tog. g. lowen,/city college of new york a f. r. tepper,arradcom daac-29-76-g-0058 9. performing...
TRANSCRIPT
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TOApproved for public release, distributionunlimited
FROMDistribution authorized to U.S. Gov't.agencies only; Test and Evaluation; JUN1978. Other requests shall be referred toArmy Armament Research and DevelopmentCommand, Large Caliber Weapons SystemsLab., Dover, NJ 07801.
AUTHORITY
USARDC ltr, 10 Apr 1984
THIS PAGE IS UNCLASSIFIED
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0 DYNAMICS OF THE PIN ALLET
:RUNAWAY ESCAPEMENTL
'CiTY COLLEGE OF NEW YORK
ARRADCOM DDCI~Jf~AUG 21 M
UARMY ARMAMENT RESEfC ANDDEVELOPMENT COMMANDtLARGE CALIBER
WEAPON- SYSTEMS LABORATORDOVER, NEW JERSEY
V
Distribution- limited to, US Government agencies only because of testandevaluation, J une 19'S te e e frhi dobuxieiit iust;1
) referred'to Commander, ARRADOOM, A TTN,: DRDAR-TSS, Dover,New Jersey 07801.
78 07 26 55
isN
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The view, opinions, and/or, findings contained
inthis report are those of the author(s) 'and -
should not be -Odnstrukd; as, an offidial, Depart-ment of the Arnmy position, Policy or decision,.unless ,So designated by other documentation.
Destroy-this report When no'longer needed. Donot return it to the originator.
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UNCLASSIFID ..SECURITY CLASSIFICATION OF T.IS PAGE (When Date nt.red)
REPOT DCUMHTATON AGEREAD INSTRUCTIONSREPOR DOCMENTAION AGE EFORE COMPLETING FORM
1. REPORT NUMOER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMULER( Technical Report ARLCD-TFR-77062
4. TITLE (and Sublttle) 5. TYPE OF REPORT & PERIOD COVERED
DYNAMICS OF THE PIN PALLET RUNAWAYESCAPEMENT 6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(a) 5. CONTRACT OR GRANT NUMUCR(s)
G. G. Lowen,/City College of New York AF. R. Tepper,ARRADCOM DAAC-29-76-G-0058
9. PERFORMING ORGANIZATION NAME AND ADDRESS I0. PROGRAM CLEMENT. PROJECT. TASK
ARRADCOM AREA & WORK UNIT NUMBERS
Fuze Division, LCWSL
II. CONTROLLING OFFICE NAME AND ADDRESS 12, REPORT DATE
ARRADCOM JUNE 1978ATTN: DRDAR-TSS I,. NUMBER OF PAGES
Dover, NJ 07801 10914. MONITORING AGENCY NAME A ADORESS(If different Iront Controlling Office) IS. SECURITY CLASS. (of thla report)ARRADCOM
Fuze Division, LCWSL UNCLASSIFIEDATTN: DRDAR-LCF 15,. DECL ASSI'FICATION/DOWNGRADING~SCHEDULEDover, NJ 07801 _
I6. DISTRIBUTION STATEMENT (of thi. Report)
Distribution limited to US Government agencies only because of test and evalua-tion, June 1978. Other requests for this document must be referred to the CommanderARRADCOM, ATTN: DRDAR-TSS, Dover, New Jersey 07801.
17. DISTRIBUTION STATEMENT (of the abttact entered In Block 20. It different from Report)
I:IS. SUPPLEMENTARY NOTES
AUG 21 19"78
19. KEY WORDS (Con~tinue on reverse aide If necessary end Identify by block numnber) tRunaway escapement Safing and armingPin palletB
Escape-wheelM525 fuze
;= -?!AC"r
dCnfcfwu# an ,ersa, sla It treemwa sy an Idonltfy by block number)
)JThis investigation developed a workable computer simulation uf the pin paAt
runaway escapement which can provide the basis for the analysis and synthesis ofvarious safing and arming devices. The dynamics of the various regimes of escape-ment motion are modelled. Coupled motion, impact, and free motion are considered.Sensing equations for the determination of the instantaneous positions of the palletpin and the escape-wheel form the basis of the controls in the computer program.
DO I FJAN 1473AVr1t'I o INOV 6IS ODLETE UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Date Kntered)
it
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SECURITY CLASSIFICATION OF TVIS PAOt(Whn D&I. Se..4
20.14rhe simulation is applied to the timing of the mechanism of the M525 fuze withan assumption of constant input torque to the escape-wheel. The influence ofchanges on such paramoters as escape-wheel torque, pallet moment of inertia.center distance, pallet radius, etc., are explored in detail by appropriate com-puter runs. Agreement has been found with existing experimental data. Finally,recommendations for continued work are given.
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SEtCURITY CLASSIFICATION OF Tt4IS PAGE(Wbheo Data riverod)
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ACKNOWLEDGMENT
The first author gratefully acknowledges the support of the res6archdone, under Grant No. DAAG-29-76-G-0058, by the Army Research Office.The assistance rendered by Picatinny Arsenal and the City College compu-tation centers is also greatly appreciated. Last, but not least, the first authorwishes to thank his students, Mr. R. Van Nostrand and Mr. R. Chin,
for their invaluable help.
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TABLE OF CONTENTS
Page No.
Introduction 1
Simulation of Pin Pallet Runaway Escapement 1
Coupled Motion 2Free Motion 2Impact 3Escapement Nomenclature 5
Description of Computer Program 7
Coupled Motion (Location 100) 7rree Motion (Location 200) 10Impact (Location 300) 12
Example Mechanism 12
Dimensions of the Mechanism 12Gear Train Driving Escapement 13Standard Torque Used in the Simulation 13Other Dimensions Associated with Standard Configuration 13
Results for Standard Configuration 15
Influence of Various Parameter Changeson the Total Fuze Time 21
Influence of Escape-Wheel Torque 21Influence of Pallet Moment of Inertia 21Influence of Pallet Escape-Wheel Center Distance 21Influence of Pallet Radius 25Influence of Coefficient of Friction 25Influence of Coefficient of Restitution 25
Discussion and Recommendations for Continued Work 29
References 32
I
Appendixes
A Kinematics of Coupled Motion 34
B Differential Equation of Coupled Motion 38 f
C Moment Arms 42
D Contact Sensing Expressions 44
E Free Motion Equations 47
F Impact Equations 49
G Computer Program 55
H Some Geometric Relationships 76
I Computer Output for Standard Configurationwith p = .3 and e = .25 73
J Computer Output for Standard Configurationwith p = .3 and e = O 93
Distribution List 105
Tables
1 Summary of events in the first 2 cycles forstandard configuration with [L = .3 arid e .25 17
2 Maximum reversal angles and associated valuesof g for standard configuration with p = .3and e = .25 18
3 Summary of events in the first 31. cycles for standardconfiguration with p = .3 and 6 = 0 19
4 Maximum reversal angles and associated values ofg for standard configuration with p = .3and s =0 20
5 Escape-wheel reversal as a function ofcenter distance a 33
Figures
1 Coupled motion 1
2 Free motion 3
3 Impact 4
4 Impending top contact 4
5 Escapement nomenclature 6
6 Flow chart for simulation of runaway escapement 8
7 Top free motion test 12
8 Influence of escape-wheel torque on fuze delay time 22
9 Influence of pallet moment of inertia on fuzedelay time 23
10 Influence of pallet escape-wheel center distance
on fuze delay time 24
11 Influence of pallet radius on fuze delay time 26
12 Influence of coefficient of friction of coupledmotion on fuze delay time 27
13 Influence of coefficient of restitution on fuzedelay time 28
[I
Ii1
IS
I NTRODUJCTION
It was tho aim of tho prosent investigation to dovelop a1 realistic andworkablo computer simulation of the dynitc behavior ot the pill palletruna way escapemetnt which can be used in the analysis and 'ynthosis ofvarious safing and arming deviceis. Tis goal hans been achieved.
parameter changes Is explored, Pirevious oxperimentation and practicalexperience with this me1chanisml confirmed tho results ot the comlputations.Finally., recommendations for continued work are given. The previous workdone in this field Is listed in references 1-15.
