AD-771 987

SPRING DRIVEN PRIMER STRIKER STUDY

Gerard G. Lowen

City College Research Foundation

Prepared for:

Harry Diamond Laboratories

1 December 1970

DISTRIBUTED BY:

National Technical Infonation ServiceU. S. DEPARTMENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151

I I

IISPRING DRIVEN PRIMER STRIKER STUDY

FINAL REPORT

I By

O • Dr. Gerard G. LowenProfessorMechanical Engineering Department

j The City CollegeNew York, New York 10031

/ THE CITY COLLEG-EZ LA RESEARCH FOUNDATION

L THE CITY COLLEGE ofTHE- CITY UNIVERSITY of NEW YORK

This research was conductedunder contract No. DAAG-39-68-C-0026 with the HarryDiamond Laboratories, Branch420. Washington, D. C. 20438

. 4NI( A[' s , < [ F ,/IJ

ITIlE CITY COLLEGE RESEARCH FOUNDATION

THIE CITY CO LL EGE

T1H"1 CITY UNIVERSITY OF NEW YORK

I SPRING DRIVEN PRIMER STRIKEIR STUDY

I FINAL REPORT

II BY

Dr. Gerard G. Lowen

Professor

Mechanical Engineering Department

'The City College

i New York, N.Y. 10031

IThis research was conducted December 1, 1970

under contract No. D)AAG-39

-68-C-002b with thc Harry

Diamond Laboratories, Branch

g 420. Washington, D.C. 20438

Approved for public release, distribution

unlimited.

II Uw

Contents Page No.

Illustrations 3

List of Symbols 5

Abstract 8

I. Introduction 9

II. Theoretical Investigation 10

III. Experimental Investigation 1s

IV. Optimization Procedure 19

V. References 2;

VI. Acknowledgements 29

Appendix A: Iielical Spring as a Distributed System A-i

Appendix B: Determination of Velocities of Freely Elxpanding Springs and B-1Spring Driven Masses by Means of Wave Propagation Theory

Apperdix C: Distance Travelled by Mass M between Time of Spring Release C-1

and Time of Separation of Mass from Spring

Appendix D: Experimental Investigation D-1

Appendix 1i: Space Optimization i--I

Appendix F: Optimum i)esign rables F- 1

1 ,itriations,

No.

I Helical spring without driven mass

- Helical spring with driven mass M3 M42G primer firing test results

4 Mass-velocity combinations for firing pins which lead to

100% firing of M42G primer

,; Typical optimum design table

oi Typical application of optimum design table

I. Helical spring terminology

A._2 Spring element of length dx

A-3 Rotation about wire axis due to axial deflection

B-I Initiation and transmission of pressure wave

3-2 Two compression waves meet

B-3 One compression and one tension wave meet

B-4 Helical spring

11-5 Spring driven mass

13-b Free body diagram of mass after deflecting force has been removwd

- 2a

Ii 7 Tabulation of factor (-I + 2 e 2 )

1)-I M42G Primer firing curves (Erwood)

l). 2 Test program for M42G primer firing tests

1- ) Primer firing test setup

J)..4 Firing pin designs

9 Springs used for various firing pin groups

,i( Single probe method of measuring velocity

1 11 Two probe method of measuring velocity

-3

Illustrations (cont.)

D-12 Number of test firings

D-13 Typical data sheet

D-14 M42G primer firing data (test results)

D-IS M42G primer firing curves (C.C.N.Y. tests)

D-16 Correction factor CD

E-1 Firing pin length of initially considered test masses

E-2 Optimization of overall length of system with respect to spring index c

E-3 Typical optimum design table (other than Figure 5)

-4

List of Symbols

i2

A = cross-sectional area of bar or sprirg wire (in2)

a velocity of wave propagation or surge velocity (in/sec)

0 m s /M, the ratio of mass of spring to the driven mass

c = D/d, the spring index, ratio of coil to wire diameter

cl W Spring constant per coil of spring

Cf = coefficient of expansion

C = design correction factor

d = wire diameter of spring (in)

D = mean coil diameter of spring (in)

D = outsiue diameter of firing pin and spring

E = modulus of elasticity (lbs/in )

EK = kinetic energy (in-lbs)

E = strain energy (in-lbs)

suE su = strain energy per unit volume (in-lbs/in 3

= strain associated with stress o (in/in)

f = total spring deflection (in)

f = deflection of spring per active coil (in)

F = contact force between mass and spring (lbs)

Ft total distance travelled bN mass N between t = 0 and t = TA (in)

= pg, density (lbs /in3 )

g = gravitational constant (386.05 in/sc 2)

G = modulus of rigidity (lbs/in2 )

J - area polar moment of inertia of spring wire (in )

k = spring constant per unit length of bar or spring wire (lbs /in/in)

-5-

List of Stmbols (cont.)

K curvature correction factor for helical springs

L = length of bar or spring wire (in)

L e Length of spring when firing pin attains max. velocity

L = free length of spring (in)

"", = length of firing pin (in)

Lt = overall required system length to attain design velocity (in)

X = spring constant of helical spring (lbs/in)

m = mass per unit length of bar or spring wire ('b-sec2/in 4

m = mass of active coils of spring (lb-sec2/in)

NI = mass of driven mass (firing pin) (Ib-sec 2/in)

MN = d'Alembert moment

N number of active coils of helical spring

N = n total number of coils of helical spring

t

P = applied force (lbs)

P = force due to deflection f (lbs)

2 4,P = mass density (lb-sec /in )

C = compressive or tensile stress (lbs/in-)

TD D'Alembert force

t = time (sec)

T = surge time

T = 2T s

T = time at which spring flies off support, counted from springZ release (sec)

T A time of separation of mass from spring, counted from springrelease (sec)

T funcorrected shear stress corresponding to deflection f, (lbs/in2 )Tf

I

T KTf corrected shear stress corresponding to deflection f,

(lbs /in2)

v = particle velocity (in/sec)

v maximum velocity attainable by freely expanding spring withoutVH end load as it flies off support (in/sec)

VT 0 Vmsx, the maximum theoretically attainable velocity of the driven

mass (firing pin) for a given system and spring compression (in/sec)

vN = nominal firing pin velocity (in/sec)

W = work (in-lbs)

I W = work due to force P (in-lbs)]P

I

-7-

r

Abstract

A space optimization procedure is developed which allows the design of100 percent reliable spring-firing pin combinations for the M42G per-cussion primer. Optimum design tables list complete data for approximatelytwo hundred systems ranging in overall length from 0.500 to 2.000 inchesand in diameter-from 0.093 to 0.375 inches.

j The optimization is based on a combination of results of the theoreticaland experimental phases of the investigation. The theoretical workemploys wave propagation theory for the determination of the maximumat tainable velocities of spring driven masses. The experimental workconfirmed existing 100 percent firing data (Erwood curves) for the M42Gprimer. In addition, new 100 percent firing data were obtained, at variousenergy levels, for firing pin velocities up to 1200 inches per second.All firing tests were conducted at ambient temperature and at -40 degreesFahrenheit.

i

-8-

I. Introduction

1. Purpose of Investigation

The present investigation had the following two aims:

a. It was desired to confirm and to extend to higher firing pin velocitiesthe existing firing curves (Erwood Curves) for the M42G percussion primer.This experimental investigation was to be conducted both at ambienttemperature as well as at -40 degrees F.

b It was further des~red to devise a design method which furnishes 100percent reliable spring driven primer striker systems with minimumoverall space requirements for the M42G primer.

2. Outline and Results of Investigation

a. M42G percussion primers were test-fired with firing pin velocitiesranging from 50 to 1200 inches per second at energy levels of 16,14,12,10, 8, and 6 inch-ounces. These firings took place both at ambienttemperature and at -40 degrees F.

The lFrwood Curves were partially confirmed. That is, one hundred percentfiring of the primers at ambient temperature started at the same minimumfiring pin velocities for energy levels of 16 and 14 inch-ounces. For12, 10, 8, and 6 inch-ounce levels the minimum firing pin velocities for100 percent firing were somewhat higher than those of the correspondinglErwood data. Increases of the velocities beyond the above minimum valuesand up to 1200 inches per second caused consistent 100% firings at allenergy levels. While in most cases a slightly higher minimum velocitywas required to fire the primers at -40 degrees F, one may generallystate that there is no significant difference between firings at ambienttemperature and those at -40 degrees F. The attempt to conduct firingtests at an energy level of 5 inch-ounces was not successful.

b. The theoretical phase of the investigation concerned itself with thedetermination of an expression for the maximum attainable velocity ofa spring driven mass in the absence of friction. Existing earlierwork was confirmed to this end. In addition, an expression for the axialspace need of a helical spring which accelerates a mass to its maximumattainable velocity was derived.

c. An optimization prucedure was devised wlich combines experimental andtheoretical results. It allows the design of :;pring striker combinationsof minimum overall lengths and practical diameters which employ identicalfiring pin masses and produce identical firing pin velocities as systemswhich have been found one hundred percent reliable during the test phaseof the investigation.

The overall length of the spring striker system, i.e. the combined lengthsof the firing pin and the length of the helical spring when impartingthe prescribed velocity to the firing pin, has been found to be a functionof the spring index (the ratio of the coil diameter to the Aire diameter).

Computer searches for the optimum spring index for various combinationshave resulted in optimum design tables. These tables contain all necessaryphysical data for springs and firing pins and overall dimensions for more

-9)-

than two hundred .ystems with overall lengths of less than 2.000 inchesand diameters of less thian 0.375 inches. (One of the smallest designsrequires for u diameter of 0.12r inches a total nominal length of 0.763inches.)

Sections It to IV give the hLghlights of all work which is described indetail in Apple:idices A to Ii. The use of the optimum design tables ýf Appendix Fis also Illustrated in section IV.

During the Initial phase of the investigation the literature dealing withpercussion primers and explosive initiation [1 - 15]*1, as well as thatdealing with penetration of plates by impacting bodies [16-24], was examinedwith the intent of establishing some theoretical criteria concerning firingpin masses and velocities required to effect primer firing. This effortproved fruitless and had to be abandoned because of the scarcity of inform-ation on percussion primer initiation.

1I. Theoretical Investigation

The following outlines the results of the theoretical investigationwhich are described in detail in Appendices A, B, and C. Figure 1 shows ahelical spring of coil diameter D and circular wire diam.,ior d which is fixedon its left end Z to the support S, while its right end A is deflectedthrough the distance f by

WIRE AXIS

z

SUPPORT S

p

L 0

Figure 1

Helical Spring Without Driven Mass(Deflected through distance f by force 1f)

Sf

force P) The free lengthi of' thec spring i s 1. . pplendix A gives thoderivatron of the part i al diEfe runt ial equation for the a.\ i a di splIacement>,as a functi on of the posit ion x along the wi re axi-s and of time.

'I Numberis iii briackets des ignutQ re'fe rences in Section '

- 10-

(See also [32,33,40, 42]). The constant coefficient of the differentialo equation A-16 contains an expression for the velocity of wave propagation,or surge velocity a, with which a relieving tension wave moves from end A

of the spring wire axis to end Z when the force P1 is suddenly removed. Thediscussion preceding Eq. (A-19) shows that whenever the spring index D/d>5,the influence of the rotational inertia of the spring wire will sufficientlyaccoumted for in the following expression for the surge velocity:

a = 0.99 2

I V (4) (d)

SI where

A j spring constant [see Eq. (A-la)]

I L = length of spring wire along x-axis, and embracing all active turns

I y density of spring material

g acceleration of gravity

[I See also List of Symbols]

The solution of the partial differential equation indicates that the maximumvelocity of the free end of the spring, (point A), is attained according toEq. (A-41) at the time

s a (2)

after the force Pf has been removed. Ts is defined as the surge time.

This maximum velocity v H of the expanding helical spring is given byI Eq. (A-48):

vH 0. 9 9 "rf 2 yG (3)

SI where

Tf = uncorrected shear stress associated with deflection f [See Eq. (A-45)]

G a modulus of rigidity

Equation (3) points up the fact that the maximum velocity attainable by thefree end of a given spring is only a function of the shear stress due to itsdeflection. This velocity is thus clearly limited by the maximum permissible

1 tuncorrected shear stress. 2

2The corrected shear stress T, w K Tf includes the effect of the curvaturecorrection factor K, (see (42] and Eq. (D-8) in Appendix D). It must beconsidered t6 avoid spring set. Therefore Tf must be low enough so thatTC does not predict set.

-11-

"WITI

[See also Eq. (A-49) and subsequent discussion on theoretical velocitylimit for steel helical springs.]

When the spring drives a mass M, as shown in Figure 2, the maximum velocityof point A, and ,with that of the mass M, can also be obtained by way of thesolution of an appropriate partial differential equation. Since a separatenumerical determination of the eigen-values must be made for each occurring

WIRE AXISz0

SUPPORT S

"M Pf

fI

L0

Figure 2

Helical Spring With Driven Mass(Deflected through distance f by force Pf)

-12-

ratio of driven mass M to spring mass in , this approach is not very practtiL.,and was not further pursued. In additign, it is limited to cases where theend Z of the spring remains fixed to the support S at all times, while umas-M is firmly attached to the spring. In fuze applications it iS not custol tiYto attach the spring to its support and the mass to the spring. Thus ifenough room is given the spring may fly off of its support and the wass fiyiVseparate from 'the spring.K. Maier [28,35,36,37] has shown without proof that wave propagation theor'yfurnishes relatively simple analytical expressions for the behavior of sprii,driven masses. Beyond this it allows for the possible separation of thespring from its support as well as the separation of the mass from the spri))1,Appendix B gives derivations of appropriate expressions along these lines.Starting from the basic concepts of wave propagation in thin prismatical bai-[2(,.31,39,41] the response of springs without driven end masses is considit.:first. (Sec also [25,27,29,30,34,38,43]). Equations(B-25) and (B-20) indicate that as the relieving tensile wave, which is initiated by the ienmoV;:of the compressive force Pf, travels from point A to point Z (see Figure I,the particles along the wire axis acquire the velocity vri as given by Eq. (3.Once point Z near the support has attained the above velocity the springwill fly off from the support. This will occur at the time Ts after theforce is removed. [T. is also given by Eq. (2) above.]For the determination of the velocity of the spring driven mass of Figure 2it is necessary to distinguish between two time intervals of duration 2'I'since the wave behavior in each of them differs from the other. AppendixB gives detailed discussions of these phenomena.When the compressive force Pf is suddenly removed at t 0 0, the velocity otmass M, for the interval 0 < t <2Ts, is given according to Eq. (B-S0W) by:

--2 12 '

v1 = vIf ( - 2 T t

where

T = 2 TS

Sms mass of the active coils of the helical spring

Eq. (4) is valid for all values of the mass ratio cA.The maximum velocity of mass M is attained during the interval 2T I i- ifor values of 1/a < .569. Since this range of mass ratios covers allpossible applications to primer systems, no other intervals were considcj0l,The maximum velocity occurs when the contact force between mass M and tht.spring vanishes, and the mass separates from the spring. Iquation (B-h2gives the time of this event for a given system as

T ( 2 1 - 2aA s (2+ @ )

-13-

after the compressive force is suddenly removed. The mass ratio restric-tion mentioned above is shown by Eq. (B-86) to be based on the conditionthat TA < 4T.

Equation (B-84) furnishes the expression for the velocity of the mass inthe time interval 2T < t < TAv2 vH e- 2aT- 1) + 2 - 4a- 2e -2a 1

2 If[ T (6)

The maximum attainable velocity of the mass M is given by the evaluationof Eq. (6) at t = T as given by Eq. (5). According to Eq. (B-85):

e - 2u

Vmax = v 11 ( - + 2e 2

(7)

Figure B-7 on page 13-23 of Appendix B lists values of the term inparenthesis of Eq. (7) for mass ratios M/mrs from 0 to 5.4.

Appendix C shows now Eq's. (4) and (6) may be integrated to obtain thetotal distance Ft travelled by mass NI from the time the restraining forceP is suddenly removed at t - 0 until the maximum attainable velocity isreached at t = TA. Equation (C-13) gives this distance in the followingform:

Ft = f Cf (8)

where

f = deflection due to the force PfC and-2a 2(

Cf 2 + 4 3 -f 2 4 2 )

(9)

The mass ratio restriction NI/m S 5.69 also applies to the above, sinceEq. (6) is involved in its derivation.

Equation (7) which allows the determination of the maximum velocity ofmass M for a given system, and Eq. (8) which indicates the minimum spaceneeds to attain this maximum velocity play a major part in the optimizationprocedure shown in section IV.

Appendix B also indicates, (see pp. B-15 and li-16), how wave propagationtheory may be used for the determination of the time Tz at which a spring whichdrives a mass will separate from its support at point Z.

-14-

III Eaxeriinental Invest igat ion

Appendix D reports in detail on all phases of the test program, testsetup, and test results.

