/(1.-£RR(J») :0^F=CO£F-2. "ONTINUE ÄM = QV
iRs.l <X/QY DMÜX .. 'Jrf*<l.-ER»)
ETUf. :ND
123
hi MOMMMIki tauu! MM«« ■ÜHMi^ttii ... ._-.-.. .■■„,..-■■*..
m\»mm "«"»>"« " mm ■m^mr-- -rrrrr. i ii, mp,.,,,, , ,, j.,,,,, .^j,, ■■» '< """»»I"-- mwnii li "
■jtwi'W'-J',«;»^«1«'>wwwi»«iiiiiiiL»i»»K>v.t>...,
53 52 *1 1
C C
10
12
11
7
2D
3 33
5:
71
72
ZC2^)f Zl(2<») XI, XC« X« XI, X2 LC, HP, HC, Hi, HZ
XI, OXI, OXH, XC, EMIN, EMAX . NZ I, VCII, I, zm
SUBROUTINc PRNT (K) PROVIDE OUTPUT AS REQUESTED BY OPTIONS
CÜHNON YtZl»l, VKZi»), Y2(Z^), Y3(2<>), Y(»(2<»), CONHON Z2(2<»), Z3I2<*), Z«t(2<»), YIUUI, E(2<»), COHHON X3, XV, DXI, OX, OXH, EHIN, EHAX, KONT, COHHON H3, H<», N, IT COHHON CPO,RAO,R90,XU,YQ,R,AK,AH,AL<»0,ALSO,CP,ALII,NZ,DHDX,V COHHON U,CAPl,R2,R3,CAP6,CAP7,RlO,CAPi2,Ri3,NHAX COHHON UO,CO*CURRI,NCI,ZAP,RSEC CONHON RHKl(tt),RMK2(B)
TEST PRINT ON INITIAL AND INTERHcOIATE POINTS IF (K - 3) 30,3,60 IF (K - 1) 52,1,52 IF (HP)60,5i,60 IF (K-l) 1,1,2 IF (HT) 10,10,11
PRINT 2J0 PRINT 206 PRINT 206 LF = 1 PRINT 231, DO 6 I = 1 PRINT 208, :0NTINUE PRINT 209 LC=11*NZ 30 TO 60 PRANT zaz PRINT 206 PRxNT 206 LF = 2 GO TO 12 IF (LC ♦ NZ - :>RINT 203, X, 30 <t 1=1,NZ PRINT 235, I, CONTINUE PRINT 206 LC = LC ♦ 2 ♦ GO TO 60 IF (LC ♦ NZ - PRINT 20^, XC ÜO 5 1=1,NZ PRINT 2i.C, I, CONTINUE PRiNT 207 RETURN GO TO (71, PRINT 2w0 PRiNT 239 LC = 0 GO TO 23 PRiNT 202 LC = 6 PRiNT 209
58) OX
20,20,73
Yd), i, zm
NZ
56) 30,33,73
y(i), I, Zd)
Ed)
72), LF
124
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^^J,^W'm^w^fmt^^^^m'^r■■ ^^^ 1 ■ .^-'. •*'•> ■ ■ ;»*( ■ ■ ,- '"' >i>. ,-,*« .•■,v-,.
