1/ R-TR-77-017

A MATHEMATICAL MODEL

OF THE 30 MM ADVANCED

MEDIUM CALIBER WEAPON SYSTEM

(AMCAWS-30)

MICHAEL R. KANE

APRIL 1977

FINAL REPORT \.

SMALL CALIBER WEAPONS SYSTEMS LABORATORY

Distribution Statement.

Approved for public release; distribution unlimited.

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Disclaimer:

The findings of this report are not to be construed as an officialDepartment of the Army position, unless so designated by otherauthorized documents.

DISPOSITION INSTRUCTIONS:

Destroy this report when it is no longer needed.

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REPORT DOCUMENTATION PAGE READ INSTRUCTIONSR BEFORE COMPLETING FORM

1. REPORT NUMBER 2. GOVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

R-TR-77-017TTTLE (n Subt,,itle TE o1 COVERED

A CATHEt&TiCAL ODEL FTE MDVN Final ' p,--ASI) i 6. PERFORMING ORG. REPORT NUMBERZ ABCAWS-:d 30" e .... _

8. CONTRACT OR GRANT NUMBER(s)

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Commander , Rock Island Arsenal'/ _ARE & WRK NIT NUMBERS

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1I. CONTROLLING OFFICE NAME AND ADDRESS

Sameas 9April 77a 'I-NUMBEROF PAGES

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16. DISTRIBUTION STATEMENT (of this Report)

Approved ase; Distribution Unlimited.

17. DIS o btract entered In Block 20, If different from Report)

18. SUPPLEMENTARY NOTES

Report fulfills DARCOM thesis requirement for the author's DARCOM sponsored

SI. long-term CAD-E training at the University of Michigan.

19. KEY WORDS (Continue on reverse side If necessary and Identiy by block number)

1. High impulse 6. Computer program

2. Externally powered 7. Weapon model3. Mathematical model 8. FORTRAN

4. Numerical integration- 5. d'Alembert force method

2 , APTsRACT (tCrrou am roveros ela Itf neceseafY 4Kid tdentfytr b, block number)

- A mathematical model for the AMCAWS-3OMM weapon is ieveloped using the gen-

eralized d'Alembert force equations. The development cf the one degree offreedom differential equation of motion for the weapon is shown. The equationaccounts for operations including feed, eject, chamber locking, round crush--up,

chamber translation, face cam rotation, and drum cam rotation. The resultantequation is numerically integrated to obtain the time response of position,

cvelocity, acceleration, and force for the major components. The solution isbased on the known drive ,motor characteristics.

D FOR" 1473 EDITION OF I NOV 65 IS OBSOLETE UEH CR Y UNCLASSIFIED

SEUIYCASFCTO FTHSPG We aaEtrd

UNCLASSIFIEDSECUR, ,. _ CLASSIFICATION OF HIS PAGE(. he, Data .nt._ed)

'The numerical integration is done using the HPCG subroutine out of the IBMSSP Math Library. The total program is modulaiized and inclusions of addition-al parts or design changes in the weapon can be incorporatt without extensiverevision of the program. The program is written in FORTRAN.

o4'

ii UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE(nhen Data Pnearod)

A MATHEMATICAL MODEL OF THE 30 MMADVANCED MEDIUM CALIBER WEAPON SYSTEM

(AMCAWS-30)*

by

Michael R. Kane

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ACKNOWLEDGMENT

This work was undertaken to meet the DARCOM project requirementwhile the author was engaged in DARCOM sponsored long term CAD-Etraining at the University of Michigan. The project was completedafter the author returned to the advanced concepts group of Gen.T. J. Rodman Laboratory at Rock Island srk. .

Prof. M. A. Chace (Chairman, CAD-E Department) of the Universityof Michigan was the faculty advisor for the project. His guidancewas invaluable.

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ABSTRACT

A mathematical model for the AMCAWS-3OMM weapon is developed using the

generalized d'Alembert force equations. The development of the one degree

of freedom differential equation of motion for the weapon is shown. The

equation accounts for operations including feed, eject, chamber locking,

round crush-up, chamber translation, face cam rotatio., and drum cam

rotation. The resultant equation is numerically integrated to obtain the

time response of position, velocity, acceleration, and force for the

major components. The solution is based on the known drive motor character-

istics.

The numerical integration is done using the HPCG subroutine out of the

IBM SSP Math Library. The total program is modularized and inclusions of

additional parts or design changes in the weapon can be incorporated without

-Atensive revision of the program. The program is written in FORTRAN.

W T-ne AMCAWS-3OMM weapon is currently under development in the AdvancedConcepts Group, Aircraft and Air Defense Weapons Systems Directorate,General Thomas J. RPoman-Labomt yRock Island, Ill. (SARRI-LW-A)..CAWSVT s -530 mifineter single barrel weapon that utilizes an aluminumcased, fully telescoped, and consolidated propellant round withballistic characteristics slightly better than GAU-8 rounds. Therototype weapon fires ten round bursts at a nominal rate of 120 spm.he second prototype weapon has a nominal rate in excess of 400 spm,

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TABLE OF CONTENTS

SECTION PAGE

1.0 OBJECTIVE 1

2.0 AMMUNITION/WEAPON DESCRIPTION 3

3.0 GENERALIZED d'ALEMBERT h)RCE METHOD 31

4.0 MATHEMATICAL MODEL FOR AMCAWS-30 38

5.0 MODEL INPUT 44

6.0 CONCLUSIONS 57

REFERENCES

A-I FUNCTIONAL RELATIONSHIPS COMPUTER PROGRAM (FRCP)