Figure D-2 shows the original test program, giving all planned testpoints at 16,14,12,10,8,6, ond S inch-ounces of firing pin kinetic energy.The associated firing pin masses and velocities, from 50 inches per secondto 1200 in/sec, are indicated together with the respective IdentificationNumbers. Figure D-3 depicts the test setup. The test firing pins aredescribed in Figures D-4 to D-8c, and the final design of the test springsis shown to be based on the results of the theoretical investigation. Thephysical data of these springs are listed in Figure D49. In addition, thetwo methods of measuring firing pin velocity are discussed together withall test procedures. Figures D-12a and b report on the number of firingsI at the various test points, both at ambient temperaLate and at -40 degreesF, while Figure D-13 represents a typical data sheet.I1. Results of FirinR Tests

Figure 3 gives the results of all firing tests, which were completedsuccessfully, by indicating the percentage of primers which fired at eachenergy level, firing pin velocity, and temperatur( (The test resultswithin the heavy outlines serve as the basis of tl,,.. Optimum Design Tablesof Appendix F.) The 5 inch-ounce tests proved to be too inconsistent, andwere not completed.

The Erwood data were only partially confirmed. A comparison ofFigure D-1, the original Erwood Curves (which refer to ambient temperature),with the comparable data of Figure 3 indicates that for energy levels of16 and 14 inch-ounces both tests showed that 100 percent firing of theprimers starts at a minimum firing pin velocity of 100 in/sec.

For energy levels of 12 inch-ounces and less, the present tests show some-what higher minimum firing pin velocities for 100 percent firing at ambienttemperature than the Erwood data. For 12 inch-ounces the Erwood data givea minimum velocity of approximately 100 in/see, while the present testrequired 200 in/sec. At 10 inch-ounces the comparable velocities areapproximately 140 and 200 in/sec. The Erwood data show 260 in/sec for9 in-oz, while present tests indicate for 8 in-oz a reliable value at500 in/sec. Finally, at 6 in-oz the comparable numbers are approximately340 in/sec for the Erwood data and 600 in/sec for the present tests.

Just as in the Erwood tests it was found that once a minimum velocityhad been reached which produced 100 percent firing, all higher velocitieswould also lead to the same result.

I Figure 3 shows also that in most cases a slightly higher minimumvelocity is required to fire 100 percent of the primers at -40 degrees Fthan at ambient temperature. The differences are small enough to concludethat when one uses a sufficiently high velocity to be certain of firing atambient temperature, one also may be certain of firing at -40 degrees F.

SFigure 4 gives such safe combinations of firing pin masses and velocities

which assure 100 percent firing both at ambient temperature and at -40I degrees F.

-1S-

- * - - p- *n M CD *- -

IzI n 0ý In ; ; c-r LLn i- c J) o m c- ,

0ý 0 Iý cý

oN . oc C-1 0

cZ ~ ~ ~ ý CO(4 ~' 'OZ Q00000

-- 1-4 -Io .'