9 GO TO 2a H 73 GO TO ( 7*,
1 7k PRINT 2üC GO TO 30
1 75 PRINT 202 GO TO 30
1 HL GO TO (Alt 1 ei PRiNT 200 PRINT 20 9 LC = 6 GO TO 63
82 PRINT 202 PRINT 209 LC = o JO TO 60
75) , LP
82), LF
20C FORMAT (5oHl RUNGE-KÜT TA-GKL / ADAMS-MOULTON INTEGRATION ROUTIN If»
?01 rORMAT (23H INITIAL CONDITIONS/16X1HX ,17X5HOEL Xt12X9HMAX DEL X, 113X,5rtCUT X,12X,9HMIN £«RüR,10X,9HMAX tRROR/5X,6E19. 7)
2D2 FORMAT UJH1 RUNGE-KJTTA-GiLL INTEGRATION ROUTINE) 2C3 FIMMAT (2E15.7) 2:«» FORMAT (/3«*H FINAL RESULT AT A CUT-OFF X OF,tl5.7) 2i5 FORMAT (lQX,2rty (,I2,i»H) = , E15 .7 .UX^HOERV (,12 ,<»H) = ,E15,7,<»X,
15H£RR0R(,I2,'»H) = ,tl5.7) 216 FORMAT (1H I 2u7 FURMAT (1H1/1H1) 2ifl FORMAT (9X,3Hiri (,I2,i,H) = , E15 . 7 ,3X ,6HOEkV I (, 12 ,<,H) = ,E15.7) 219 FORMAT < / 12X19HlNT£GkATI ON RESULTS/ ÖX, IHX ,13X , 5HDEL X/2ÜX ,«»HY ( i ) ,
123X,7HOERW(I),2 3X,dHERROR(I)/) 210 FORMAT (iax,2Hf (,12,«*MI = ,E15 . 7 ^X SHOERV (, 12 ,<»H) = ,£16.7)
ENJ
SUBROUTINE MOVE TC REPOSITION DATA ARRAY FJK NEXT POINT
COHMON y(2'»), Yl(2i»), Y2(2(»), Y3(2H), Y<*(2<*), COMMON Z2(2>«), 23(?<*), Z^(2<i), Vllch), E(24Jf CO.IMOM X3, Xi,, DXI, OX, DAM, EMIN, EMAX, KONT, COMMON 113, Mi», N, IT COMMON CP0,Rö0,R9C,Xt(,Ya,R,AK,Ah,AL«»C,AL60,CP,ALll,NZ,OHOX,V COMMON U,CAPl,R2,R3,C«D6,CAP7,Rin,CAP12,Rl3,iNMAX COMMON Uü,C1,CURRI,NCI,2AP,RSEC COMMON RMKKo» ,RMK2(8)
Z<2W», 71(2'+» XI, XC, X, XI, X2 LL, MF, MC, Ml, M2
Kk - X3 = X2 = XI = OO i Yi»{I) = Y3(:) = Y2(I) = YKI) = 7MI) = 73(1) = Z2(I) = ZKI) = CONTINUE RETURN ENO
X3 X2 XI X I = 1, N = Y3(I) = Y2(I> = YKI) = Yd) = Z3(I) = Z2<I) = ZKI) = Z(I)
125
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'"" "I' ■ ' ■ ■-••■— »" II' " -l.^. ■■-.— r.i'Pvwii "i in!'
c c
n c c
SUBROUTINE RKGI OBTAIN VALUE OF NEXT POINT VIA RUNGE-KUTTA-GILL INTEGRATION
COMHON Y(2^l, Yl(2l»)f i2{2k) * Y3(2<»)* Y^(2<»>, ZiZki , ZHZk) COMMON Z2(2HI, Z3iZk) , ZklZki, YI(2<»), E(2<»), XI, XC ♦ X, XI, X2 COMMON X3, X^, 0X1, OX, OXM, EMJN, EMAX, KONT, LC, MP, MC, Ml, M2 COMMON M3, M(», N, HT COMHON CP3,R80,R90,XU,YülR,AK,AM,AL<tO,AL50,CP,ALll,NZ,DMDX,V COMMON U,CAPl,R2,R3,CAP6,CAP7,R10,CAPi2,R13,NMAX COMMON UO,CO,CURRi.NCI,ZAP,RSEC COMMON RMKliai,RMK2(8) DIMENSION A(2<»)
RKG Di * 02 « 03 - Cl = C2 * C3 = Ok = C5 = C6 = 00 1 A(i) CONT
INTEGRATION CONSTANTS SORT 10.5QC0)
(2.0C00) 01 01 02 03 01 02 03
1, N
SQRT 3.0 ♦ l.J - 2.0 - 2.J - 1.3 ♦ 2.0 ♦ 2.0 ♦
0 1 = = Z(I)
INUE
♦ » #
00 1 I = 1, N Yd) = Y (I) ♦ CONTINUE X = Xi ♦ OX / 2.0 CALL OERV 00 2 I» 1, N Y(I) = Y(II ♦ Cl Ad) = C2 ♦ Zd) CONTINUE CALL OERV DO 3 I = 1, N Yd) = Yd) ♦ Ok Ad) = C5 • Zd) CONTINUE X = XI ♦ OX CALL OERV DO <» I = 1, N Yd) = Yd) ♦ OX CONTINUE CALL OERV RETURN END
OX • Zd) / 2.0
OX C3
OX C6
(Zd) Ad)