SIMULATION OF PIN PALLET RUNAWAY ESCAPEMENT
The present effort onl the pin pallet escaponment represents ain ox-tension of the work of M. E. Anderson and S. L. Redmond (ref. 7) Newmethods of contact kinematics for coupled motion, of contact sensing . andof computational controls are developed.
P'allet 9Escape wheel6V
0
F19gure 1. CouIpled motion
The following outlines the overall assumptions and derivations forthe variou:3 regimes of motion. A detailed description of the actual-computer program and its controls is then given.
Coupled Motion
Figure 1 shows the upper pallet pin' being driven in coupled motionby tooth no. 1 of the escape wheel (,op contact). The escape-wheel angle
i is defined by the line from the escapk-wheel pivot 0 to the tip (with-S
out radius) of the contacting tooth (or the one about to make contact) andthe line connecting 0 to the pallet pivot 0 . Similarly, the angle4,
s pwhich is defined by the line from 0 to the activa pallet pin center (topPor bottom) and the center line, describes tie motion of the pallet. Theescape wheel is driven by the constant moment T in the positive directionof rotation. While it is assumed that friction acts on the pallet pin/escapewheel tooth interface, it is neglected at both pivots since investigationshowed that its effects are negligible when the pivots are of the usualsmall diame-ter.
The quantity g, wnich represents the distance from the contactpoint to the tip of the escape-wheel tooth, is used to determine the end ofcoupled motion. Appendix A contains derivations for this expression,as well as all other kinematic quantities associated with coupled motion.Appendix B, with the help of appendix C, gives the derivation of the dif-ferential equation of coupled motion in terms of the escape-wheel angle (P(eq. B-10).
Free Motion
When coupled motion is finished (i.e.,g = 0) or when separationof contact occurs after impact, the escape wheel and pallet move indepen-dently of each other in free motion. Figure 2 shows this free motion forthe bottom phase of the action, i.e. the bottom pallet pin is about to makecontact with tooth no. 2 of the escape-wheel. The constant torque T con-tinues to act on the escape-wheel, while the motion of the pallet dependsonly on its initial conditions. Again, any frictional retarding moments atthe pivots are neglected. ',)sition sensing during free motion is based ong'. the distance of the pallet pin center from the tip of the escape-wheel
[This report uses the word upper when referring to entrance motion andlower when referring to exit mo-tion.
2
t
tooth (parallel to the face of the tooth) and f, the distance between thepallet pin and tooth surfaces normal to the tooth). Expressions for thesequantities are derived in appendix D. The differential equations for thefree motion of the pallet and the escape-wheel are derived in appendix E.
Diretion
If
" Figure 2. Free motion
Impact
Impact foilows free motion whenever f equals zero, g' is smallerthan zero, and the relative velocity between the contacting surfaces war-rants it. Such an impact usually reverses the motion of the pallet (fig. 3)and under certain circumstances also reverses the motion of the escape -wheel temporarily. While the impact equations, derived in appendix F,allow for both normal and tangential impulses to produce changes in theangular momenta of the pallet and escape-wheel, the computer programpresented here neglects the tangential impulse for the sake of simplicity.The classical coefficient of restitution formulation is used to account forthe energy loss during impact.
Figure 4 shows free motion for the subsequent top phase of themotion, i.e. the top pallet pin is about to make contact with tooth no. 3
3
Direc tion 2of :otion
cjT
I Figure 3. Impact
Fiur 4. Impedin to conac
4
W
-
All kinematic and dynamic expressions are derived in such a manner
that they are valid for top and bottom action. The simulation recognizes
only contact on the front faces of the escape-wheel teeth. Pathological
conditions, such as impact on tho tips or the back faces of the escape-
whael teeth are not considered. (The control quantities g and g' make
it clear that when such a condition exists the computation can be discon-tinued. )
Escapement Nomenclature
Figure 5 shows a schematic representation of the pin pallet escape-
ment and indicates its basic geometric nomenclature.
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Paletb Ecae Wee
.04
btA0I I~ "t
bi Bslptu whool vidhwi
o 1'illo n~du4 oquil onWP ~d o bottom for %,;Ifplluty)
Pulo rdl; oqM R OP111C
r Pallet-pin radius (equal for top and bottom)
a= Escape-wheel tooth half anglo
8 = Angle between escape-wheel tooth
q) Angular position of escape-wheel
Angular position of pallet
DESCRIPTION OF COMPUTER PROGRAM
The following gives the essential stops of the computer program aslisted In appendix G. The flow chart of the program is shown in figure 6.The choice of variable designations was made in such a way that they differas little as possible from the nomenclature used In- the various derivationsin appendixes A to F and H.
The main- program starts the simulation with top contact coupledmotion at a starting anglo q) = 1351 (called PHID in- the computer program)and a cumulative-escape-wheel anglo PHITOT of 00. The choice of thisstarting angle is-connected with the g&,omtry of the example mechanismused (yM = 132.4871 for the example mechanism).
Coupled Motion (location 100)
To solve the differential equation of coupled motion (see equation.B (-10) of appendix B)- the main program calls on-an available fourth-order
Rungd-Kutta routine. The subroutine FOT computes all needed valuesand presents the-second-order differential equation-in terms of two first-order ones to RKGS. 13111 (1) and PIl (2) represent the angle q) and theangular velocity p, respectively. The associated subroutine OUTP isresponsible for printing out the results of the Integration together withthe current valus of time, g, it, ,, and PHITOT. In addition, it hasthe task of deciding whether coupled motion is to-be continued. Coupledmotion is continued
a. as long as g < 0. (See equation (A.12) and note, thatbecause of the nature of the coordinate system, g is always negative whilethe pallet pin makes contact with the escape-wheel tooth.) And,
The program shown is written in FORTRAN for the CDC System at ARRADOOM,Dover, NJ. It is also available In a slightly modified form for the IBMSystem/360 at the City College of Now York.
3 RKGS Routine, IBM Systm/360 Scientific Subroutine Package,(360A-CM-OX3) Version-Ill,
7
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b, for a positive (counterclockwise) rotation of the escape-whoel, if the succeeding absolute valuo of f (DPSI) Is larger than the oneobtained from the preceding computation (called DPSIP) Those values of
are computed with the help of equation (A, 18). which assumes closure.This condition is necessary, since in coupled motion when p is positivethe escape-wheel can only drive the pallet, not slow it down. If such aslowdown is indicated, it means that pallet and escape-wheel have separatedand free motion will take place. Or,
c. for a negative rotation of the escapeowhoel (which mayoccur after Impact), a succeeding absolute value of rust be smaller thanthe preceding one. If for some reason the escape-wheel should speod upand cause the computation to show a larger succeeding value of i, i wouldalso serve as an indication that closure has terminated and free motionhas started,
To make this process loss sensitive, the-last two conditions will only beviolated when the absolute magnitude] -I :. I red/soc.
Thus, when-control Is returned to-tho main program-, It Is eitherj-because the pallet-pin has left the end of the tooth and there Is no further
-possibility of coupled -motion or the pin has disengaged from -the insideof the tooth, In either case, free motion results and control-is eventuallyshift3d to the subroutine FRE13Ii (location 200) . This is done dlrectly if g< 0. In the case that g < 0, the main program must deide whether thepreceding computations have boon made for top or for bottom action andwhether the next contact will occur on top or on bottom. In-the samplemechanism, g = 0 when qo is approximately 146V for top action- and approx-
imatoly 2070 for bottom action, (The values of (p0 for top and-bottom action
are located In tho section on the examplo-mechanism) . If q( < 1500, allpossibility for top contact is ended and q, must be Incremented- by thetooth angle 8 (see -figures 2 and 5), while * must be incremented by theangle (2 n - X) . For (p > 1501 , top contact Is expected at th -end of bottomaction and y must-be decremented by the-angle 2 8 (seo figure 4, wherethe new top tooth no. 3 comes into antion) . At the same timea, the palletangle * must be decromented by (- 2 71 + X). Those indexing operations-have no effect on thu-continuous computation of the cumulative escape-wheelangle PIIlTOT,
9
il: 1V
Free Motion (Location 200)
After transferring the initial values for time, angles, and angularvelocities from the main program, the subroutine FREE computes the sub-sequent positions and angular velocities of pallet and escape-wheel in freemotion according to equations (E.2), (E.3), and (E.7) and (E.8), respec-tively, for time increments of 10- 5 seconds. In addition to the above vari-ables, PHITOT is also continually computed and caused to be printed out.The decision, whether or not to remain in this subroutine, is made withthe help of the sensing variables, f (according to equation (D-7)) and g'(according to equation (D-11) and now called GP).