N

9

0

93~

0" P" -4 P" P"Co ~ I~ 0.0 L~ C30 C. 0

0 .. .N 0.co

00 n33o 0 oC( 0 0000000 0~- - 4- -4 r-- F" f -

t0 Co Co 0 000 00 0 C)4:t O 00~ NO 141O0 0 0 0

4.~~~~4 -4 -4-"4- 4 q-

~~~.t 0 000~0 0

p.4 - .

w in (D 000 0 0:~ 43~r 00 00 000 000

N o 01 0 0C 0 0 0 0 0 0M C: 0 0 0F" 4 4-4 - -4 - 4 -4 f4U-

M 00 Co 0 0 0 00 00 v0 (z 0C

0 0 CIt 2L C 0 0 0 0 0 0C : 0 0 0 0-4 14 -4 -4 .-1 "-4 -4 -4 -4 04 - .-

0 0I cm 0 0 0000ý; C0 00

a1 c

~~ Co . 0 0 00 0 00

- -4 -4 14-4 1-1 -4

N~~n 0 f 00000 00Co ~ O 0 0 0 0 0 0 000

.-4 44.4

4.

:1 •

I~4i

m oco

0

-4 opl

" 0 40ca 0

00

,'4 CD )

N O

0o

m 0-

p 0D

-r-4

.00

co -

AC1 - '-l ( 03-17-

They were chosen in such a manner that fur each energy level the lowestvelocity is at least two subdivisions higher than the minimum one whichproduced 100 percent firing during testing.

Finally, Figures 15a and lSb present the results of the tests in thesame form as the Erwood curves.

2. Verification of Theory of Spring Driven MassesSince the experiment used spring driven firing pins, the data also made

it possible to compare the actually attained velocity of each test point,which was held within + 2.5 percent of the required nominal velocity v., withthe theoretical velocity vT. (See top of page D-28 for discussion on vN).

As the data sheet of Figure D-13 indicates, the spring compression f wasrecorded for each test firing, along with the actual mass of the firing pin.Appendix D-4b shows how, together with the spring data of Figure D-9, theabove data may be used to compute the theoretical velocity vT according toEq. (7).

v was found for all test points to be somewhat higher than the corres-ponding vN. This is to be expected since the theoretical expression doesnot make prjvisions for friction in the system or for certain pecularitiesof the test setup. (See bottom of page D-33)

For design purposes the above means that to attain a certain nominalSvelocity vN, the spring deflection must be such as to produce a somewhathigher theoretical velocity vT. To this end the design correction factor

CD - VT VN (0

N (10)

was introduced. [See Eq. (D-13)]. This permits the following relationshipbetween the theoretical veiocity of a given design and the actually desirednominal velocity:

vivT vN( + CD) (1

(See Eq. (D-12)].In order to attain a generally applicable value for CD which can be used

in the optimization procedure, its individual values were computed for alltests. Figure D-16 shows this factor to vary between 0.01 and 0.25 fornominal velocities above 100 in/sec. For the sake of safety, it was decided toassume always

CD 0 .25, (12)

and accordingly to make Eq. (11)

VT a 1.25 VNr t(13)

for the optimization procedure.

-18-

•_ -- • - _. i L • '''•: . .. . .... ,,.:. ,, .- • •= . -. ,,• -•. • • • _ • • • • • 1 =• • ,•. .... ... r .......... .•L

IV. Optimization Procedure

Appendix F gives details of the optimization procedure which leadsto the optimiun design tables presented in Appendix F. The optimizationis based on a combination of the theoretical and experimental results ofthe present investigation. The optimum design tables give complete physi-cal details of spring striker systems with overall lengths of less than 2.000inches and diameters equal to or less than 0.375 inches. These spring firingpin combinations reproduce the reliable 100 percent firing points given within the heavy outlines of Figure 3.

In addition to reproducing proven firing data with the use of identicalmasses and velocities as used in the tests, the optimization uses the follow-ing criteria:

a. The nominal velocity vN associated with each test point isto be attained by using Eq. (11) which takes the design correc-tion factor CD into account.

b. In order to make the springs as short as possible the firingpin velocity v is to represent the maximum attainable velocit/of the particuTar system. To this end the spring is tc be com-pressed to its solid height, and the associated corrected shearstress Tc is to be the maximum allowable one for the material.

c. Only systems with diameters of 0.375 inches are to be initiallyconsidered, and only those resulting systems are to be retainedwhich have overall lengths of less than 2.000 inches.

1. Determination of Spring Mass

According to Eq. (E-12) one may write Eq. (7) for the theoreticalfiring pin velocity vT with the help of Eq. (11):

2msTc e M

vN (1CD) = 131 K I(- 1 + 2 e)

(14)whenever the spring is made of steel.

With v and M determined from the test point involved, and with I andCD given, •he required mass of the active turns of the spring is showni byEq's. (E-13) and (E-14) to be obtained from the above expression:

ms In 2 In (8]

where

8 131 K vN (I + CD)

2 T 'V 0Sc r (c (

K is the curvature correction factor (42]:

1 •-19-

K 4c- 1 0.6154c - 4 c

(17)and

D= d(18)

The above shows that the mass of the active turns of the spring dependson the spring index c when M, vNo tcI CD and D are known.

2. Free Length of Spring and Expanded Length of Spring Correspondingto Maximum Velocity of Firing Pin.

Since the spring is to be compressed through the distance f to itssolid height, its free length L is determined by:

L = f+ Nd + 2d0 (19)

where N stands for the number of active turn's and 2d stands for two fullend turns of the spring.

The number of the active turns may be related according to Eq. (E-8)to the mass of the spring in the following way:

N4m g

S2yDd 2 (20)

When the above is substituted into Eq. (19) and the deflection f isexpressed ir, terms of the maximum allowable corrected shear stress, oneobtains for Eq. (19) according to Eq. (E-10):

24 ms g c T c n

L =C+) +4 2-O = 2y D2 K G c C

(21)

where

y = density of the spring material

G = modulus of rigidity

At the instant when the maximum velocity of the firing pin has beenreached, its total tray 1 is given by Eq. (8). With its help one obtainsthe expanded length of the spring Lex at this instant from

Lex O fCf + Nd + 2d

(22)where C is given by Eq. (9). With the appropriate substitutions for f andN, as fgr Eq. (21), one obtains:

-20-

I L - 2 yDG 2 K G c )

1 (23)

3. Overall Length of System at Instant of Maximum Firing Pin Velocity

The overall length L of the spring- firing pin system at the instantof maximum velocity of thS firing pin is given according to Eq. (E-16) by

SLt = Lex + L (24)

I where L represents the length of the firing pin.

4. Choice of Firing Pin Dimensions and Coil Diameter of SpringWhile each successful test point has a definite mass associated with its

firing pin, there are many possibilities for the firing pin dimensions. Toavoid unnecessary complexity it was decided to assume the use of cylindricalIfiring pins of constant diameter for the optimization procedure. The .030 inlengthof the hemispherical tip of the firing pin as well as any possible re-duction in diameter, which might accommodate the seating of the pin within thespring, was disregarded.

Figure E-1 shows the results of computations for the length Lp of suchsteel firing pins for all eligible I.D. numbers, and outside diameters be-tween 0.375 and 0.093 inches. In general, firing pin lengths of less than0.125 inches and more than 2.000 inches are omitted from the table.I5. Optimization of Overall Length of System

I Appendix indicates that once a choice has been made concerning the out-side diameter D of the spring and the firing pin, the length Lp of the pin,as well as the Fpring material, the overall length Lt of the system for agiven I.D. number depends only on the spring index c.

While it is not possible to find an analytical expression for the springindex which produces the shortest system, an appropriate computer searchwhich varies the spring index will produce the desired result. The springindex which gives the minimum overall length Lt for a given set of conditions,also furnishes the shortest free length Lo and the shortest expanded length Lexof the spring.

The optimum design tables of Appendix F give the results of such computersearches. Each table represents a single test point (I.D. number) wherereliable 100 percent firing occurred, and enumerates the following data for

various firing pin outside diameters Do and associated firing pin length L:

a. Mass of active turns of the spring, according to Eq. (15).b. Wire diameter corresponding to optimum c for given Do.c. Number of active turns of the spring, according to Eq. (20).d. Free length of the spring, corresponding to optimum c, according to Eq. (21).e. Expanded length of spring, corresponding to optimum c, according to Eq. (23).f. Overall length of system, corresponding to optimum c, according to Eq. (24).g. Solid height of spring, including two end turns.

-21-

All computations are based on the assumption of steel springs and firing

pins, and the following data are used:

y = 0.283 lbs/in3

"Tc = 200 000 psi

G = 11.5 x 106 psi

g = 386.05 in/sec2

CD = 0.25

[See also discussions in Appendices E and F.]

Figurc 5 is typical of the optimum design tables of Appendix F. It treatsI.D. No. 67 which had a firing pin velocity v of 700 in/sec and a firingpin mass M of 4.0816 x 10-6 lb-sec'/in at an energy level of 16 Inch-ounces.

The following example makes use of one of the designs of I.D. No. 67,and illustrates the use of the tables:

Example

When one chooses an outside diametex Do = 0.156 in., the following dimensionsare given by Figure 5:

Length of firing pin (L) 0.291 in. -6 2Mass of active turns ofPspring (m ) 3.4957 x 10' lb-sec /inWire diameter (d) 0.03319 in.Number of active turns of spring (N) 14.3Number of total turns (N + 2) 16.ZFree length of spring (L ) 0.786 in.Expanded length of sprin? (Lx) 0.816 in.Overall length of system (L tx 1.109 inSolid height of spring 0,541 in.

It must now be recalled that the tables give the overall length Ltby considering firing pins where neither the hemispherical tip nor areduced diameter for seating the pin inside the spring has been accountedfor. Let these design factors now be considered. With a spring O.D. of0.156 in. and a wire size of 0.03319 in. the inside diameter of the springis approximately 0.089 in. Allowing for sufficient clearance, one lets theseating diameter of the firing pin be 0.080 in.and its length 0.125 in.Figure 6a shows the resulting firing pin design which maintains the massof the pin at 4.0816 x 10" 6 1b-sec2/in, while allowing for the addition ofthe seating diameter as well as the hemispherical tip. The resulting lengthof the 0.156 diameter is 0.255 in., and together with the 0.030 length of thetip the active length of the firing pin becomes 0.285 in. This length mustbe added to the solid height of the spring to determine the nominal spacewhich the system requires when the spring is compressed. (It serves to locatethe release mechanism.) This length must also be added to the expanded springlength to determine the space requirement at maximum velocity firing.Figure 6b shows the first dimension as 0.541 + 0.285 a 0.826 in., (not takingsome necessary clearances into account). The actual overall length of thesystem has to be 0.816 + 0.285 a 1.101 in.

-22-

3:u~r! rIu

0 c

1. %0 _4UC) -- a-4

> L)4 -4 ,,-

S1- .Y co N N It 10 0 0Z LDCL f- 4 0 -Ol e

I0. LU 0 MNL 0

W.) ' - -Ij 0 10 (N10 '- C.0n - V

Z rr. u. u . A-10 1.- x.- t I A Cn 0L

U'%- Q 7 i"

to ofi LLJJ F4

LAU -L X: 4 1 - -r- cL

.* 't fnMONN N

S0 10 -r 7c 1.11n-AulPS-OV 10 , O N Cl 9 a X l'-

0n rn-r4 4 t It CL0 0 -A 0 '.LCL X~ (*@ @ / A -. 1 07z

on on fn( nf U.) -a U -0U. -.1 W: > 40

on 0 ZO Ft- 04 V L z w

- Z .4N N N (nI %t n LA LA 0AZ

0 -0 M r- - 4 4 N

t -23-

_0.2855 0.030 R. SPHERE

0.12.5 -0. 255I-

0. 080 ± 0.10.060 +0-00

EDJ OA.O3I

a.

1.101

b.

Fi gure 6

Typical Application of Optimum Design Tables

a. Firing pin design

b. Space requirements for system both when spring is compressed and when expanded

-24-

Minor size dcvi at ions from the opt imu[I d(s i ',n tals di. t necssar S cd Ia r-antes and part tolerances, (includin4g change' iln wi re si :e of 1..0()03 in.will not influence the results to any extent since the factor C has lecnIchosen sufficiently high.

It has to be kept in ;,iind though that in order to avoid spring set durnassembly the corrected shcar stress ic should never exceed its iunoer limit 1t"approximately 200,000 psi at solid heiglht.IV References

I A. Material liealing with Percussion Primer Perform:nce

1. Bowden, F.IP. and Yoffe, A.D). Initiation and G;rowth of Lxplosivcsin Liquids and Solids.Camnbridge University P ':;,.1952

2. Boyars, C. and Levine, ). Ilrop-ci ght Impact S:n:; it ivity V lestilli

ofo Ioxt losiives. Pyrody'iiamics. 1938,Vol. 0 , pp. 53-77G;ordon and Breach, S;ci nCc Puil. Ltd.

3. Kniss, J.R. and Wenger, W.S. Factors Affecting Sensitivity lestinitof Primers.. Balllit ic Icese-arch latorratories Memorandum Report No. 1692.Aberdeen Ptroving G roiid,,, .M:ir vland lI1,

a 4. Nelson, R.W. Velocity Sensitivity of the ,.12i1SPrinir. Nat ional Bureau of Standards.Ordnance leveiopmenllnt lDi xiý; i on.

Report No. 13-4-32R. ]'.l,3.

I . Army Material Command Pamphlct I:ngi neui rng lDesign IlindijL, 1. Ixll •J i

AM(iI' 706-179 Series ,xplos ive Trains. 1965 .

0. Air-..3 22.1 I')eaicli on, l the! llilt 1 11 (!l , ! i , I 1,IAU-54 113 i ig of' Smilla l Arm PII; Irii r,, \ i\rl(

A'- 50 2415 1.r1¶IlU -il l)',u~ kcs( rrcf In tittit r. DI

NI i cli yi [0n. Prugl!, ciwslj' t ALOIdt Ilt I- lI'7. A 0 2 tll ;iChi,. l. I' 111ot, 01,'< PI'llII('l ~ i~ l'I

t i p ay ,I •I ' o S , tUx iNv ; , I,, tvt' Iitc

tor.yJ, Wh te(larchry~w 1% i vo r (I

-Pri 222c i i5n .M ,7i. AD /,tl 562{ FtIT i:;t o 'Ica i ud P 1 Iril c q'l, t \ I

tcpori t i ol..R, 58 l 1915.

M. l.o 1 6No. IR-580. 19 5..

I). A!-208 563 Prinier Sensitivit v Ic i riý PkiIFccent ri ci tyv. F:rallm t'ord ,t'• nlb ql

Report No, 1\ lo2,. 1913,i

I Ill~1 . All-.240 059 Inivest i ), t io)n of' Scillsi t ivil h v' It' ' ( it

IPrimcr IPcrcir;,;ioli H',I. I'., , I )., .A.i I'~~icatinny A1T-:;cnlal TC' l jlC.tti li C I te Ol't ',O .

l, I, o| > 1 900.i!)

11. AD-268 S34 Powders and Explosive Substances:Means of Ignition and Initiation.Translated from: Porokha i VzryvchatyeVeshchestva Pt. 4, Gos. Tzd. Ob. Prom.(3-4), Moscow, 1957 USSR.

12. AD-425 612 Malfunction Investigation of Cartridge,57MM, TP, M306A1 (Misfires)by R.J. Seraphin and F.G. Ladd, Jr.Picatinny Arsenal Technical ReportNo. 3132. 1963.

13. AD-4S1 450 Information Pertaining to Fuses.Volume IV. Explosive Components.National Development Mission BranchAmmunition Engineering Directorate.Picatinny Arsenal. 1964.

14. AD-464 842 Percussion Primer Systems. BermitePowder Co. Saugus,California. 1965.

1S. AD-476 513 Standard Laboratory Procedures forDetermining Sensitivity, Brisance,and Stability of Explosives. By A.J.Clear. Picatinny Arsenal TechnicalReport No. 3278. 1965.

B. Material Dealing with Penetration of Plates by Impacting Bodies

16. Allen, A.W., Mayfield, B. E. Dynamics of a Projectile PenetratingMorrison, II.L. Sand. Jnl. of Appl. Phys. Vol. 28,

No. 3,1957, pp. 370-376.

17. Goldsmith, W. Impact, The Theory and Physical Behaviorof Colliding Solids. Edward ArnoldPublishers, London, 1960.

18. Goldsmith, W., Liu, T.N. Plate Impact and Perforation by Project-Chulay, S. iles. IExp. Mechanics, Vol. 5, Dec.

1965, pp. 385-404.19. Huth, JJI[., Thompson, J.S., Some Now Data on Iligh-Speed Impact

Van Valkenburg, M.E. Phenomena. Jnl. of Appl. Mechanics.March 1957. pp. 65-68.

20. Kucher, V. Penetration with Optimal Work.Ballistic Research Laboratories, Aberdeen 2

Proving Gounds, Md. Report No. 1384,1967. (AD-664 138).

21 Masket, V.A. The Measurement of Forces Resisting ArmorPenetration. Jnl. of Appl. Phys.Vol. 20, Febr. 1949 pp. 132-140.

22. Nishiwaki, Jen Resistance to the Penetration of a Bulletthrough an Aluminum Plate. Jnl. ofPhys. Soc. of Japan. Vol. 6, No. 5,1951, pp. 374-378.

-26-

23. Phillips, J.W. Impact of a Rigid Sphere on aCalvit, H.H. Viscoelastic Plate. Jnl. of

Appl. Mechanics. Paper No. 67-WA/APM-6. 1967

24. Rinehart, J.S. Behavior of Metals under ImpulsiveLoads. American Soc. for Metals. 1954

C. Material Dealing with Wave Propagation, Elastic Impact Phenomena,Helical Springs, and Rigid Body Mechanics.

2S. Anon. Valve Springs. Investigation into theProblems of Surge and Vibration.Automobile Engineer. March 1942.pp. 95-100.

26. Burr, A. H. Longitudinal and Torsional Impact ina Uniform Bar with a Rigid Body at OneEnd. Trans. A.S.M.E. Jnl. of Appl.Mech. 19S0. pp. 209-217.

27. Bourdon M.W. Valve Spring Surge. Bus and Coach.v. 12, Nov. 1940, pp. 240-243.

28. Chironis, N.P. ed. Spring Design and Application.McGraw-Hill Book Co., Inc. New York1961.

29. Dick, J. Surging in Helical Valve Springs.Jnl. Royal Aeron. Soc. vol. 37,July 1933, pp. 641-654.

30. Dick, J. Shock Waves in Helical Springs.The Engineer, Aug. 1957, pp. 193-195.

31. Donnell, L.i. Longitudinal Wave Transmission andImpact. Trans. A.S.M.E. vol 521930. (APM-52-14) pp. 153-167.

32. Gross, S. Lehr, E. Die Federn. Ihre Gestaltung und"Berechnung. (Springs, their formsand computations.) VDI-Verlag GMBH,Berlin 1938.

33. Kozesnik, J. Dynamics of Machines. ErvenP. Nordhoff, Ltd. GroningenHolland, 1962.

34. Lehr, E. Schwingungen in Ventilfedern(Vibrations in Valve Springs.)Z-VDI, v. 77, No. 18, May 1933pp. 457-462.

35. Maler, K.W. Dynamic Loading of CompressionSprings. Product Engineering.January 1954. pp. 162-167.

36. Maier, K.W. Dynamic Loading of CompressionSprings. Product Engineering.March 1955. pp. 162-174.

-27-

37. Maier, K.W. Surge Waves in CompressionSprings.Product Engineering. August 1957pp. 167-174.

38. Marti, W. Vibrations of Valve Springsof Internal Combustion Engines.Sulzer Technical Review. No. 21936. pp. 1-12

39. Ramsauer, C. Experimentelle und TheoretischeGrundlagen des Elastischen undMechanischen Stosses. (Experimentaland Theoretical Foundation of Elasticand Mechanical Impulse) Ann. d. Physik.No. 13, vol. 30, 1909. pp. 417-494.

40. Timoshenko, S. Vibration Problems in Engineering.(2nd Edition) D. Van Nostrand Co.New York, 1937.

41. Timoshenko, S. andGoodier, J.N. Theory of Elasticity. McGraw-Hill

Book Co. New York, 19Sl.

42. Wahl, A.M. Mechanical Springs. (2nd Ed.)McGraw-Hill Book Co. New York 1963.

43. Wittrick, W.H. On Elastic Wave Propagation inHelical Springs. Int. J. Mech.Sci. Pergamon Press Ltd. 1966.Vol. 8, pp. 25-47.

-28-

VI. Acknowledgements

The author acknowledges gratefully the help of the many people whoassisted him in the design and building of the test setup as well as therunning of the experiment.

Special thanks are due to the College Engineering Technicians:

Mr. Thomas Ferro,Mr. Gerard !1. Noll,

and to the graduate students:

Mr. W. Jandrasits,Mr. M. Ramaih,Mr. G. Reddy,Mr. N. Srivastava,Mr. F. Topper,

and to the undergraduate students:

Mr. K. Fassi,Mr. B. Feldman,Mr. U. Rieff,Mr. J. Strahl,Mr. A. Veca.

-29-

S I APPENDIX A

HFPLICAL SPRING AS A DISTRIBUTED SYSTEM

1. Derivation of Partial Differential Equation

I a. Definition of Terms

3 Figure A-I shows a helical spring which is held on one end against the fixedsurface S while its other end is deflected through the distance f by theaxially applied force P The indicated x-coordinate is measured along thehelically shaped centerfine of the spring wire. The y-axis is coincident withthe coil axis, and the coordinate y = y(x,t) describes the instantaneouspositions of all points along the x-axis.

I WIRE AXIS D

SUPPORT 5

Pf Y y y(x~t)

: Figure A-1

SHelical Spring Terminology

SmThe following terms must now be defined:Sd z wire diameter

SD a mean coil diameterL s length of spring wire (along x-axis and embracing active turns)

iN w number of active turns

SL0 a free length of springV' w f/N, deflection per coil due to force Pf

!•A - cross-sectional area of wire (wd /4)

JO polar moment of inertia of wire cross-section (wd4/32)i •y density of spring materialg 386.05, acceleration of gravity (in/sec2)

•' G • modulus of rigidity

IA-1

I'r

The spring constant (,) of a helical spring is given by:

S= Pf = Gd'# (A-la)

-F 8D3N

The spring constant per coil of the spring becomes:

cl "' Pf = XN = Gd4 (A-lb)• - 8D3

The spring constant per unit length of spring wire is given by:

k = XL = Gd4 L = Gd4 (iTDN) = Gd4n (A-ic)8D 3 N 8D 3N 8D2

b. External Forces Acting on Element of Length dx

ni

mm - - - 1

I-XSI. , t.-' -- 9 / /\I . .// _/_i - -- / .

IdN

S, v-AXISYI~n

Figure A-2

Spring Element of Length dx

If the element of length dx of Figure A-2 is deflected so that the axialdisplacement corresponds to dy, one may use the following proportionality

dx = d , (A-2)L/N f'

A-2

SI i.e. dx bears the samne relationship to th- length of a single coil as dv has tothe deflection of a single coil due to an applied force. Therefore usivr',j partialderivatives

ay f'N (A-3)

Tx -L

now consider the force in the y-direction at section m according to Eq.(A-lb)

Pm =f (A-4)

I and substituting the value of f' according to Eq.(A-3) gives:

I py _ (A-S)m N )x

IThe force at section n is obtained by a Taylor Ser.es expaits 4on of P1o, i.,ý.

pP = P + a (P)dx = P + c'L _2_ dx (A-6) Sin m - m -I

IAssuming that Pn acts in the direction of the positive y-axis. the sum of theexternal foices EF, acting on the element is given by:

IF = P-n P = P - cla2vdx - P

Ior

EF c'l, 2 dx (A-7)I N x2

c j.c. D'Alember. Forces cting n Element of Length dx

Translational D'Alembert Force

I The D'Alembert force due to the translational acceleration of the element ofmass dm is given by:

F Fvt -- dm 32y (A-Sa)

•' I at2

With

dm * ywd- dx

4g

A-3

the above becomes:

=Ft ynrd 2 dx a2

4g -b

FDt (A-b

D'Alembert Force Derivable from D'Alembert Moment

Figure A-3 illustrates that the application of an axial force in the directionof the positive y-axis does not only lead to the axial deflection dy of thewire, but also causes it to rotate about the x-axis through a counterclockwiseangle do.

¢¢W

ROTATION

do

AXIAL DEFLECTION dy D

D IRECTION

OF FORCE

Figure A-3

Rotation About Wire Axis Due to Axial Deflection

This rotation leads to the angular acceleration d 2o/dt 2 , and therefore to thefollowing D'Alembert moment per unit length dx:

MD - J y dx d2_ (A-9)- dt 2

g

Since any moment may be expressed in terms of the product of a force and adistance, one introduces a "rotational D'Alembert force" (TDr) which isdefined by:

MD T D.

where D represents the mean diameter of the coil. Then

A-4

S.. . -- ' - •, .. " -•.. . =• • I• .. ••, . . . . ... . I•"['• [• • :• • .. • = . .. ...... j

TDr =MD 2r D

Equation (A-9) then becomes:

T 2J0Y dx d2o (A-10)r gD dt2

The angle do is related to the axial deflection dy by:

dy = 2 R sin- E do (A-li)

J Using partial derivatives one obtains:

'"-2 -2 • 2 y (A.-12)20 = DDt D t2

Substitution of Eq.(A-12) into Eq.(A-10) results in:

TDr 04J dx ý2y (A-13)Dr 2

d. Partial Differential Equation of Motion

The differential equation of motion of the helical spring with circular wireis obtained with the help of D'Alembert's Principle-

I F+ ZTD = 0 (A-14)

Substitution of Eq's.(A-7), (A-8b) and (A-13) into the above yields:

I c' L dx - Uld2 + 4J0o] y dx - 0 (A-15)

Y~ -47 D2 J it2

Rearrangement leads to the following result:

I a 2 Dy 2 •y (A-16)I •X2 3t2

where

A-5

a 2 c'Lg 1 Ag (A- 17)NY [4d2 + 4 JO1 Y [rd + 4 JO

e. Influence of Term Containing Polar Mon,,ht of In a J on the Velocity of

Wave Propagation

The expressiona

V yU_ +4Jo] .- 8)

as given by Eq.(A-17) represents the velocity of wave propagation or surg(velocity of a helical sprin'; with circular wire. (See also Appendix B-3b.)It represents the velocity with which a disturbance is propagated along thex-coordinate of the spring. In order to examine the influence of the term4J/D 2, which accounts for the rotational inertia of the spring wire, letEq.(A-18) be written in the following form:

a =I ,Xg (A-18)+ 0.5 (1•_ • (Rd

The term .i 1 equals 0.990 for a spring index D/d = 5, and it becomes

equal to 0.994 when [)/d = 10. Sincle the above range of spring indices isusually encountered in practice, one may write Eq.(A-18) for all springs in thefollowing form:

a = 0.99 Y (L.d_ (A-19)

4

A-0

2. Determination of Maximum Attainable Velocity at Free End of Helical Spring

The following uses the partial differential equation (A-16) to determine ther displacement y(x,t) as well as the velocity y(x,t) of a helical spring which is

fixed on one end to the support S (see Figure A-1), and whose other end isinitially compressed a distance (-)f before it is released.

The solution of the above makes it possible to find the maximum attainablevelocity of the free end of the spring.

a. Boundary Conditions

I •Since the spring wire remains fixed to the support(s) at all times

y(Ot) = 0 (A-20a)

The spring is free at x = L. This is expressed by saying that there is no forceat this point and therefore there is no strain. The force at any element wasdescribed by Eq.(A-5)

m -N ax

-i; I Since c'L/N , 0, for Pr(x = L) = 0, the second boundary condition must be:

o (L,t) 0 (A-20b)ax

b. Initial Conditions

Let it be assumed that the initial deflection of all elements along the x-axis!I is proportional to the deflection (-f) at x = L. Thus

y(x,O) = - f x = f(x) (A-21a)L

Since all parts of the spring are standing still when t :, the secondI initial condition becomes:

I(tx,0) = 0 = g(x) (A-21b)

at

c. Solution of the Equation of Motion

I Using the method of separation of variables for the solution, one lets

y(x,t) = X(x)T(t) (A-22)

To satisfy Eq.(A-16), one must letj X(x) = ClCosXX + C2sinXx (A-23)

IA-7

whilu

'I(t) = C3 cOs~at + C4 sin) at (A-24)

The coefficients Cl and C2 are now found from the boundary conditions ofEq's. (A-21a) and (A-21b)

X(O) = 0 = C1 (") + C2 (0) (A-25a)

and thus

C1 = 0 (A-26)

Furthermore

X'(L) = 0 = C 2cosAL (A-25b)

For a nontrivial solution, where C2 0 0, one must satisfy

cosXL = 0

This occurs when

XL = nit (n=l,3,S,.. odd)

and thus the cigenvaluc A must be

A = fitit (A-27)2 L

Equation (A-23) then becomes:

X (x) = C sin -jx (A-28)

i.e. there will be an Xn(x) for each i.Similarly Eq.(A-24) takes the form:

T (t) = A cos -r-t + B sin -at (A-29)n 11 2L- n 2

The total solution is a sununation of all Xn(x)Tn(t) according to Eq.(A-22):

nra nira .nniy(x,t) = (nDcOs --t + EntLt )sinVLx (A-30)

n=l,3,... odd

Since Eq. (A-21b) shows that all initial velocities are zero, i.e. g(x) = 0, itfollows that

En = 0 (A-31a)

A-8

Equation (A-21a) serves for the evaluation of DlI1, Thus:

y(xO) AfX) D si) Slax (A-311))

1 3,... odd

To evaluate Dn multiply both sides of Eq.(A-31b) by sinm-Ex dx and integrate

between 0 and L:

xLr , ni1 m. dxT

sNX dx D i n1)-ix dx (A-32)-~ jx 20nxd JJ~~I 2LS1,3 odd 0

J For m # n the integral on the right hand side of Eq.(A-32) vanishes, i.e.

LL L 111 11 Isillsn ""' L1~ dx = cs(n-in) ni x dx - cos(n+m)ii x dxJ T 2 J 21.

= L siln(n-m) lix - sin(n+mn) 'nx = 07 1 ( ii-m ) 2 L 1 0 (n +m ) 2 L 0

The above is. proven as follows:a. For in = odd: Since n = odd, -ill (n-m) as well as (n+m) will be even,

sin(n-rn) n/2 and sin(n+m) '/2 will havc the form sin(kT),where k = 1,2,3,....

b. For w = even: Since n = odd, all (n-m) and (n+rn will be odd, and thedifference between (n+m) and (n-n) will be equal to 2m.Thus (n+m)ii/2 and (n-re)ir/2 are angles which are mtr radiansout of phise, and the signs and values of the sinefunctions are identical.

For mi n Eq.(A-32) bcomes:

.stn-Tx dx 1 il Lsin 2 -- x dx (A-33)

0JI 1,3,...odd 0

Since

sn 21X dx = f( 1 - X) dx L (A-34)

A-9

one obtains

-D 2J x siný--{x dx (A-35)n L2

Integration by parts of the above leads to:

n 2 T 8f 1in- (A-36)

Equations (A-31a) and (A-36) are now substituted into Eq.(A-30), and thesolution is given by:

8f nir . n n-ny(x,t) 12,2 sin- sin•n-x cos--at (A-37)n=l,3,... odd

d. Maximum Velocity of Free End of Spring

Equation (A-37) becomes for the free end of the spring, i.e. for x =L:

8f n if (A-7ay(L,t) 8f sin 2 - C1 2.T 2 CW tL'-8a

n=l,3,... oddand since

sin2 = 1 , for n=l,3,... odd

one may write the above as

y(L,t) 8f r, 7T (A-38b)n2 7t2

n=1,3,... oddThe velocity at x = L is given by differentiation of the above:

Z4fa . n--- (Lt) nnL Sn--s at (A-39)at

n=1,3,... oddTo find the maximum attainable velocity at x = L, one differentiates againwith respect to time and sets the result equal to zero:

a2y(L,t) = 0 = y4fa 2 0 (A-40)

Dt .7ý L cos2L-at•t•2L 2 2n=l,3,... odd

The series will vanish for all n for the first time when

at = 1LA

A-ID

i.e. when,

t = TS_- L (A-41)

t = Is is known as the surge time. 1his is the time it takes for thereleasing tension wave to travel along tile full length (L) of the wire withthe surge velocity (a). (See Appendix B.)

Substitution of Eq.(A-41) into Eq.(A-39) gives the maximum velocity at x = Lduring the first cyclic motion of the spring:

D (LL) Z4fa . nvVMAXL a :iL s1n-2 (A-42)MAXa nt 2

n=l,3,... oddTo evaluate the above consider lsin(nii) for various values of n odd:

n 2n = 1: lsin(ni7) = I

n 2

n = 3: Isin(nII7 ) -1n 2 3

n= 5: lsin(nii) _1 ; etc.n 2 5

Since

sin nn/2 7_n 4

n=l,3,... odd

equation (A-42) may be expressed asfa

V (A-43)

Now if Eq.(A-19) is substituted for (a) in the 'ibove, one obtains:

V A V 0.99f Xg (A-44)MAX HI ly (1d24.) jA4

L -4 )

The symbol VH is used throughout this report for the maximum velocityattainable by the free end of a helical spring (which does not drive any mass).

Now consider that the deflection f may be expressed in terms of the springconstant A, and the respective load Pf, i.e.

Pf

Further, the shear stress Tf which corresponds to f is given by

A-11

8P f D (A-45)£f ,d 3

so that

Tfl~d3

f = 8D3 (A-46)

Substitution of the above into Eq.(A-44) leads to

V1 = 0. 9 9 T1f (A-47)

Further substitution of Eq.(A-lc) for the spring constant per unit length ofwire (AL) gives

Vif = 0.99T f/7I (A-48)

NOTE: For a spring made of steel with y = 0.283 lb/in3 and G = 11.5x10 6

TfV Ii 'x (in/sec) (A-49)

With a permissible Tf = 200,000 psi, (in absence of curvature correction), thevelocity of the free end of a spring becomes 1527 in/sec.

A-12

APPENDIX B

DETERMINATION OF VELOCITIES OF FREFLY EXPANDING SPRINGS AND SPRING DRIVEN

MASSES BY MEANS OF WAVE PROPAGATION THEORY

K. Maier [35-37] ha.s shown without proof that waive Proravation theory may beused to excellent advantage for the determination of spring velocities undervarious circumstances. This approach furnishes simple design equations withoutrequiring the knowledge of eigenvalues of the partial differential equation."(See Appendix A.)

The following work gives the derivations of the above mentioned design equations,starting with the underlying principles of wave propagation in prismatical bars[311.

1. Longitudinal Pressure Waves in Thin Pristiatical Bars

Concepts of Particle Velocity and Velocity of Wave Propagation

K. at r ] _ _ _ __

P 1(a)

Figure B-1

Initiation and Transmission of Pressure Wave

a. Pressure wave of length Z is initiated.b. Pressure wave travels across bar.

Figure B-1 shows a thin long ,,niform bar of elastic material. If a uniformlydistributed load P is suddenly applied to the end of the bar, and acts for atime t, then a pressure wave will be started at this end, and it will movealong the bar. This pressure wave represents a zone in which the material ofthe bar is under compression.

In front of the zone as well as behind it the material is entirely uncompressedand at rest. At each instant new material in front of the zone is about to becompressed, while at the same time material at the rear of the zone is losingits compression.

B-1

Thus the zone can move along the full length of the bar, although individualparticles have only the small motion involved in compression and decompressionas the zone passes.

The velocity with which the pressure zone advances along the bar is termed thewave propagation velocity (a), while the velocity of the individual particleswithin the pressure zone is called the particle velocity (v). This phenomenonmay be explained as follows: When the force P is applied it compresses thematerial near the end of the bar. The compressed portion is shortened slightly,i.e. some of it experiences a displacement, and in being displaced is given avelocity. (This velocity is towards the right in Figure B-I.) It strikes thestationary material in front of it and compresses it, and starts it moving. Inso doing it is stopped itself, and the zone of compression moves foward. Thusthe material in the compresbed zone always has a certain velocity, while thematerial in front of it and behind it is at rest.

Expressions for the particle velocity as well as for the velocity of wavepropagation will now be derived. Let the following magnitudes be defined:

P = magnitude of compressive forcet = time during which the compressive force actsL = length of bark = length of compressive or tensile zonev = particle velocitya = velocity of wave propagationE = modulus of elasticityA = cross-sectional area of the barm = yA(l), the mass per unit length of bar

gy = pg, density of bar material (lb/in3 )g - gravitational constant PLk = EA, stiffness constant per unit length of bar, from E =

P EAone obtains the stiffness of the total bar as 3= UL "WhenL equals unity, the above expression results.

a. Particle Veloity

When the zone of length k, (see Figure B-1), is compressed by the force P, itis foreshortened by the amount

d Pi Pi (B-1)

EA rF

During this period force P does the work

a *Pd -p2t (B-2)P -I-

The kinetic energy of the mass in the zone of length k is given by:

E * 2-ilv 2 (B-3)K 2

The strain energy of this zone is found from the unit strain energy incompression or tension:

B-2

]": li..... !. .. / = =• =• = .... 1 '1• " ... .. m~ . . .• I ... i .... T .... I• • = . ... -- • ,.... .. . ....

,7........ ... .. r --~r--- -. . .. . = . ... .. --- ,, •.' ..........- •, •, ,: -..... n w•. ,.Z, I JJ• jllJ . • !iIj.~

Esu a C (B-4)

where= compressive or tensile stress

S= compressive or tensile strain due to aThus for the zone of volume At one obtains

S1 P dZ (A p 2k (B-5E =( T- A) -k~(2.A) _P2s 2 tA (I k - (=-)

The energy balance of the compressed zone is given by:

Wp = EK + Es (B-6)

or with the help of Eq's.(B-3), (B-4) and (B-5)

f p22 m + v P2

k 22 (B-7)

Note that the above shows that the total energy in the zone is half potentialand half kinetic. Equation (B-7) is now used to determine the particlevelocity:

v =P (B-8)

When the above is expressed in terms of the stress and the mass density p,one obtains:

v = a (B-9)

Equation (B-9) shows that the particle velocity depends entirely on the stressapplied to a given bar.

b. Velocity of Wave Propagation

During a time dt the pressure zone advances the distance adt, where a is thevelocity of wave propagation.

The associated change of momentum experienced by an element of length adt isgiven by:

m (adt) (v - 0) = m (adt) Pjl (B-10)

According to the Principle of Linear Momentum, this momentum change equals theimpulse experienced by the mass involved. Since the compression P representsthe only force acting, one obtains:

P dt = m (adt) (B-li)

The velocity of wave propagation: may now be determined as

B- 3

a = (B-t2)

ur iih different terms:

a=VL (B-13)

Equation (B-13) indicates that the wave propagation velocity depends entirelyon the material properties of the bar.

It should still be rointed out that wghen dealing with the above describedcompression wave, lie :locity of the individual particles has the samedirection as that of the advancing wave. In a tension wave the velocity of theindividual particles is in thi opposite direction to that of the wavepropagat ion.

2. Superposition of Waves

Wave Reflections at Free and Fixed Ends of Bars

Superposition of wi'ves is valid as long as the material follows Ilooke's Law,and there is no friction. The resulting force or strL:;s level, as well as theresulting particle velocity are the vectorial sums of Ahe respective:omponents. In the following discussion it is assumed that the bar is ofconstan. cross-section and of the same material throughout.

a. Two Compressiont (or Tension) Waves Meet

Figure B-2 depicts the conditions of the forces and the particle velocitieswhen tvo compression waves meet. It is assumed that both ends of a prismaticbar e>perience suddenly applied and uniformly distributed com, ression loads ofdiffeient magnitudes and durations. Similar conditions would prevail if bothwaves were prKoducedby censile loads. Part (a) shows the two wzves ofcompression Ill and F2 traveling towards each other with identic,, absolutevalues of wave propagation velocity a. The particle velocities .F both waveshave the same dircctio-is as their associated wave propagation veocities.Sinc,, v1 is dcfined as having a positive direction, the direction of v2 iscourted as negative. 'In the case of two tension waves these sign! would bereversed.)

Whcn the w-ves meet, as shown in part (b), the resulting compressio.i forcePR becomes:

)R = - + P2 = P1 + P2 (B-14)

The resu ting particle velocity is obtained with the help of Eq.(B,-8):

PI - PVR a VI + V2 ;; v1 " v2 = (B-is)

Thus irae stress level is increased, while the particle velocity becomessmall.r than the largest component velocity.

11-4

Va

a V1

1 2 (a)

PRFv

L 2 (b)PH lip.}{{ x{{ x

V V2

x x

V2

Figure B-2

Two Compression Waves Meet

a. Before meeting

b. At meeting

c. After meeting

B- 5

Part(c) shows that once the waves have passed each other, each continues withits original stress and particle velocity.

The above allows a certain deduction concerning the stress at the fixed end ofa bar or a beam. Due to the fixity there cannot be any particle velocity atsuch a point. To assure this condition, the stress of the reflected wave mustbe of the same magnitude and sign as that of the incident wave. This causesthe particle velocities of thd reflected and incident waves to be equal inmagnitude but of opposite signs. While the particle velocity at such an endpoint becomes zero, the stress doubles.

b. One Compression and One Tension Wave Meet

Figure B-3 indicates the conditions of the forces and the particle velocitieswhen a compression and a tension wave meet. Part (a) shows both waves travelingtowards each other with identical absolute values of wave propagation velocitya. Load T1 is compressive, while load P2 is tensile. The particle velocities71 and v2 have identical signs and are positive.

Part (b) shows the meeting of these waves. Because of the signs of thecomponent forces one finds the resultant force FR smaller than either of thecomponent forces, i.e.:

PR = F1 + P-2 = PI - P2 (B-16)

On the other hand, the resulting particle velocity vR is larger than that ofeither of the component waves:

P1 + P2

vR = 1 4 V2 a V I + v = (B-17)

Part (c) again indicates that once the waves have passed each other, eachcontinues with its original properties.

The presently discussed case allows r deduction concerning the particlevelocity at the free end of a bar or heam. Since there will he no opposingforces, the stress at such a free end must be zero. TR assure this condition,the stress of the reflected wave must be of the same magnitude, but of oppositesign, as that of the incident wave. As a consequence the particle veiocity ofthe reflected wave is equal to that of the incident wave both in magnitude aswell as in sign. While the stress becnmes zero in this manner, the particlevelocity doubles at the free end.

Up to now only waves produced by coistant forces were considered. The stress aand the particle velocity v were constant along the wave. In the case of avariable force, a wave will be produced in which a and v vary along the length.Conclusions obtained above regarding propagation, superposition and reflectionof waves can also be applied in this more general case.

B-6

I

(a)

V2

, (b)

F1 7

SI i2

Ii Cc)

i 7TV2I

_ _rrrl_

II F •igure 8-35

I One Compression and One Tension Wave Meet

a. Before meeting

b. Afte meetingI

5-7

3. Application of Wave Propagation Theory to Helical Springs

a. General Relationships

The expressions for particle velocity and wave propagation velocity derived forthin prismatical bars, as given by Eq's.(B-8) and (B-12), can be applied tohelical springs, if one adapts the parameters m and k to the situation at hand.Thus one must define:

m mass per unit length of spring wire along x-axis as defined byFigure A-1 of Appendix A.

k = stiffness constant, or spring constant per unit length ofspring wire (also along x-axis), (see Eq.(A-lc)).

m is obtained with the help of the cross-sectional area A of the spring wireand the unit length along the x-axis. Thus:

m =Td 2y (B-18)g 4g

The parameter k is derived with the help of the spring constant A for thewhole spring as given by Eq.(A-la) in Appendix A:

Gd4A= - (B-19)

8D 3N

Since the above is the spring constant for the full length L along the x-axis,any shortening of the spring increases the stiffness, one obtains k from:

k = AL = Gd4L Gd% (B-20)8D 3N 8D2

b. Velocity of Wave Propagation or Surge Velocity in Helical Springs

When Eq's.(B-18) and (B-20) are substituted into Eq.(B-12) one obtains thefollowing expression for the surge velocity:

al i . g A (B-21)

As in Eq.(A-18) this represents the surge velocity when the rotational inertiaof the wire is neglected, This expression furnishes slightly large values, andit is best to use the corrected form of Eq.(A-19), i.e.

B-8

Ia 0.99 (B-22)IY y•jd 2

I c. Particle Velocity in Helical Springs Without Driven Mass

Substitution of Eq's.(B-18) and (B-20) into Eq.(B-8), together with the.expression relating the load Pf of the spring to its associated maximumshear stress Tf, i.e.

*Tf lTd3

f 8D

I leads to an expression for the particle velocity as a functiot f the maximumshear stress:

v f F2•yG (B-23)

I.The same expression results when one substitutes Eq.(B-21) into Eq.(A-43) toobtain Vmax L. If one includes the rotational inertia term, the particlevelocity becomes identical with the maximum velocity at x = L as given byEq.(A-48):

! 0. 99Tf IN (B-24)

Equations (B-22) and (B-24) are used throughout this report as the applicablei design equations.

SB-

iSi

I

B-9

d. Maximum Attainable Velocity of Helical Spring Without Dr-,,,,a "',ss.

Time of Separation from Support

A

SUPPORT I I

WIRE y-AXIS

AX IS

Figure B-4

Helical Spring

