AD-All7 593 CITY COLL RESEARCH FOUNDATION NEW YORK F/G 19/1AUTOMATED STRENGTH DETERMINATION FOR INVOLUTE FUZE GEARS. (LfFEB 82 6 6 LOWEN. C KAPLAN DAAKIO-79-C-0251

UNCLASSIFIED ARLCO-CR810 NL

13flffllfllff lI.ImIEEEIIIIIIIEEohmhmhhmhEmhEEhmhmhmhmhuEhmmmhmmhlEIIIEIIIIIIIIIEEEIIIIIEIIIII

111.0 128 111122III __ IIII ~L136

440 1111120

11111.2 11111-.111 11.8

MICROCOPY RISOLUTION TST CHARi

WX

--. --

4,. 'M';- :

MOM?.

TO

T"A

A AGM-a Qo

, 4 0.MA

Mw,

A-0

VaNn"'.

rw"

V a R %k: IMan

.. 2., oi l

o 1 OSLO%

Iwo

- tK4.'

mom

WWI w Wwo,

too,AM am An

441"6.'Aw

i.. glel'4t,",

em 4.. ..........

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Date Entered _________________

REPOT DCUMNTATON AGEREAD INSTRUCTIONS______ REPORT___DOCUMENTATION_____PAGE_ BEFORE COMPLETING FORM

1. REPORT NUMBER 1__2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

Contractor Report ARLCD-CR-810 0 /,7 g ____________

4. TITLE (and Subitl~) S. TYPE OF REPORT A PERIOD COVEREOD

AUTOMATED STRENGTH DETERMINATION FORINVOLUTE FUZE GEARS

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(q) S. CONTRACT OR GRANT NUMBER(*)

Gerard G. Lowen and Clifford Kaplan DAAKlO-79-C-0251City College Research Foundation

Frederick Tepper, Project Engr, ARRADCOM9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

AREA & WORK UNIT NUMBERS

City College Research FoundationNew York, NY 10031

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

ARRADCOM, TSD February 1982STINFO Div (DRDAR-TSS) -IS. NUMBER OF PAGES

Dover, NJ 07801 21214. MONITORING AGENCY NAME & ADORESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report)

ARRADCOM, LCWSLNuclear and Fuze Div (DRDAR-LCN) UnclassifiedDover, NJ 07801 15a. DECL ASSI FICATION/ DOWNGRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of Cho Abatract entored in Block 20. If different hrom Report)

10. SUPPLEMENTARY MOTES

19. KEY WORDS (Continue an, revere* side if naoaaaay and identify by block nwmber)

Involute gears Fuze gearsLewis factorGear strength

I& ABSYNAC? (CMt0a m gwate. S N&*" umaa h Iimiff by block amiber)

This investigation developed two computer programs which allow the determina-tion (if the tooth profile coordinates, together with the associated Lewis gearstrength and AGMA geometry factors, for all types of involute gears includingfuze gears. These two programs differ only in the manner in which the rootportion of the teeth is obtained. The origins of the involute and trochoidcoordinate expressions and of all programming schemes are discussed in detail.Sample runs and operating instructions are provided.

DD , PO 13 ROMiON OF I uIOV 451619 fT9 UNCLASSIFIEDSECUIT CLAMIFICAIION OF THIS PACE (Ifen Do&a Sneered)

ACKNOWLEDGMENTS

The authors gratefully acknowledge the help of Messrs. I. Siegel

and R. Lobou of the City College Computation Center and Ms. Denise

Duncan of the Mechanical Engineering Department of the City College

of the City University of New York.

Aocession For

NTIS GRA&IDTIC TABUnannouncedJustificatio

Distribution/

AvallabilltY Codes

vall and/orDint Special

m ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ s I . ...........-.. '-...

CONTENTS

Page

Introduction 1-5

Description of Computer Programs 13

Sample Runs of Programs LEWIS CIRCLES 36

and LEWIS ENVELOPE

Discussion 51

Appendixes

A Determination of Involute Coordinates 59

B Determination of Coordinates of Path of 63

Effective Rack Point

C Generation of Tooth Root in Presence of 75

Fillet Radius on Rack Cutter

D Initial and Final Angles for Trochoid and 95

Involute Generation

E Gear Strength Calculations: Determination 111

of Lewis and AGMA Factors

F Program MATRIX 129

G Program LEWIS CIRCLES 143

H Program LEWIS ENVELOPE 181

Distribution List 213

TABLES

Page

1 Selected computations of Lewis and AGMA

geometry factors 50

2 Comparisons of results of programs LEWIS CIRCLES

and LEWIS ENVELOPE 52

FIGURE

1 Tooth profile and Lewis parabola for maximum

load F 6

1. INTRODUCTION

This report describes the development and use of

certain computer programs which allow the determination of rele-

vant information with respect to strength calculations for all

types of involute gears including fuze gears.

Both programs,LEWIS CIRCLES and LEWIS ENVELOPE, furnish

the coordinates of the left side of any involute tooth (See Figure 1),

together with the associated Lewis and AGMA geometry factors. The

determination of these factors is based on the assumotion that the

maximum contact force F is applied at the tip of the tooth and that

therefore the origin V of the inscribed Lewis parabola is located at

the point of intersection of the line of action of this force and the

tooth centerline. The point of tangency between this parabola and the

tooth profile, together with the x-coordinate XTAN required for the

determination of the above factors, is found by an iterative process.

The involute portion of the tooth profile is obtained in

both programs by a mathematical description of the unwinding of the

base circle tangent. The trochoid or root portion of the tooth

profile is obtained by two different methods.

Program LEWIS CIRCLES finds the associated coordinates

by a simulation of the hob cutting or drafting process, while program

1-5

Y

Tooth Centerline

Left Tooth

Profile

LewisParabola

Pon ofTagec 'T

Po0 ofX

Figure 1. Tooth profile and Lewis parabola for maximum load F

-6-

LEWTS ENVELOPE makes use of an analytical method, based on

the relationship between generating curves and envelopes.

While the results of both programs are identical, program

LEWIS CIRCLES cannot be used when the hob fillet radius is

zero.

To assist the user as much as possible with the details

of operation of these programs, the following outline of input

and output parameters, is given: (The input and output parameters

of both programs are identical).

a. Input Data (Algvnfor unity diametral pitch)

1. Number of Teeth

2. Effective distance (see Appendix B)

3. Unity diametrical pitch

4. Hob pressure angle

5. Circular tooth thickness at standard

pitch circle

6. Hob fillet radius

7. Gear tooth addendum.

8. Angular increments for involute and

trochoid generation

b. Output Data

1. Results of various intermediate computations

7 -8

2. Lewis factor

3. AGMA geometry factor

4. Various parameters associated with

the point of tangency between Lewis

parabola and tooth profile

5. Coordinates of Tooth profile

As a further aid to the user, material on which this

report is based is compiled in the appendixes.

Appendix A: Determination of Involute Coordinates

This appendix gives the derivation of the x and y-coor-

dinates of the involute portion of the tooth profile of an arbi-

trary tooth in terms of the coordinate system shown in Figure 1.

Use is made of the angle a which describes the unwinding of the

base circle tangent.

-9-

Appendix B: Determination of Path Coordinates of Effective

Rack Point

After defining the concepts of effective rack point

and effective distance for hobs with and without fillet radii, this

appendix gives derivations for the path coordinates of the effective

rack point for undercut and non-undercut gear teeth. In order to

accomplish this, the generating angle ip, which describes the rolling

of the rack pitch line on the gear pitch circle, is introduced. Again,

the results are given in terms of the tooth centerline coordinates

shown in Fiicure 1.

Appendix C: Generation of Tooth Root in Presence of Fillet

Radius on Rack Cutter

Two methods of obtaining the coordinates of the tooth

root are shown. The first method simulates the cutting or drafting

process and is used in program LEWIS CIRCLES. The second method, which

is based on certain relationships between generating curves and

envelopes, forms the basis for the root geneiation in program LEWIS

ENVELOPE.