If f > 0 and gI < 0, free motion is continued without indexing. Iff > 0 and g' > 0, free motion is also continued, but since contact is nolonger possible for the component pair for which the previous computationswere-made, indexing takes place. This is accomplished in the same manneras described before.
If f < 0, control is returned unconditionally to the main program.If it finds that g' > 0, indexing takes place and control is given back tothe subroutine, FREE. When f < 0, contact is about to take place or hasjust occurred. The program must decide whether this contact Just repre-sents-a close approach, which will be followed by further free motion,whether it represents an impending impact, or whether it is the beginningof coupled motion. To this end, the quantities V and V (see equations(F. 22)- and (F.23)) are computed for the top andP bottoms free-motion tests.The first three cases of the top free motion test of the main program areillustrated in figure 7. With-both angular velocities (rp is PHI (2) and
is DPSI) positive, the following three possibilities exist:
1. If I V I > I[V I , the contacting surfaces will separate
again, free motion wilyresult, and control must be transferred to subrou-tine FREE (location 200).
2. If [ V I = I , the escape-wheel will start-driving thep
pallet in coupled motion, and-control must be transferred to subroutine|i' RKGS Ilocation 99--initiates RKGS).
4Under the present circumstances, if p < 1500, only top contact can
follow; if p > 1500, bottom contact will occur.
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7-!
3. If VPI I< V I ,impact will occur. and control must
be given to subroutine IMPACT (location 300).
The remainder of the top free motion tests and-the bottom free motiontests are constructed along similar lines for different combinations ofangular velocity directions.
VP I
S I
(4o and ' are positive. Distance f is enlarged.)
-Figure 7. Top free motion-test
•Ii
ElI
-;al ,-11
Impact (Location 300)
The subroutine IMPACT uses the cur-rent values of the angularvelocities *Pi and 1 and computes the post impact angular velocities
and Yf, applying equations (F.20) and (F.21). (Note that the tangential
impact has been neglected and, therefore, E = D, and F 2 A1).
After control is returned to the main program, it is decided whoth rfree or coupled motion follows the Impact. This is accomplished by con-sidering the post-Impact contact point velocities, V and V. in the Impactp stests, which are similar to the free motion tests.
If the contact velocities are vectorially equal to each other or if theabsolute value of the difference of their absolute magnitudes is less than2.0 in/sec (5.08 cm/sec), control is transferred to coupled motion. Ifthese velocities are such that they indicate a subsequent separation, whichis more usual, computation -is transferred to free motion.
The computation Is terminated when~either t = .1 seconds or whenPHITOT >_ 3150. These conditions are related to the sample mechanismwhich runs less than .1 seconds foi 3100 of escape-wheel rotation.
-EXAMPLE MECHANISM
The pin pallet escapement of the M525 fuze Is used as the examplemechanism. The following, first, gives the dimensions of the basic escape-ment (standard configuration) and then discusses certain other data andcomputed values which are of importanco in the computer simulation.
Dimensions-of the Mechanism
The standard geometric configuration has the following dimensions:
a = .1931 inches (mean center distance) (.4905 cm)
b .15838 incies (.40229 cm)
c .09683inches (.24595 cm)
r = .01365 Inches (.03467 cm)
a 400
12
S= 600 (see figure-It-1 in appendix H)
X = 109.337'
= 40'
N = 9 (number of teeth of escape-wheel)
I = .91 x 10- 7 lb-soc 2 -in. (moment of inertia IPAL of pallet)P (1.03 x 10-oN - sec2 -m)
I = .17 x 10 - lb-sec2 -in. (moment of inertia ISTAR ofs (1.92 x 10 - 9 N - sec 2 - m) escape-wheel)
Gear Train Driving Escapement
The escapement of the M525 fuze is driven by a clock spring througha step-up gear train with a ratio of 45.98. The timing function of the fuze,which involves a delay of between 2 to 4 seconds, is accomplished oncethe spring driven input gear has rotated through 3100 , This correspondsto a 45. 98 times greater escape-wheel rotation. Since-the -motion becomesstablilized after one cycle of the pallet (corresponding to approximately400 of escape-wheel rotation), one-obtains an excellent idea of the totaltime by computing the response time for 3100 of escapement rotation andmultiplying this result by the aforementioned gear ratio (see the sectionon results for standard configuration).
Stanc'3rd Torque-Used in the Simulation
Measurements on actual fuzes showed that the initial torque on theescape-wheels varied between .0177 (2.000 x I0 - 3 m-N) and .031 in-lb (3.503x 10- m-N). Since the angle of rotation of the input gear is small, the de-crease in torque during the rotation is also relatively small. Therefore, aconstant torque was assumed in the simulation. Its standard value was-chosen to be .0177 in-lb (2. 000 x 10" m-N).
Other Dimensions Associated With Standard Configuration
For purposes of control in the computer program the following otherdimensions are-of interest.
The maximum absolute value-attainable for the dimension g for thegiven escapement is computed according to equation -(H-4) of appendix H:
gMAX 05467 Inches (- 1369 cm)-
13
The, ,-nciated values of the escape-wheel angle yM are obtained
with the use of equation H-10. For top-action
TM =132.4870
Because of this value, Initial coupled motion is started at 1350.(See the section on the computer program,) For bottom action, this angle'becomes:
(PM = 187.5180
The corresponding values for the pallet angles M are found through theuse of equation (A, 8) appendix A and-are computed with g -. 05467 anda = 400. For top action,
*M = 39.4910
and for bottom-action,
*M-= 320.3960
These values serve as valuable checks-on the computer output.
The escape-wheel angles p corresponding to g = 0. i.e. -the position0
when th.,. pallet pin leaves the tooth in coupled motion, are obtained accordingto equation (H-12). For top action,
P0 = 146.3280
Because of this angle, decisions concerning indexing, the free motionand impact tests are based on whether the angle (p is smaller or larger than1500. (See the section on the description of the computer program.) Forbottom action, this angle becomes
T = 206.5121
The corresponding pallet angles are
60.1380 for top action, and0
299.8690 for bottom action, respectively.0
14
-jt
Finally, It Is important to know the center distance between pallet
and escape-wheel for which disengagement will occur. According toequation (G.14)
adi s .208 Inches (.528-cm)
RESULTS FOR STANDARD CONFIGURATION
Appendix I shows computer output for the first, second and eighthcycles of a run for the standard configuration with a coefficient of friction,p = .3, and a coefficient of restitution, , = .25 (run no. 46). Table 1 sum-marizes the results of the first two and one half cycles. In appendix I, Ecycle is defined as the interval between first contacts on top. In tables 1and 3 the interval is between the last contacts on top.) As discussed earlier,top motion in the coupled mode initiates the program. In the subsequentfirst bottom action, the first impact is followed by free motion while thesecond impact leads to-coupled motion by virtue of the cut-off criterion.Starting with the second top contact, both top and -bottom action-consistsof two impacts followed by free motion. The subsequent third impact pro-duces little rebound-of the pallet pin and thus is the beginning of coupledmotion -which lasts until the pallet pin leaves the tooth.
The various impacts cause reversals in escape-wheel motion. Inthe actual mechanism, -this backward rotation is limited by the possibilityof the pin making contact with t'e backface of the next escape-wheel tooth.The program does not provide for the presence of this condition and anycontact of this type makes itself known only if during or after the firstimpact, the escape-wheel angle (p < 132.4870 for top action or (o < 187.5180for bottom action. If such a reversal of the escape-wheel takes place duringcoupledz-motion, g will be less than -. 05467 in, (-.1284 cm). (See the sec-tion on-other dimensions of the standard configuration for discussion of theabove values,)
Inspection of table 1 shows that this condition first occurs in cycle no, 3for top contact, i.e. (p = 132.4800, Table 2 gives-the values of the-maximum-revorsal angles for the first eight cycles. While these reversals-of theescapr- wheel angle never reach critical values for bottom action, they ex-coed critical values a number of times-for top action. (The geometry lendsItself more for this condition with respect to top action.)