Figure B-4 shows a helical spring which is compressed a distance f along itsy-axis by a corresponding axial force Pf. The spring is not rigidly fastenedto the support at point Z. Contact is m~aintained only as long as there is acontact force. As soon as the contact force vanishes the spring will fly offthe support. Note that L is Lhe length of the wire along the x-axis.

When the force Pf is suddenly removed, a tensile wave containing a force ()will start moving into the spring in the direction from point A to point Z.Associated with this relieving force is the particle velocity

v 1i = 0.99T f Vt2'yC (B-2S)

S'

/M

Figure 10

ST = (B-26)s a

where a is the surge velocity according to Eq.(B-22) and T is called thesurge time. Of course one starts counting time when the fofce Pf is removed

I from point A of the spring.

Since all particles have the same velocity when the spring separates from itssupport, i.e. there will be no strain energy locked into the spring, therecannot be any vibratory motion in the spring after separation.

• Ie. Maximum Velocity of Spring Driven Mass

I. Period Between t = U and t = 2Ts

I A. Derivation of Tensile Force and Particle Velocity Associated With Tension

: IWave

The following considers the events experienced by the spring mass system shownin Figure B-5 between the time t = 0, when the deflecting force P 'is removedfrom the mass M and thus a leftward moving tensile wave is started at point A,and the time t = 2Ts = 2L/a, when the front of this wave returns to point A

I after having been reflected at point Z at the other end of the spring.

II Z• I A

.I

I f!

Figure B-5

I Spring Driven Mass M

Since the effect of the removal of the deflecting force is communicated alongthe length L of the spring wire with the surge velocity a, one is justified intreating this situation in terms of the superposition of a constant compressionof magnitude Pf and a leftward moving relieving tensile wave. The magnitude of

I the tensile force P(t) associated with this tensile wave will now be derived byconsidering the law of maotion of mass M! under the given circ istances.Figure B-6 shows a free body diagram of mass M at the time t a 0 when the

I deflecting force Pf has just Leen removed.

( B-11

SF=P- IP(t)M!, " f •._M lowI

y-AXIS

Figure B-6

Free Body Diagram of W1ass After Deflecting Force has Been Removed

The only force acting on the mass is the contact force F, which conasists of thevector sum of a force equal and opposite to the deflecting force Pf, (i.e.acting along the positive y-axis) and the relieving tensile force P(t) of thetension wave, which may be thought of as acting on the mass in the negativedirection of the y-axis. Thus:

F = P f- P(t) (B-27)

The differential equation of motion according to Newton's Law becomes:

Pf - P(t) . d(B-28)

Now one considers that the velocity of the inass must be identical with thevelocity of the particle at the contact point A as long as contact existsbetween the mass and the spring. The particle velocity is only a functionof the tensile force P(t), and therefore it is expressed according to Eq.(B-8):

dv = dP(t),F (B-29)- =dt Vimk

Substitution of Eq.(B-29) i.-to Eq.(B-28) leads to:

P P M dPmk dt

or

dP _ mIT aP (B-30)Tt M f M

rho compiementary solution to Eq.(B-30) is obtained by means of the trialfunction

st

c

B-12

II

Substitution of this trial function and its derivative into the homngeneous5 part of Eq.(B-30) leads to the solution for the characteristic value s:

=- (B-31)

The complementary solution is given by:

-ATPC = C e M (B-32)

I where C is a constant which must be determined from the initial conditions.The particular solution of Eq.(B-30) is now obtained by the Method of5 Undetermined Coefficients. With the assumption that

dPPp = C2, and theretore atkP= 0 (B-33)

one obtains after substitution into Eq.(B-30):

P p = Pf (B-34)

The general solution is then given by the addition of Eq's.(B-32) and (B-33):IP(t) = C1 e + P (B-3S)

f

It remains now to solve for the constant C1 with the initial condition foriP(t) ,:

P(0) = 0 (B-36)

i.e. at the instant of the release of the deflecting force P the relievingtension is still zero. Substitution of Eq.(B-36) into Eq.(B-3S) leads to:

I C1 = - Pf (B-37)

I and the expression for the relieving tension force at point A is finallygiven by:

P(t)=Pf -a ) (B-38)

SThe velocity of mass M, as well as the particle velocity at point A, is thenfor the period under consideration according to Eq.(B-8):

P f NI) (B-39)

This velocity is along the positive y-axis. The contact force, according toEq.(B-27), becomes:

9-13

M 7

F Pf" P(t) =Pf e M (B-40)

B. Desirable Change in Notation

To conform with the notation usually found in the literature dealing with wavepropagation, one considers that according to Eq. (B-12):

a 2 = k (B-41)

and thus

k =la2m (B-42)

and accordingly

vmT = am (B-43)

Now consider the exponent of Eq's.(B-38) to (B-40):

am = am(LM "- M(L)

and since

mL = fs, the mass of the spring (B-44)

and further according to Eq.(B-26):

L--a Ts

one may write:

Sms (B-45)M T

If one introduces S

T = 2Ts (B-46)

where T represents the time it takes for a surge wave to traverse the springtwice, then Eq.(B-45) becomes:

AM 2ms (B-47)

Now one introduces

m s (B-48)

M

B-14

Thus finally one obtains

Vm-i 2a (B-49)M T

and Eq.(B-38) becomes:

2a

P(t) = Pf f - e T (B-SO a)

and Eq. (B-39) together with Eq. (B-25) becomes: v, = vji(l-e T-t) (B-50 b)

C. Separation of Spring from Support at Point Z

While the tensile wave travels from point A to point Z the total force Pt(1)at any location will be given by

p = - P + P(t) (B-Si)c~l) f

A negative sign has been chosen for compression, i.e. PP, while P(t) whichrepresents tension, has been given a positive sign.

Whenever the wave arrives at an arbitrary place between points A and Z, themagnitude of P(t) corresponds to that at point A at t = 0, i.e. P = 0. Themagnitude of P(t) at the same point at any time At later is found by computingthe value of P(At) at point A.

Wheq the tensile wave arrives at point Z, it is reilected with the same signsince the particle velocity at this point must vanish as long as contact ismaintained. (See also Section 2a of this appendix.)

The total force at point Z before separation has occurred, and after the wave1as been reflected is given by:

PtZ = - Pf + 2P(t) (B-52)

If one now starts counting time at the instant the tensile wave arrives at

point Z, Eq.(B-52) becomes with the help of equ. (B-S0):

S2a -- t

P Pf + 2Pf(l ) (B-53)

The spring will leave the support at the time tZ, after the tensile wave hasarrived at point Z, when the force Ptz of the last equation becomes zero. Thusone obtains for Eq.(B-S3):

2at

0 Pf[-l + 2(1 - e tz)] (B-54)

or

2a_2 t

2c -T z

This leads to

2at- tzeT z 2e =2

With the help of Eq's.(B-26) and (B-48) one obtains:

= In 2 !T In 2 (B-55)tz 2 m s

To find the time of separation of the spring at point Z when time is countedfrom the moment the deflecting force is removed from the mass, one considers:

T= T + t = T ( + ±- In 2) (B-56)Tz s z s m -m

Equation (B-56) is valid only for the mass ratios M/m < 2.89 and for t < 3T .

The time restriction is based on the fact that at t = 3Ts, counted from theremoval of the deflecting force, a further reflection of the wave will takeplace at the point Z and Eq.(B-52) ceases to be a valid description.

This consideration is also responsible for the restriction on the mass ratio.Equation (B-55) is only valid for t < 2r, with the time counted from theinstant of arrival of the wave at point 2. Therefore Eq.(B-55) must satisfy:

2T = 2> I ln 2 (B-57)SS a -rm a

Then

M 2 (B-58)i - n = 2.89

This means that whenever M/m > 2.89 the separation at point Z will take placeat a time later than t = 3Ts5 s counted from the instant of the removal of thedeflecting force Pf, and a new set of equations must be derived.

B-16

.11. Period Between t = 2T. and t = 4T,

Separation of Mass from S

A. Determination of Contact Force Between Mass and Spring

Separation of mass M from the spring at point A will occur when the contactforce between them becomes zero. It is the purpose of the following to determinethis time TA, counted from the removal of the deflecting force. Since at thistime the particle at point A will have attained its maximum velocity, it alsorepresents the instant of maximum velocity of mass M.

Examine now the magnitude of the contact force at point A just an instant beforethe tensile wave, which has been reflected at point Z, arrives again at pointA This occurs at t <2L/a = 2T, = T. According to Eq.(B-40), the magnitude ofthe contact force for 0 < t < t:

2a

F = Pf e T (B-59)

(With the above sign it represents the force of the spring on the mass in thedirection of the positive y-axis.) Substitution of t = T, or rather a value oft just a little smaller than T, indicates that the contact force cannot vanishbefore the tensile wave is once again ref'.cted at point A.

One need now to examine what happens at point A when T < t < 2T. The reflectedtensile wave arrives at point A when t = S Since there cannot be an abruptchange of the velocity of the mass, this incident tensile wave will bereflected as from a fixed end. This type of reflection causes a doubling of thetensile force of the incident ware at point A.

Note that there are then two waves moving from point A towards point Z:

a. the original tensile wave which was started by the removal of thedeflecting 'orce Pf, and

b. the wave which is reelected from A at t - T.

Furthermore there is the tinsile wave moving crom point Z toward point A.

In order to determine the contact force at A it is not only necessary toconsider the action of all three of the above waves, but one must also take theinitial compression of magnitude Pf inti account.

Now let P1 stand for the tensile forc;e of the two waves moving away from pointA. The wave which advances towards point A is merely the wave sent out frompoint A at t - 0, delayed a time T due to its travel across the spring and back.The tension produced due to this latter wave at point A is obtained bysubsttuting (t - T) for the time in Eq.(B-50), which represents the expressionfor the tension produced at point A in the preceding interval, (i.e. 0 < t < T).Let this force be termed P0 :

(B-60)

B-17

The total tensile force at point A for the interval T < t < 2T will thus begiven by:

P(t) = Pl(t) + PO(t - T) (B-61)

The associated particle velocity is obtained as the difference between theparticle velocity due to the tensile wave going away from point A and the onedue to the wave moving towards point A. Thus according to Eq.(B-8):

v =- [Pl( t ) - P0(t - T)] (B-62)

In order to determine the contact force F for the period under consideration,one proceeds in a manner similar to the one leading to Eq.(B-40). This meansone must determine Pl(t).

One starts by writing Newton's Equation of Motion for mass M again:

F = M dv (B-63)

(Refer to Figure B-6 for a free body diagram.) The contact force F consists ofthe original rightward acting force P as well as the sum of the leftwardacting tensile forces given by Eq.(B-61). Equation (B-63) becomes:

F-P - P(t) =. M . (B-64)

Now substitute Eq.(B-61) and the derivative of the Eq.(B-62) into the above:

Pf - [Pl(t) + Po(t - T)] -t[P(t) Po(t - T)] (B-65)

With the help of Eq.(B-49) one obtains:

d[lt - O(t - T)] + 2a[Pl W + PO(t- T)] 2a P (B66

Equation (B-66) must now be solved for P(t) in order to find an expression for

the contact force F. a2act

Miultiply all terms by e

2at 2at 2 at 2at 2at-T- dPI (t) 2 y T P~ ) 2 a_

t- - P1 (t) e dt + - T)e -XT Po(t -T)e

2ai; 2at (0-67)a P(tT)+ -L -PfT 08 T

B-18

S 2ctt2aT

Note that ±--e Po(t - T) has been added to the right hand side of the above.

Equation (B-67) may now be written as:

2at 2at 2atd ' P~)Id eT4a Tdt[e T P(t)] [e T P(t T)] -- Te P 0 (t T) (B-68)

2at2a TT6 Pf

Integration of the above leads to:

2at 2ut f2at

e (Plt) (e P0 (t - T) T JeT P0 (t - T) dt(B-69a)

f2at+2a f T dt + C

2at

where C is a constant of integration. Now divide by e T

2cat ('2at

Pl(t) = P0 (t - T) - Le Tje P0 (t - T) dt1 0 T(B-69b)

2at r 2ct 2at2r% T T Te -+ - "T -e-e -dt + Ce

Now substitute Eq.(B-60) for %0(t - T):

t2t 2attT- 4 T 2a(cF 1P1 (t) f P- Pf e -T e [f Pe ] dt

2cz z(t 2B-70)

2 7 PJe' dt + Ce'F-

Rewrite the above:t 2atfp 2at-2aCT ) 4 T p --

P (t) =Pf Pf e - 4a ef e dt

4a Td ~d~ J_~e tTe f

2atT

+ Ce

After the indicated integrations are carried ou*, one obtains:2c2t

p (t) P e T - 1] + Ce (B-72)

B-19

To evaluate the constant C in the above one needs PI(T). Io obtain it thefollowing reasoning is used: The wave which was started at t = 0 has justreached pLint A, and it has been reflected. Since the magnitude of P(O),according to Eq.(B-50), is zero, both the incident as well as the reflectedwave have zero tension at t = T. The only force present is due to the P(T),

i.e. again according to Iiq.(B-50):

2otTPI(T) = P(T) = Pf(1 - T (B-73)

f)Substitution of the above into Eq.(B-72) gives:

pf -) 14TT•.- -2a (B-74)P f Pfe TT j C

at t = T. Solving for the constant one obtains-

-2cx

C =p (2- 4a - e ) (B-75)f-2a e

Finally:

2-t 2a= Pf[e 1) -- 4t e 2 a - T tp M--2 4 e )e ] (B-76)

The contact force F may now be determined with the help of the left hand side

of Eq.(B-65), i.e.:

F = Pf - PI(t) - P0 (t- T) (B-77)

Substitution of Eq's.(B-76) and (B-b0) leads to:

tF aP t e - 1)[0"2a - 4( 1)] (B-78)

B. Determination of the Separation '1inJe TA

The separation time TA is determined by means of setting

F(TA) - 0 (B-79)

into Eq.(B-78). One obtains:

TA0 * e - 4 -- 1) (B-O)

B- 20

-2C TA - 1)

since the factor Pfe cannot be zero at t = TA. Finally fromEq.(B-80):

TA = T(l + e- ) (B-81)

According to Eq's.(B-46) and (B-48) respectively:2m

T = 2T and 2a =

so that Eq.(B-81) becomes:

2ms

M M (B-82)

C. Determination of Maximum Velocity of Mass

Substitution of Eq's.(B-60) and (B-76) into Eq.(B-62) furnishes an expressionfor the velocity of the mass M while contact is maintained with the spring atpoint A. Thus

P -2u(- 1) eV =f [e T 4at + 2 - 4a - e -2a 1] (-3

Now consider that according to Eq.(B-8), and its modified form Eq.(B-25) whenapplied to a helical spring:

f 0.99f A u. vH

Thus Eq.(B-83) becomes:

-2a{I-- 1)v2 VH[e (-4 t + 2 - 4a-e -2 ] (B-84)

To obtain the m.aximum velocity of the mass one evaluates the above at t = TAIthe time of separation as determined by Eq.(B-81):

.2ms

_ __ " -(B-8S)

Vmax V H [-I + 2e 2

B-21

Figure B-7 shows a tabulation of the term in brackets in Eq. (B-85).

Since the expression for the contact force F of Eq. (B-64), which formsthe basis oZ the present section, is valid only for 2Ts<t<4Ts there willalso be a limitation on the ratio M/ms. TA must be less than 4TS, andtherefore substitution into Eq. (B-82) leads to:

2ms4Ts > Ts(2 M M

This in turn requires that:

_2m_

4 -I > e M (B-86)

This condition can only be fulfilled for W/ms < 5.69.

B-22

Kj

M Value of M Value ofm ms Factor s Factor

0.0 1.00000 2.8 0.56577

0.1 1.00000 2.9 0.55624

0.2 0.99995 3.0 O.54719

0.3 0.99873 3.1 0.53858

0.4 0.99327 3.2 0.53038

0.5 0.98177 3.3 0.52257

0.6 0.96464 3.4 0.51512

0.7 0.94338 3.5 0.50801

0.8 0.91958 3.6 0.50121

0.9 0.89452 3.7 0.49471

1.0 0.86914 3.8 0.48848

1.1 0.84409 3.9 0.48252

1.2 0.81977 4.0 0.47681

1.3 0.79641 4.1 0.47132

1.4 0.77415 4.2 0.46605

1.5 0.75304 4.3 0.46099

1.6 0.73307 4.4 0.45613

1.7 0.71423 4.5 0.45144

1.8 0.69647 4.6 0.44693

1.9 0.67974 4.7 0.44258

2.0 0.66397 4A8 0.43839

2.1 0.64911 4.9 0.43435

2.2 -0.63510 5.0 0.43045

2.3 0.62187 5.1 0.42668

2.4 0.60938 5.2 0.42303

2.5 0.59757 5.3 0.41951

2.6( 0.58639 5.4 0.41610

2.7 0.57581

- 2m,Figure B-7 "2ms

Tabulation of Factor - 1 + 2eg

(See equ.B-85)

5- 23

APPENDIX C

DISTANCE TRAVELLED BY MASS M BETWEEN TIME OF SPRING RELEASE AND TIME OF

10 SEPARATION OF MASS FROM SPRING

"The results of Appendix B are now used to determine the distance through whichmass M must travel until it reaches its maximum velocity. This is accomplishedby integrating the velocity of the mass with respect to time between t = 0,when the spring is released, and t a TA, when the maximum velocity of the massis attained.

Equation (B-39) of Appendix B shows that the velocity of the mass during thetime interval 0 < t < 2TS = T is given by:

-Pf ( e- (C-1)SvI =- (1- e )

S~VW

Using the change of notation indicated on page B-14, and letting

Pf kC-2)-k= vi

according to Eq's.(B-8) and (B-25), one may express Eq.(C-1) in the followingform:

- 2ar

v v (I e- (C-3)

wherea f ms the ratio of spring mass and driven mass

T = 2Ts, twice the surge time TS

The velocity of the mass for the time interval T < t <2T =4TS is given byEq. (B-84) of Appendix B:

-2c%(t - 1) (C-4)

V= -l(C (TTt + 2 -4a e-e2a) - I]

1. Distance 'Travt,.J be Mass M Dluring !nterval from t = 0 to t - T

Integration of Eq.(L-3) furnishes the distance Fl, which is traveled by themass between t a 0 and t a T. Thus:

Fl1 = vuIf (I - e ) dt (C-S)

Performing the integration between the indicated limits leads to:

C-I

F H + 1- 24e -1)] (C-6)1 VH*T1l 2-(e

2. Distance Travelled by Mass M During Interval from t - T to t * TA

Integration of Eq.(C-4) between the limits of t & T and t = TA furnishes thedistance F2 travelled during this time interval. After rearrangement oneobtains:

2 TA 2ct . 2 a tF a V e2 4a - -2a 1- (C-7)2 1 H -ite (2 - 4a - e' )e e 2 dt

where, according to Eq.(B-82):

1 -2a.TA = T (1 + TO )

Term by term integration furnishes the following results:

TA 2avHT e- 2a

2ci4f A ;--ftpdt vHT-2~le -2aF2(a) = vHe T te dt• -T-[2c+l-e (4a+e +2)] (C-8)JaT

andTA. 2a

v~ ~(2 - • ••F2(b) vHe2 2 4 - e 2a )fT dt

(C-9)

vHT -ZciS2a-(4a + e - 2)(e - 1)

Furthermore:

Adito 2(c) o '- dt (C-10)•: teT

Addition of Eq's. (C-8), (C-9) and (C-10) leads to:.-2*

-2 (2 - 2e - (C-11)

C-2

I3. Total Distance Travelled by Mass M During Interval from t = 0 to t TA

The total distance Ft travelled by mass M during the interval from t =0 tot TA is now obtained by addition of Eq's.(C-6) and (C-i1):

e2a1 i - 2a 2 .Ft F1 + F2 =V T[1 + -L(3 - ye - 4e 2 (C-12)

It ý,, convenient to express the product vH1T in terms of the spring deflectionf. Equation (A-43) gives:

fa f 2f-H - •L T

Therefore Eq.(C-12) may be written in the following form:

SFt = fCf (C-13)

where

I • •° °2aee-

C =2 + (3 - - 4e ) (C-14)

I!!III

I

C-3

I APPENDIX D

EXPERIMENTAL INVESTIGATYON

1. Aims of Experiment

3 The present appendix describes the experimental phase of the investigation.This experimentation had the following aims:

5 a. Verification of the "Erwood Curve." (see Figure D-l) which give thereliability of the M42G primer in terms of percentage of primersfired versus firing pin velocity (at ambient temperature). Thesecurves are plotted for firing pin velocities b',tween 50 and 300in/sec with firing pin kinetic energy levels of 16, 14, 12, 10, 9, 6and 5 inch-ounces respectively.

I b. Extension of the above mentioned M42G primer firing curves to thehighest possible velocities attainable by firing pins of comparableenergy levels which arc driven by helical compression springs. Thiswas to be obtained at both ambient temperature as well as -40degrees F. The energy levels were to be 16, 14, 12, 10, 8, 6 and 5inch-ounces. In addition the "Erwood" range was also to be tested at-40 degrees F.

I c. Since the test setup was to use helical compression springs, theexperiment could also serve as a verification of the theoreticalresults concerning the velocities of the spring driven masses. (SeeI Appendix B.)

The following sections discuss the test program, the test setup, as well asthe test results.

2. Test Program

Figure D-2 represents the test program. It gives the various levels of kinetic

energy of the firing pins as well as their masses and velocities at which the~: 5 primers were to be tested at both ambient temperature as well as -40 degrees F.(The firing pin identification numbers are indicated in the upper left handcorner of each box.) The velocities range from 50 to 1200 in/sec. The topI velocity of 1200 in/sec was chosen because computation, using the theory ofAppendix B, had indicated that this speed was close to the ultimate oneattainable for such a system. (See computations in Part b below,)

The determination of the firing pin masses will now be shown on hand of anexample: Let it be required to find the mass of a firing pin for an energyI level of 10 inch-ounces and a velocity of 200 in/sec. Consider that

K.E. V2 (D-0

whereK.E. a kinetic energy in inch-ouncesi a firing pin mass (lb-sec 2/in)v a firing pin velocity (in/see)Then according to the above:

I)

CL

L)

4A.+r

teoCL

M.4. ,,'-. OT

10

(.3 to A

0 -4 U t

.-4 0 I

UN .,q Cd94. 41 1

4 0 r

0e

010li uaauLdD)-2

Velo- Mass of Identcity of Firing No.Firing Pin atPin 16 14 12 10 8 6 5(in/sec) in.oz, in.oz, in.oz. in.oz. in.oz. in.oz. in.oz.

IJ iso L..J ._ZJ -3.j600.0 500.0 ,400.0

_4 _ _7 j _5 .7100 200.0 175.0 150.0 125.0 100.0 75.0 62.5

11 12 1 1 4 _15 1_6 17188.8889 77.7778 66.6667 55.5556 44.4445 33.3334 27.7778•oo8 ý,O 21=J,• l

200 1819 _ 21 272 2=3250.0 43.75 37.5 31.25 25.0 18.75 15.625

25 S 26 27 28 291 332.0 28.0 24.0 20.0 16.0 12.0 10.0