-10-

Appendix D: Initial and Final Angles for Trochoid

and Involute Generation

This appendix first gives a review of existing procedures

for the determination of the inner form radius for undercut and non-

undercut teeth. Subsequently, expressions for the limits of the

generating angles a and 'p are derived for both types of teeth.

Appendix E: Gear Strength Calculations. Dletermination

of Lewis and AGMA. GeometryFactors

Initially, the origin of the Lewis gear strength

formula is reviewed and expressions for the determination of various

associated parameters are derived. Subsequently, the iterative

procedure for finding the point of tangency between the Lewis parabola

and the tooth profile, which is used both in program LEWIS CIRCLES

and in program LEWIS ENVELOPE, is explained.

Appendix F: Program MATRIX

This appendix describes and lists program MATRIX, which

is used to determine the size of the arrays in the two computer

programs. It also contains sample outputs of program MATRIX

which pertain to the sample runs of both LEWIS programs.

(See Section 3.)

Appendix G: Program LEWIS CIRCLES

A complete listing of the program LEWIS CIRCLES is

given, together with outputs of the two sample runs.

Appendix H: Program LEWIS ENVELOPE

This appendix contains a complete listing of program

LEWIS ENVELOPE. In addition,the results of the two sample runs

are shown.

-12-

I1

2. DESCRIPTIONOCOPTRRGAM

The following section gives descriptions of the

essential steps of the programs LEWIS CIRCLES and LEWIS ENVELOPE.

Program LEWIS CIRCLES is listed, together with sample runs,

in Appendix G. Appendix H1 furnishes the same for program LEWIS

ENVELOPE. The specific sample runs of both programs are discussed

in Section 3.

The input and output parameters of both programs are

identical and will therefore only be discussed in detail in

conjunction with program LEWIS CIRCLES.

Since the vector Y and the matrix X, as used for the

determination of the root profile in LEWIS CIRCLES (see also Appen-

dix C-2: Iterative Method of Obtaining Root Profile), may be

very large, it is necessary to use care in the dimensioning of

these arrays as well as in the request for computer storage space.

Appendix F gives program MATRIX which allows the determination of

the sizes of Y and X, as well as of the vectors Xl and Yl which are used

for the initial storage of the involute coordinates.

I. Program LEWIS CIRCLES (See Appendix G')

a. Dimensioning of Arrays

(For more details on the use of the various arrays

see description of computations below).

-13-

The vectors Xl and Yl are used for the initial storage of

the involute coordinates of the gear tooth (see Appendix A).

The minimum number N of locations for either of these arrays

is given by: (See also Appendix F)

N fin ain (-1)

where

Ctin initial involute generating angle [See equ. (D23)

of Appendix D].

afin final involute generating angle [See equ. (D24)].

S = increment of involute generating angle.

The arrays X2 and Y2 are used for the final storage of all

values cf the tooth profile coordinates. The required number of

locations is represented by the sum of those needed for the

vectors Y1 (or Xl) and Y. It can only be determined once the

length of the vector Y is known. The later is obtained with the

help of the aforementioned program MATRIX, which also finds the

size of matrix X.

-14-

b. Input Parameters

The following parameters represent the required

input data for the program. (They are also printed out on

top of the first output page).

N = number of teeth in gear under consideration

BEFF = beff for unity diametral pitch (See Appendix B)

PD = Pd = 1.000, unity diametral pitch of hob

THETAD = 0, the pressure angle of the hob (the D at the end

expresses that an angle is given in degrees)

CAPTCS T cs the circular tooth thickness of the gear at theN

standard pitch radius Rp = yd, in terms of Pd = 1.000

RC rc , the rack fillet radius for unity diametral pitch

KADD = Addendum constant of gear, with the magnitude of the

addendum defined by KADD/PD , in terms of Pd = 1.000

DELPSI = A , the increment of the roll angle of the rack

pitchline on the pitch circle of the gear (See note below)

DELAL = Ac, the increment of the involute generating angle(See note

below)

CONB = Initial value of constant B of Lewis parabola[See equ. (E32)]

-is-

DELCON = AB, the increment of the constant B of the

Lewis parabola (See note below)

NRITE = Test constant. When NRITE = 1, all the tooth

profile coordinates stored in X2 and Y2 are

printed out. These coordinates are not printed

out when NRITE = 0.

NOT'F: The choice of the magnitudes of Aa, AP and AB is discussed in:ction 3.

c. Computations

Starting with line 29 of the program each location

of the matrix X is filled with the value -10.00. This is necessary,

since the subsequent sorting process must determine the largest

value in rows whose locations are either unfilled or contain

negative numbers. If such an unfilled location contains a zero,

it will be found as the answer ralher than the actual x-coordinate

of the left root profile.

Subsequent to line 33 various gear parameters, such as

the pitch radius R the base radius Rb and the outside radius Ro

are computed. Further, the following quantities are determined:

BEFFB = b ff PD the actual distance to the effective

point for the given diametral pitch

THETAB = 0b see equ. (Al)

-16-

The later is computed with the help of the

involutomery formula given by equ. (E16) of Appendix E.

For this case:

1 TCS , 1 =

and

T = T b R2 = Rb

so that

R cosOT b = T cs P- 2 Rb [INV (0) - INV (8)]

or

Tb = c oO+ 2 Rb tan (e) - 61

and finally,

T b T ____s + 2( tane - 0 (2)b Rb Rb

-17-

Following line 41 one finds:

EPS b + a, for use in eou's. (B8) and (B9)

beff - tane, see equ. (B22)DELTA =

Rb sine

BALLOW = beffmax according to equ. (BI)

Lines 45-47 represent a test which decides with the help of

equ. (Bl) whether or not the tooth is undercut.

Lines 48-60 are applicable for undercut teeth. The following

computations are performed:

A = a, according to equ. (B2)

TAU = EPS, from line 42

The inner form radius RF = Rf is obtained with the help of equ.

(D9). This non-linear algebraic equation is solved by subroutine

DEKKER. The radii Ro and Rb represent the limit within which the

root Rf is determined.

ALPHIN = cin' according to equ. (D23)

ALPHF 'fin' according to equ. (D24)

GAMT Ytot' according to equ. (D16)

- 18-

PSIIN = in' according to equ. (D18)

PSIF *fin according to equ. (D14)

Lines 62 to 82 are applicable to teeth which are not under-

cut. The essential computations are given by:

A = a, according to equ. (BIO)

TAU = EPS + DELTA, from lines 42 and 4 3 [This irs T in equ's. (B22),(B24)

and (B25)]

RF = Rf, according to equ. (D12)

GAMT = tot' according to equ. (D16) with the above RF

ALPHIN = cin' according to equ. (D23). The test for ain

has the purpose of avoiding an almost zero number

when a zero should be obtained

ALPHF = afin, according to equ.(D24) with the RF of line 66

PSIIN = 0, according to equ. (D20)

PSIF = 'fin' according to equ. (D19)

-19-

The computations represented by lines 82 to 88 are

applicable to both types of teeth. In addition to converting

certain angles from radians to degrees for printing out, the

following parameters of the center of the rack fillet are

computed:

A2 a2, according to equ. (Cl)

B2 b2 , according to equ. (C2)

Lines 89 to 104 concern themselves with the determina-

tion of the czordinates of the involute portion of the tooth. It

is at first most convenient to store XI = xinv and YI = Yinv which

are obtained from eqiations (A5) and (A6), respectively, in thevectors Xl(I) and Yl(I). For I = 1, the angle a = in. When this

counter has attained its maximum value, the angle a is just a

little larger than afin" Note also that this maximum value of the

counter I is.stored as IPREV for future use. For purposes of

subsequent computations it is desirable to have the counter of the

coordinate locations of the tooth profile start at the tip of the

tooth and end at that root location which corresponds to * fin'

For this reason, the sequence of the involute coordinate counter

is inverted in the operations of lines 100 to 103. These coordinates

are now stored in the vectors X2 and Y2. (Note that the counter

-20-

________ _____

number corresponding to IPREV now denotes the location of

the involute coordinates associated with the angle ain).