:15
I
Tci e fact that the absolute value of gA will be exceeded in certainconfigurations will be accepted or more dissipative values for p and c
will have to be used'. The discussion in the section-on the influence ofparameter changes in the total fuze time will show that a slight increase ofthe centor distance a decreases the absolute value of g at maximum reversalto well below g This correlates with the results for It = .3 and c = 0
given below,
Table 1 shows that the stability of motion is essentially establishedduring the first cycle. The time interval per cycle, now counted betweenthe Instants when the pallet pin leaves the top tooth, is shown to be .00774and . 00773 seconds, Subsequent intervals (not shown here) are .00773,
,00775, .00785, .00784, and .00773 seconds. The total time of the fuzemay be obtained with the help of the time at PHITOT , 3100. Appendix Ishows that t(310.3620 ) = .06058 seconds. Thus, the total fuze delay timebecomes .06058 x 45.98 = 2.79 seconds. (See the section on the examplemechanism,) This result is well within the fuze requirement of between2 to 4- seconds.
Appendix J gives portions of the computer output when the standardconfiguration was run with c = 0, while t = .3 was kept (run no. 70).Table 3-summarizes the events-of the first 3 cycles and table 4 lists themaximum reversal values of the escape-wheel for eight cycles.
As expected, the fully inelastic coefficient of restitution causes theinitial Impacts to be followedby coupled motion. Now, g < gMAX at all
times. (Again, the reversals-are smaller for bottom action.) As in thefirst-program above, the motion stabilizes immediately with cycle Intervalsof .00725, .00730 and .00725 seconds. Subaequent intervals, not shownhere, are .00726, .00729 and .00726 seconds. The total fuze delay timeis computed with t(PHITOT 310.6001) = .05699 seconds (see appendix J) .This results in 2.62 seconds.
Since no high speed motion-pictures of the M525 are available, it is notknown whether contact with-the backside of the escape-wheel tooth isever made.
16
II
Table 1. Summary of events In the first 21 cycles for standard configuration
with it at ,3 and a a ,25 (Program No. 46)
Escape-wheel Time Total escape-anjle q, Time Interval wheel angleType of action (deg) (sec) (sWc) (deg)-
Top: start of coupled motion 135.000 0.00000 0.000start of free motion (end of tooth) 146.491 0-003401 11,491Bottom: impact followed by free motion 194.179 0.00382 19,179maximum return motion (free)continued free motion 192.911 0,00402 17,911impact followed by coupled motion
(cut-off criterion, I VlJ - IVS1j <2) . 193.366 0.00415 18.366maximum return motion (coupled) 193.281 0.00425 18.281-start of free motion (end of tooth) 206.844 0.00645 31.844-
Top: impact followed by free-motion 135.522 0.00688 40.522maximum return motion (free)
continued free motion 133.361 0,00714 38.361impact followed by free-motion 133,435 0;00720 38-435maximum return motion 133.214 0;00724 38.214impact followed by coupled motion
(cut-off criterion) 133,214 0400724 38,214maximum return irotion- (coupled) 133.540 040753 37.540-start of free motion (end of tooth) 146.514 0M011141 0.007741 51.514Bottom: impact followed by free-motion 193.802 0.01154 58,802
maximum return motion (free)continued free motion 192.321 0.01176 57,321impact followed by free-motion 192.803 0.01189 57.803
maximum return- motion (free)continued free motion 192.749 Ml01193 57.749impact followed by coupled- motion(cut-off criterion) 192,790 0;01197 57-790start of free motion (end of tooth) 206,768 001417 71:768,
Top: impact followed by free-motion 135.519 0.01460 80.519maximum return motion -(free) icontinued free motion 132.297 0;01487 78,297impact followed by -free motion 133.394 0-01493 78,394maximum return motion -(free) 133.165 0.01497 78,165impact followed by coupled motion
(cut-off criterion) 133.165 0-01497 78.165maximum return motion (coupled) 132.480 0;01526 77-480-start of free motion (end-of-tooth) 146.496 0.018871 0,007731 91.496'Start of first cycleE; end -of first cycle, start of second- cycle
'Time- interval for first cycle'End -of, second cycle'Time -interval for second cycle
173|
- - - - ------ '--------...
Table 2. Maximum reversal angles and associated values of g forstandard configuration with i = .3 and c = ,25
(Program No. 46)
Top action Bottom actionMaximum Maximumreversal reversalangle, p- g (in.) angle, p g1 (in.)
Cycle No. (wpM = 132.487) ( = -. 05467) (cpM = 187.5180) (gM -. 05467)
1 192.910
2 132.540 -.0547 192.321(-.1389 cm)
3 132.480 -.0549 192.303(-.1394 cm)
4 132.312 -.0555 192.320
(-.1410 cm)
5 132.305 -.0555 192.485
-(-.1410 cm)
6 132.094 -.0562 191.837.1427 cm)
7 131.593 -.0580 192.078(-.1473 cm)
8 132.634 -.0543 192.315(-.1379 cm)
'Not applicable. Occurs during free motion. V
18
- - _
40 r- -C" co C Dt- -cc-)H c wC) o0 DHC
v C C )0, -t - c OrCO O O " M c tO~ - . w ~ C D w ( " C D C t
0m
L..
t- -
C)0H
00 C C'14 1 HL Y m~-U to tU -- V H t V H
C ) czC = C CD 1- ~T-4 HV- r-4 H H 1 T4C C-4 C4~ 0q
o2 o z0 0 0 00- 40-0
0 co
.2 D I t t o t0) CD CDr4 rI) cn r W-o t ) (
0.0 0 . .
0 .0 w
:j 04-00 .4J0 -0 0.o. 0 0 0 10 0
A. 4-'4 1-4-
4.1 4 *0-+ -U 10 8v 8. 10 QHiu1
0- 0 0 0 0 0- 0 44
190
-77 7~ -
Table 4. Maximum reversal angles and associated values of g forstandard configuration with i = .3 and c = 0
(Program No. 70)
Top action Bottom actionMaximum Maximumreversal reversalangle, (p g (in.) angle, (p g1 (in.)
Cycle No. (PM = 13 2 .4 8 70 ) (m -. 0567) (pM = 18 7.5180) (gM =-05467)
193.776 -.04011 (-.1019 cm)2 133.957 -.0495 193.083 -.0418
(-.1257 cm) (-.1062 cm)
3 133.504 -.0512 193.698 -.0403(-.1300 cm) (-.1024 cm)
4 134.171 -.0487 192.883 -.0423(-.1237 cm) (-.1074 cm)
5 133.466 -.0513 193.659 -.0404(-.1303 cm) (-.1026 cm)
6 134.170 -.0487 192.883 -.0423(-.12537 cm) (-.1074 cm)
7 133.466 -.0513 193.659 -.0404(-.1303 cm) (-.1026 cm)
8 134.170 -.0487 192.884 -.0423(-.1237 cm) (-.1074 cm)
A
20
INFLUENCE OF VARIOUS PARAMETER CHANGES ON THE TOTAL FUZE TIME
The following reports on the results of numerous computer runs inwhich a single input or geometric parameter was varied in order to de-termine its influence on the total fuze time. In all cases, the individualchanges were made with respect to the standard configuration in the sec-tion on the example mechanism, with p = .3 and s = .25.
Influence of Escape-Wheel Torque
Figure 8 shows the influence of the escape-wheel torque on the fuzetime. One may compare these timing results with those obtained from thewell-known empirical expression:
If t, and T, represent fuze time and torque, respectively, as associatedwith-the standard configuration, one obtains from the above:
For T, = .75 T1. t 2.79 - 3.22 seconds
12
For T2 = 1.25 T1 t2 = 2.79 -= 2.49 seconds
The results of the simulation (runs 47 and 48 of table of figure 8) showexcellent agreement with the above. The empirical relationship has beenconfirmed time and again-by experiment.