30 32 _33 34J 351 301 371 38122.2223 19.4444 16.6667 13.8889 11.1112 8.3334 6.944435 40 41 421 431 44 451

16.3265 14.285 12.2448 10.204 8.1632 6.1224 5.1020

400 46 471 481 4 501S12.5 10.9375 9.375 7.8125 6.25 4.6875 3.90625oo S4 55J 561 577 581

8.0 7.0 6.0 5.0 4.0 3.0 2.560 611 6 62 05 6

600 5.5555 4.8611 4.1666 3.4722 2.7777 2.0833 1.7361700 67 681 69 -70 7Z1 72 73

4.0816 3.5714 3.0612 2.5510 2.0408 1.5306 1.2755

741 5 76 781 . 893.125 2.7343 2.3437 1.9531 1.5625 1.1718 0.9765

900 81 82 83 84 85 86 872.4691 2.1604 1.8518 1.5432 1.2345 I 0.9259 0.7716

1000 8.9 901 92 9 9412.0 1.75 1.50 1.25 1.0 0.75 0.625

1100 97 8 9 100 01.6528 1.4462 1.2396 1.0330 0.8264 0.6198 0.5165

1200 10 0 14 10 .107 1 01.3888 1.2152 1.0416 0.8680 0.0944 0.5208 0.4340

Figure D-2

Test Program for M42G Primer Firing Tests(Masses of Firing Pins in lb-scc 2 /in x 106)

I)-3

KE. _ 0 -0-6 (D-2)

m 8 (200 )1 = 31.25 x 10 (lb-sec 2 /in)8v2 8 (200)2

3. Test Setup

Figire D-3 shows an exploded view of the test apparatus. The M42G primers aremounted in a .38 caliber cartridge, and this cartridge is inserted into thecartridge holder. This holder may be fastened into the front of the tube.Before its rel.ease, the firing pin is held against the pull pin by thecompression of the spring. This compression is made adjustable by the leadscrewslider. A scale (not shown) which is mounted on one of the runway cover platesmakes it possible together with the micrometric dial to repeat the springsetting accurately.

When the firing pin is released, the spring is allowed to expand freely untilit flies off the supporting lever. To prevent the spring from following theseparated firing pin it is stopped by the end of the slot in which the padrides, (The spring leaves its support some time after the firing pin separatesfrom the spring. See Appendix B.) It is still to be noted that *the spring padis fixed to the spring.

Since it was necessary to divide the firing pins into five groups of differeýntdiameters, all parts which depend on the firing pin diameter such as tubes,levers, etc. had to be made in sets of five. In addition, in order to make thetesting as fast as possible thirty cartridge holders were made.

The velocity of the firing pin is measured with the help of a Fotonic SensorModel KD-45 Serial 65 (Mechanical Technology Inc., Latham, N.Y.). The sensingprobe(s) of this'unit consist of 0.082 inch bundles of approximately 800optical fibers, One half of these fibers transmit a light source, while theother half may again receive the light source once it has been suitablyreflected. The output voltage of this unit is proportional to the quantity ofthe light reflected, and is used to trigger an electronic counter (BeckmanUniversal Eput and Timer, Model No. 7360).

Depending on the size of the firing pin, a single probe or a two probe techniqueof measuring the velocity of the firing pin is used. Both methods of measuringare described in Section b below.

a. Design of Firing Pins

In order to obtain the firing pin masses described by the test program it wasnecessary to divide them into five groups of different diameters:

Group 1: 0.500 inch diameterGroup II: 0.312 inch diameterGroup III: 0.187 inch diameterGroup IV: 0.125 inch diameterGroup V: 0.093 inch diameter

Further subdivision according to shape became necessary in order to accomodatethe velocity'measurements. Figures D-4, D-Sa, D-Sb, D-Sc, D-5d, D-6a, D-6b,D-6c, D-7a, D-7b, D-7c, D-8a, D-8b and D-8c show all the firing pins used inthe experiment. The masses are given in lb. sec 2 /in and in grams and allrelevant dimensions are given in inches. In all cases the pins were made of

D-4

I~cc1

41l

at >Ju I

/ ~ i''~/i/

IV . 1t.- .I- -1 .II - - . I I-, . -,- - -T-- r ".- , - - - ' ', -' 1 -- -- - :I-- .-. , . ' I-v~ - -'Y .YW W . .

steel. Their tips consisted of the usual hemispheres of .030 inch radius.All firing pin masses were held within * 0.5 percent of their theoreticalweights.

D-

,i ' I

i5 tI t

II

.0.750

1.000 -L L

I.D. MassNo. L Mlb-sec2 /in gram

1 600 x 10-6 105.077 4.042

2 500 87.564 3.346 -

3 400 70.051 2.648 -

4 200 35.025 2.305 2.930

S 175 30.647 2.020 2.645

Figure D-4

Group I Firing Pins

D-7

0.2500.200020

---

M

I.D. MassNo. L Mlb-sec 2/in gram

6 150 x 10-6 26.269 2.817 3.127

7 125 21.891 2.372 2.875

8 100 17.513 1.927 2.427

9 75 13.134 1.483 2.000

10 62.5 10.945 1.263 1.763

11 88.8889 15.567 1.730 2.230

12 77.7778 13.621 1.532 2.032

13 66.6667 11.675 1.335 1.835

14 55.5556 9.729 1.137 1.637

15 44.4445 7.783 0.937 1.439

18 50.0000 8.i56 1.037 1.537

Figure D-Sa

Group 11-1 Firing Pins

U-8

0218-1

L 0 . 6 .56 -sonO. 093F"- - ---'

M

I.D. MassNo. L M

lb-sec 2 /in gram

16 33.3334x10"6 5.837 0.902 1.417

19 43.75 7.661 1.087 1.602

20 37.50 6.567 0.976 1.491

21 31.25 5.472 0.865 1 .380

25 32.00 5.604 0.878 1 .93

Figure D-5b

Group 11-2 Firing Pins

Di- 9

S... ... ... .... ... ... + .. . ". .. • •. . .• ' ... I °I ... • • I ... . I • • ... . ... ... ...

S

0.218 -- "- -- w"00.. . ÷ . 0.312

I . ..O,=DR 1 L L

0.156 0.656 NO-0.093

I.D. MassNo. LMSlb-sec2 /in gram

17 27.778x10"6 4.864 1.061 1.581 1.488

22 25.000 4.378 0.995 1.515 1.422

23 18.75 3.283 0.846 1.366 1.273

26 28.00 4.903 1.066 1.586 1.493

27 24.00 4.203 0.971 1.491 1.398

28 20.00 3.502 0.876 1.396 1.303

32 22.2223 3.891 0.928 1.448 1.355

33 19.4444 3.405 0.863 1.383 1.290

Figure 0-Sc

Group 11-3 Firing Pins

0j- 10

S / 0.205-

I .D. Mass

N. lb-sec2 /in grainLGroup

1 14

iin-PnI DR IIIII

24 15.625xi0"6 2.736 0.760 1.260 I1.167

29 16.000 2.802 0.769 1.269 1.176

34 16.6667 2.918 0.785 1 .285 1.192

39 16.3265 2.8S9 0.776 1.276 I1.183

Figure 0-5d

Group 11-4 Firing Pins

0.1870.125 0.12

o-0.656 wo0.093

• ~L

I .D. MassNo. lb-sec 2 /in gramL M

30 12.OooxlO"6 2.101 0.902 1.402

35 13.8889 2.432 0.995 1.495

36 11.1112 1.945 0.858 1.358

40 14.285 2.501 1.015 1.515

41 12.2448 2.144 0.914 1.414

46 12.500 2.189 0.926 1.426

47 10.9375 1.915 0.849 1.349

Figure D-6a

Group Ill-1 Firing Pins

D-12

I

4¸

"--0.062 0. 531--o- 0. 093

- n

I.D. MassNo. L MSlb-sec 2/in gram

31 10.000x,0O6 1.751 0.8.3 1.353 1.260

37 8.3334 1.459 0.760 1.260 1.167

42 10.204 1.787 0.864 1.364 1.271

43 8.1632 1.429 0.751 1.251 1.158

48 9.375 1.641 0.818 1.318 1.225

49 7.8125 1.3LC 0.731 1.231 1.138

53 8.0000 1.401 0.741 1.241 1.148

Figure D-6b

Group 111-2 Firing Pins

1•-13

_ _ 0_ .125 -0o.187 --_

• F L

M

I.D. MassNo. L M

lb-sec 2 /in gram

38 6.9444x10"6 1.216 0.315 0.690

44 6.1224 1.072 0.275 0.650

50 6.250 1.094 0.281 0.656

54 ."0 1.225 0.318 0.693

.55 6.000 1.050 0.269 0.644

Figure D-6c

Group 111-3 Firing Pins

SI'- 14

0.070 ~0.0931012

0.656 wap-fI0.093

M

I.D. MassNo. L... . Mlb-sec2 /in gram

60 S.5SS6xlO" 6 0.973 0.885 1.197

Figure D-7a

Group IV-I Firing Pin

.. .....

1+-,E- b. - i

m L

I.D. MassNo. LMlb-sec 2 /in gram L M

45 5.1020;:10- 6 0.893 0.548 0.860

Si 4.6875 0.821 0.502 0.814

52 3.9062 0.684 0.415 0.727

56 5.000 0.876 0.537 0.849

57 4.000 0.700 0.425 0.737

58 3.000 0.525 0.314 0.626

59 2.500 0.438 0.258 0.570

61 4.8611 0.851 0.521 0.833

62 4.1666 0.730 0.444 0.756

63 3.4722 0.608 0.367 0.679

64 2.7778 0.486 0.289 0.601

67 4.0816 0.715 0.434 0.746

68 3.5714 0.625 0.378 0.690

69 3.0612 0.536 0.321 0.633

70 2.5510 0.447 0.264 0.576

74 3.125 0.547 0.328 0.640

75 2.7343 0.479 0.285 0.597

Figure D-Tb

Group IV-2 Firing Pins

1-lb

I .D. Mlass-1. 530 0. 268 -- 0.4

7i65 2.0343710- 0.410 0.639 0.579

77 1.9531 0.342 0.582 0.522

; 78 1.5362 0.274 0.524 0.404

81 2.4691 0.422 0.658 0.598

82 2.1647 0.378 0.612 0.562

""837 1.85318 0.324 0.567 0.507

S84 1.5325 0.270 0.521 0.461

88 2.1000 0.350 0.588 0.528

89 1.7500 0.306 0.S67 0.491

_-2'1/

041.062.7 051 .6

-i -- N -

65 2.08x000 0.350 0.601 0.541

66 1 .7360 0.304 0.541 0.489

90 1.5306 0.263 0.514 0.454

Figure 0)-7c

Group IV-3 Firing Pins

D-17

S0.050

+ ~0.0937

L

M__-__ _____/____

I. D MassNo. 2g L MNo Ib-sec2/in gramL

79 1.1718xi0"6 0.205 0.204 0.391

80 0.9765 0.171 0.165 0.352

85 1.2345 0.216 0.216 0.403

86 0.9259 0.162 0.155 0.342

91 1.250 0.219 0.219 0.406

92 1.000 0.175 0.170 0.357

95 1.6528 0.289 0.299 0.487

96 1.4462 0.253 0.258 0.445

97 1.2396 0.217 0.217 0.404

98 1.0330 0.181 0.176 0.363

99 0.8264 0.145 0.136 0.323

102 1.3888 0.243 0.247 0.434

103 1.2152 0.213 0.212 0.399

104 1.0416 0.182 0.178 0.365

105 0.8680 0.152 0.143 0.330

Figure D-Ba

Group V-1 Firing Pins

I1- 18

•!i I

I

II

I... ,0.09375DR I LL ... - - -

0.052

!SI M

I.D. MassNo. M Slb-sec 2/in gram

87 0.7716x10- 6 0.1351 0.317 0.287

93 0.750 0.1313 0.311 0.281

94 0.625 0.1094 0.275 0.245

100 0.6198 0.1085 0,274 0.244

101 0.5165 0.0904 0.244 0.214

106 0.6944 0.1216 0.295 0.265

1 07 0.5208 0.0912 0.245 0.215I _ __o•o _ _o, o•• o

I Figure D-8b

I Grop V-2 Firing Pins

iII!

[1-19

-. ... ..- -- 0.09375

DRILL I

0o0.060 L

0.200

0.232

I.D. MassNo.o lb-sec2 /in gram

11080 0.4340x1l" 6 0.0760

Figure D-8c

Group V-3 Firing Pin

U-2U

g b. Selection of Springs

Figure D-9 lists the springs which were used with the various groups offiring pins.

f Group I II III IV V

Lee Cat. LC- LC- LC- Special SpecialNumber 063G-12 045D-18 024B3-1S

Wire Dia. 0.063 0.045 0.024 0.022 0.018I (in)

OutsideDiar.meter 0.470 0.300 0.180 0.117 0.088(in)

FreeLength 2.500 2.250 2.000 1.500 1.625(in)

Active 16 22 30 31 48Turns approx. approx. approx. approx. approx.

Load atSolid fit. 29 19.75 5.4 8.5 6.5

SI (Ibs)

Mass ofSpring 8.988 3.888 0.903 0.476 0.35S(grams)

iFigure D-9

I Springs Used for Various Firing Pin Groups

IIII

i: D- 21

The following computation serves to illustrate the manner in which the springswere selected. In addition it shows that a velocity of 1200 in/sec representsthe upper limit of the possible velocities for the types of mass spring systemsinvolqed.

Given: A spring for Group V firing pins:

Outside diameter = 0.088 in.Wire diameter (d) - 0.018 in.Mean coil diameter (D) - 0.070 in.Number of active coils = 48Free length = 1.625 in.Spring mass (ms) = 0.355 gramsSpring material = music wire

Firing pin corresponding to identification number 102, i.e.

M = 1.3888 x 10.6 (lb-sec 2/in)0.243 (grams)

Required firing pin velocity for identification number 102:

v = 1200 (in/sec)

Find: Is it possible for the present system to impart a velocity of1200 in/sec to the firing pin?

To obtain the answer to the above question it is first necessary to determinethe uncorrected shear stress which the spring must experience in order toproduce the prescribed firinn pin velocity. After that it must be determinedwhether the necessary spring deflection caa be obtained with the existingsolid height of the spring.

Finally the curvature correction factor must be determined for the spring andwith it one may obtain the corrected shear stress. If the resulting shearstress is much beyond 200,000 psi it is likely that the spring will take a setand cannot properly perform.

Equation (B-85) together with Eq.(A-49) gives the maximum velocity attained bya spring driven mass:

2mS

e (D-3)

Vmax = 1-31-(-1 + 2e 2

This holds because the spring is made of steel. For a mass ratio M/m.s0.243/0.355 - 0.685 the factor in parenthesis becomes according to Figure B-7approximately 0.943. Substituting this factor as well as the desired velocityof 1200 in/sec in the above one may solve for the required uncorrected shearstress:

1200 x 131Tf = 0.943 - 166,700 psi (D-4)

The required deflection of the spring is determined with the help of Equations(A-46) and (A-la):

D-22

I

f Gd (D-5)I = Gd

With G = 11.5 x 10- 6 and the above Tf one obtains:

f = 0.595 (D-6)

Since the free length equals 1.625 inches and the solid height equals(48 + 2) x 0.018 = 0.900, the spring can be deflected a total of 0.725 inches.Therefore f = 0.595 is well within the range and the free length need not bechanged.

So far it would seem that the spring is perfectly satisfactory. Now considerthat the spring index

C D 0.070,_ 3.888 (D-7)d 0.018

and that therefore the curvature correction factor

K = 4c I + 0.615 = 1.417 (D-8)-4c4 C

The corrected shear stress now becomes:

Tcor = TfK = 236,200 psi

The above becomes marginal when one considers that the shear stress for musicwire should not exceed 200,000 psi by too much.

This high stress means that the spring will take a certain set when assembledthe first time and strain hardening will take place. While this raises theelastic limit somewhat and allows the newly formed spring to do its job if thefree length remains sufficient, it is not at all desirable when the spring isto be repeatedly used.

There was a certain amount of trouble with this sprint during the performanceof the experiment. It was overcome by such brute force means as stretching the

i spring repeatedly.

Since all efforts at designing a better spring failed, it can be safely statedthat 1200 in/sec does not represent a practically attainable firing pin speed.(This design difficulty had its reason in the fact that a single spring typehad to accomodate many firing pins.)

c. Determination of Firing Pin Velocity

The following describes the manner in which the velocity of the firing pin isI measured. Single as well as two probe methods are involved.

g 1. Single Probe Method

[igure D-1O shows the essentials of the single probe method of measuring thefiring pin velocity. The firing pin consists of two reflective surfaces(polished steel) and one nonreflective surface between them. This surface waspainted with optical black. As the reflective surface no.1 passes the probe

I 02

PATH OF FIRING PIN

REFLECTIVE SURFACE 1

REFLECTIVE SINGLE PROBE NONREFLE'TIVE SURFACE

SURFACE 2 C-TICAL DISTANCE

Figure D-1O

Single Probe Method of Measuring Velocity

the output voltage of the instrument rises a predetermined amount and triggersthe counter. As the nonreflective surface passes the probe the voltagedecreases, while the counter continues to count. As reflective surface no.2passes the probe, the voltage rises again, and is used to shut off the counter.

The so called "optical distance" between the two points which cause sufficientvoltage to switch the counter is determined with the help of a speciallymodified micrometer. It has been found that the optical distance is usuallywithin less than one percent of the actual distance between the steps.

The velocity of the firing pin is determined by the time t which passes as thefiring pin travels through this single probe optical distance 1.op

LV . OP (D-9)

t

ll.Two Probe Method

The two probe method of measuring the firing pin velocity is applicable to theshorter firing pins, where a reasonably long optical distance of the typeillustrated in Figure D)-10cannot be realized on the pin itself. (Conversely,to use the two probe method on the longer firing pins would require too longan optical distance,) Figure D-11 gives a schematic of the two probe method.

The firing pin is reflective throughout its length. As it passes probe no.1the output voltage rises and starts the counter. Once the pin has passed probeno.1 the voltage drops but the counter continues. When the pin passes probeno.2 the rising voltage shuts off the counter. Again the optical distance mustbe established by a mechanical measurement of the distance of motion requiredto operate the counter. This distance conforms closely to the center distancebetween the probes. Equation (D-9) is also applicable.

[-24

SI

1140 PROBF

M pOR(E -1 OPTICAL DIST. PROB-E 2

PATH OF FIRIN'K, PIN

I"REFLECTIVE SURFACEI

i Figure D-11

T'o Probe Method of Measuring VelocityIIn both methods the optical distance is ot the order of 0.750 inches, and thevclGcity of the firing pin is measured approximately 1.0 inch from the pointof impact. This general method of measuring the velocity of the firing pin hasbeen checked independently with a peripherally slotted disc rotating at various

I known speeds. Excellent correlation was attained (t1 percent).

d. Test Procedure

i !he tests were rur both at an ambient temperaru, of approximately 70 to 75degrees F and at -40 degrees F. In order to make sure that handling would not%wIrm up the primers, the cartridges were mounted in the cartridge holders andk, oled together to -4-1 degrees F. The number of firings per ,nergy level,velocity and temperature were planned to be 30I. in a few' ca:,es there were only'I :5 firings per identification number, while wlicii there a•w: ;iy doubt concerninr,ý thvresult considerably mIure Itirngs were made. Iigures b-12a and lD-12b show then'amber of test firings fto ambient temperature and -40 degrees respectively.

I gure D-13 gives a typical data sheet from an actual reiu. !'Iis test coversD.D. No.60. It represents an energy level of 16 inch-ounccs at a velocity of

S-90 in/sec gt -40 degrees F. The required mass of the firingw in is".555 x 1J lb-sec2/in, while the actual mass ij 5.5t,7 x 10"Y lb-sec2 /in.Mie average of the optical distance is obtained from four readings as 0.75925

.Inches. (The firing pin is rotated and readings, ire taken 90 degrees apart.)l The required time for the velocity of 600 in/sec , computed to be 0.001265mconds.

I!lie 30 data sets record the time indicated by the,. tiectronic counter, the sprinp& compression and whether the primer fired.

I.)- 25

Ident. Number of Ident. Number of Ident. Number of

No. Firings No. Firings No. Firings

1 3C 37 60 73 --2 30 38 -- 74 303 30 39 30 75 304 30 40 30 76 305 57 41 30 77 306 120 42 30 78 307 118 43 120 79 308 240 44 30 80 309 85 45 -- 81 30

10 90 46 30 82 3011 A0 47 30 83 3012 120 48 30 84 60is 119 49 30 85 3014 210 50 120 86 30IS 360 S1 30 87 3016 330 52 -- 88 3017 480 53 30 89 3018 30 54 3l 90 6019 30 55 30 91 3020 60 56 30 92 3021 30 57 30 93 3022 240 58 3P 94 3023 120 59 -- 95 3024 90 60 30 96 3025 30 61 30 97 3026 30 62 30 98 3027 30 63 30 99 3028 30 64 30 100 3029 120 65 30 101 --

30 60 66 -- 102 3031 30 67 30 103 3032 30 68 30 104 3033 30 69 30 105 3034 30 70 30 106 3035 30 71 30 107 3036 30 72 30 108 --

Figure D-12a

Number of Test Firings at Ambient Temperature

D-26

1 .~

Ident. Number of Ident. Number of Ident. Number of

I No. Firings No. Firings No. Firings

1 25 37 30 73 302 25 38 74 303 25 39 30 75 304 55 40 30 76 30A S 55 41 30 77 306 ',0 42 30 78 307 90 43 120 79 308 00 44 30 80 - -9 60 45 - 81 30

10 3(0 46 3. 82 30i 11 30 47 30 83 30

12 30 48 30 84 3013 210 49 30 85 3014 120 so 120 86 90I1 150 51 30 87 --I 270 52 -- 3017 30 53 Ž0 89 3018 30 54 M0 90 3019 30 55 30 91 30

1 20 10 56 30 92 3021 150 57 30 93 301 22 60 58 30 94 --

23 15U 59 -- 95 3024 120 60 30 96 3025 30 61 30 97 3026 90 62 30 98 3027 90 63 30 99 3028 3() 64 30 100 30I19 150 65 30 101 --330 0 6b - 102 3031 07 30 103 30

62 68 30 104 3033 36 09 30 1 0S 30

i 34 30 70 30 106 30is 60or 71 30 107, 301;6 90 72 3 0 108--

I ~lL61urv D-Il)-

Number of T,•t Firings at -10 )ckgrecs V

i I

1'- 27

Throughout the tests it was made a rule to count only those firings where theactual time did not deviate more than ±2.5 percent from the computed time. (Thestars on the data sheet indicate that the tube was cleaned preceding theparticular firing.)

The 5 in-oz tests were discontinued as it became apparent that it was impossibleto attain reproducible results.

4. 'rest Results

a. Results of Firing Tests

Figure D-14 is a tabulation of the results of all the firing tests at bothambient temperature and at -40 degrees F. It shows the percentage of primersfired at the various energy levels and firing pin velocities. As was statedearlier, the 5 in-oz tests were discontinued since it was not possible toobtain reproducible results.

Figures D-1Sa and D-15b presents the identical data in a graphical form similarto the "Erwood Curves". Figure D-1Sa gives the results for ambient temperaturewhile Figure D-1Sb deals with -40 degrees F.

b. Verification of the Theory Concerning Velocities of Spring Driven Masses

The spring compression associated with each test velocity was recorded for alldata points. (See sample data sheet in Figure D-13.) This makes it possible tocompare the measured nominal velocities (vN) with the theoretical velocities(VT) which correspond to the actual spring compressions.

This procedure, which allows the determination of a design correction factor,will now be outlined by way of an example.

Figure D-13 gives the following test results for I.D. 60-B (-40 0 F):

Firing pin velocity: 600 in/secFiring pin mass: 0.975 gramsSpring compression (f): 0.445 inches

"This sprinW belongs to Group IV (see Figure D-9), and has the followingdimensions:

Outside diameter: 0.120 inchesWire diameter (d): 0.022 inchesMean coil diameter (D): 0.095 inchesNumber of active turns (N): 31Mass of spring (ms): 0,476 grams

The velocity v which corresponds theoretically to the above spring compressionf is now computed with the help of Eq.(D-3):

-2ms

(D-10)f 2vT = vmax = -M(-l + 2e

"The shear stress associated with the deflection f is given by rearrangement of

Lq. (D-S) :

D-28

= . ..... -----------

COLD TLES TEMP. (F) 4-'( '1LSTJrR: _____.

NOMINAL LNERGY (IN-OZ) _/j , NOMINAL VELOCITY (IN/SEC)NOM. MASS (#-SLC /IN) -.'_ I ACTUAL. MA',. 2 6-SEC

ACTUAL MASS (g)j/