The determination of the coordinates of the tooth

root, i.e. the trochoid, starts in line 105. Initially, the

vector Y(I) and the matrix X(I,J) are created in the manner

described in Appendix C-2. Subsequently, a search procedure

determines the largest value of the x-coordinate X2 associated

with a given y-coordinate Y2. It is to be recalled that, the

parameters appropriate to the type of tooth under consideration

have already been stored and that, therefore the same basic

expressions can be used in the program whether the tooth is under-

cut or not.

The following explains the manner in which the program

fills the vector Y(I) and the matrix X(I,J):

(1) The angle 4 is made equal to in and the counter I,

for che y-coordinate, and the counter J, for the

number of the circle, are initialized.

(2) Starting with location 25 (line 111), the fillet center

coordinates XC = xc and YC = yc are computed according

to equ's. (C3) and (C4) or equations (C5) and (C6).

Further, the y-coordinate YT = Ytr of the effective noint

of the rack is obtained with the help of equ. (B9) or

-21-

equ. (B25). If these computations were made for the

first position of the fillet circle, i.e. for J = 1 (in

the program I = I is used), control is

transferred to location 80.

(3) Starting with location 80, the value for YT is multi-

plied by 1000 and the result is truncated such that the

integer IYT results. This operation, which in essence

multiplies a y-coordinate given to one one thousandth of

an inch by 1000, is necessary for subsequent identificagion

purposes. In addition, IYT is stored in vector Y(I) in

location 90. (Thus, rather than storing the actual

y-coordinate its thousand-fold value is used). If the

the effective point of the first circle is involvedI = 1.

With the above accomplished, IYT is again made into a

real number and divided by 1000 ( giving the y-coordinate

to the nearest one one thousandth of an inch) for further

work. It is now labelled YTTT [The y of equ. (C7) I. In

order to be able to stop the computations for points on

a given fillet circle,once

Y YC Y- r c 0 (3)

the test of lines 127 and 128 has been devised.

The x-coordinate associated with YTTT is then obtained

-22-

.. . . .. . .... .. . . . .. . .. .. . ... . . . . .

with the help of equ. (C9) of Appendix C and stored

in X(I,J), where J is the circle number and I is the

counter of the y-coordinate. This sequence, which starts

on line 124 and ends with line 132 is continued until the

limit of equ. (3) is reached for a given circle. In

each step the counter I is incremented by 1, the integer

IYT is decreased by 1 and stored in Y(I), and finally the

associated x-coordinate is stored in the appropriate X(I,J).

When the computations for any fillet circle are completed,

control is transferred to location 150.

(4) Starting with location 150, the stage is set to compute the

x and y coordinates of points on subsequent circles: The

angle P is decremented by Al, and a test is made to check

whether *fin has been reached. In addition, the circle

counter J is incremented by 1 as long as < *fin* To

obtain the data for a new circle, control is transferred to

location 25 again. In case the limiting angle 'fin has

been reached, the value J corresponding to the last circle

is stored as JFIN, and control is given to location 500 to

commence the search for the largest value of X(IJ) in a

given row of the array.

(5) If control was transferred to location 25, the steps

described in (2)above are repeated for the new position

of the fillet circle. In order to be able to form an array

-23-

as shown in Table C-I of Appendix C, it is

necessary to establish that location Y(l) of any

previous circle which contains an IYT of the same

magnitude as that which corresponds to the effective

point of the new circle.

This is accomplished by the sorting procedure starting

in line 115 and ending in line 121: The counter I, which

serves to call on the already existing Y(I), is initialized

and YT is made into the integer IYT. The desired location

Y(I) is found by comparing the content of each of the

existing Y(I) with the value of the above IYT. When the

test

Y(1) - IYT < 0 (4)

is met, control is transferred to location 90 and the

present values of I and IYT are reestablished as the

counter and the content of this Y(I).

At this point all operations for a given circle may be

performed as described beginning with the second paragraph

of section (3) above.

(6) As stated earlier, once all the possible Y(I) and X(I,J)

are filled, control is transferred to location 500 to

-24-

search for the x-coordinates of the root

profile.

The preparation for the determination of the largest

value of X(I,J) corresponding to any y-coordinate Y(I) starts in

location 500 (line 139). YFIN, the smallest value of the

y-coordinate for which X(I,J) has been found earlier, is

obtained by way of YCFIN. This variable is obtained with the

help of the fillet center expressions (C4) or (C6), evaluated at

S= fin Then, similar to equ. (3):

YFN = YCFIN - RC (5)

Subsequently, the following operations are performed:

(1). The counter I is initialized. It is used to keep

track of the Y(I) and X(I,J) of before.

(2). The counter K = I + IPR.EV is introduced in location

501. This counter makes it possible to add the

final values of the root coordinates to vectors

X2 and Y2, which presently only contain the involute

coordinates in locations from I = I to I = IPREV.

(See line 98).

-25-

(3). KFIN = K is introduced, in order to be able

to label the last values of X2 and Y2.

(4). The sorting process for each row of the array

X(I,J) starts in line 144 with the initialization

of X2(K) = -10.00. The loop between line 145 and

147 starts by comparing the above value with that of

the content of X(l,l) i.e. the first value in the

first row. If X(l,l) is larger than -10.00, X2(K)

is made equal to X(l,l). If it contains the

initially read-in value of -10.00, X2(K) remains as

-10.00 (See line 31). In the next comparison, the

content of X(1,2) is compared with the current value

of X2(K). In case X(1,2) > X2(K) , this value of

X(1,2) is made the current value of X2(K). This

sorting process is repeated until J = JFIN , the

number of the last fillet circle (See line 137).

(5). The outer loop of the sorting process includes

lines 142 to 151. Once the largest value of any

row of X(I,J) has been stored in X2(K), the content

of the associated Y(1) is divided by 1000.00 and

stored in Y2(K). This operation is repeated until

the last value of Y2 becomes eaual to or smaller

than the previously determined YFLN.

-26-

Control is transferred to location 21, whenever

the vectors X2 and Y2 contain a complete set of coordinates foi.-

the left tooth profile.

The determination of the various parameters, needed

for the computation of the Lewig factor, starts in location 21

with the following:

PHIRO =, according to equ. (E22)

BETA = , according to equ's. (F21)

THETAO = 0 , see equ. (E29)

RV = R , according to equ. (F28)

v

The search for the point of tangency between the

Lewis parabola and the tooth profile is conducted according to

the outline in Appendix F-4. It is first necessary to find the

value of the counter K, used in X2(K) and Y2(K), which corresponds

to that of the y-coordinate RV of point V of Figure E-4. The

range between this point and the already known YFLN, for which

K = KFIN, is the range in which the tangency will occur.

For computational reasons, the point START rather

-27-

than point V is chosen. Thus,

START = RV - .005 (inches) (6)

The location of that Y2 which corresponds to START

is found in the loop between lines 165 and 169, Once the number

of this location is found, it is labelled BEGIN.

The determination of the tangency point is carried

out in the loop between lines 172 and 179. Using the counter J,

the value of K is varied between

K = BEGE + J- 1 (7)

and

K = BEGN + KFN-i , (8)

where KFIN is the difference between the KFIN of line 143 and

BEGIN of line 170.

The operation consists of the following steps:

(1). The value of XPAR = x par is computed according to

-28-

equ. (E32), using the initial value of CWB B

[See equ. (E31)], for the full range of values of

Y2(K). In each step, the difference

DIFF2 = IX2(K)I - XPAR (9)

is computed [See also equ. ('E33)]. If this difference

is less than or equal to zero, the point of tangency

has been found and control is transferred to location

109 (line 180).