Influence of Pallet Moment-of Inertia
Figure 9 shows that the total fuze delay time increases with an in-crease of the pallet moment of inertia. The ratio of any two fuze periodsis approximately proportional to the ratio of the square roots of the asso-ciated pallet inertias. These results are confirmed by the experiments ofAnderson and Redmond (ref. 7).
Influence of Pallet Escape-Wheel Center Distance
Figure 10 indicates, for the range explored, that the fuze time in--creases as the center distance a is increased. For a total increase of .007inches (.018 cm), the time increase is approximately 7%. This result is [
21
- ~-~ ~-k-
Time 4.(sec.) Standard
configuration
3
2
I3a)
) ,-
.O1 .02 .03
(1.13 x 10-1) (2.26) (3.39)
Torque, lb-in. (m-N)
Run no. Torque Total Fuze Time
(lb-in.) (sec.)
48 .0133 (75% of St.) 3.23(1.503 x 10 - m-N)
46 .0177 (St.) 2.79(2.000 x 10- 3 m-N)
47 .0221 (125% of St.) 2.50(2.497 x 10- 1 m-N)
Itt
Figure 8. Influence of escape-wheel torque on fuze delay tine
22
Time 4(sec.) standard
configuratio)n
3
2
It
0 •.5 x 10-7 1.0 1.5
(.56 x 10-') (1.13) (1.69)
Pallet moment of inertia, lb-sec -in. (N-m-sec2)
Run no. Moment of inertia Total fuze timeof pallet
(lb-sec2-in) (sec.)
57 .68 x 10- 7 2.46(7.68 x 10-1 N-m-sec')
46 .91 x 10- 2.79(std. conf.)
(1.027 x 10- 0 N-m-sea2 )
58 1,137 x 10-7 3.07(1.283 x 10 -' N-m-sec2)
Figure 9. Influence of pallet moment of Inertia on fuze delay time
23
Time-*
(eec.) --
2.86 -
2.82 -
2-.78 - - -
2.74 ad d Lf ui
II I
.190 .194 .198(i483) (.493) -(.503)
Center- distance "Al in. (cm) -
Run no. Center-distance "a"l Total-fuzo -time(15.) (e.
55 .1901 2.070(.4829 cm)-
54 .1911- 2.74(.4854 cm)
53 .1921 2-M(.4879 cm)-
46 .1931 (-std. conf.) 2.79(.4905 cm)
49 .1941 2.81(4930 cm)-
50 .1951 2.82-(4956 -cm)-
51 .1961- 2.84(.4981 cm)-
52 .1971 2.86______ (5006-cm)
Figure 10. Influence of pallet-escape-wheel center distance on fuze delay time
24
generally confirmed by the experimentation of Anderson and Redmond(ref. 7), where it is shown that an increase of time is effected as the centerdistance is increased from a dimension somewhat below nominal.
Influence of Pallet Radius
Figure 11 shows a continuous and quite dramatic decrease in fuzeperiod as the pallet radius c is increased by .008 inches (.020 cm): It:isbelieved that this effect is similar to that observed in connection with adecrease in center distance a, i.e. an increase in pallet radius representsan effective decrease of center distance. The experimentation performedin reference 7 gives good correlation with this-result of the simulation.
influence of Coefficient of Friction
Figure 12 indicates that the fuze time increases as the coefficientof friction, associated with coupled motion, is-increased. One would ex-pect that an increase of energy dissipation -will slow the mechanism.
Influence of Coefficient of Restitution
I According to figure 13, the fuze time increases considerably as thecoefficient of restitution is varied from completely inelastic to partially
- elastic; i.e. from zero to .5. When e = 0 (see description of run no. 70 =
in the section on results-for standard configuration), coupled motion followsimmediately after impact. In run no. 75 (which is not reproduced in thisreport), where e = .5, top action consists of four impacts, three of whichare followed by free motion while the fourth is-followed by coupled motion.Bottom action shows two impacts with the last-one followed by coupled--motion. Each of the impacts is followed by considerable escape-wheel re-versal. These multiple impacts and associated motion reversals seem toaccount for the observed increase in fuze time.
25I
Time - I,(see.)
2.80 - -
2.70
2.6o ..-
2.50 z.092 .094 .096 .098 .100
(.234) (,239) 4.244) (.249) (-.254-
Pallet radius 110, in,. (cnv)
Run no, Pallet-radius 1"ct Total fuze time=zUn,) (soc.-)
66 .0928 3.04(.235 cm)-
65 .0938 2.972383-60)-
64 .0948- 2.90(.2408-cm)-
63 .0958 f!.811(, 2433-cin)-
46 .0968 -(std. conf.) 2.?9(,2459-cm)
59 .0978 2.73(.2489 cnm)-
60 .0988 2.70(.2510 cm)-
61 .0998 2.67(,2535-cin)
62 .1008 2.60• (.2560 cmi)- _,,,
-Figure 11. Influence ofpallet radius on fuze delay-time
26-
Time
(sec.)
2.90
2.80
2.701 standi ri config'uration
2.60
.20 .25 .30 .35
Coefficient of friction, i
Run no. Coefficient of Total fuze-time-- friction, i (sec.-)-
68 .20 2.68
67 .25 2.73
46 -.30 (std. conf.)- 2.79
69 -.35 2.92
-Figure 12. Influence-of coefficient of friction-of coupled motion on-fuze delay time
27
Time(see.) -
3.20 - - -..-
3.10 .... ; /
3.00o .... . . .. .,,2.90
2 .80 . t
2,70 - -
2.6o - -. sta idard conf guration2.6o05
0 • .2 .3 .4 .5
Coefficient of restitution, c
'Run no. Coefficient of Total fuze timerestitution, C (sec.)
75 .5 3.15
74 -4 3.01
73 .3 2.87
46 .25 (std. conf.) 2.79
72 .2 2.78
71 .1 2.66
70 0.0 2.62
Figure 13. Influence ofcoefficient of restitution on fuze delay time
28
3 I t,
DISCUSSION AND RECOMMENDATIONS FOR CONTINUED WORK
There is no doubt that-the goal of a workable computer simulationof the pin pallet runaway escapement has been attained. This is mainly
due to the fact that every effort was made to keep the program and its con-trols as simple as possible.
As a consequence of this, there is presently no built-in sensingmechanism that indicates whether the pallet pin makes-contact with thebackside of an adjacent tooth-during free motion. While-this abnormality
did-not occur for the M525 mechanism, there is no assurance that it willnever occur.
Secondly, there is the unresolved problem of the somewhat excessiveescape-wheel reversal for the standard configuration unless fairly unrealisticvalues are used for the coefficients of friction and restitution. (Recall the dis-
cussion in the section on the-results for the standard configuration.) Table 5shows the extent to which this escape-wheel reversal depends on the centerdistance between the pallet and escape-wheel pivots.
Clearly, the reversal increases with a decrease in center distanceand-deureases as the center distance is enlarged. Note, -that for a = .1941inches (.4930 cm), which is .001 in (.0025 cm) abovenominal, I g <I gMA
Note-further, that the disengagement center distance of .208 inches (.528 cm)-(see-end of the section on the-example mechanism) is-much larger than thevalues which have been explored- in table 5.
Excessive escape-wheel reversal may be indicative of the fact thatthe simple impact formulation is not sufficiently descriptive, as shown byAnderson and Redmond (ref. 7-) or that the escapement- only operates in
the-expected manner when the center distance is somewhat enlarged fromnominal.
To resolve this problem, -high speed motion pictures must be taken
in order to be able to observethe actual motion. Following this, certainmodifications of the impact model may have to be made.
Even if the question-of -the motion reversal cannot be fully resolved,the-present simulation is sufficiently descriptive to undertake the followingextensions:
29
-. --.-- -.-
1. Adaptation of the pin pallet runaway escapementsimulation to a centrifugally driven mechanism, suchas the M577 safeseparation device (SSD) .
2. Adaptation of the pin pallet simulation to a spring-driven timing mechanism.
3. Modification of the present model to accommcdate thesimulation of a plate pallet (verge) type runaway escapement.
"successful simulations, incorporating the present escapement model,of both the M577 SSD and the M12SA1 booster have been recentlycompleted. The results will be given in a future report.
30t
430
T;F
Table S. Escape-wheel reversal as a function of center distance a
Value of g atmaximum reversal
Center distance, a* during top action Program number(in.-) (in.)