A-2 GENERALIZED dALEMBERT FORCE PROCEDURE AND AMCAWS-30

EQUATION OF MOTION DEVELOPMENT

A-3 PROGRAM LISTING, FRCP

A-4 PROGRAM LISTING, DYNAMIC MODEL

A-5 PROGRAM LISTING, DATA SMOOTHING

DISTRIBUTION LIST

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No. Figure Page

ANCAWS-30 Cutaway Round ... .......... 4

2 Drum Cam Assembly and Drive Train ... ....... 5

3 Internal Drum Cam Campath .... ......... 6

4 AMCAWS Assembly Drawing 73F40101-3 ...... . 8

5 AMCAWS Assembly Drawing 73F40101- . . .... 9

6 Face Cam-Eject Side ...... ............ 10

7 AMCAWS Assembly Drawing 73F40101-4 . ...... 11

8 Feed Mechanism ..... ............. 12

9 Eject Mechanism .... ............ 13

10 AMCAWS Assembly Drawing 73D40098 . ....... 15

11 AMCAWS Assembly Drawing 73D40101-2 . ...... 16

12 AMCAWS Assembly Drawing 73D40099 . ....... 18

13 AMCAWS Assembly Drawing ... .......... 19

14 Round Function Sequence ... .......... 20

15 Ready to Fire Assembly Drawing ... ........ 21

16 Weapon Showing Follower Stud .. ........ 23

17 Receiver Drawing 73F40270-2 .. ........ 24

18 Buffer Assembly Drawing 72C40288 . ....... 26

19 Mount Assembly Drawing 73F40007. ....... 27

20 Gear Drive Assembly Drawing 74F0066 . ...... 29

: 21 Input Motor Torque. . ...... .. ... 30

22 Simple Pendulum ..... ............. 34

23 Cam Contact "Pinned Linkage" .......... 41

24 Dynamic Program Block Diagram .. ........ 44

( 25 FCT Block Diagram ................. 45

No. Figure Page

26 OUTP Block Diagram ... .......... 46

27 Drive Motor Position vs Time .. ....... 48

28 Drive Motor Velocity vs Time .. ....... 49

29 Drive Motor Acceleration vs Time . ...... 50

30 Smoothed Feed Data . .. .......... 52

31 Smoothed Eject Data .... .......... 53

32 Smoothed Lock Data .............. 54

33 Drum Cam Data ............... 55

34 Weapon Timing . ........... . 56

Al-I FRCP Block Diagram ..... .......... Al-2

AI-2 Drum Cam Angle vs Input Angle ... ....... Al-3

A1-3 Face Cam Angle vs Input Angle ... ....... AI-4

Al-4 Feed Pawl Angle vs Input Angle ... ...... A-5

Al-5 Eject Pawl Angle vs Input Angle ... ...... AI-6

AI-6 Lock Ring Angle vs Input Angle . ..... . AI-7

AI-7 Chamber Displacement vs Input Angle .. ..... AI-8

Al-8 Graph of Chamber Drawing Data. . ..... AI-9

Al-9 Graph of Eject Drawing Data ... ....... Al-9

Al-10 Graph of Feed Drawing Data ... ........ Al-lI

Al-il Graph of Clevis Model Data ... ........ AI-12

Al-12 Feed Mechanism ...... ............ Al-13

AI-13 Eject Mechanism . ..... . .. ....... Al-14

Al-14 Complete Weapon Timing ... ......... Al-15

No. Figure Page

A2-1 AMCAWS-30 Black Box ............. A2-3

A2-2 Drive Coordinates ............. A2-5

A2-3 Drum/Face Cam Coordinates .......... A2-8

A2-4 Feed Coordinates .... ........... A2-11

A2-5 Eject Coordinates ... .......... . A2-15

A2-6 Lock Ring Coordinates .. ......... A2-18

A2-7 Chamber Coordinates ... .......... A2-21

A2-8 General Translating Mass .. ........ A2-23

Table

1 1 Simple Pendulum 34

A2-1 Mass a'ca Inertia A2-26

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1. OBJECTIVE

The "first prototype AMCAWS-30 weapon capable of operation in an automatic

mode has existed since 2 May 74. That prototype has since shown a high degree

of reliability in many ten round bursts and other lesser firing schedules.

Those first automatic firings were the result of a design process historic-

ally similar to the design of other medium caliber weapons. A new concept,

approach, or need leads the designer to develop the design, using tools

generally available to the draftsman (assemblies, sections, blowups, etc.).

Parts are sized by a coarse static force analysis or by the intuition of the

designer. Operational or dynamic forces are not investigated extensively.

The drive motor, for example, on the AMCAWS was chosen because it was available

and it was felt that it was "big enough". Throughout the complete design cycle

including the initial layouts, a few part redesigns, and a very circumspect

assembly there were several questions that begged answers. The questions

included:

(1) What is the complete position description of all the major

components during a firing cycle?

(2) *'hat -,s the response of the weapon as a whole to different

'4 drive motors?(3) What are the forces between parts during weapon operation?

(4) What is the effect on weapon performance when parts are redesigned?"aThe AMCAWS.-30 mathematical model provides the ability to answer these

questions. The model is one degree of freedom and accounts for inertial,

translational, and dissipative forces. Modeled components include the feed,

2

eject, chamber, lock, and the drum/face cams. The model employs generalized

d'Alembert forces to develop the differential equation of motion and uses

functional relationships developed by a preliminary program to establish a

component's positional dependence on a single coordinate, the motor input

angle.

The purpose of this report is threefold. First, it is intended to document

the computer programs that make up the AMCAWS mathematical model package. The

programs are discussed and listed in the appendices and contain excellent

internal documentation. Second, the report is to describe the operational

characteristics of the weapon. A brief ammunition description is in the

next section and a detailed ammunition description can be found in the

ammunition final report (1) Third, the report will document the actual

modeling procedure.

The model package discussed in this report treats only the first proto-

type weapon. The procedure employed, is, however, general and can be

applied to a wide variety of weapon systems and subsystems.

*,umbers in parentheses designate References at end of paper.

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2. AMMUNITION WEAPON DESCRIPTION

The AMCAWS 30 weapon is the result of an advanced development program for

a high performance 30mm automatic weapon system. The weapon has been designed,

developed and fabricated in-house at Rock Island Arsenal. The weapon is exter-

nally powered and various cams accomplish the feeding, firing, and ejecting of

the round. The weapon treated by this report is the first prototype weapon

which fires up to 'en round bursts at a 121 shots per minute (spm) rate.

Parenthetically, the second prototype weapon has a design rate of 500 spm.

The second prototype is, as of the date of this report, some 90% fabricated.

While the detailed description of the first prototype weapon may seem long

and involved, the weapon itself can be characterized as clean and simple.

Various cams ensure positive motions and the lateral feeding and ejecting of

ammunition permits the absence of some of the more complicated extracting

mechanisms used on more conventional weapons.

2.1 AMMUNITION

The AMCAWS 30 anmunition (Figure 1) is an aluminum cased and fully

telescoped round. The case is one piece and has a .75 degree radial taper

to the base. The main charge is consolidated and concentric about the

projectile. The full weight of the round is about 9595 rains (1.37 pounds).

Extrartion forces after firing are very low, A complete ammunition

description can be found in the amimunition final r',ort (1).

2.2 DRUM CAM

The drum cam assembly (Figure 2) has several functions. Torque .from the

drive unit is transmitted to the weapon through the 151 tooth external

gear on the outside diameter of the drum. An internal cam path (Figure 3)

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controls the chamber motion of the weapon. A follower stud cantilevered

from the chamber and passing through a receiver slot follows the drum cam

path, thus providing the proper chamber motion. A lump can. fixed to the

inside diameter of the drum initiates the lock and unlock sequence

(Figure 4). The drum cam is concentric to the weapon centerline

(Figure 5) and is located over the rear half of the receiver. The face

cam is locked to the front of the drum cam at weapon assembly (Figure 4).

2.3 FACE CAM

The face cam (Figure 6,7) has cam paths that control the feed and eject

functions for the weapon. The face cam is fixed to the drum cam at weapon

assembly and is thus timed to the chamber and locking motions. The feed

cam path (Figure Al-io) is followed by the feed rocker arm which transmits

rotation to the feeding pawls through the feed shaft. The feed shaft is

the center of rotation of the rocker and the pawls and is fixed via supports

to the receiver. The total feed mechanism (Figure 8) places a new round of

ammunition at the center of the chamber. While the new round is being placed,

the fired case is being ejected from the system. The eject campath (Figure

. I-9 1 is followed by the eject rocker arm which transmits rotation to theeject pawl through the eject shaft. The center of rotation of this rocker

and pawl is the eject shaft which is fixed to the receiver. The total

eject mechanism (Figure 9) must allow the fired round to escape the chamber

centerline. This is accomplished by waiting until the chamber is fully

rearward and swinging the pawl up into an exposed chamber area so that the

fired round can pass under the pawl, As the fired round is passing under

the pawl the pawl moves down into a dwell position that will cause a positive

stop for the new round being presented. After the stop is made the pawl

swings out of the chamber area so that the chamber can be brought forward for a

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2.4 LOCKING/SEAR ASSEMBLIES

As the cam drum rotates, the lump cam, located on the cam drum inside

diameter (Figure 4) contacts the actuator. The unidirectional rotary

motion of the cam drum is changed to a bidirectional oscillating motion of

the actuator by their cam/follower relationship. The motion (Figure Al-13)

is controlled by the profile of the contact surface of the actuator. As

the lump cam moves along the actuator, it forces (cams) the actuator to rotate

about its own pivot point. As the actuator rotates, it in turn rotates

the lock ring to its locked position via a set of gear teeth on the actuator

and mating teeth on the lock ring. Once in the locked position, the lump

cam rides on the dwell portion of the actator profile. Since the lump cam

and actuator are in constant contact during this period no motion can occur,

thus positive locking during the total lock dwell results. As the lump cam

moves along the actuator, it contacts the unlock portion of the actuator

profile. The actuator is forced (cammed) back to its original position and

through the gear teeth it returns the lock ring to its original unlocked

position.

Located on the inside diameter of the lock ring is a small cam path

(Figure 10). Riding on this small cam path is a bean-like object called a

sear extension (Figure 11). As the lock ring is rotated, the small cam

path lifts the sear extension. When the lock ring reaches its fully locked

position, the sear extension is at the top of the small cam path and just

enough lift has occurred so that the sear extension releases the weapon's

main sear in the bolt assembly. If the lock ring does not turn to full lock,

the sear extension cannot reach full lift, This feature prevents the gun

from being fired in any position other than the desired full lock position.