(2). If the point of tangency cannot be found for a given

value of CCNB, this value is decremented by DELCCN

(see line 178) and equ. (9) is used to test a new

series of XPAR values. This process is continued

until the values of X2(K) and Y2(K), associated with

the tangency point, are found.

Finally, with X2(K tan ) and Y2(K tan ) found, EX = x is

computed qccording to equ. (35), and the Lewis factor as well as

the AGMA geometry factor are determined with the help of equ's. (Eli)

and (F36), respectively.

-29-

d. Output Parameters

(All lengths are given in inches and all angles in degrees).

The output of the program first states the numerical

values of the input parameters N, BEFF, PD, THETAD, CAPTCS, RC,

KADD, DELAL, DELPSI, DELCN and CCNB. Subsequently, it is stated

whether or not the tooth is undercut. In addition, the output

gives numerical values for the following quantities:

RP Rp = Nd the pitch radius of the gear

RB = R cose , the base radius of the gearP

RO = R + KADD the outside radius of the gearP P d

RF = Rf , the inner form radius, either according to

equ. (D9), or according to equ. (D12)

EPS = b + 0, in degrees

THETAB = 0 , according to equ. (2), in degreesb

DELTA b eff - tanO , according to equ. (B22), inRb sine

degrees

-30-

TAUJ E PS or EPS + DELTA

GAMT - to , according to equ. (D16)

AL PHIN a in according to equ. (D23)

ALPHFIN a fin 'according to equ. (D24)

FSIN = i 'p according to equ. (D18) or equ. (D20)

FSIFIN = fin ,according to equ. (D14) or equ. (D19)

PHIRO = according to equ. (F-72)0o

BETA = 3,according to equ's. (F21)

CAPTO = T 0 tooth thickness at R 0 , computed according to

equ. (E16)

THETAO = R ,computed according to equation (ElS) and0 1

furnishing a result identical to that of equ. (E29)

RV R Rv , according to equ. (E28)

-31-

FX x , according to equ. (E35)

YLI'WIS Y , the Lewis factor, according to equ. (Ell)1

YAGMA Y AGMA I according to equ. (E36)

TANGENCY K = value of counter K at point of tangency between

Lewis parabola and tooth profile

Y2(KTAN) = y-coordinate of tooth profile associated with

point of tangency

X2(KTAN) = x-coordinate of tooth profile associated with

point of tangency

XPAR TAN = x-coordinate of parabola associated with point

of tangency

CON1 = final value of constant B of Lewis parabola

NRITE = Test constant (see section b above)

If NRTTE = 1, the program causes the printing of the

tooth profile coordinates Y2(K) and X2(K) for all values of K,

starting from the tip of the tooth down towards its root. Thc

transition from the involute to the trochoidal part of the profile

-32-

LL

occurs where the values of Y2(K) are given to one one-thousandth

of an inch. Recall that, since the program computes the left tooth

profile, the values of X2(K) are negative.

II. Program LEWIS ENVELOPE (See Appendix H)

a. Dimensioning of Arrays

Again, the vectors X1 and Yl are used for the initial

storage of the involute coordinates of the gear tooth. The sizes

of these vectors are also determined with the help of equ. (1), or

by way of program MATRIX (see Appendix F).

The vectors X2 and Y2 must again hold the values of the

involute coordinates as well as those of the root coordinates. The

required dimensions of these arrays may be ottained by addin , the

number of rows of vector Y1 to the following number of trochoid compu-

tation intervals:*

N = in '4fin

Nt -fi (see below) (10)r

where

*in initial trochoid generating angle [For undercut teeth

use equ. (D18), for non-undercut teeth equ. (D20) is

a The number of these intervals is identical with the column numbcrfor the tooth root interval, as given in the output of programMATRIX in Appendix F. See also equ. (FM).

-33-

applicable. iin is positive or zero.]

'fin final trochoid generating angle. [For undercut teeth

use equ. (D14), for non-undercut teeth use equ. (D19).

fin is always negative. This explains the form of

equ. (10).]

At = increment of roll angle of the rack pitch line

(See earlier.)

b . Input Parameters

The input parameters for program LEWIS ENVELOPE are the

same as those for program LEWIS CIRCLES.

c. Computations

With the exception that MATRIX X does not exist in this

program, LEWIS ENVELOPE is parallel to LEWIS CIRCLES up to line

98: Various initial gear parameters are computed. It is decided

whether or not the gear tooth is undercut and the associated para-

meters are determined. The involute coordinates are computed and

stored in vectors Xl and Y1, with the last computation corresponding

to the outside radius of the tooth. In order to be able to obtain

-34-

Ud

tooth profile coordinates which start with the tip of the tooth,

the order of the vectors X1 and Yl is again reversed and the

result is stored in the vectors X2 and Y2. The location number

of the last pair of involute coordinates (now associated with the

angle cin ) corresponds again to the value of IPREV.

The determination of the tooth root coordinates starts

in line 99 and proceeds in the following manner:

(1) The angle P is made equal to in' and the counter I

is initialized at

I = IPREV + 1 (11)

(2) A loop, starting with line 104 and ending with line

112, computes the coordinates XENV = x and YENV = venv "env

according to equ's. (C21) and (C22), respectively, until

= Wfin (PSIFD). The results are stored in vectors X2

and Y2, between the locations I = IPREV + 1 and I = KFIN.

The computations for the LEWIS and AGMA factors start in

line 114 (statement no. 21) and are identical to those of program

LEWIS CIRCLES up to and including line 151.

d. Output Parameters

The output parameters of program LEWIS ENVELOPE are identical

-35-

to those of program LEWIS CIRCLES. The print-out of the tooth

profile coordinates differs from that in program LEWIS CIRCLES

somewhat. For the present case, it was possible to separate the

involute coordinates from the trochoid coordinates by appropriate

headings, i.e. INVOLUTE PLOT ar TROCHIOD PLOT.

3. SAMPLE RUNS OF PROGRAM LEWIS CIRCLES AND LEWIS ENVELOPE

This section first discusses the results of two identi-

cal sample runs which were made with each of the two programs. The

first type of run involves a non-undercut 36 tooth gear. The second

concerns itself with an undercut 12 tooth pinion.

Appendix G lists program LEWIS CIRCLES on pages G-2 to

G-9. The associated output for the 36 tooth gear is given on pages

G-10 to G-18, while that for the 12 tooth pinion covers pages G-19 to

G-37.

Appendix H lists program LEWIS ENVELOPE on pages H-2 to

H-9. The associated output for the 36 tooth gear is given on pages

H-1O to H-14. There are two outputs for the 12 tooth pinion. The one

with the larger angle increment A5 is given on pages H-15 to H-19, while

that for the smaller increment Aip is reproduced on pages H-20 to H-32.

Finally, the results of certain selected computations of LEWIS and

AGMA factors are given in tabular form.

-36-

I. Sample Runs of Program LEWIS CIRCLES

a. 36 Tooth Gear

1. Input Parameters (See also Section 2-1b)

With one exception, all input data of this run are printed

out on top of the first output page (See p. G-10).

N =36 teeth

BEFF = 1.215 in.

PD 1.0, diametral pitch

THETAD = 20 degrees

CAPTCS = 1.846 in.

RC = .04 in.

KADD = 1.425 in.

DELAL = .250 degrees

DELPSI = .125 degrees

CONB = 4.0

DELCON = .001

-37-

Further, on the bottom of the first output page:

NRITF 1

2. Array Dimensions

The output of the program MATRIX on p. F-12 gives the

following minimum dimensions for the various arrays associated with

the 36 tooth gear:

Length of rows for involute

vectors XI and YI: 94

Length of rows and colunirs

for tooth root arrays: POW = 370, COL = 85

This leads to the minimum number

of locations for vectors X2 and Y2: 94 + 370 = 464

As a consequence of the above, the following dimension

statements are included in the program of this run (see p. G-2):

X1(100), Yl(100)

Y(400) , X(400, 100)

X2(500), Y2(500)

-38-

LA

3. Output Data (See also Section 2-Id)

The first output statement on page G-10 conveys the

fact that the "TOOTH IS NOT UNDERCUT". The subsequent data

block is useful for checking purposes:

RP = 18.000 in.