(Zd) Ad)
- Ad) )
- Ad) )
Zd) / 6.0 - OX * Ad) / 3.0
126
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m*m^mww»m*~~™~~~~~r~m*m<»m ipi»!'-"^M ll|lllll«lmw•l<■■l«pw)>f■ll|li>l"l,l■l ii wwup^iniMipiw^w^w^nwn i. ■« i - -^miwmmr^rrrw^m
I 2 3
SUBROUTINE ITRO <K)
TO INTERROGATE »URR£NT VALUE OF X FOR CUT-OFF X
COMMON COMMON COMMON COMMON :OMMON COMMON COMMON COMMON IF (X ♦ IF (Mt) < = 0 RETURN
lidki* nuki, iziZh), mzku tkUkit z(2<>)« nizki Z2(24). 23(2«*)« Zklik), YHZkl* EiZki , XI, XC, X, XI, X2 X3, *kf OXI, OX, OXM, EMIN, ENAX, KONT, LC, MP, MC, Hl, H2 M3, Ml», N, MT ÜP0,R80,R90,XQ,rQ,R,AK,AM,ALV0,AL&a,CP,ALll,NZ,OHOX,V U,CAP1,R2,R3,CAP6,CAP7,R10,CAP12,R13,NMAX UO,CO,CUKRI,NCI,ZAP,RSEC RNK1(8),RMK2(6)
OX - XC) 2*1,1
OX s XC - KONT s i Hk - I GO TO 3 K = 1 GO TO 3 END
127
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m ■ i "< wmm mm " mtmm Mam mi iiiiiiwif)m|in)iinj| i^umiwi III.U'IHI^ i.i. p.iMpin^mci
WW—W ——— i Hin Hi«— iw,
SUBROUTINE AOMI OBTAIN TENATlVc VALJE OF NEXT POINT VIA AOAHS-MOULTON INTEGRATI LHctK ERROR FOR POSSIBLE CHANGE IN SUP-SIZE
Z(2(»)t lliZk) XI, XCt X« XI, X2 LC, MP, NC, Ml* M2
COMMON U2%), Yl(2i*)f r2(2t»), Y3(2<*), Y<»(2W), COiMON Z2(2<*), Z3(2t), Z<*(2H), Yl(2<f), E(2^», COMMON X3, X(>, 0X1, OX, OXM, EMIN, EMAX, KONT, COMMON M3, Mt», N, MT COMMON OPa,R0]*k?O,XU,YQ,R,AK,A*,AL*O,AL5O,CP*ALll,NZ,OHOX,V COMMON U,CAP1,R2,R3,CA'6,CAP7,R10,CAP12,R13,NMAX UOMMON üO,Ca,CURRI,NCI,ZAP,RSEC COMMON RMK1(8) ,RMK2(6) DIMENSION YY(2<»)
IF (KF) 50,51,5C 5Ü IF (KFF> 52,53,52 52 KF = ü
<FF = 3 GO TO 5i
53 KFF = 1 51 CONTINUE
C STEP X AND PREDICT A Y X = XI ♦ OX MM = 0 ji = ox/21».a Oc s 19.0/270.0 DO 5 I = 1, N Y(I) = YHI»*01»(55.0»Z1(I)-59.0»Z2(I>*37.0*Z3(I)-9.0*Z'»(I)»
YY(I) = Yd) 5 CONTINUE
CALL OERV
C c
c c
CORRtCT Y AND PREDICT CHECK ON ERROR DO 6 I = 1, N Yd) = Yld) ♦ Dl*(9.0*Z(I»-d9.0»Zl d)-5.0*Z2CI)+Z3d) ) CONTINUE CALL OERV
00 7 I = 1, N Ed) = ABS (02 ♦ (Yd)
CHECK ERROR RANGE IF (EMAX - Edl )l»5,l,l
I IF (t(I» - EMIN ) 8,7,7 !»E MM = MM ♦ 1
GO TO 7 S M3 = M3 ♦ 1 7 30MTINUE
IF (HM) i»ö,^6,3 «b IF ( M3 - N ) 1C,2,2
« « » » 4
- fYCI) ))
; ERROR LESS THAN MINIUM - DOUBLE THE STEP-SIZE 2 IF (MC)10,21,i0 21 IF (X ♦ k,Q »OX - XC) 22,22,10 22 IF (DX+ÜX -OXM) 23,23,10 23 CALL PRNT (2)
128
■■^•M. mum
•: iwmmffi-wtv w ^tfßv^^^irm^^imvm^ßm^trmtvt **wfv}nfnmmmw*,w****www*> •■»" ■ ' ^w*r IWH P VW W w^
«r»
i
2Ö1 250
?u
25
26
OX = ÜX ♦ OX IF (LC ♦ N " CALL PRNTC») PRINT 2J1 10 tLU I - 1, PRINT 20J. i» CONTINUE PRINT 20»» LC = LC ♦ 2 XI = X 00 25 I Z3(I) = ZKI) = mi» - CO^TINUL
♦ ♦ OALL liK&I HO 26 I = It Hi) = J .o CONTINUE Ml = 1 &0 TO 13
581 230,230,231
N E(I>
♦ N
= 1, N Z4(I) Z(I) Yd)
3 3u 31 3i 33
j
351 350
37
Wl
36
31.