Since the lock ring is concentric about the longitudinal axis of the gun,

the load on the receiver is also concentric about this axis, and therefore

induces no bending moments to the receiver.

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2.5 BOLT

The bolt assembly is pictured in Figure 12. The firing pin is contained

in the bolt housing and is driven by a spring with a rate (at assembled

height) of about seven pounds per inch. The firing pin is held in a ready-to-

fire position by the sear extending through the sear hole in the bolt housing.

The firing pin is seared off as discussed in the Locking/Sear explanation.

The firing pin can only be released from its ready-to-fire position when the

weapon is fully locked. The bolt has about one-half inch of total travel

relative to the receiver along the centerline of the weapon. In operation,

the firing pin is seared off by the action of fully locking the lock ring.

The firing pin drives forward and discharges the round. A fixed time elapses

that allows the projectile to exit the barrel and the high pressure exhaust

gases to bleed off. The lock ring unlocks and the chamber begins to move

back. The fired round's case, the chamber assembly, and the bolt assembly

move rearward as a unit for about the half inch indicated in Figure 11 and 13.

At this point the bolt housing impacts the backplate and the force of this

impact frees the round case from the chamber wall. The chamber assembly

continues to move over the now stationary bolt assembly, thus exposing the

fired round (Figure 14). The chamber-lock ring assembly moves rearward so

far as to interfere with the bolt crosspin (Figure 12) and move it from the

fully forward position (because the firin- pin haq been seared off) to its

3. fully rearward position (as shown). In the fully rearward position, thefiring spring is recompressed and the sear drops into the sear hole. The

chamber is also fully rearward and load/eject operations take place. The

bolt is ready for another cycle. The shoulder of the chamber interferes

with the shoulder on the bolt head to bring the bolt away from the baseplate

(Figure 13) and secure the round for the next firing. Figure 15 shows a

round and all the components in a now ready to fire position.

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2.6 CHAMBER

The chamber is pictured in Figure 10 with the lock ring in place. The

chamber is translated by the action of the drum cam path through the follower

stud. The stud is cantilevered from the chamber to the drum cam path. The

stud is constrained to move within the receiver slot shown in Figure 16.

In the automatic weapon assembly the handle seen in Figure 16 is replaced

by the follower stud.

The follower stud is cantilevered as shown in Figure 4. The duckbills of

Figure 10 are essentially extensions of the top and bottom of the chamber

and serve to keep the round very close to the weapon's centerline during

feed and eject. The matching tapers on the chamber and round cause line to

line fit of the round and the chamber duripg the lockup phase of Figure 14.

The chamber provides the support during the peak pressures, although the round

itself performs all obturation functions (1). Figures 13 and 14 show the

chamber's relationships to the other components just prior to firing.

2.7 RECEIVER

The receiver (Figure 17) essentially acts as a housing for the chamber

and bolt assemblies. The front end of the receiver has a set of lugs which

h-hold the barrel, The rear of the receiver has flanges cut that match the

lock ring lugs (Figure 10) and against which the lock ring lugs seat during

firing. In the unlocked position the lock ring lugs (thus the chamber) can

move down the receiver, but the locking operation turns the lock ring 150,

which lines up the lock ring lugs and the corresponding receiver flanges.

The receiver is slotted (Figure 16 and 17) to provide a guide for the

chamber-to-drum cam path follower stud. The forward area of the receiver,

near the barrel lugs, is cut to allow brackets that connect to the buffer

packs. The receiver window, just to the rear of the barrel lugs, is cut

to allow the ammunition to be fed and ejected.

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2.8 BUFFERS

The buffer assembly drawing (Figure 3) shois one of the two identical buffers.

The buffers are mcunted on the top and bottom of the weapon, as can be seen in

Figure 19. The ring seen in Figure 16 onto which the front end of the buffer

packs fasten clears the barrel and allows the barrel to move up and back within

it. The ring has trunion mounts on both sides which allows the gun to fasten

onto its fir;ng platform. As can be seen from the assembly drawing (Figure 18)

the buffers have the same preload and spring-rate in both recoil and counter-re-

coil. These two spring factors are variable simply by changing the belville

spring packs. Currently the weapon has nineteen sets of double springs in

each buffer pack which gives a preload and rate per buffer of 3500 pounds and

10,700 pounds per inch respectively, although Figure 18 indicates triple

spring sets in an elongated buffer.

Upon firing, a peak force of up to 200,000 pounds is developed against

the flange lugs on the receiver. This force (more properly, the firing

impulse) is transmitted through the receiver to the buffers which lengthen

to absorb the recoil energy of tie firing. The buffer begins to counter

recoil and eventually damps out prior to the next shot. Typically recoil

is (for the preload and rate discussed) 1.20 inch and the counterrecoil is

.75 inch. The system is completely damped in .120 seconds or 1/5 of a cycle

at 121 spm.

2.9 BARREL

The barrel joins the receiver with a set of lugs as shown in Figure 15.

Mechanically the barrel acts only as a recoiling mass although rifling

torques (2) might be significant in a multiple degree of freedom model. Note

the cut away portion of the barrel at the muzzle end in Figure 13. This gives

clearance to the chamber duckbills (Figure 10) during the chamber forward

portion of the firing cycle.

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2.10 DRIVE TRAIN

The gear drive assembly (Figures 2 and 20) allows four final shots per

minute rates, 90, 121, 18'i, and 242. The prototype weapon is limited to

121 spm because there is insufficient lock time (time necessary to sear

off the firing pin, obtain projectile exit, and bleed of the high-pressure

gasses) designed into tie drum cam path to allow the higher rates. At

321 spm the 59 tooth gear from the motor drives a 120 tooth pickup gear.

The 16 tooth pinion gear which is part of the shaft for the 120 tooth gear

then drives the 151 tooth gear on the outside diameter of the drum cam

(Figure 2). The torque-speed curve for the motor is shown in Figure 21.

This curve is from data supplied by Aeronutronic-Ford. The drive motor

itself is from an XM-140 system. Since a firing occurs once every 3600

rotation of the drum cam, a quick calculation indicates in the 121 spm

configuration a firing cycle completes every 69100 rotation of the 59 tooth

motor gear. The 59 tooth gear angle is the input angle for the mathematical

model developed. Zero degrees input angle is defined so that the drum cam

is also at zero degrees (as a reference, The weapon actually fires whEn

the drum cam is at about 330 rotation, or 541' rotation of the 59 tooth

[ I gear).

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3. GENERALIZED dALEMBERT FORCE METHOD

Obtaining the differential equation of motion for a dynamic system is

obviously one of the important things that must be done in order to achieve

the position description (the solution) of the system. Utilization of

d'Alembert's principal (F-MN=O) and virtual work arguments allows a derivation

of the generalized d'Alembert equation, with constraints (5). The equation

N E L o (3.1)

j=l aqi k=l aqi

explained more fully in Appendix 2 and Table 3.1, allows the methodical

generation of the differential equation of motion. The d'Alembert equation

as expressed in Eq. (3.1) handles, for a generalized coordinate set and any

degree of freedom, external forces applied to the system, d'Alembert forces,

and closed loop constraints. The formulation is not limited to linear or

"linearized" motion, in fact, DRAM (6) is based on Eq. (3.1) and the DRAM

program development is partly based on a need for a general program to

facilitate computer aided design of large, linear or non-linear, displacement

systems of the type found in most machines.

The d'Alembert equation (3.1) reduces toM aM i = 0 (3.2)

' I for the AMCAWS 30 system (Appendix 2). The AMCAWS system is somewhat simple,having only one degree of freedom. Representation of the AMCAWS system as

single degree of freedom is achieved by the representation of the various

weapon cams as motion generators. Another simplifying factor is that AMCAWS

is essentially a two dimensional system.

Since there are over twenty effective forces that must be considered, an

explanation of the procedure used to obtain the differential equation of motion

33

using the AMCAWS weapon as an example would be unnecessarily detailed. The

example chosen for illustration is the simple pendulum shown in Figure 22.