RB = 16.91447 in.

RO 19.425 in.

RF = 17.11373 in.

THETAB 7.58392 degrees

GAMT = 0.0 degrees

FPS 23.79196 degrees

DILTA -8.82052 degrees

T'AIJ =14.97144 degrees

ALPHIPN 8.82052 degrees

ALPHIFIN 32.35465 degrees

PSIIN : 0.0 degrees

PSIFIN -10.6258 degrees

-39-

PSIRO 29.4532 degrees (This is actually PHIRO acc. to

Section 2-Id)

BETA 28.56269 degrees

CAPTO = .60383 in.

THETAO .89052 degrees

RV = 19.25831 in.

The first line of the lowest block of results on page

G-10 deals with the essential output of the program, i.e.:

EX .55278 in., the Lewis dimension x

YL EW IS = .3685, the Lewis factor

YAGMA = .36221, the AGMA geometry factor

The second line of this data block gives further values

which are of interest for checking purposes:

TANGENCY K 237, i.e the point of tangency between the tooth

profile and the Lewis parabola was found

by the program to occur at Y2(237)

Y2(KTAN) 1C.937 in. , y-coordinate of tooth profile

at point of tangency.

-40-

X2(KTAN) -1.13277 in., x-coordinate of tooth profile

at point tangency

XPAR TAN 1.132784 in., absolute value of x-coordinate

of Lewis parabola at point of

tangency

CONB 1.809, final value of constant B of Lewis

parabola

The listing of the coordinates of the left tooth profile,

i.e. of the values stored in the associated vectors X2 and Y2,

begins on top of page G-11. Recall that the values of these

coordinates are printed out in such a way that they start at the

tip of the tooth and end at its root. The involute portion of the

profile begins with K = 1 and ends with K = 96. While the first

set of values of the trochoidal portion of the tooth profile at K = 97

is not specifically marked, it may be recognized by the fact that

Y2(97) represents the first y-coordinate which is given in terms of

one one-thousandth of an inch. This trochoidal portion of the profile

ends with K = 460. Note that the po'int of tangency at K = 237, is

well above the last point of the profile.

b. 12 Tooth Pinion

(Appendix 6 shows only the output of this run.)

-41-

IU

1. Input Parameters

With the exception of the NRITE statement, all input

data are printed again on top of page G-19:

N = 12

BEFF = 1.0526 in.

P DI

TIlETAD = 20 degrees

CAPTCS = 1.57079 in.

RC .300 in.

KADD = 1.000 in.

DI.AI. = .250 degrees

I171,PS = .500 degrees

CONB = 4.0

DILCON = .001

Further, as shown on the botton of page G-19:

NRITF =I

2. Arra Dimensions

(The program listing containing the dimension statement

-42-

given below is not reproduced in this report.)

The output of program MATRIX on page F-13 gives the

following minimum array dimensions for the given pinion data

(For rationale, see parallel section for the 36 tooth gear.).

XL(lSO) , YI(150)

Y(995) , X(995, 52)

X2(1145), Y2(1145)

3. Output Data

The first output statement on p. G-19 indicates that

the "TOOTH IS UNDERCUT". The subsequent data block may be

interpreted as discussed in section 2-Id and is parallel to the

one shown for the 36 tooth gear. The first line of the lowest

block of results again represents the essential output of the

run, i.e.:

EX .34446 in., the Lewis dimension x

YLEWIS = .22964 , the Lewis factor

YAGM'A : .2364 , the AGMA geometry factor

The second line of this data block gives:

TANGENCY K = 729

-43-

Y2(KTAN) = 5.019 in.

X2(KTAN) .780016 in.

XPAR TAN = .78003 in.

CONB = 2.903

The listing of the tooth profile starts on p. G-20.

The involute portion of the tooth profile begins with K = 1 and

ends with K = 152. The trochoidal part of the profile extends

from K = 153 to K = 1133. Note that the point of tangency, at

K = 729, is well above the last computed profile point.

IT. Sample Runs with Program LEWIS ENVELOPE

a. 36 Tooth Gear

1. Input Parameters

With the exception of CONB = 5.0, the input parameters

of this run with program LEWIS ENVELOPE are identical with those

used in the run with program LEWIS CIRCLES (See section 3-Ia). As

before, these data are listed principally on top of the first output

page of the program. This output page is reproduced on p. H-10 of

this report. (Note that the format of the first output page of

program LEWIS ENVELOPE is the same as that of the first output page

-44-

of program LEWIS CIRCLES.)

2. Array Dimensions

A certain part of the results of program MATRIX may

be used to determine the dimensions of the vectors X1 and Yl as

well as those of the vectors X2 and Y2. Thus, from p. F-12:

Length of rows for involute vectors Xl and YI: 94

Since there is now one set of root profile coordinates for each

position of the angle i, the number of rows of these coordinates

is equal to the number of columns given by program MATRIX under

the heading of "Number of rows and columns for tooth root arrays".

[See equ's. (10) and(F8)]. Thus, the number of root coordinate

pairs equals 85 according to p. F-12.

The minimum dimensions for X2 and Y2 then become

94 + 85 = 179. Actually, the following larger dimensions were

included into the program for this run (see program LEWIS

ENVELOPE on p. H-3):

Xl(180) , Yl(180)

X2(l000), Y2 (1000)

-45-

3. Output Data

The central output data block, starting with "TOOTH

IS NOT UNDERCUT" and ending with RV = 19.25831 is identical with

that obtained by way of program LEWIS CIRCLES, as shown on

p. G-10. The first line of the lowest output data block again

furnishes the principal results of the run. (Recall that the

root profile coordinates are computed in a different manner in

program LEWIS ENVELOPE.) Thus,

EX = .55277 in.

YLEWIS - .36851

YAGMA = .36221

These results are identical with those obtained by way

of program LEWIS CIRCLES. The second line of this lowest block

is useful for comparison purposes (see Section 4):

TANGENCY K = 117

Y2(KTAN) 16.93642 in.

X2(KTAN) -1.132901 in.

XPAR TAN 1.132924 in.

CONB - 1.809

-46-

I ..... ... ... .... ... ..

The listing of the tooth profile coordinates starts

on top of page H-11. As for the comparable run with program

LEWIS CIRCLES, the involute portion of the profile is given

from K = 1 to K = 96. The root part of the profile extends

from K = 97 to K 182.

b. 12 Tooth Pinion

Program LEWIS ENVELOPE was run twice for the 12 tooth

pinion. The input data of the first run are identical with those

used earlier in conjunction with program LEWIS CIRCLES (See section

3-1b). In the second run the increment A was reduced to .05

degrees in order to observe the influence of the more closely

spaced y-coordinates of the root profile.

1. Input Parameters for Run with A = .5 degrees

The input data for this run, as shown on p. H-15,

are the same as those given in section 3-Ibl.

2. Array Dimensions for Run with A = .5 degrees

The minimum array dimensions for this run may again

be obtained in the manner discussed in section 1I-a2 above with the

help of the results of program MATRIX on p. F-13.

-47-

Since the lengths of the rows of involute vectors Xl and Y1

must be at least 150,

Xl(150) , Yl(I50)

The number of columns for the root array X of program LEWIS

CIRCLES was determined to be 52. Therefore,

X2(202) , Y2(202)

3. Output Data for Run with AP = .5 degrees

The central output data block of p. H-15 is iden-

tical with that obtained from program LEWIS CIRCLES (see page G-19).

fhe principal results, as given in the lowest output block, are as

follows:

EX .34443 in.

YLEWIS - .22962

YAGMA - .23643

TANGENCY K = 177

Y2(KTAN) = 5.01112 in.

-48-

X2(KTAN) -. 781767 in.

XPAR TAN = .781767 in.