.1901 -.0662 55(.4829 cm) (-.1681 cm)-
.1911 -.0635 54(.4854 cm) (-.1613 cm)-
.1921 -.0600 53(.4879 cm) (-.1524 cm)
.1931 (std. conf.) -. 0580 46(.4905 cm) (-.1473 cm)
.1941 -.0536 49(.4930 cm) (-.1361 cm)-
.1951 -.0520 50(.4956 cm) (-.1321 cm)
.1961 -.0471 51(.4981 -cm) (-.1196 cm)-
.1971 -.0441 52(.5006- cm)- (-.1120 cm)
*Dimension-a is variable, otherwise dimen3ions- are standard config-
uration with - .3, e = .25 and g -. 05467 in. (-.13886 cm).
: I31
F,
REFERENCES
1. "A Study of the Dynamics-of an Untuned Clock Mechanism,"Contract No. CST-1224 with NBS, Department of Physics, VirginiaMilitary Institute, Lexington, VA, September 1953
2. "A Study of the Effect of Geometrical Factors upon the Behavior ofan Untuned Clock Mechanism," Contract No. CST-1224 with HarryDiamond Ordnance Fuze Laboratories, Department of Physics,Virginia Military Institute, Lexington, VA, June 30, 1954
3. "A Study of the Non-Uniformity in Running Rates of a Certain Typeof Time Delay Mechanisms," Contract No. DAI-49-186-502-ORD (P) -186with Harry Diamond Ordnance Fuze Laboratories, Department ofPhysics, Virginia Military Institute, Lexington, VA, August 31, 1954
4. "The Effect of Unbalanced Verge Torques, Acceleration FrictionTorques, and Verge-Starwheel Material upon Running Rates ofCertain Time Delay Mechanisms," Contract No. DAI-49-186-502--ORD (P) -186 with Harry Diamond Ordnance Fuze Laboratories,Department of Physics, Virginia Military Institute, Lexington, VA,January 1955
5. "Part I: An Untuned Clock-Mechanism of Special Design-. Part I:An Experimental Study of-the Motion of an Untuned Clock," Contract-No. DAI-49-186-502-ORD-(P)-186 with Harry Diamond-Ordnance FuzeLaboratories, Department ofPhysics, Virginia Military Institute,-Lexington, VA, June 1956
6. 'Untuned Escapements" Session-5, Picatinny ArsenalHorologyFamiliarization Course, Bulova Watch Co., Valley Stream, NY
.7. M. E. Anderson and S. L. Redmond, "Runaway (Verge) EscapementAnalysis and Guide for Designing Fuze Escapements, -'1NWCCL TP 860,Naval Weapons Center, Corona Laboratory, Corona. CA 91720,December 1969
8. R. X. Brennan, "Several New Mathematical Models Incorporated-into the Picatinny Arsenal Version of the NWCCL Runaway Escapement-Digital Program, "Information Report No. 50, Fuze Development Branch,Picatinny Arsenal, Dover, NJ 07801, October 1973
32
.]A
-~It -'M R r ~
9. L. P. Farace, "Runaway Escapement Redesign-M125A1 ModularBooster," FA-TR-74045, Frankford Arsenal, Philadelphia, PA19137, December 1974
10. W. Goldsmith, "Impact-The Theory and Physical Behaviour ofColliding Solids," Edward Arnold (Publishers) Ltd, London,England, 1960
11. G. W. Hemp, "Dynamic Analysis of the Runaway EscapementMechanism," The Shock and Vibration Bulletin, Bulletin No. 42,Naval Research Laboratories, Washington, DC, January 1972
12. W. Horenstein, "Summary of Reports on an Untuned Clock
Mechanism," Technical Memorandum, Diamond Ordnance FuzeLaboratory, Washington, DC 20438, December 1954
13. D. L. Overman, "Analysis of M125 Booster Mechanism," HDL-TR-1550, Harry Diamond Laboratories, Washington, DC 20438,June 1971
14-. J. N. Shinkle, "A General Study of Verge Escapement Performance,"SC-RR-69-495, Sandia Laboratories, Albuquerque, NM, October 1969
15. J. P. Sirpal, "Rocker Mechanisms," Institution of Engineers(India), Vol 45, No. 5, pp 132-138, January 1965
33
APPENDIX A
KINEMATICS OF COUPLED MOTION
Figure Al shows the kinematic relationship-of the escape-wheel andthe pallet during coupled motion.
UNIT VECTORS
The unit vectors nt and nn are along and perpendicular to the con-
tact surface of the escape-wheel tooth in the indicated directions, respec-tively. Thus,
nt = cos (p% -a)i+sin (y -a) j (Al)
n = - (sin {o-a i+cos {(p-a} W)2n
In addition, the unit vectors nb# for the escape-wheel, and nc, for
the line connecting the pallet pivot and the center-of the pallet pin, areintroduced:
- =cos (p i +-sin J A3)nb
n cos -T+sin-'T (A4)
INPUT -- OUTPUT RELATIONSHIP
- The mechanism loop-equation is used to determine the pallet angle' and the pallet pin locationg with respect to the tip of the escape-wheel
as functions of the escape-wheel angle (p and the-applicable mechanismconstants, i. e.
0= b(cos -y-+ -sinoTj + g[cos(y - a)i + sin(p - a)j-
4, + r[- sin(o - ai + cos(p - a)j-] - c(cos i 1 + sin *- j-'
+ a (A5)
The above is rewritten in component form:
b cos y + g cos({p -x) - r sln(cp-- a) -ccos*+ a =O(A6)
34
-%-
h
n
b
_ _ _ _ _ _ _
Figure At. -Coupled motion kinematics-
35
b sin qp + g sin(Wp-a) +r cos((p-a) -c sin*= 0 (A7)
The angle jip is obtained from (A7):
and
cos 4,(A9)Cos c' - (b sinyp+ g sin(p- a) + r cos(p - a)1
Substitution of equation (A9)- into equation (A) and subsequent squaringof both sides of the resulting expression leads to:
(b cos (p + g cos (q- a) - r sin (p -a) + a] 2
= c2 - [b -sin p( + g sin(p- a) +r cos(p- a)]2 (AO)
Rearrangement gives:
-g + g [2b cos a + 2a-cos (y - a)
+ -[a + b + r2 - c2 + 2br sin a + 2ab cosqp - 2ar sin(p -a)]
=0 (All)
Finally, solution of equation (All) leads to:
-H+ rHT - 4
g= 22
where-
H= 2[b cos a +a-cos(q - a)) (A13)
K= a2 +b2 +r - _c + 2br sin a + 2ab cos-o-2ar sin (p-a)
(A141
The correct value of g, as obtained from equation (A12), must have thesmaller absolute magnitude. -
36 -
!I
OUTPUT VELOCITIES
Implicit differentiation of equation (All) furnishes the velocityof the instantaneous contact point on the pallet with respect to the coinci-dent point on the escape-wheel during coupled motion:
g -Y(A15)
where
P =bsin (p + gsin (y - a) + rcos ((p - a) (AI)
S =g + bcos a + acos((p - a) (A17)
Differentiation of equation (A8) leads to the pallet angular velocity duringcoupled motion:
Q= + Asin (qp - a)(A18)(c) cos4
where
Q =bcos (p + gcos-((p -- a) -rsin ((p - a) (A19)
OUTPUT ACCELERATIONS
Differentiation of equation- (A15) results in an expression for therelative acceleration of the-contact-point- on the pallet with respect to thaton the escape-wheel:
- t i _4; Pa + 4P _Q a- + 2to a sin_((p - a) -- -s( (A 20)S
Further, differentiation of equation- (AiB) furnishes an expression for theangular acceleration of the pallet during coupled motion:
e=Q- P +-2g-qtcos((p - a) + sin (q - a)_-ccos ,
37.