The pendulum is a one degree of freedom system in two dimensions.

Eq. (3.2) holds and using the terms indicated in Figure 22 Eq. (3.2) can be

written

3 +P *.D0 o (3.3)j1 ao 2

There are three effective forces 'rotational and translational d'Alembert forces and

gravity) acting on the pendulum bar, hence j - 1,2,3. aj is a vector from come point4.

in ground to the point of application of the F being considered. 02 is the angle of

the bar measured as indicated in the figure and is, of course, the degree of freedom.

The blow-by-blow procedure of determining the differential equation of motion is a

relatively straightforward application of vector analysis.

For J=l, is the rotational d'Alembert force -120 2 k.

F -1 0 k (3.4)

is the second time derivative of 0 (angular acceleration in two dimensions). k22

is the unit vector about which the rotation takes place and is oefiacI in FIVAre 2 2

(all the coordinates in this report are right-handed). 12 is the moment of inertia of

part 2 about the center of rotation, point 0. Note that part 1 is, by convention,

ground.

The point of application from some point in ground to the point of application of F1o+

is a with

4. - 0a, const+ 2 k (3.5)

cl-i

CNJ

t~ N 11w~ w LU C zC

(UN NCjLD c /

LL. : _ DLL) L)

D tri tN

LU __N OcD

C-

-U I- I

42 14tCD

35

The partial derivative with respect to e2 is-ai (3.6)

-~k.

2

The dot product is then

aa .. (3.7)FI • I = _I**

for 1=2, F2 is the translational d'Alembert force -nip2 c P2c is the second time

derivative of a vector from a point in ground to the center of mass of the bar.

Since

const + P (3.8)P 2c 2c

+ 4

2c is identically r2c. The point of application of F2 from the ground point is a2,

with

4 4- 4a2 const + r2c (3.9)

and aa 2 , (3.10):[. . a2 - 2c"

r must be expanded, eventually into i and j components. In two dimensions,, r2c

S r r - 2-2c02 A .2+rc r 2c+ 2 c (k x r2 c) -02 r2c + 02 (k x r2c) (3.11)

Since the bar is of fixed length there is no change in length in time and

Sr~ 0 (3.12)

* r2c c 0 r = k r2c) (k x where is inextensiablea 2 2 2 02 2c 2c

and moves entirely in a plane.

36

yielding for Eq. (3.11)

r2c 02 (kx r 2 c) - Or 2 c* (3.13)

Thus

2 -m2 { 2 (k x r) - r(3.14)

and the dot product becomes-). , a a2 2F - _ -mr 2"

(3.15)2 H2 2c2'

2

for 3, F3 is the gravity force. The gravity field is considered to act at the mass

center with

3 = -mgj. (3.16)

The point of application (a3) is vector a2 . The dot product for j=3 is then+ -aa3 . aa2

i F3 O 3 =0 (-mg j) .(k x rg2c =-r2eMg sin O (3.17)2 2

The sum of the right hand sides of equations (3.7), (3.15), and (3.17), when set to

zero, is the differential equation of motion for the pendulum system of Figure 28.

3aa.

J=l Do -2_2 - 2r 02 sin 02 0 (3.18)2

2 "2-I + mr2c)0 2 r-r2cmg sin 02 0 (3.19)

Lie

37

Eq. (3.19) is the differential equation of motion. Generally, in development

of such an equation, translational d'Alembert forces and gravity forces

are lumped, since they have the same point of application. A tabular form

of bookkeeping, such as used throughout Appendix 2 and in Table 1, is useful

in documenting the procedure without undue space or verbiage. Table 1 is

the development of the differential equation of motion for the system of

Figure 22 with the translational d'Alembert force and gravity force combined.

The equation of motion for this particular example is easily obtained

with any number of other approaches. The simplicity of the generalized

d'Alembert Force procedure claimed is not overwhelmingly apparent in a

trivial example such as the pendulum, but for additional degrees of freedom

or a single degree-of-freedom system with as many effective forces as the

AMCAWS 30 the method provides an efficient well defined procedure for

generating the differential equation of motion for dynamic systems.

L

38

4. MATHEMATICAL MODEL FOR AMCAWS 30

The AMCAWS 30 MM weapon system, while seemingly difficult to describe

(Section 2), is easily modeled. This is because the model need not be as

detailed as the description. Many parts can be lumped, other parts can be

ignored, and some complex operating characteristics can be simplified (as

long as accuracy is maintained).

The component actions and interactions of major interest are those

associated with feeding, ejecting, chambering, and locking. Drive motor

torque requirements are also of major interest. A discussion of the

simplifications and a defense of why some parts and components are not

included is pertinent.

The greatest simplification in the model evolves from the fact that

AMCAWS is one degree-of-freedom. The specification of the input angle

(which is, again, the degree-of-freedom) in turn specifies all the positions

of the major components listed. The specification of the first an( second

time derivatives of the input motor in turn specifies all the component

velocities and forces. This fact can be exploited by creating a table

that dllows each component's position to be described as a function of the

input angle. This has been done and is the Functional Relationships Computer

Program (FRCP, Appendix 1). The FRCP, using the geometric constraints of the

cams and followers, generates an extensive table of each component's position

versus motor input angle. While the FRCP is more fully discussed in Appendix 1,

briefly the program is a FORTRAN description of the weapon that has the

positional table as primary output. The program traces through one firing

cycle in 360 steps. For a given weapon geometric configuration (cam rises,

follower arm lengths as opposed to a given weapon mass configuration), the

FRCP need be run only once. The single degree of freedom allows an

39

"uncoupling" of the major components and allows each one to be considered

separately. The FRCP develops the positional response of each component

with respect to the degree-of-freedom (input angle). The tables enable the

dynamic program "Appendix 3) to treat each component as essentially a separate

problem completely unrelated to any other component. Appendix 2, which is

the detailed development of the differential equation of motion for the

AMCAWS-30, does exactly tnat. The simplification is that each component has

a "local" degree-of-freedom which is explicitly a function of the motor

input angle. The terms contributing to the equation of motion are easily

identified and calculated in terms of the local coordinate. As a final

step the dot product terms of the components are expressed in terms of the

motor input angle. The dynamic effects of an individual component on the

rest of the system are correctly accounted for when the terms from each

component are summed, but this interaction need not be considered when

developing the terms for the individual component.

A second simplification occurs in the FRCP itself. The feed system

(Figure 7) is composed of a face cam path of some width, a roller bearing

follower of diameter slightly le-s than the cam path width and the rocker

, arm which transmits motion through the shaft to the dual feed pawls. The

cam path to rollerbearing contact is an example of contact between higher

pairs (as contrasted with lower pairs) and position, velocity, and force

solution are not trivial (3,4) in terms of difficulty of incorporation

into the dynamic model itself and CPU time of running. While the higher

order pairs (all the cam contacts) could all be simulated with the

generalized d'Alembert force procedure (4) the increase in accuracy of the

positional solution and the velocity and force solutions is not felt to be

significant enough to warrant inclusion at this time. The simplification

40

made is, for the feed and eject tides, a pinned linkage (shown in Figure 23).

The length of bar s corresponds to the rise of the cam for the given rotation.

The internal angles of the linkage are computed and eventually all the

angles of Figure 23 can be computed. The positional solution for the feed

and eject components are achieved in this manner. The velocity and acceler-

ation solutions are accomplished in the dynamic model. Part to part forces,

such as cam path to roller bearing are determined as the result of force

equilibrium. The recasting of the higher pair feed and eject contacts into

lower pair pinned joints is an important simplification.

There are other higher pair contacts that have been recast into more

easily solved problems. The drum cam-chamber motion is through a follower

or the stud. The FRCP treats chamber motion entirely as a result of a dis-

placement function, the function itself being the rise versus rotation data

on the drum cam drawing. The gear sets in the drive assembly are higher

order pairs. The common simplification of assuming no losses through an

individual set and assigning a total loss proportional to the power trans-

mission through the entire assembly is made in the dynamic model. The live

round being fed into the chamber is a higher pair contact, since the round

4 slides on the feed pawl surface during the load cycle. This pair has been

ignored by placing the actual ammunit;on mass at the pawl end during the time

the round is being placed into the chamber and setting that mass to zero while

the feed pawls retract. The lump cam-clevis pair for the locking ring is a

higer order pair. The position solution for this pair was achieved by a lOx

cardboard model of the cam surfaces. Quite sophisticated.

