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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I1NCTASSTFTRDSECURITY CLASSIFICATION OF THIS PAGE ("en Date Enteree)
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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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D FOR" 1473 EDITION OF I NOV 65 IS OBSOLETE UEH CR Y UNCLASSIFIED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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A99
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