CONB = 2.903

These results are essentially identical with those

obtained from program LEWIS CIRCLES. The difference in the

y-coordinate of the tangency point is due to the different

manner of computing the tooth root coordinates. (See discussion

in section 4 below.) The listing of all tooth profile coordi-

nates is given on pp. H-16 to H-19 of Appendix H.

4. Output Data for Run with AW = .05 degrees

The input/output data of this run with a reduced

angular increment are listed on pp. H-20 to H-32 of Appendix H.

The principal results are as follows:

EX = .34447 in.

YLEWIS = .22965

YAGMA = .23635

TANGENCY K = 385

-49-

N BEFF CAPTCS RC KADD Lewis Factor AGMA12 (in.) "( 7. (in.) (in.) Factor

12 1. 000 1 .5F7n79 .239 1. 000 .24189 .248 07

'13 .25493 .25922

.33822 .32876

30 '"" .36134 .34784

Is .39494 .37542

5O .40209 .38121

100 .43683 .40976

-;0 j-____ ______ .45828 .42 337

12 .800 1.57079 .304 .800 .33517 .325981

13 .34831 .3371215 ... '37016 3SS21

4S ' "... .47858 44417

300 ."5-370 .48946

12 1.0526 1 57079 .300 1.000 .22964 .23640l1i3 ~ .24321 .24817Jf if i _ _

(For all runs: Diametral pitch is unity and hot) pressure angle is

200. All runs made with program LEWIS CIRCLES.)

Table I

Selected Computations of Lewis and

AGMA Geometry Factors

-so-

Y2(KTAN) = 5.02859 in.

X2(KTAN) -.777906 in.

XPAR TAN = .777909 in.

CONB 2.903

III. Selected Computations of the Lewis and the AGMA Geometry

Factors

Table 1 shows the results of selected determinations

of the Lewis and the AGMA geometry factors. In all cases, the

computations were made with the help of program LEWIS CIRCLES

and a diametral pitch of one, together with a 20 degree hob

pressure angle were used.

4. DISCUSSION

This section first presents a discussion of certain

differences in the results for the two sample gears of section 3,

as obtained with programs LEWIS CIRCLES and LEWIS ENVELOPE. (Table

2 juxtaposes the essentials of these results.)

-51-

!4

Subsequently, a number of special features and

peculiarities of both programs are pointed out.

I. Comparison of Results for 36 Tooth Gear and 12 Tooth Pinion

When comparing the results of the two programs for a

given set of data, it must be kept in mind that the point of

tangency of the Lewis parabola is generally located in the root

portion of the tooth profile and that the programs produce the

root profile in different ways.

Program N " YLEWIS YAGMA Y2(KTAN) IX2(KTAN) XPAR TAN CONBam l|_N 1 (inches) (inches) (inches)

CIRCLES 3 .36852 .3622 16.937 1.13277 1.13278 1.809I

At = .1251 i

1ENVELOPE I 361.36851 .36221 16.9364 1.13290 1.13292 1.809

AiP -.12.

CIRCLES 12 .22964 .23640 5.0190 0.78002 0.78003 2.903

IA = .500

ENVELOPE 11 121.22962 .23643 5.0111 0.78173 0.78177 2.903

A = S00

ENVELOP]' 121.22965 .236 5.0286 0.77791 0.77791 1q03

A = .050 _

Table 2

Comparisons of Results of Programs

LEWIS CIRCLES and LEWIS ENVELO-L

-52-

Program LEWIS CIRCLES furnishes one x-coordinate

of the tooth root for every one one-thousandth of an inch of

the y-coordinate. The accuracy of these x-coordinates improves

as the increment Ap is made smaller, i.e. as more circles become

involved in this simulation of the drafting process. (See Appendix

C-2.)

Program LEWIS ENVELOPE produces accurate x-coordinates

of the root profile, because of its analytical origin. To obtain

these x-coordinates in one one-thousandth of an inch increments,

the increment AW has to be quite small.

Let the results in Table 2 for the 36 tooth gear he

compared first: Both sets of results were obtained with Aq, = .125,

and they are essentially identical. The spacing of the y-coordinates

due to program LEWIS ENVELOPE may be seen on p. H-13 of Appendix H.

The point of tangency is at Y2(117) = 16.9364 in. The preceding as

well as the succeding ialues of the y-coordinates are approximately

.006 inches away. Because of the magnitude of these distances, it is

probably fortuitous that such close agreement could be obtained.

While the results for the Lewis and AGMA factors as

well as for the constant B are also essentially identical in the

-53-

r

case of the 12 tooth pinion, there are some small differences

in the x and y-coordinates of the root profiles. Even though

the increment Ai = .500 is quite large, it had to be used in

running program LEWIS CIRCLES, since the associated array X

already had a 995 x 52 dimension and with that required a very

large amount of computer core. (See output of program MATRIX

on p. F-13.) When this increment was used to run program

LEWIS ENVELOPE, the magnitude of the spacing of the y-coordinates

near the point of tangency was approximately .020 inches. (See

K = 177 on p. H-19.)

When program LEWIS ENVELOPE was run with AqP = .05.

the spacing of the y-cooridnates surrounding the point of tang-

ency was of the order of .002 inches (See K = 385 on p. H-27.).

This result appears to be the best of the three.

Aside form these small discrepancies, both programs

furnish excellent results.

II Special Features of Program LEWIS CIRCLES

a. All input data must be expressed in terms of unity

-54-

diametral pitch.

b. The program cannot be run for a hob fillet radius

RC = 0, since its simulation of the drafting process

requires such a radius. This limitation can be over-

come by choosing a very small radius for the case of

a sharp cornered hob. (By letting RC = .040 in. for

the unity diametral pitch computation, an actual gear

of 60 diametral pitch will be cut by a cutter with a

radius of .00067 in.)

c. The angular increment Aa must be chosen sufficiently

small if one desires a fairly close spacing of the

y-coordinates of the involute part of the tooth profile

for drawing purposes. A Aa = .25 degrees caused a spac-

ing of between .015 and .030 inches for the 36 tooth

gear and a spacing of between .003 and .015 inches for

the 12 tooth pinion. The larger spacing will always

occur near the tip of the tooth.

d. The choice of the root angular increment is especially

important. The smaller AiP, the larger the number of

circles for a given y-coordinate of the root region,

-55-

and with that the greater the accuracy of the

resulting root profile. This becomes crucial

when the hob fillet radius RC is small. For

RC = .040 in. an increment Ai = .125 degrees

gave good results. With RC = .300 in., the

increment Aip = .500 deg. was satisfactory.

(There is a trade-off between accuracy and core

space requirement.)

Because the number of overlapping circles

decreases near the bottom of the root profile,

it is a good idea to check whether the v-coor-

dinate of the point of tangency is at least one

quarter of the magnitude of the hob fillet radius

away from the final Y-coordinate of the root

profile. If this is not the case, it is best

to diminish the angle PSIFIN, i.e. continue

further with the drafting simulation. (The above

serves only as a cautionary note, since during

all trial runs the point of tangency was well

away from the end of the tooth root.)

e. Since the root profile coordinates are stored at

-56-

.001 inch intervals of the y-coordinate, the

point of tangency will be accurate as long as

the root profile is accurate.

III. Special Feature of Program LEWIS ENVELOPE

a. All input data must also be expressed in terms

of unity diametral pitch

b. The program can be run for all values of the hob

fillet radius, including RC = 0.

c. The above remarks concerning the angular increment

Aa also apply here.

d. While the y-coordinates of the root portion of the

tooth profile are given at intervals of .001 in.

by program LEWIS CIRCLES, the spacing of these

coordinates by program LEWIS ENVELOPE depends

entirely on the magnitude of the angular increment

AW. If this increment is too large, the y-coordinates

of the root profile may have a fairly wide spacing.

-57-

Since the point of tangency of the Lewis

parabola can only be found for existing

y-coordinates, this point will only then be

accurately determined if the spacing of

these coordinates is of the order of .001

inches. Since there is no core problem with

program LEWIS ENVELOPE, A* ' .05 degrees can

rea(dilv be used.