Two major assemblies are not explicitly in the model. The bolt assembly

appears as a translating mass in the chamber routines and the sear spring

compression is treated as a spring force acting on the chamber. The firing

CAMV' ecfNN 6Abour G eou N 0

//vo

CAM1 fbiE&f,y 6NI(0

5irvpUlpeo CAM

42

pin travel after sear off is not pertinent to the overall model and is not

modeled. The buffer assembly is not treated at all. This, in effect, ignore

any recoil of the weapon due to the round impulse. The recoil of the weapon

is an independent degree-of-freedom that could be included at a later date

if deemed necessary. Firing pin motion could also be treated as even another

degree-of-freedom. Neither of these two possible additional degrees of

freedom is important to the major parameters of current interest (feeding,

ejecting, chambering, locking, torque).

The power input into the weapon system is modeled as a torque versus

RPM curve (Figure 21) for the drive motor. A table lookup yields the torque

input to the system from the drive motor for any specified RPM. Evaluating

different motors is oniy a matter of substituting the different torque

curves.

An extremely important aspect of the model is the drag forces and

other sources of power loss through the weapon. These are not yet implement,

The mathematical model is the differential equation of motion for the

system described. The complete equation for that system is159 - C1120

2-(es)2(Idrum+I face )

-(E)[-FIpawl + Fl rock Fshaft tMPawil F cm)

+ FMrock(FRcm)2 + FMamo(FPe)2]

V.-o)[El El + El. + EM (EP )pawl + rock shaf pawl cm

* +EMrock (ERem)2],,~ ~ -e)llock'

,+(R71) 2(VCHMBR)

43

+622+ - 3 drum + face)"0 --F F I + FI

- 04[ Ipawl + Flrock shaft

2+FMoawl(F~cm) + FMrock(FRcm)2

+FMammo (PE) 2]I II

-.C 'rI +EI + EI5 pawl rock shaft

+EMpawl (EPcm)2 + EMrock (ERcm)2]

-606(lock)

+RR7 (Vchmbr)

+ Tmotor - Cmotor

-0[EMpawl (EPcm) g COS04-EMrock(ERcm) 9 COSO)rf + Tfeed)

-O[EMpawl (EPcm) g cosO5+ENrock(ERcm) 9 cOSO rf]

"oY(Tlock)

-R [CRUSH - SEAR + Cfchb (R'62/ABS(R' 62))]

= 0 (4.l.a)

which is rewritten

aO+b 2 =0 (4.l.b)

This ordinary differential equation is integrated numerically using the HPCG

routine out of the IBM-SSP library (all the routines used by the FRCP and the

dynamic program are included with their respective listings). HPCG uses Hamming's

predictor-ccrrector method coupled with a Ra~ston modified R .ge-Kutta procedure

d for start-up values (7). HPCG is quite general because it requires two user

supplied external subroutines FCT and OUTP. FCT is the routine that must

evaluate the constants of Eq. (4.1.b) and OUTP is the output vehicle for HPCG.

The construction of the dynamic program is shown in Figure 24. The OUTP

and FCT blocks are expanded in Figures 24 and 25, respectively. Both the

dynamic program and the FRCP are reasonably well documented and so a more

+ LU

U--

C.:,1 13 C-__ _

Ii H ' ow

(TIME, Y(2), DEiW(2))

SY(1) elY(2) - e.

EVALUATE TERMS OF EQ. 4.1.ai.e. aez + be

+ DETRMNE

D (1 )

FIGURE 25FCT BLOCK DIAGRAM

45

suBMirIN. cUrT'(TIM, Y(2), DE_(2), Tn=, STEP)

TNET - MEMP + 1SP

Y(1) -. ezY(2) -+e 2

DERY(2) -- a

i UPDATE ALL PART PO SIT1I0tVELOCIIES, ACELRAICU-

WITH 9z VAUE

STORE UPDATES IN TABLE

FIGURE 260WTP BLOCK DIAGRAM

46

47

detailed description of the program can be found in Appendix 4 and 1. The

program has some interactive capability in input which was used during

initial development on MTS (Michigan Terminal System, University of Michigan)

but difficulty of doing interactive work and limited disk availability at

AVSCOM almost demands the dynamic program be run with its batch default.

The batch default causes the dynamic program to have the following

initial conditions:

Initial motor input angle position = 0

Initial motor input velocity = 0

Initial motor input acceleration = 0

Initial time = 0

Final time = 2.5 sec

Output steps every .01 seconds

Figures 27 through 29 show the response of the motor input angle for a typical

run.

i i

FOSITI t CF DRIVE ['TOR (jEGRES)

..p.oeo pO po~oo 1oo.OO 2opo o. O. ' o oo a oo~o 36o.oo, 4o.oo1 • ----

0

0

N 0

-1 0

H

VE1IY OF DRIVE MOTOR(DEGREES/SECOU)

~11'0.0O 2000 40.00 60,00 80.00 O0.00 120.00 140.00 100.00

-7 '

LO N

-I -

0t

Nz

ACMI ION( CF DRIVE MTOR(DEGREES/SECctMJJ2)

e- 000 -.10.00 0.00 10.00 20.00 30.00 403.00 50.00 6NO.0

0

0t

0vH

- a)

F

r 5. 1ODEL INPUT

The mathematical model for the AMCAS-30 uses as input a table of position

solutions for each major component versus input motor angle. The range of

values for the table is such that one complete firing cycle is described.

The table is shown in Table 5.1 along with a description of the element

entries. The table, although on disk file, is constructed as if it were on

80 character cards and thus the resulting 2 card groups seen in Table 5.1

Format for the table is format (' ', 15, 3F16.4) and format (' ', 15,4F16.4)

for the two lines. This position table is the only input necessary to the

A4CAWS-30 mathematical modeling program.

The table itself is, as discussed, generated by the functional relation~hips

computer program (FRCP) which is listed in Appendix A-3. The inputs here

consist of smoothed drawinq data for the single turn cam AICAWS weapon. The

data required for the FRCP is data for the feed cam, eject cam, drum cam

and lock cam. This data is taken from the engineering drawings fcur these

components. The drawing data was tabulated and read into the FRCP with a

Format (2F17.4)

stateamel.

RTere arp no other inDuts to either program.

Since the mathematical model program is based on its numerical inteqration

routine, it is essential that the input data have no discontinuities aLout

'which the integrating routine would cycle and ultimately stop. This did

occur with the original drawing data and thus the "sharp" corners were smoothed

with a cubic fitting routine that forced a match in the zero and first

derivatives at the end points of each region to be smoothed. The smoothing

is done on the drawing data and thus the table output is also smoothed.

Comparisons between original and smoothed drawing data can be seen in Figures

30, 31 a.(! 32. The drum cam data was well behaved enough to use without

smoothing and is shown in Figure 33.

The smoothi, routine and program listing can be found in Appendix A-5.