APPENDIX B
DIFFERENTIAL EQUATION OF COUPLED MOTION
Figure B-1 shows free body diagrams of the pallet and escape-wheel,as they are found during coupled motion. The torque T acts on the escape-wheel. The forces between the escape-wheel and the pallet are representedby the normal forces ±P in the direction of nn and the friction forces ± (pIPn)n -nin the direction of n . The sign of the friction forces must be such that whentthe contact point K on the pallet pin moves in the direction of the positivevelocity g with respect-to the coincident point L-on the escape-wheel, thefriction force on the pallet pin must act in the negative direction of g. (Notethat g, as given by equation (A15) in appendix A, has the same directionas the relative velocity VK/L•)
The frictional resistance at the pivots O and O is neglected, sincethe pivot radii are commonly held very small Pcausing the friction momentsto become insignificant.
The differential equations of rotation will first be written separately:for the pallet and the escape-wheel. Since the output angle * is a functionof the input angle y -throughout coupled motion, both expressions may-becombined by way of the common force magnitudes.
Equation of Motion for the-Pallet
nnh nI t p~
Equation of Motion for the-Escape-Wheel
Aix (-P )n +-Bj1x 4( L n + kY- (B2)
The-moment arms 3,1, , 1, and D1 are given by equations (C6), (C7),-(C4)-, and (C5), respectively. I and I represent the moments of inertia
of the pallet and the escape-wheel with respect to their pivots.
Appropriate vector operations on equations _(Bi) and (B2) furnishthe-following scalar expressions:
38 1
I . . .
- V' #rtr.~..~ ~ ~'rE-4.
-~~~~C ,--1 - . ,4- .
N0
L
p4 4-
U.
IMI
P (DI-CL )IpV (B3)n p
and -P- (A, + B, LK) + T = IsV (B4)
When the above expressions are-equated in terms of Pn, one obtains:
I (A,1-(DI Ca- + "" T (B5)
The angular acceleration is now expressed in terms of 4i and qwith the help of equation (A21) of appendix A. Equations (Al5), (Ai8),and (A20) are utilized to replace - , 4 and g , respectively. After con-siderable simplification, one obtains the following expression for qI:
(B6)-
where
Q + 21 sin (y --a)
U° =" c)cos, B7)
andI [2aP . +2a =p
= [-. - .cos( t( - a) -P --T sin' (p--a)
S~~~ sn(p) a2 P2+ .sin - sin (p- - a
c-tan + Pasin -((P - a) (B8)-• cos I L S
40
ti
Equation (B6) is now substituted into equation (B5). This results in:
I U (A, + jBj_ L) A, + Bl_ L [
• + I + Iv (F1 (DI~ 1 ~( RC, (DI+ BI
ST (B9)
For computational purposes, equation (B9) is rewritten in the followingform:
(p +Wp' =Y (BI0)
where I v
El
W (Bl)
E1 F,
TF-1
Y= I(B12)
p + _.El Fl
and
D- -C (B13)
Fl- =-A1 - + i'(B14)
iii t
41
APPENDIX C
MOMENT ARMS
DETERMINATION OF MOMENT ARMS C1 AND D
Figure C-1 suggests the- following loop equation for the determina-tion of the vectors q andT :
cn - rnn - D n C n 0 (Cl)
When the components in the nt and n directions are separated, oneobtains:
ccos* + rsln(cp - a) + D, cos (y - a) - C1 sin (cp- a) = 0 (C2)
and
csin* - rcos ((p - a) + Drsin (y - a) + C1 cos (y - a) = 0 (C3)
Simultaneous solution of the above expressions leads-to-the followingvectorial- expressions:
[r + csin(yo- a - *)In (C4)n
and
= ccos (q -- a - nt (C5)
DETERMINATION-OF MOMENT ARMS A AND-
Inspection of figure C-1 leads to the following expressions for the
vectors A, and B1 :
= (bcosa + g)nt (C6)
(Note that g-is negative during contact.)
Further,
B1 = (bsina)n (07)
42
i,,
,
E1
I.m
EA
0
4-)-
VLL
43
APPENDIX D
CONTACT SENSING EXPRESSIONS
When the mechanism is in free-motion, i e . y and ~iare independentof each-other, it is necessary to know-the distances f and gt, in figure D-1,to determine the relative positions. The quantity f, which indicates thedistance-of the pallet pin from the escape-wheel face, vanishes at-the in-stant of impending impact. When contact between the pallet and the escape-wheel becomes impossible, i.e. the pin has left the tooth, g' > 0.
By inspection of figure D-1, the loop equation for free motion isgiven by:
b + gn t + (r + f)n - cn + ai = 0 (D1)n c
(Note that p and *i are independent variables and are assumed-to-beknown.)
In componentform, the above becomes:
bcoso + g'cos(V - a) - rsin(9- - a) - fsin (y - a) - ccos* +-a = 0
_(D2)
bsino + g'sin ( - a) + rcos( -- a) + fcos (y - a) - csin* = -0
(D3)
Multiply equation (D2) by sin( -- a)
bcoss- sin (p -a) + sna)cos (y - a) - rsin (qp - a)
-fsin2 (9 - a) ccossinE(y - a) + asin (9 - a) + asin-((p - a)
-O (D4)
44-
_ __
- ~~-- - ~-'- - -
tt
VI
'A.9.
.0 CI,
0*
C
C*'A
-~
-4------I ~~-A
-. 4'-1 L
-' LILL H
0~ 0p4 1't1*
~1£
II
'I
45t
- - -- -- e -~- t 11
F urther, multiply -equation -(D3)- by cos ((p- - a)
bsirnpcos(yp-- a) + g'sin(cp -- a)cos((p -a) +-rcos2 ((p - a)4 + fcos -(y- - a)- - csinVcos ((p - a) (5
=0I-Subtract -equation- (D4) from- equation (D5)
bsin-[p - (yp - a))- + r + f +-csin [(9p - a) -4 asin-(C - a)
=0 (D6)
This furnishes
f =asin-(p - a) - bsina- - -csin [-(y - a) -4] r (D7)
-Multiply (D2) by cos (y- --a)-
bcospcos-(q -- a)- +- 'cos 2 _ (cp - -a)- - rsinlcp - -a) cos-(c -- a)-
-fs inp (P- -a) cos-(y - a)co ccos4'cos-(y- - -a)
+ -acos ((p -a)-
A ~ ~~~~Multiply -(D3) by sin:-;;:i2 p- )+ sn ( -acoa -a (8
bsinypsin(cp -- a-+gsn-T-a-+ri-y-a-o-(--)
Fialy add n (y-a)tos-(q - a)- - csin4'sin~p-a
Finll, dd-(D)-and--(D9)-
Ibcos((p ly -al+g-cos* +-acosfy-a
=0O -(DID)-
Thtus,g' =ccos[(q,--a)- -' -bcosa -acos(y - a) -(Dli)
46-
APPENDIX E
FREE MOTION EQUATIONS
During the froo motion -phaso of the meohanism, the pallht imovosindopundontly of the usoapo-whool.
Apain, thu frictional, rossitanco ati the pivots 0 and 0 tire nogloctod.p
FREE MOTION OF PALLET
In tha ubsonco of all uxtornal torquos, tho diffurontial oquation ofmotion for the pullot Is giv n by:
With an Wniitil angular velocity jo tit the limo to , whon the froomotion-starts, tho angular vloolty of the pallet tit any timo Is:
If '41 (t ) lfurthur Intogration givos thu aingular -displacuinunt of0 0
tho-pallat at tiny thu t to bo:
.- " *0 (t - to) +I 0-o (1333)
I :! ~~~Th -Whito t 4s otuod fromi tiht stalrt of" the mlotion of the tiscuptuiont,
FREE MOTION-OF THE ESCAPE-WHEEL
Whon a constant torquo T acts on the oscapo-whoot, its--diffurontialoquation of motion Is givon by:
p; U (1E4)T
J, 8
and tho applicable initil conditions tiru:q) (to) Rs
* I4
5,I5:; 47
5 1'
and
P (to ) (4) (E6)0 0
Integration-of equation (E4) gives the following expression for theangular velocity of the escape-wheel at any time t:
K T (t (t-t 0 ) +q)0 (E7)
s
Further integration furnishes the angular displacement:
Tp= (t - to) + (t - t o (138)0 0- 0 0
s
-48
I, "$
APPENDIX F
IMPACT EQUATIONS
Before the impact equations can be given, certain velocity expres-sions associated with the impact points on-the pallet and the escape-wheelmust be derived. These velocities occur during free motion preceding im-pact when the angular velocities ( and * are independent of each other.