-58-

LL1

APPENDIX A

DETERMINATION OF INVOLUTE COORDINATES

Figure A-I shows an involute gear tooth, originating

at its base circle, in the standard vertical position as

defined for this report.

It is assumed that the circular tooth thickness tb =

at the base circle and the associated angle Ob are known. To

determine the x and y coordinates of the arbitrary point T on

the involute, the unit vectors nou and nut, in directions OU

and UT, respectively, must first be determined. Thus,

b bnou = cos(90 4-- c) I + sin(90 + 2 -- ) (Al)

where

= the roll angle of the generating tangent UT

tb-b = b 'where Rb represents the base radius

Equation (Al) may be simplified to:

0b %b (A2)n-ou sin(-- - a) i + cos(-- - c)

The unit vector nut is found by cross multiplication of nou

by the unit vector R:

-59-

IYO}

VV.- U

Point V is

origin of involute

at base circle _b2 Rb (Base Radius)

Figure A-1

Involute Coordinates

0 X(j)

A-2-60-

eb ebnut = n x ou - cos( -- c) 3- sin(--- ) } (A3)

The vector f- may now be written in the following form:

U-T Rb n ou+ Rbnut (A4)

Substitution of equations (A2) and (A3) into the above and

subsequent separation into x and y-components, gives the

following coordinates of an arbitrary point T on the left

involute profile:

x.R i b ci bXv = R sin ( - a Cos(-- - X)] (A5)

and

R b 0beb

Yinv = b[ cos( --- a) - a sin(- - a)] (A6)

A-361-/- a'1

APPENDIX B

DETERMINATION OF COORDINATES OF PATHi OF EFFECTIVE RACK POINT

In order to be able to plot the root of a gear

tooth, as cut by a rack cutter, it is first necessary to

obtain the x and y coordinates of the trochoidal path described

by the effective point of the rack cutter tooth.

Figure B-I shows the effective points of rack cutter teeth

with and without fillet radii as well as the associated effective

ACT'ING ACTINGPITCH LINE PITCH LINEOF RACK OF RACK

bff bf

a. ITEiICTI\'E POINT 1. I;I'ICTIVI; POINTFOR SIHARP (P()RNRI:I) (1 RACK W1TIIRACK 1IA,I:T RAI)IIS r

Figure B-1

Effective Point and Effective Distance beff of

Racks with and without Fillet Radii

B-1

-63-

dis--nces beff. When there is no fillet radius,(see

Fig. B-la) the corner becomes the effective rack point

and beff is represented by the distance from the acting

pitch line of the rack to this corner. If the rack has a

fillet, the point of tangency of the rack flank and the

fillet circle becomes the effective point and beff is

measured to this point.

The magnitude of beff decides for a given gear or

pinion whether the teeth are undercut. Figure B-2

illustrates the applicable criterion.

If the effective point of the rack cutter tooth

moves horizontally along a line which passes through point

U(the origin of the involute) or above it, the involute

tooth will not be undercut. Thus the limiting distance

beffmax is given by

beffmax - Rb cose

beffmax = R(l - cos 2 ) = R sin 20 (Bl)

Thus, generally,

If beff < Rpsin 20 , the gear tooth will not be undercut. The

effective point of the rack initially contacts the gear tooth-

to-be-cut above its base circle.

If beff > R sin2O, the gear tooth will be undercut. Thep

effective point of the rack initially contcrats the tooth-

B-2

-64-

Rc CutrTooth RakPitch

Rark Pitch

Rby Rack Cutter

Figure B-2

Maximum Allowable bef f which

avoids Undercutting

B-3

-65-

• I N -. .. -

to-be-cut below its Vase circle.

The following first gives derivations for the trochoid

coordinates of the path of the effective rack point for both of the

above cases. Appendix C shows how to obtain the associated

coordinates of the path of the center of the fillet circle.

1. COORDINATES OF PATH OF FFFECTIVIF RACKPOINT

FOR beff > Rp sin20

Figure B-3 shows a rack cutter, which is about to cut the

root of the indicated tooth, in its standard position. The

flank of the rack tooth is tangent to the initial involute

point V.

The angle W, which defines the rolling of the rack

pitch line on the gear pitch circle, is zero when line OP

is at the rack pressure angle 0 to the initial involute

tangent OV.

As the angle increases in a CCW direction, a part

of the undercut root of the gear tooth is generated. A clock-

wise decrease in the angle i causes the comDletion of this

root. In actual gear hobbing, it would also cause the

generation of the involute portion of the tooth. The deter-

mination of the required maximum positive and negative values

of the angle W, which are necessary to form a sufficient por-

tion of the root profile, together with the determination of

the inner form radius of undercut teeth, is shown in APPENDIX C.

B-4

-66-

'

Rack Pitch Line

Y(J Gear Pitch Circle

t/\ a

\ v

P2

bef f 2-

J Rb Figure B-3

Coordinoites of 1'ath

of luffective V,161:

Point foi-R p b ef f R Rp sin2O

90 + b

+, X(i)

-67-

When beff Rpsin 2O, the distance from the

instant center P to point R. becomes for the indicated

standard position, i.e. for W = 0

a = (P - beff) tan 0 (B2)

For non-zero values of p. the distance PR is given b,

PR = a + R p (k3)

To define the position vector OS, from the center

of the gear to the effective rack point S, it is necessary

to define unit vectors for the directions of lines OP andI-

PR, respectively.

n = COS( 90 + - + 0 + ) + sin( qO + + 6 + (B4)2

or

b - b

nsin( + 0 + p)i + cos( - + 0 + q)i (B5)op2

Further,

6b 0bnpr = -k x n op cos(2--+ 0 + )i+sin(2-+0+,)j (B6)

The position vector OS may now be defined as:

OS = R pi op+ ( a + Rp)fipr - beff nop (B7)

Substitution of equ's (B5) and (B6) and separation of x and y

components gives the following coordinates for the path of the

effective rack Foint:

B-6

-68-

i 4 -

bXtr = -(p - eff)si (--+ + )

+ (R p + a) cos (-- + 6 +) (B8)

and

Ytr =(R beff) cos (--+ +)

0b+ (Rp + a) sin (- + O +) (B9)

p

2. COORDINATES OF PATH OF EFFECTIVE RACK POINT

FOR beff < R psin 20

Figure B-4 shows the effective point of the rack making

initial contact with point Q on the (future) involute profile.

(This is considered standard position for the non-undercut teeth.)

This point must be located on the line of action U-P. Note that,

point V again represents the initial point of the involute. The

angle , which again defines the rolling of the rack pitch line

on the pitch circle of the gear is zero in the indicated standard

position, when line OP makes the rack pressure angle 0 with the

instantaneous base radius U01 of length Rb. A clockwise decrease

in 4 starts the generation of both the involute and root profiles.

For a discussion of the limits of angle W see APPENDIX C. (Let the

rack in Figure B-4 be considered to be in standard position.) For

the present case, with beff < Rpsin 20, the distance a becomes:

B-7

-69-

Rack Pitch Line

beffe Effective

Coo~ Rd i rclee

fP tI

of ff Effj e Rctiv

PoPoit fo

FigurB- 8-

-0

bef

a= bff (BlO)tan 8

This represents the distance PR when ' 0. In general

PR = a + Rp (BlI)

To define the position vector U-S, from the gear

center 0 to the effective rack point S, it is necessary to

introduce unit vectors for the directions of lines 'P and

P-R, respectively, Now, (see Figure B-4):

nlop = cos( 90 + ri + p )I + sin( 90 + n + ' )j (B12)

or

flop = -sin( n + ' )7 + cos( n + ' )j (B13)

further,

= -k x n = cos( n + P )7 + sin( n + 'p )7 (B14)pr op

The angle n is defined as follows:

a b

n - + (B15)

where

=0 - V yOU, (B16)

and 0 is the pressure angle of the rack

B-9

-71-

Further,

~VOU VU Lu(B17)-' Rb

by the properties of the involute curve.