IF IF IF IF IF
TX IF
iRROR GREATER (M2) 30,<,C,3C (MG)ia,31,10 (Ml)3<4,32,3^ (X fjX -fOX - XC) (KFF)3b,35,36 » » » *
= OX / 2.Q (LC ♦• N - 36)
:ALL PRNTH») PRINT 2i2 00 37 I = 1, N PRINT 2Ü3, I, Ed) CONTINUE PRINT 231. LC = LC + 2 + N ÜO ^2 I = i, N Yd) = Yl (I) Z(I) = Zld) CONTINUE CALL RKlil DO 36 I = 1, N '(I) = 0.0 CONTINUE KONT = 2 <F = 1 <FF = 0 GO TO 13
♦ » »
XI = X2 f'i = a OÜ 39 I = 1, N Y(I) = Y2d) YKI) = Y2(i)
THAN rAXiMUM - HALVE THE STEP-SIZE
33,33,10
asc-iSi^si
129
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wft^w>»ti*niNt'/aiW
r
^Ml^w mmmm** -«rr-- 1. iilUKIIil ■I l«"" "I "«I^FWnFWPFfll
3b
II 13
36
hi
'«0
231 232 ?C3 2D4
zm s Z2(i) ZKI) s Z2(I) COWTlMUb GO TO 35
s 1 (Mill!,13,11 = 0
IF Ml M3
= 1. N Y4(I) Y<»(I) Z<*(I) Z<*(I)
RETURN •
KFF = 0 XI = X<» 00 ki I Yd» =
YKI) = Z(I) =
ZKI) = CONTINUE GO TO 35 Z2(i) - 1.QE12 GO TO 13
♦ ♦ »
FORMAT i51H FORMAT (50H FORMAT FORMAT END
DOUBLING THE INTERVAL« HALVING THE INTERVAL,
(8X,6HERR0R(,I2,5H) s ,E15.9) (1H )
ERROR ERROR
ARkAY ARRAY
♦
AS AS
FOLLOMS) FOLLOWS)
130
MMMMM ■ I -
mmmt^imimmmmmm " '•'■■'Pi ip i ii i nil. mtmwm i mmw 1" "•■"» -
AFWL-TR-74-205
RLFERLNCES
Bunker, R. B., Doran, J. A,, Impedance-Mismatch High Stress Transducer (IMHST) . AFWL-TR-73-45, AF Weapons Laboratory, Kirtland AFB, NM, March 1J73.
B
>
I
Baum, N. , Shunk, R. , Bunker, R, • (T3) Development, AI tory, Kirtland AFB, NM, April 1973,
Thermoelectric Thermopile Transducer (T3) Developmentj AFWL-fR-73-24 , AF Weapons Labora-
Danek, W. L., Jr., Schooley, D. L., Jerozal, F. A., Particle Velocimeter for Use Close-In to Underground Explosions, DASA Report 1431-3, Defense Atomic Support Agency, Wash, DC 12 October 1967.
Liberman, P., Newell, D., Piezoresistive Dielectric/Dielectric Microwave Waveguide, AFWL-TR-73-37, AF Weapons Laboratory, Kirtland AFB, NM, February 1973.