VELOCITIES AT IMPACT POINTS JUST BEFORE IMPACT
Figure F1 shows the velocities of the contact points K and L on thepallet and the escape-wheel, respectively, just before impact. The com-ponents in the direction normal to the escape-wheel tooth are:
V =4'ikx D1 n for the pallet (Fl)pn1 I t
andVsni = Pk x A, nt for the escape-wheel, (F2)
where the subscript i stands for the angular velocities prior to the instant
of impact.
The components of velocity in the tangential direction are given by:
V t -k x Cln (F3)pt i r nand
Vst =cikx B I n (F4)
The expressions for the moment arms are obtained from appendix C.
To determine the direction of the tangential impact, the sign of therelative-tangential velocity Vt must-be known. This velocity is given by(refer to equations (F3) and- (F4)): V
VtV =4'-x Cn - ik xBnn (F5)
49
i
II
to 0
0
VIPA4c
503
Substituting equations (C4) and (C7) into equation (FS) produces:
V = ( [r + csin (p - - I)] + qbsina) nt (F6)
IMPACTS ON PALLET AND ESCAPE-WHEEL
The impact simulation is based on the classical angular impulse-momentum model shown in figure F-2 for the pallet and the escape-wheel.
P represents the normal impulse between the components. Further,n
a frictional impulse pP which is tangential to the contact surface and
has the direction of the velocity V t (see equation (F6)), is assumed be-
cause of the angular displacements during impact.
The angular impulse T on the escape-wheel, which is due to the
torque T and has the duration t of the impact-phenomenon, is disregardedsince it is felt that its-magnitude is small when compared to that of the otherterms involved. The angular impulse cn-the~pallet then becomes:
VJp =DxP[--- nt -(F7)
p n n nI~
After the cross product has been executed, the following equation isobtained:
Vt
Jp = Pn - D l- - (CI tI) kF8)
Similarly, the angular impulse on the escape-wheel reduces to:
VJ =A i x-Pn)n +B 1 xpP -n (F9)
This becomes
VtTs P n + tB, tk 10)
51
- -" - U.--- ---- ~--
1 r4
Nt
0S
.
AL
52L
Equations (F8) and (F10) are now substituted into the following angularmomentum equations:
Ip -of - i) ( p F11)
and
Is((Of -(F12)
The subscript-f indicates the angular velocities after impact. Substitutionof the scalar portions of equations (F8) and (F10) into equations (Fl)and (F12), respectively, as well as the subsequent simultaneous elimina-tion of P from both of the resulting expressions, leads to:n
IpF2 *f + IsEf =-Ip F 2 i + IsE2 i (F13)
: where Ewh Dr XI V t (F14)
and: F2 = A1 + pB1 t;(FiS)Vt
To solve-for the angular-velocities after impact, it is now necessaryto make use of the concept of the coefficient of restitution. This coefficientis adapted to-the-present situation in the following manner:
V -Vpnf snf
V -V (Fi6)pn i sn.
:1
Using equations (Fl) and (F2) -and letting
V = kx Dl-n t for the pallet (F17)pnf
53
-- _|
_ HI'-
V qof k x A, n for the escape-wheel (F18)
where P6f and Vf are the respective angular velocities of the pallet andthe escape-wheel after impact, produces
(F19)0 iD, _ Al )
Simultaneous solution of equations (F13)- and (F19) furnishes the desiredangular velocities after impact.
Thus,OfAl - c(*iDl - OiAl)l
f -1 D(F20)
f D
and I F2 F i AtpF2 i + IE 2 Pi D-
I FA,D + I E2 -Dt s
GENERAL EXPRESSIONS FOR THE NORMAL VELOCITIES ATTHE IMPACT POINTS
The-normal velocities at the impact points of the pallet and-theescape-wheel are also used in the logic-of the computer program. If qand * represent-the angular velocities-of the escape-wheel and the-palletjust before-or after impact, expressions are derived for these velocitiesusing (Fl)-, -(F2), (05), and (C6):
For the pallet
V = *c cos (y -a -) (F22)p
For the escape wheel
IV =q (bcosa + g) (F23)-
-(Note the simplification of the- notation.)
54-
,71
k ll__ __ _
IiAPPENDIX G
COMPUTER PROGRAM
: 55
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-' ---- -----'
APPENDIX H
SOME GEOMETRIC RELATIONSHIP
DETERMINATION-OF gMAX
Figure H-1 shows the geometrical relationship between the pallet pin ofradius r and the right hand escape-wheeltoth when the quantity g (equation(A12) in appendix A) reaches its maximum possible absolute value. Thefollowing relationship holds between the angles a and P:
8(H1)
where8 = 360 (H2)
N
and N stands for the number of teethon -the escape-wheel.
The distance ST of the triangle OST is determined with the help ofthe sine law:
' bsin(8/2)sin (Or - P) (H3)
where b represents the escape-wheel radius.
Finally, g is obtained from:
gy __ r(H4)! g~MAX = Ttanp -4
To use gMAX in connection with expressions derived in appendix A, it
must be furnished with a minus-sign to conform to the origin of the
St - nn system.
-
9MAX
T
I 0\ I /
• \ I7 360 /
N
0
Figure Hi. Configuration for gMAX
67
: I-r - -
DETERMINATION OF ESCAPE-WHEEL ANGLE (pM CORRESPONDING TO g =gMAX
The-angle y = PM. which corresponds to the maximum possible
value of g = gMAX' is obtained with the help of equation- (All) of appendix A:
-9 gt + gM [2bcos a + 2acos ((oM - a)]MAX
+ a + b2 + r 2 - c2 + 2brsina + 2abcos M -- 2arsin(qM - a)]
S-= (H5)
After the expansion of the trigonometric terms cos (pM - a) and sin (pM - a),
the following expression results:
LsinpM + McospM + N =0 (H6)
where-
L 2a (g xsin a- - rcos a)
M = 2a (gMxcosa + b + rsin a)
(MAXcN= gMX +a'- +-bV +-r' -c2 + 2b (gMAxcOs a + rsin a)
Equationi (H5) may now be rewritten with the help of the following trigono-metric identities:
2tan 1'-sin y M =- (H7)
1 + tan -- )
, an C-){ -Cos TPM (H8))
68
!;!,_ I'
This leads to:
(N - M)tan' (--) + 2Ltan (--) + (M + N) =-O -(H9)
This expression may now be solved for the angle %0M using the solutionfor a quadratic equation. Accordingly,
T[-L A IL2 - N3 + MzM 2 tan L N-M (H0)
The two solutions correspond to top and bottom action.
DETERMINATION OF ESCAPE-WHEEL ANGLE q CORRESPONDING TO g 0
The-angleO =- , which is associated with the instant when thepallet pin leavesthe-escape-wheel tooth during coupled motion, is obtainedfrom equation (All) with~g =O.
An expressionsimilar to equation (H6) results:
Losinqyo +-M-cos(p + N 0Q (H.1)0 0 0 0 0
where
L -2arcosa
Mo = 2a(b-+rsin a)
N O =-a b 2 - c + 2brsina
0-
Equation (H11) is solved-in the same manner as equation (H6)-.
This results in:
= tan-1 (HI 2)
Again, top and-bottom action results are provided.
69
H" I-
Ort
DISENGAGEMENT DISTANCE BETWEEN PALLET AND ESCAPE-WHEEL PIVOTS
Figure 11-2 shows the disengagement configuration of the symmetrical
pin pallet runaway escapement. The pallet angle * equals - when both
pallet pins -together are least advanced with respect to the- escape-wheel.The center distance a is such thatthe escape-wheel radius-band the palletpin radius- r are collinear.
The -disengagement center distance a is determined with the helpdis
of the-cosine law:
(b + r)= a2 + c' - 2 adis (c)cos (T) (H13)dis i
From the- solution of the quadratic-equation, one finds:
adis= (C) Cos 4CCS X)]- [c_ - -(b-+ r)']
(H14)
Li
70
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-APPENDIX I
COMPUTER OUTPUT FOR STANDARD CONFIGURATIONWITH = .3 AND = .25
73
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DISTRIBUTION LIST
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