FOE CM~ PJIUS (IIOE)

.50 3,.70 3,490 4,*0 4e30 4.-50 4.70 4,-90 ,1

OIw rtlA RADIUS (lirmt)

3,,,OD 3i-20 3z.4O 3.*6O 3,40 4,00D 4..20 4%0

?7*

I(~j ,r

LOCK RMh~ ROTATIONI MOjB3E)

A ,oo 1P.0 0 t1P 1 .oo 2p.o 0 256.00 3P.00 3t.o 4p0O0

.78.---i-

8'

I - ~ ~~ ~ ~ Pi 1 ~RISE (IMBE) * .0~000 too zCro 4 ,0 9'00 l 00 1 00 860

CR

Li1

N"* CI4RIBER nlSPIr'E11ENT I.II7IS

0 I0 O,. 2,.00 3.l ,~

,0 .0 70 ,0

* LOCK CRA1 RNOLL (DEGREES)

0.00 2.00 400 ,.00 a,0 ip0 ig.0 14.00 S.0

4- ~ CCT Cih At GLE DEGREES)20-0200.00~F 2p .O ,

1 3fl 220.00 22.0 () 20 00 2 60 0 01 60~~~ ~ ~~.s 7.00 0.00 21.00 (.I 4.0 100 3.0 100( RU RM RNGLE ODEGREES) 12.00 30.00 400.00

.00O 50.00 100.00 150.00 200.00 2500

0

0;

IaAncInc

c- O

Ll0

i7

j I6.0 CONCLUSIONSIne three primary objectives of this report have been largely met.jlThe computer programs that comprise the model package have been well doc-

umented. The modeling procedure itself has been explained in detail.

The operating characteristics of the AICAWS-30 weapon have been document-

ed. As a bonus, the dynamic model itself seems to work well and the

results seem to be similar to the actual weapon.

Figures 27, 28, and 29 are graphical output showing the response

of the input drive motor gear over the 0.0 to 2.5 second time range

considered. The spikes in the acceleration curve are certainly not

present in the actual weapon and might be induced in the interpolation

routine or by the highly discreet nature of the various fast acting cams.

The model is not "correct" in the purest sense until the source of these

spikes are tracked down and the source is either eliminated or justif-

ied. The smoothed data line for the acceleration curve represents the

more realistic situation.

The computer model package operates at some disadvantage. Since

the cancellation of AMCAWS funding in late 1976 there has been under-

• standably little interest or enthusiasm in verifying the model against

the actual weapon. As such even the masses and inertias (Table A2-1)

are the result of calculations and not measurements. While the calcul-

ated and actual values probably do not differ greatly that is a known

source of model error. The weapon has not been built up and fired since

58

this modeling project was started, although a testing program to aid in

the model verification was planned.

This report then cannot document an extensive verification, al-

though the position, ,elocity, and acceleration curves match rough data

that could be found from early 1975 gun firings.

j Hopefully, the d'Alembert procedure might serve as a basis for mod-

eling efforts for other Army developmental weapons or mechanisms. The

procedure is relatively easy to apply and program and the cost is reas-

onable. The dynamic program in this report was run during prime hours

at AVSCOM S&E computers for about fifteen dollars.

REFERENCES

1. Scott, L. AMCAWS-30MM Ammunition Development and Evaluation, Final Reportunder Contract DAAA25-73-C-043 Hercules Inc., 1 August 1973.

2. Kane, M. R., Rotating Band Torques and Stresses on AMCAWS-3OMM CopperBanded Projectiles, Technical Report, Aircraft & Air Defense Weapon'sSystems Directorate of Gen. T. Rodman Labortorary, Rock Island Arsenal,Report R-TR-75-022, May 1975.

3. Chace, M. A., Dynamics of Machinery Systems - A Vector/ComputerOriented Approa'ch, Prentice-Hall, 1971.

4. Kass, R. C. and Chace, M. A., 'An Approach to the Simulation ofTwo-Dimensional Higher Pair Contacts," Proceedings of the FourthWorld Congress on the Theory of Machines & Mechanisms, Newcastleupon Tyne, England, September 1975.

5. Chace, M. A., Published and unpublished Notes presented to Universityof Michigan M.E. 540 students, Fall 1975.

6. Chace, M. A. and Angell, J. C., Users Guide to DRAM (Dynamic Responseof Articulated Machinery), Design Engineering Computer Aids, Departmentof Mechanical Engineering, University of Michigan, 3rd Edition, April 1975.

7. System/360 Scientific Subroutine Package, Versian III, IBM Application Program,

Publication GH20-0205-4, Fifth Edition, August 1970.

8. Carnahan, B., Luther, H. A., and Wilkes, J. 0., Applied Numerical Methods,John Wiley & Sons, 1969.

9. Rothbart, H. A., Cams, John Wiley & Sons, 1956.

10. Tann, P. W., Cam Design and Manufacture, The Industrial Press, 1965.

11. Merritt, H. E. , Gear Engineering, John Wiley & Sons, 1971.

I

APPENDIX 1

FUNCTIONAL RELATIONSHIPS COMPUTER PROGRAM

•, I

A-1-1

The Functional Relationships Computer Progrdm (FRCP) specifies, as output,

the positional relationships versus the input drive motor angle of the components

in the AMCAWS-30 weapon. Since the AMCAWS-30 model is a one degree of freedom

system and the degree of freedom is the input drive motor angle, specification

of an input angle also specified the position of all the other gun components.

Given that there are no part failures (the weapon model uses fantastic materials

that cannot fail) these various positional relationships are unchanging.

The FRCP uses the various part geometries, cam paths, and assembly angles

to evaluate each part position given the input motor angle. Figure Al-l is

the basic flow chart for the program. The programs needs as input the drawing

data for the drum cam, the eject and feed cam paths for the face cam and the

various offset angles at which the components were assembled. A incremental 0

is chosen and the program loops through the six function calls until one

complete firing cycle is over. The results of each loop are tabulated.

The program itself is listed in Appendix 3 while the output from the

program, in graphical form, is shown in Figures Al-2 thru Al-7. Figures

Al-8 thru Al-11 are the unaltered cam drawing data in graphical form and are

included here for report completeness.

V, Figures Al-12 and Al-13 illustrate the basis of program functions FUN43F

and FUN53F.

INPUT:

0 2 MAXDEL 02

JOUT = 0READ DATA FROMDWGS

71 DSTOREVALUS INTABLU I s-4es

62 62+ DE02 e262MAI?V0

Lv-3 (2

0D

0

00l

zz 0

N

K--

d0(0

uQz 0 L

z -0

UJ 0jYw Lfr'1 S A

I I I I

*.

(g2~UJ ~7~Yv AM232V

JLO

6I

0

t-4Q

L .

MOWN

Qv-4 v-4 " r-

i0I,0

L0

C3

LO C;

C)

< CF)

lit W-J >

CDU 0 0

z--V4 -

w 0wzJ 0

03

10

0 i-0 D

C3

z C)

9 W AlPcr- L)zw

LL 0Fo -j Z .CLO

u 0 >

0

C),

0

U << I- d

fed

NS4

cs L

IH~ 0 IU0

I) ILIIILO r0< 0( C3A (Y ~Y) U

CS-,2V? /ZG~/HZ~A~? LN "

Ii 7 __N

MEO '7 7 -- & -

I>z

coo

0<

x 04

w I

!,

w ;rn

SOD WN6 .vi . . ... ... . .

17O &VtO0-9

Li

Z 00I- -J

so F

No XV~Ze LL~CLJO

LOV-1-

2fl)

CD. 0 0 C0 0 0 0 0 0

AOT V2 6_uoj 5Aw1.rtq

CA ,-

I 'I%

I wjIiL

ec.1

GA) 6A

4 + (to

1

U

c~rmp PPM

SID

OB 'p 3

)KE C KPERM UPS LREMER1" (IMHES)S1'.00, 2,.flO 3go 4 "0 .D 3, OO' 721 . 8

)x ) LOCK-CRfl RNDCE CUEWREES),LMD 2,.oo 44-00, ' ,Oo 8-po 1P.00 1 DOo 1.wLO s 1.00

I&-4 EJECT JR RN10CE(EELS)2OO 2pt.OO 2p 40, 2 u.DO 2?O.00 2FILO ZPOd)G 46-00 24w..00

(D (DrJ 7 ® FEED CRII RN9E ( EMU)

PJ4f DRUM1 CRd RNSLE (DEGREES)I .fO 5p.D po -O lpo we 2P0.00 2se. o c SanJ 35.0D 4fl0.00

CC

7C 2,

APPENDIX 2

Generalized d'Alembert ForceProcedure and AMCAWS-30

Equation of Motion Development

2 4

'I:

L'

iv

A2-1

A2.1 - INTRODUCTION

The systematic derivation of the differential equaiton of motion for

the AMCAWS-10 weapon was accomplished by the application of the generalized

d'Alembert force equation

iS +• 0 (A2.1)il qj k= aqj

where,

M = number of d'Alembert forces in the total system

= the d'Alembert force

p = the position vector from a point in ground to the point of

application of the d'Alembert force

i = the index of the particular d'Alembert force being considered

qj the degree of freedom being onsjlvdered

'Pi = the partial of Pi with respect to the degree of freedomqj

= closed loop chord contact forces

= positional vectf' spanning the closed loop chord]~ ii

! L= number of independent closed lopps

loop being considered

j degree of freedom

Two facts allow a significant simplification of an already simple equation.