* ~~~OPTICAL DI54TANCE (N L=x / 2 ~t L~<sjL2I :TAKEN 9 A NOMINAL 'lAIPA L--/V o= . , / 22 (, j (SEC)

Ho TIME NSPI F SHOT T IME SPRING F NPi NO. (SEC) CO,', NO, (SEC) Comp.

I(IN) (IN)_. . .... ... . ... . ... L

_-._..._ '!~_i~~ -A~ 7.11 i ,,, ' /

13 21

% ... - ... ... - -. .. . .- ,,.. .. , !-i -- --i ... . "LL _ _ _

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I 'r.913

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I !~~~~~~~~~--,_ ,< _,,-st _, a. S... I--- ' ...... C-f ...Ct! :- • ... .,. .. - . ...

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- -- a4 - 4 1 4 ý

m0 ~4)(3000 0 CD 0 0 0 0 0co r,4 00 cm m) 0 0) 0 0 0 0 0

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1 C 00 0 0 0 000

000000000000;

-- 4 -4 -4 -4 -4 -4 -4 - 4 -41 -4 -4 -41 1-4 1-41

LJ* C I 0 U) 0 0f 00 00 00 00 0Cd ~ ~ ! z4 LA 0 LI) 0 '. 0 0 0 0 0 0

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00

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00 CD NL

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40 CD

D- -2

II

Gdf (11.5 x 106) (0.022)(0.445) 128,500 psi (D-)

With 7= D (3.1416)(0.095)2(31)

SN_ = 2.Usm mS

the factor in parenthesis of Eq.(D-10) becomes according to Figure B-7

approximately 0.656.

Substitution of the above into Eq.(D-lO) leads to:

V1, = 128,500 x 0.656 = 643.6 in/sec.

This is larger than the actual velocity of 600 in/sec, The percentage increasewith respect to the nominal velocity is piven by:

percentage increase = VT V N x 100 = 643.6 600 x 100"vN

= 7.26 percent

This means that in order to actually attain a certain nominal velocity (vN),I. the spring deflection must be computed for a theoretical velocity (VT) which• iis higher than the nominal one by a certain percentage. Thus:

VT = vN(I + CD) (D-12)

where the design correction factor is given by:SvT - v N

C = N (D-13)

Figure D-16 gives a tabulation of the correction factors CD for all test runs.Except for I.D. numbers 4 and 5, and whenever the spring took a set, i.e. thespring characteristics changed, the value of the correction factor does notexceed 0.25. In the case of I.D.'s 4 and 5 it was found that high frictionbetween the firing pin and the tube was responsible for the need of increasedspring compression to attain nominal velocity. (The use of a Teflon sprayproved very helpful in reducing friction.) It needs to be pointed out that inthe experimental setup the firing pin traveled as far as 4 inches afterseparation from the spring in order to facilitate the velocity measurement.In an actual situation the firing pin travel would be much shorter and thus thepossibility for speed reducing friction work would be very much smaller.

The spring set can definitely be avoided by limiting the corrected shear stressin the spring to approximately 200,000 psi.

Lastly, the presence of the spring pad (see Figure U-3) contribuLed to theslowing down of the firing pin.

For design purposes one may safely assume the getieral correction factor

CD = 0.25 (D-14)

D)-33

Appendix iE, which deals with the space optimization of spring driven primerstriker systems makes use of the above general correction factor.

D-34

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*- 4 ý go' VON, Z LI) 00- N 00 t - '0 LI) RT N

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I

APPENDIX E

SPACE OPTIMIZATION

The following appendix gives the derivation of the various design equations andoutlines the combination of experimental and theoretical results which leads tothe final optimization methods.

The resulting optimum design tables, which are listed in Appendix F, allow thedesign of 100 percent successful spring striker systems for the M42G primerwith minimum space needs.

1. Free Length of Optimum Spring

The free length of the optimum spring is obtained with the assumption that thisspring is deflected to its solid height and that the spring reaches at solidheight its maximum allowable corrected shear stress.

The deflection (f) may then be defined by:

f = L0 - solid height (I3-i)

where L0 is the free length of the spring.

When one defines solid height as the product of the wire diameter (d) and thesum of the active turns (N) and two extra turns:

solid height = d(N + 2), (F-2)

then Eq.(E-1) becomes:

f - d(N + 2) (E-3)I From the above the free length of the spring is given by:

1I0 = f + d(N +2) (E-4)

As was stated above, the spring is to be designed in .vch a manner that itreaches its maximum allowable corrected sheor stress at solid height. Inaddition its mass inu:st hie sufficient to produce the desired velocity of itsassociated firing pin of mass SI.

When the deflection f is such that it produces the maximum allowable correctedshear stress T at solid height, then

c

-= K_ (--5)

where= uncorrected stress associated with deflection f

fi| i4c - 1 0.615K f - + -c1, the curvature correction factor [4214c -4 C

c = the spring index relating mean cutl diamcter to wire diameter.

Equation (U-5) is now to be expressed in termrr of tile deflection f.According to Eq.(A-45):

F - Ii' ' • -1

8Pf D

" •Td3

I and according to Eq.(A la):

d G4 f!~ Pf = ,f = G4-•

* 8D 3N

Thus Eq.(E-5) becomes:

f= KGd

Sc ND2 7r

* And finally:

* - NnD2

f= c (E-6)i KGd

The mass of the spring, in the sense of Appendix B, is due to the active coils

only:

ig = NCnD) (•d2 (E-7)

and the number of active turns may then be expressed by:

4m g

,T2 yDd 2 (E-8)

Substitution of Eq's.(E-6) and (E-8) into Eq.(E-4) gives for the free length ofthe spring:

L " 4ms g Tr c D 1 + 2d (E-9)0 7T 2~d Dd"

When the above is expressed in terms of the spring index c, it becomes:S4m sgC2 Tc D

ff:: fiL0u-•• + -) +2

0ii2yD2ii c (EIO)

2. Optimum Spring Mass m,

The optimum spring mass is now obtained with the help of the maximum velocityequations (D-10) or (B-85):

E-2

rj 21m

V, 2tf 2e 2

*r - 131(-1 + 2e (l-I)

Of course the theoretical velocity vT must be attained without exceeding thecorrected shear stress rc at solid height. In addition, the theoreticalvelocity(v ) must be expressed in terms of the corrected nominal velocity (vN)according To Eq.(D-12). The latter is necessary since the design procedure isbased on the experimentally attained 100 percent firing points as well as onthe experience with the springs.

Thus Eq.(E-11) is written in the form:

2ms

C (E-12)VN(I + C)I + 2e 2

The required spring mass m. is now obtained from the above:

ms In [-2 In B] (E-13)

,where

131 KvN(l + CD)B - + 0.5 (E-14)

c

Eiquation (E-13) is valid only for certain ranges of the factor B: First, asshown in Appendix B the ratio Ni/ms < 5.69, anid in addition, to obtain apositive value for ms it is necessary that 0.606 < B < 1.000.