Specification Sheet for Velocity Transducer, Crescent Engineer- ing and Research Company, 15 November 1964.
Perret, W. R., Wistor, J. K., Hansen, G. J, "OX Pendulum Velocity Gage," in Four Papers Concerning Work on Ground Motion Measureme n tj , SC-R-65-905, Sandi Albuquerque, NM, pp 2 5-^8, July 1965.
Palmer, D. G., Recent
a Corp.
7. Liberman, P., Theory and Operation of the IITRI Pressure-Time Gages in the 20-2 00 Tn"loba'r Region, DASA Report 2161, Defense Atomic Support Agency, Wash, DL , November 1968.
8. Barker, L. M., Laser Interferometry in Shock-Wave Research, SC-DC-70-5465, Sandia Laboratories, Albuquerque, NM, December 1970.
9. Danek, W. L., Jr., Reily, D. M., Gushing, V. J., Particle Velocimeter for Use Close-In to Underground Explosions, Phase I, DASA Report 1431, Defense Atomic Support Agency, Wash, DC, 30 November 1963.
10. Danek, W. L., Jr., Particle Velocimeter for Use Close-in to Underground Explosions, Phases II and III, DASA Report 1431-1, Defense Atomic Support Agency, Wash, DC, 29 October 1964.
11. Danek, W, L., Jr., Dargis, A. A., Particle Velocimeter for Use Close-In to Underground Explosions, Phase IV, DASA Report 1431-2, Defense Atomic Support Agency, Wash, DC, 31 December 1965.
12. Uanek, W. L,, Jr., Private Communication.
13. Danek, W. L., Jr., Schooley, D. 1.., Jerozal, F. A., Close- In Particle Velocity, Operation DISTANT PLAIN, Event 6, Project 3.09, DASA Report 2021, Defense Atomic Support Agency, 15 November 1967.
131
,■,1. ■if.i'.ni.>6ilV M ■■^-.. l.] ,, | . HllBllMBMII
PPP^WIPI^W»"^«^"'^ "^P ■ I' I
REFERENCES (cont'd)
Keough, ü. D., Constitutive Relations from In Situ Lagrangian Measurements of Stress and Particle Velocity, DASA Report 2685, Defense Atomic Support Agency, Wash, DC, 27 January 1972.
15. Smith, C. W., Grady, D. E,, Seaman, L., Peterson, C. F., Constitutive Relations from In Situ Lagrangian Measurements ot Stress and Particle Velocity, DNA Report 28851, Detense Nuclear Agency, Wash, DC, January 197 2.
16. Grady, D. E., Smith, C. W., Seaman, L., In Situ Constitutive Relations of Rocks, DNA Report 3172Z, Defense Nuclear Agency, Wash, DC, January 197 3.
17. Garcia, P. R., IMHST and Particle Velocity Gage Measurements, MIDDLE GUST IV and V, Report No. AL-881, EGSG, Inc., Albuquer- que, NM, 3 November 197?.
18. Renick, J. D., et al., Close-In Ground Stress Measurements, MIXED COMPANY III, Project LN -SjM, Report No. TÜR 6633, AF Weapons Laboratory, Kirtland AFB, NM, November 1973.
19. Stern, T. E., Theory of Nonlinear Networks and Systems, An Introduction, Addison-Wesley Publishing Company, Inc., Reading, MA, 1965.
20. Seshu, S., Reed, M. B., Linear Graphs and Electrical Networks, Addison-Wesley Publishing Company, Inc. , Reading, MA, 1961.
21. Terman, F. E., Radio Engineers' Handbook, McGraw-Hill Book Company, Inc., New York/London, pp 53-57, 1943.
22. Test Facilities Handbook (Tenth Edition), Arnold Engineering Development Center, Arnold AFS, TN, May 1974.
23. Bridgeman, P. W., "The Resistance of 72 Elements, Alloys, and Compounds to 100,000 kg/cm2," in Proceedings of American Academy of Arts and Sciences, 81, 1952.
2 4. Neutron Cross Sections, Vol I, Resonance Parameters , BN L 325, National Neutron Cross-Section Center, Brookhaven National Laboratory, Upton, New York, June 1973.
132
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