First, the AMCAWS-30 is a one degree of freedom system. This sets the

index j = 1. Second, the development of functional relationships describing

the response of the major components to the motor input angle, 02, allowed

the analysis to proceed with no closed loops (Appendix 1). This sets the

index L=O. The generalized d'Alembert equation then becomes

A2-2

MZ" 1i "01i 0 (A2.2)

i~l(E) 2)

rThe problem at hand is then the cranking cut of the t.'s and i's

for the six major components involved. The detailed development follows.

The black box idealization of Figure A2-4 is somewhat altered in that the

reference frames for each component are oriented as they actually are on

the weapon. Throughout, the development the 1,J,k reference frame holds

where k is aligned in the direction of projectile travel after a firing,

the i axis is described by a horizontal line, and j corresponds to a

vertical line (Figure A2-1). A single dot (i.e. 03) refers io the derivative

of 03 with respect to 02 (the degree of freedom). Similarly, double dots33

and primes (i.e. 03 and 03") are the second derivatives with respect to time

and with respect to 0(2* The interpolation routines scan the tabular data

(coordinate of interest vs 02) and returns a local 5th degree polynominal

that fits the data in the region of interest. If 0 is the coordinate of

interest, for any specific 02'

!,,' iI O= ao + a102 + a202. + a e2 + aO 2 as2(23

V = 0. al+2a62 + 3a,,2 + 4a4O0 + Sase (A2.4)3)02 (25e'l - )20 =2a2_ + 6a362 + 12a40ei + 20abe (A.5

aT 0 = " (A2.6)

O6 +" ell (0 2)2 (A2.7)

SWith these relationships the development of the differential equation of

motion can proceed by applying the "reduced" d'Alembert equation to each major

component and tien summing all the terms into a single equation.

~~771L~~L L___________

L~U)

LL

IT

A2-4

A-2 - LOCAL COORDINATE 2, DRIVE MOTOR INPUT

The gun parts associated with this coordinate are the input motor

(,4001 , 80, WeStern Gear Motor) and the gear unit transferring torque

from the motor to the weapon itself. The parts and reference frame

are shown in Figure A2-2.

TERMS:

159 = Mass moment of inertia of the 59 tooth drive gear

1120 = Mass moment of inertia of the 120 tooth transfer gear

(including the 16 tooth output gear shaft)

0, = Motor input angle measured as indicated

0120 = Angle of 120 tooth gear measured as indicated

T = Torque drive applied at the 59 tooth gear (a function of 62)

jf = Frictional losses through the drive train but considered to act

on the 59 tooth gear.

F X

Jj_ _ _ _ _

-^r - -jr

t6 -ror Cffm T77

oaLJ CAP^

FIGURE A2-2DRVECORIME

A2-5S

A2- 6

Rel ationshi ps

02 = 02

01 e2 = (59/120)02 = - CO2

0120 = -CO2

0120 = -C6 2

cf = Cfk

Development:'Pi ]ri- i

Fi p -

1 -1596k 02k k -15902

A A A

2 -1 1 23 0 2k 0 1 20 k -Ck CI1206-C6 2

3 Tk 02k k T

4 -C k 02k k -Cf

in terms of 02,

4 1 )

i=l~ • e-I5902 -C . e02 + T -Cf

602 (-15 - C2I 1 ) + T -Cf (A2.8)

59

In

A2.3 - LOCAL COORDINATE 3, DRUM/FACE CAMS

The gun parts associated with this coordinate are the drum cam and the

face cam. Although in practice they are assembled so that they have no

relative motion between them, they are treated separately in keeping with

the practice of trying to do the analysis on very primary level so any

program revisions necessated by part changes are minor. The parts and

reference frame are shown in Figure A2-3.

Ii

", F GREA-

A-

FIGURE A2-3DRUM/FACE CAM COORDIMT\ES

A2-8

A2-9

Terms:

IDRUM = mass moment of inertia of the dnmi cam

IFACE = mass moment of inertia of the face cam

= angle made by the zero point of the drum cam with the

i axis, measured as shown.

0F = angle made by the zero point of the face cam with the

i axis, measured as shown.

Relationships:

OF = 03 + constant angle = 03 + A

=OF= 0

03 = ao + a,02 + a260 + a302" + aa0+as0 for a given 02

Development:-4o 4.. ..

i Fi Pi 902

5 -IDU' 03k 03'1 -O3'IDRUO 3

6 -IFACEOFk OFk 6 3 i -03'IFACE03

*- in terms of 02

6iP= i5 =2

= -03 IDR J(0; 2 + 03 2:

~-03 11AC (03" 02 + 0

= 02 [-(0 3 ) (IDRUM + 1AC)]

-03 -- 2 (I)RUM + IFACE) (A2.9)-0 ,_

A2-1o

A2.4 -LOCAL COORDINATE 4, FEED MECHANISM

The gun parts associated with this coordinate are the feed pawls, the

feed shaft, and the feed c" follower which is called a rocker. The parts

and reference frame are shown in Figure A2-4.

Terms:

IPAWL = mass moment of inertia of the pawl about the shaft

IROCK = mass moment of inertia of the rocker/follower about the shaft

ISHAFT = mass moment of inertia of the shaft

MPAWL = mass of the pawl

MROCK = mass of the rocker/follower

MAMMO = mass of the ammunition

02= pawl angle measured as indicated

Sr =rocker angle measured as 'indicated

AF _-

FIGURE A2-'ir

FIUR 2-4

1A2-12

POI = Distance from shaft to PAWL center of mass

RC4 = Distance from shaft to rocker

PE = Distance from shaft to end of PAWL

= positional vector from ground to pawl center of mass

A = postional vector from ground to ammo center of mass

R = positional vector from ground to rocker center of mass

T = torque needed to pull anmo belt.

Relationships:

0r = 04 + constant angle = 04 + C6 r = 64

o r = 04

= + pCii

consi~ + PP, + (DA44/2) i

o= coist + Re,

api

7 -IPAqL64k qk -IPALO 4O4

8 -MPALCOFgJ) PCO4V'(-sinO4I +.cosO)') -MPAWL (PCM) 2 4{'4 +cos04(g/PD1'O }

9 -IROCKOrk Ork 04--k -IROCK0404

10 -NROCK +gj) R RC r (-si~ner +COSer) -MROCK (RGM) 20{cr+cosor (g/RCq }

11 -ISHAFT64 04k 04"k -ISwAI-q 6,,04'

12 41MA v ] PE04'(-sinO4$ cos04^) -bi(P)0 4"

in terms of 02

A2-13

13 pE i ° -i=7 0

= 02(-(4 * )2{IPAwL + IROCK + ISHAFT + MPAWL(PCM)2 + MROCK(RCM)2

+ MAMMO(PE)2}

- 6 )'( 4 ){IPAWL + IROCK + ISHAFT + MPAWL(PCM)2 + MROCK(RCM)2

+ MAMMO(PEJ'}

-0 '{MPAWL(PCM)gcos04 + MROCK(RCM)gcoso r + T) (A2.10)

NOTE, however that the ammunition is only present during the portion of the cycle

during which the ammo is transferred to the chamber for firing. Thus the term

MAMMO = (mass of AMMO) Unit (64)

where unit (04) = 1 when 64 < 0, or since 62>,O, when E)* + 0

= 0 when 64 > 0, or since 62>,O, when e >,0

Also, there is currently no ammo belt system yet fabricated so that T is always

zero.