Therefore for the usual value of the corrected maximum shear stress 'c=

200,000 psi, and the general value CD = 0.25 (see lq.(D)-14)), the nominalvelocity which may be used in the above computations is restricted to thefollowing limits:

EN(min iK 260 in/sec

and

r -'

LN(maXJK < 1220 in/see

'i. Overall Space Requirement of Optimum Systetm

The overall height 1, of the mass-spring sy.;tem is now determined as thecombined length of the firing pin Lp, the solid height of the spring, and thetotal distance traveled by the mass MI until it. reaches the maximum attainablevelocity which corresponds to the deflectioi r. The latter dimension was givenas Ft = fCf in Eq.(C-13) of Appendix C.

E -3

With this concept one finds:

Lt = fCf + d(N + 2) + Lp (E-15)

Substitution of Eq's.(E-6) and (E-8) leads after rearrangement to:

4msgc 2 ric 1 D +Lt + 1 + 2 L (E-16)tz , -y2 tK- f c p

4. Design Equations and Optimization Procedure

The combination of experimental and theoretical results of the presentinvestigation, which was used to devise an optimization procedure, will nowbe explained in detail.

a. Use of Experimental Data

The design procedure aims to recreate those conditions which led to 100 percentfiring of the M42G primers during testing.

To this end the following successful test points (see also Figures D-2 and D-14in Appendix D) were initially considered:

16 in-oz: I.D.: 32, 39, 46, 53, 60, 67, 74, 81, 88, 95, 10214 in-oz: I.D.: 33, 40, 47, 54, 61, 68, 75, 82, 89, '", 10312 in-oz: I.D.: 41, 48, 55, 62, 69, 76, 83, 90, 97, 10410 in-oz: I.D.: 42, 49, 56, 63, 70, 77, 84, 91, 98, 105

8 in-oz: I.D.: 71, 78, 85, 92, 99, 1066 in-oz: I.D.: 79, 86, 93, 100, 107

b. Space Restrictions and Firing Pin Dimensions

In order to stay within reasonable dimensions for the resulting spring drivenprimer striker systems, it was decided that the diameter of the firing pin(and with that the diameter of the spring) should not exceed 0.375 inches, andthat the overall height of the system, as described by Eq.(E-16) should notexceed 2.000 inches.

Figure E-1 gives various possible firing pin lengths which correspond tofeasible firing pin diameters for the firing pin masses associated with theI.D.'s considered. (See last section.) The material is assumed to be steel(y = 0.283 lbs/in3 ).The following diameters are examined for each I.D. number: 0.375, 0.312, 0.250,0.218, 0.200, 0.187. 0.170. 0.156, 0.140, 0.125, 0.110, 0.100, and 0.093 inches.

Lengths below 0.125 inches were generally excluded.

NOTE: The firing pin is assumed to be a solid cylinder without considerationof the 0 03 inch length of the tip, or any reduced diameter forlocating it within the inside diameter of the spring.

E-4

41 - r ) J ' t\Jrn P

N LL X 0 S~ P. ?1- 00 0 0 0 0 0l 4

:j N 6r -4 m~ 0 Cf NN-' 00n

P-4 In C6- m. C,; Xa' Ia )C-~ n 7 (1 C, P- c L p-4P- Lý4 ~ t~'N -4 o rNM-

N 0 0 C X 0 0, 0n 0 0r 07 0- ?1- 0 0 ~ 0

M M C' (f) .. C'C (% - I . t c .-4 f .4F

o -Zi

aN N N p-4a)rc' Q C D F- 4 C'- -P rj

"0-- C r C C0- (71 Im -4~

0 C,

ol f~l r'l 0r r I o rNc

0*0

a, 0 X p * 0 *0 0 0 C) pl 0 I- 05

00ul)

0 z..J-~~~ 0 4' 2- N -4 t)P'- C4'. % fI- r-~~.o.fr~

c4 CCn 'r "- mt" cc pl a -4~'7 N N 4*n 0 0 0 In *- 0 t * 0 0 *

pnn

* o . .o * o

OC 0 'U

Lr 1 %tN U'% d'%

-~C C' 'P N-

UN N 0 0 0 P- 0I 0 0

COCCO0 00000

N O3L' UN-4 c NO It 0 3D 0 -4 0~ a In1N * 0 0 V% OD N 0 0 0 0 tW.n 0 0oh f0 *H 0 -00Nr"U'00U'0 - 00M0j I Nm0 0 00F-

0 j 4.~~. 0 P4P -. Mf' N4AI N4 4 ( W) --d P-4 .4 ' /

N )- 01 .O-,N- N oil'4~- %-4w .k~ 0M M %r 0 4 10 P- w w 0 0 4.)oF ww

0 0;0 E00 0 - 0

N." ~ ~ ~ N 0

000 F- i0000 ( N %

o- Nr.N '~ w 0 r- -4; PZ W0. ~NC.-j 0, 00 0 c0 90o 0

"04 4 -

0 *0 00 00 09i 0 0000 00 a " 0

w.4 00 0 0 moN 0004. . ..

N -%

p- z N 0N"

0 0 a a 0a 0 0 0 0) -D 0 0 C 0

& -4 0 0 0 0 0 0h

ry CD ~ ~ ~ 0 00 L% C. 0n 01 0, e m

04 P- 0 0t 0- 0 1 W

"4 "4 ~t~P.-"4"4 "4 .O..tNN"4%

00 0 0 0

0 m00000I0m 0

00 0

o o Of%"~~N0~~

"- .

0U 0 M 0 0MwumN0u qw04L0%WM00

0A 0' 0 - 0 MO N

0% 00 cc 0 0

0 *0 C 00 0 0

0 0 0 0 0S0 .0 0 0 6

occ c

-4 4 0~ 10 - ~ -j -4 Cm 0 * (7 10 0q 9* N0s

U. 0I 0% P0 00 6

o4 'a- 4 0 a N.

00 C

1- F- ODcr(10f f-c 7-I -d

.. ........-. ' - '.,m~..r.,,n~ f f ~ r t ~.ff

c. Design EquationsThe design equations used in the following optimization proceedure are now

recapitulated:

The free length of the spring is given by Eq.(E-lO):

4m gc 2 T cTrc 1 DLu ___ C + 2- (E-17)0 " 2yD2 (-k- + -P *

where according to Eq. (E-13):

ms H ! ln(-2 ln B) (E-18)2

and

131 KvN(I + CD)B 2T +0. (E-19)

further

D (E-20)

and

K 4c -1 0.615 (E-21)4c - 4 c

The number of active turns of the spring is given by Eq.(E-8):

4m g

N = s (E-22)w2yDd 2

Finally, the overall height of the system, i.e. the space that must beprovided so that the firing pin can reach its maximum attainable velocityfor which it has been designed, is given by Eq.(E-16):

4msgc 2 *rciC 1 2D L

Lt a -- 1 + 2- + L (E-23)2 i2yD2 c p

where according to Eq.(C-14):

eC -2. + T(-3 T 4e (E-24)

and

Ms (E-25)

Furthermore, the mean coil diameter D of the spring is related to the outsidediameter D0 of the spring and the firing pin by the following relationship:

E-8

D = D + d = G(1 +c

therefore

D D= c (E-26)1 +c

The correct application of these design equations will now be discussed.

d, Optimization Procedure

Both the free length L0 as well as the overall height of the system Lt, asgiven by Eq's.(E-17) and (E-23) respectively are directly as well as indirectlyfunctions of the spring index c. Because of the complexity of this dependence,it is not practical to determine the shortest 10 and Lt for a given conditionanalytically.

It has proven itself comparatively easy to search for that value of the springindex c which results in the shortest Lt under a given set of circumstances.Figure E-2 gives an example of such a computer search.

It was desired to find the shortest overall height L for a system employingI.D. No. 75 which has a nominal velocity vN = 800 in/sec and a firing pin massM = 2.734x10" 6 lb-sec 2/in (refer to Figure D-2 in Appendix D and to Section 4ain the present appendix).

The outside diameter Do of the spring and the firing pin was chosen as 0.125inches, and Figure E-l indicated that the length of the firing pin had to be0.304 inches in this case. The program was written to evaluate Eq's.(E-17),(E-18), (E-22) and (E-23) for LO, msi, N and Lt respectively in addition tocomputing the wire diameter d.

The above equations were computed for 3.4 < c < 6.00. The shortest overallsystem can be found from the printout to correspond to c = 4.2 with a lengthof Lt = 1.713. The associated wire diameter is 0.02404 inches, the number ofactive turns are 30.1, and the free length of the spring is 1.277 inches.

The following additional parameters were used in the program:

Y = 0.283 lb/in3 (density of steel)CD = 0.25, design correction factor (see Eq.(D-14))Tc = 200,000 psi, corrected maximum allowable shear stress for the music

wire of the s ringsg - 386.05 in/sec, acceleration of gravityG - 11.5x10" 6 , shear modulus of steel

Similar optimizations have been performed for all I.D. numbers and firing pinoutside diameters indicated in Figure E-1.

Finally, all optimum possibilities with overall lengths less than 2,000 inches,"which correspond to a certain I.D. number, were combined on one output sheet.Figure E-3 shows such a typical optimum design table for I.D. 75, (the same asused above). It now includes the least overall heights for such outside diametersas 0.200, 0.187, 0.170, 0.156, 0.140 and 0.125 inches.

E-9

c' _.

> ~0 '

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E-11

S-=--= -=-,

(iWhen comparing Figures E-2 and E-3 it can be seen that Figure E-3 showsfor an outside diameter of 0.125 inches the values which follow directly belowthe minimum value in Figure E-2. Thus Figure E-2 gives a minimum overall heightof 1.713 inches, while Figure E-3 gives 1.717 as the minimum. This discrepencyhas its reason in the needs of the final program. The error is of courseinsignificant.)

Figure E-3 also lists all other necessary design information such as: wirediameter, number of active turns, free length of spring, as well as its solidheight, (which takes two inactive turns into account).

In addition, the spring mass is printed out for checking purposes. (The valueof the spring masses are very close for all designs for a given I.D. numbersince the spring index at optimum is essentially identical for most. (SeeEq.(E-18) for dependency on c.))

Appendix F gives all optimum design tables. Their information furnishes allthose systems which are smaller than 2.000 inches in overall height, and assure100 percent firing for the M42G primer.

E-12

m m~

APPENDIX F

OPTIMUM DESIGN TABLES

The present appendix lists approximately 215 spring striker systems withoverall length of less than 2.000 inches and outside diameters of less than0.375 inches. (See Appendix E for background,) All systems correspond toactual test points and 100 percent firing is assured in all cases. (Note theI.D. numbers as well as their associated energy levels, firing pin masses, andnominal firing pin velocities [vN].) As in Figure E-3 of Appendix E thefollowing complete design information is given:

1. Outside diameter of the system2. Length of firing pin (assumed to be a solid cylinder without consideration

of hemispherical tip of 0.032 inches)3. Mass of the spring4. Wire diameter of the spring5. Active number of turns6. Free length of spring (based on assumption of two inactive turns turns of

spring)7. Solid height of spring (again including two inactive turns)8. Expanded length of spring at maximum velocity of firing pin9. Overall height of system

Similar to the tabulations of Figures E-2 and C3-3 in Appendix I', the followingvalues were used in the computations:

y = 0.283 lb/in3 (density of steel music wire)CD = 0.25, design correction factor for velocity (see Eq.{(-14) in Appendix D)Tc f 200,000 psi, corrected maximum shear stress of spring at solid heightg = 386.05 in/sec2, acceleration of gravityG = 1l.5x10 6 psi, shear modulus of steel

The computer program, (see sample printout below), was written in MAD (MichiganAlgorithm Decoder), and all computations were made on an IBM 7040 computer atthe City College Computation Center of the City University of New York.

F-i -

a. Computer Program

SPRING OPTIMIZATION

C = MEAN COIL DIAMETER/WIRF DIAMETERD = MEAN COIL DIAMETERDELCY = INCRFMENT Of CFTT = LENGTH OF EXPANi0TD SPRING AT INSTANT OF MASS SFPARATIONK = STRESS CONCENTRATIO)N OF SPRIAJGLZERO = FREE LENGTH OF SPRINGM = MASS OF FIRING PI:NMS = MASS OF SPRINGNACT = NUMBPER [IF ACTIVE COILSNTOT = TOTAl. NUMBFR OF TURNS01) z OUTER DIAMETER OF SPRINGOVRALL = OVERALL REDUIRED LENGTHPINL = LENGTH OF FIRING PINJSMALLI) = WIRE OIAmJTERSOLH = SOLL) HEIGHTVEL = EXPERIMENTAL VELOCITY

INTEGER INOZ, ID, IDIEXECUTE NOHI).READ t)ATAME6 = M*I.OF6

THE LAST DATA CARD tMIUST BE J = I-THIS IS A SIGNAL TO THEPROGRAM rl4AT ALL THE I)ATA HAS BEEN REAL)

INTEGER JWHENEVER J .E. IPRINT FORMAT N[ITLSIPRINT FORMAT NOTFS2PRINT FORMAT N()TFS3TRANSFER TO STARTEND OF CONDITIONALWHENEVER (ID .NE. II) .Ar4l). (I1)1 .NF. U)

PRINT FORMAT NOTESIPRINT FORMAT NOTES2PRINT FORMAT NUTES3

END OiF C{JNtI) TIONALWHENEVER ID .NF. IDI

PRINT FOR4MAT H)ATAl, INIOZ,IDVELMEbPRINT FORMAT HDNGIPRINT FOR4AT Ho)tqG2PRINT FORMAT HI)NG3

END OF CUNUITIDNALVECTOR VALUFS NUTESI- $Ilt ///IH II,*TOTAL TUR'JS 1F SPRING ARE OBTAINED BY

I ADDING 2 TURNS*/IH H*TO NUMBER (IF ACTIVE TURNS4**$VECTOR VALuES NOTES2 a $1HOH*ALL UNITS OF LENGTH ARE IN INCHES*/IH)HJAL

I L UNITS OF MASS ARE IN Lg-a,E*SEC/IN*E6J/IHoH*ALL UNITS OF VEI LOCITY ARE IN INlSEC**$

VECTGR VALUES NUTES3 = $IHOH*OCVERALL HEIHT OF SYSTEM IS OBTAINF.D BY AD

F-2

•'• '• .. -• •; :• -• ' = i'' ... " • • : . ..: " ' " u = I • ..• : ... , : '• . ,; ... ........ ..

I DlING EXPANDFI) LF43TH OF SIPR IN(;*/ 114 tH*To LFNc3- RI (IFI Pf N"* i

VECTOR VALUES DVATA1 = ý11QH*1U'4I-( '*I29S1U0lI*IU N0l. = *13/11H U*VtLOC"

1 ITY =*1ý6.LS3,FBF3 IN MASS= F.*VECTOR VALUES HDNIGI = IH0H*0.l).*S3,li*P)TN*S4,Hl*SPRING;*S3,H*WIRE*S,3,

I H*ACT !Vl*S3,H*FlU(:F*S3,H*gLXPANOFI)4'i)S4 t1l*JVr-ALL*S4t2 H*SOLID**$VECTOR VALUES HONG2 =SIH 6,H1*LF-4GTi*,S3,H*IMASS*S4,H*DIAM*S3,H*TURNS*

VECTOR VALUES HDN4G3 = $11 S46,14*OP SPRINý*S2,H*Of SY~fEM*S2tH*OF SPRING

FTT =lU).

FTTP 200.THR~OUGH GAMMA, FUR LCAT D IFLCY, C .G. CFI,4WHENEVFR FTTP -L. FTI

PRINT FORMAT ANS, )00, Pfl4L, MSE(1, SM-ALLI), NACI, LZEFR0, FTTP, OVKALL,I SOLH

TRANSFER If) W'XTEND flF CONDI)ITIONALii VFCTOR VALUES AN', = tH F4.3,S3,F.4.3,S3,F6.',,S2,Ib.'J,Sl,F5.1,S3,F6.3,

I S33F6.3,S6,F6.3t`;6tF-4d,3*¶FTTP = FTFV =VFL*I.2'i1) -(OD*C)/(1.() + C)K =(4.0'*C - 1.U,)/(4.0*C - 4.0) + 0.615/C

HS -O. ?~4*ELLG.(-?.0*ýLr)G,.(V*I(/1053.435 + (0.5))MSE6 =MS*1.0[6ALPHA VS/M

KF =2.0 + I 1.0/ALPH4)*( 1.0 - 0.5*EXlP.(-?.Uo*ALPHA)

+F ? +1.0/C)

SMALLD z D/CNACT =552.8eh379tVS/ W*oSNMALL.UO SMALLI)NTOT =NACT + 2.0LZERO 5(2.863?*LCk'C*MS )/((0*D)( 0(Oll4lj3'3' 1/KI+ 1.o/C) + 0*k,

OVRALL FTT PINLSOLH =SMALLI)*NTOrCONTINOE

TRANSFE'R TO !0ARI

END OF PROGRAM

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