1I

I

F

A2-14

A2.5 - LOCVL COORDINATE 5, EJECT MECHANISM

The gun parts associated w'cn this coordinate are the eject pawl, the

eject shaft, and the eject cam path fall over which is called a rocker. The

parts and reference frame are shown in Figure A2-5.

Terms:

IPAWL = mass moment of inertia of the pawl about the shaft

IROCK = mass moment of inertia of the rocker about the shaft

ISHAFT = mass moment of inertia of the shaft

MPAWL = mass of the eject pawl

MROCK = mass of the rocker

es = pawl angle measured as shown

0r = rocker angle measured as shown

PCM = distance from shaft to pawl center of mass

RCM = distance from shaft to rocker center of mass

= positional vector from ground to pawl center of mass

= positional vector from ground to rocker center of mass

Relationships:

E= e- constant angle = 05 -C

o =0r

const + PtM

_ :const + RtM

Avv\ 'UP?$PA CC

FIGURE A2-5EJECT COODIN/AffS

A2- 16

Development:+

*A A% A

14 -IPAWLOS k 05 k 05,k -IPAWLGOO

15 -MPAWL( +gj) PcmE0'(-s i nesG+cosEs)

16 -IROCKO5s k Osk os'k -RCO

17 -MROCK(0~g1 ) RcmG (-sinEo +cos0 .)r r i r

A A A

18 -ISHAFTEsk 05 k 0sk -ISHAFTOsEOii

-MPAWL(Pcm)20-5{0s+coses (g/Pcm) I

-MROCK(RCM) 2 E.{6-*r+cos0 (g/Rcm)}

in terms of' 02,

14 1i D02

--(0'5) 2{IPAWL + IROCK + ISHAFT + MPAWL (pCM)2

-(0eS*%){IPAWL + IROCK + ISHAFT + MPAWL(PCM)2

*1 + MROCK(RCM)2}627

- (O-){MPAWL(PCM)gcos05s

~j I+ MROCK(RCM)gcoso (A2.11)

'~*~ _________ _____r

A2- 17

A2.6 - LOCAL COORDINATE 6, LOCK RING

The gun parts associated with this coord n.te is the lock ring. The

part and reference frame is shown in Figure A2-6.

Terms:

ILOCK = mass moment of inertia of lock ring

T = additional torques acting on the lock ring, including friction

0 = angle from lock ring center to zero point of ring from i axis

Relationships:

6 = ao + al0z+a20+ a3+2 + a4 0 + as0 for given 07

Development:

3 0 2 a o ? .

19 -LOCK' k Qsk % k -ILOCK4 (e

20 -Tk 06k E3 k -T

in terms of 02

20 api

302

E)-IOK 2 + 62(-ILOCK e~Cog' - Tog (A2.12)

£ 2(• -ILC-e

I'

'7-

J

x~I,

FIGURE A2-6

LOCK RING COORDINATESA2-18

A2-19

A2.7 - LOCAL COORDINATE 7, CHAMBER

The gun parts associated with this coordinate are the chamber and bolt

assemblies. As discussed in Section 2.5, the bolt and chamber translate as

a unit only a small distance before the bolt becomes stationary. The

actual mass that is translating is the mass of interest. The chamber

assembly includes everything shown in Figure 9. The parts and reference

frame is shown in Figure A2-7.

Terms:

! = vector from ground to fixed point on chamber

Crush = spring force acting on chamber due to round crush up

Sear = spring force acting on chamber during researing of firing

pen spring

f = friction acting to resist chamber motion

VCHMBR = virtual (actual) translating mass

Relationships:

Rao + ai02 + a 207 a302 + a4 2 + ases= A

const +

CROSH -(CRUSH) k( StAR = (SEAR) kA

f fCk when > 0

-Cfk when < 0

R _ R, 9 02 302

A2_-20

+ pi P

21 -VCHMBR(M+T) A-Rik (VCHMBR)R'R

22 CR1USH -Rik (CRUS)H)R'

23 SEAR -Rik -(SEAR)R'

24 -kk+C f R'(

in terms Of 02

24 8P

C=21a2

-VLHMBR R'(R'Ea2 + R"O~ + (CRUSH)R' -(SEAR)RI

+C fR' (R'6,/ABS(R'62))

- 2(VCHMBR(R')2 + 2(VCHMBR RIR")

+ R' (CRUSH -SEAR + Cf(R'6,/ABS(R'62)) (As.13)

oo

CHAie?. A65E7MLytvIT( (3-F* FACE-

QOLr A566eMv~L& le~e,10C

4'l

FIGURE A\2-7

CHAMBER COOJRDINATESP2-21

A2- 2 2

A2.8 - GENERAL DEVELOPMENT OF A TRANSLATING d'ALEM]ERT FORCE

Let the positional vector be PCbI and have it acting at angle as shown

in the Figure A-8 as always, 0 is a function of Oz

Terms:

+P = pos.tional vector from ground to the center of mass

0 = angle measured as shown

PCH = magnitude of distance from origin to center of mass

g = acceleration due gravity

m = mass of part, considered concentrated at center of mass

Relationahips:

0 = a +a0 + a 20 2 + a3Ez a 40 2 + aSO2 (A2-3)

Development:

The only force considered here is the d'Alembert force associated with

- I translation of the center of mass.

A,

I

Poi'r C)

V, £a'r- feomuv A 6EPbuK0 O ~I' 10 C M

C15 V6=0i rtCjwv'A TH GP-utD P~OINT 10 T9N-C

1%'l VL'yAfl- FLv"r- 0 A~ 7IA0 07 C

FIGURE A2-8GENERAL TRANSLATING MASS

A2-23

A2 2 4

The positional vector 1 is

= + where t is a vector from some point in ground to point 0,

which is fixed relative to ground.

The second time derivative of the vector (in two dimensions) is

=0 + Cm

: (P ) (Pc) + G(kXP4 -$Pcm cm 'cm cm

+2b(kX(PCM)PCN) (A2.14)

however, the length Pcm r constant

thus P cm =0

and Pcm =0, yielding

= ;(kxP7m) - (6)2 P- (A2.14.a)cm

since

Km=P cm(C0Sei + SINei)cm cm

A A AkXPM =Pm-SINOi + COSOJ), therefore

= 0 cm(-SINGi + COSOj) -acm(COSEi + SINEj)

= {(-sino)(O)(PCM) - (6')(P )(cosO)}icm

+{(cosE)(O)(PCM) - (62)(P )(sino)}J (A2.14.b)

for this case is

-MPcm {(-sino)O - 6 2cosO}iA

+ {(cosO)O - 6sinG+ 3 } (A2.15)cm

A2-25

= + Pc~ AND

AA

'~{P (cosEi + sinEj)}j02 cm

=c dO- (-sinoi A- cosoj)AA

= Pcmol-sinei _cosE)

= (PC E0'-s inE)i + (P 0')+coso~i (A2.16)

and 30

=M(PCM) 20{sn(siE) + sinO62CoO

cm OSiE)(C+Sin)6 COSEX62 sine, + cos02-cm

= -M(P ) 2 E) {+G(sin 2+cos 2) +62(S inEcos0)-sin0cosO + cosO ccm PCM

= -M(Pm )2 0'{+; + cosORL-cm Pcm

which in terms of explains as follows1 MP ) 2 2,012 + (gIp )o0

~-M(P )m 0 (04'.1 cm {-M )2 sE

-M(P )g cose 0' (A2 .17)

- I 1 _ ____ __ ____

TABLE OF MASS AND INERTIA

Mass I MassPart Name (Lb-Sec**2/Ft) (Ft-Lb-Sec**2)

59 Gear .2094 .000949

120 Gear .4816 .01546

Face Cam .3525 .04406

Drum Cam .8540 .0997

Feed Shaft .000018647

Feed Rocker .0049232 .00002724

Feed Pawl .010888 .00032958

Ammunition .04255

Eject Shaft .000016782

Eject Rocker .0049232 .00002724

Eject Pawl .017473 .00016703

Lock Ring .10864 .0020653

Chamber 6o-29

TABLE A2-1

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