But

QU= U-P-P-P, (B18)

where

UP=R btanO (B19)

and

- b eff(B0sinG

Finally,

SVOU = tanO b eff(B1R b sinO B1

and, according to equ's (B15) and (B16):

rl= b+ 0 + b eff - ta-0(B2RbsinO

The position vector OS may now be written:

US=Rp n-op +(a+Rp )-pr -beff op (B23)

Substitution of equ's (B13) and (B14) into the above,

followed by the separation of x and y components leads to the

following coordinates of the effective point for the present

B-10

-72-

case

x r= Rp b ff)sin(i + p)+ (Y i + a)cos(n + i)(B24)

and

Ytr = (R beff)cos (n + p)+ (B25)(R p F + a)sin(fl + )

B-11

-7-i

APPENDIX C

GENERATION OF TOOTH ROOT IN PRESENCE OF FILLET RADIUS ON

RACK CUTTER

The present appendix shows two methods of obtaining

the coordinates of a tooth root, i.e. the portion of the tooth

below the involute profile, when the rack cutter has a fillet

radius r . Now, the root profile is not generated by the effective

point, as is the case for a sharp cornered rack, but it becomes

the envelope of the various positions of the fillet circle. This

fillet circle serves as a generating curve.

The first method simulates the applicable drafting

method of finding an envelope by a purely numerical computer

procedure. The center of the fillet circle is first found for

a given position of the rack pitch line. Then, the x-coordinates

of roints on the appropriate portion of the associated circle

are 'ound as functions of equally spaced y-coordinates. This

process is repeated for other positions of the rack pitch line.

For all angles ' the x-coordinates of circle points, which are

associated with certain y-coordinates , are stored in the same

row of a matrix. If the angular increment of pitch line rotation

is sufficiently small, that x-coordinate which is closest to

the tooth center line represents the desired point on the root

profile.

C-i

-75-

-I

a __

a2

/aack Pitch Line

rS cos ----

C-

b2

Nbb

I+ C -- -

r c sine

Effective Point of

Rack Cutter

Figure C- 1

Center of Fillet Radius in

Standard Position

c-2

-76-

The second method was first shown by R. G.

Mitchiner and H.H. Mabie in "The Determination of the Let. s

Form Factor and the AGMA Geometry Factor J for External Spur Gear

Teeth". Paper No. 80-DET-59, Design Engineering Technical

Conference, Beverly Hills, CA., Sept. 28 - Oct. 1, 1980. It

makes use of the fact that, the instantaneous centers of curva-

ture of a generating curve and its envelope lie on a line

through the instant center of rotation of the two planes

involved. The :ontact point between the generating curve and

the envelope also lies on this line and it represents a point on

the final envelope profile.

Before either of these procedures can be shown in

detail, it is necessary to find a method for determining the

coordinates of the center of the fillet radius as a function of

the angle 0. This is accomplished by the appropriate adaptations

of the expressions given in Appendix B.

1. Coordinates of Center Of Fillet Radius for Undercut and

Non-Undercut Teeth

Figure C-1 shows a rack cutter tooth as it appears

both for undercut and non-undercut gear teeth in the standard

position of Figure B-3 and B-4, respectively. The distance

from point R, on the rack pitch line, to the effective rack

point is represented by beff . The dist4ce P-R, from the instant

center P to point R, which corresponds to the effective rack point,

C-3-77-

is given by length a. [ As before, either equ's. (B2) or

(BlO) are applicable.] If one wishes to express the position

of the center C of the right hand fillet (of radius re) with

respect to the instant center P in this standard position,

one obtains:

a2 = P-R2 = a - rccose (Cl)

and

b = F = beff - rcsinO (C2)

With the above it is now possible to adapt the

coordinate expressions for the effective rack point to new

ones for the fillet center C by replacing the distances a and

beff by a2 and b2 , respectively. Thus, one obtains for undercut

teeth, according to equations (B8) and (B9):

xc =-(R p b2 ) sin ( + a +)

(C3)eb

+ (R p + a2) cos( + 0 + 1)

C-4

-78-

and

Y (R b) Cos ( b 0 - +p p2 2

(C4)

+ (R~ + a sin Ob+ +

Similarly, the coordinates of the fillet center C for non-undercut

teeth become with the help of equ's. (B24) and (B25):

xc ( - b ) sin (n +4') + (R + a2 ) Cos (n + ) (5

and

=c (R p b2) cos (n + p+ (R p + a2) sin (n~ + ~)(C6)

C-5-79-

2. Iterative Method of Obtaining Root Profile

Figure C-2 indicates how successive fillet circles

may be used to obtain the root profile envelope for undercut as

well as non-undercut teeth. Point C1 represents the fillet

center when the rack cutter is in the usual standard position.

The operative pitch radius is OP = Rpl and the coordinates of

point C1 , with respect to point P, are given by a2 and b2 (as

defined in the previous section). Centers C2 , C3 and C4 result

after successive clockwise rotations of the rack pitch line, with

associated pitch radii Rp2, Rp 3 and Rp4, respectively.

Figure C-3 represents an enlarged view of the

relevant segments of two such fillet radii. It will serve to

illustrate the following explanation of the computational

procedure which is used to simulate the drafting method of

obtaining the root envelope:

a. The coordinates xc and yc of centerpoint C1

of the first circle are computed with the

help of either eauations (C3) and (C4) or

equations (C5) and (C6), depending on whether

the tooth is undercut or not.

The counter J is used to indicate the number

of the circle. J = 1 for the present case

C-6-80-

Yj)

R

Root Envelope

P Center I (C )

Center 2 (C,)

Center 3 (C3 )

Center 4 (C4 )

R p4

Figure C-2

Generation of Root Envelope X(I)

by Fillet Circles 0

C-7

-81-

y

Effective Rack Points of Circles

Y (I

CC

× E(Circlee1o

Center line of tooth

Lowest PointsFillet Circle of Circles

has Radius rc ( = Yc rd

Figure C-3

Iterative Determination of

Points on Root Envelope

+ x0

C-8

-82-

lI

and generally, for that circle which

corresponds to the intial angle Pin

The total number of circles depends on

the increment At as well as on the final

angle 'fin" (These initial and final

angles are defined in Appendix D).

b. Since the center of the fillet circle

never lies at the origin, the equation

for points on the circle is given by:

(x - xc)2 + (y - y) 2 = r2 (C7)c

The x-coordinates of points on this

circle, for a given value of the y-coor-

dinate, may be obtained from

x = x. + V r (y - y (GB)x > c _ rc -

For practical purposes it is only necess~ry

to obtain the x-coordinates of points whose

y-coordinates are located between the effective

rack point of a given circle and y = yc - rc

(See Figure C-3).*) Since the center of all

(*)Points above the effective point on the rack flank are not

involved in the generation of the final root profile.

C-9

-83-

generating circles lie in the second

quadrant, their x-coordinates will always

be negative.

Further, since the circle segments of

interest will always be located to the

right of their center points, the values

of all x-coordinates will be larger than

those of their associated xc 's. For this

reason, the positive sign must be used in

equation (C8), i.e.:

x = x + 2 (y _ yc) 2 (C9)

c. The computation is continued by determining

the y-coordinate of the effective rack

point of circle 1 either with the help of

equ. (B9) or equ. (B25), depending on whether

the tooth is undercut or not.

The y-coordinates are assigned the counter I

and they will be stored in the vector Y(I).

For this specific computation I = 1. The

associated x-coordinate is then found with

the help of equ. (C9), and its values is stored

C-10

-84-

in the matrix X(I,J). Since this is the

first x-coordinate of circle 1, it will

be located by X(l,l). Subsequently, the

first y-coordinate is decremented by .001

inch and its associated x-coordinate is

again obtained with the help of equ. (C9).

The value of the y-coordinate is stored in

Y(2), while the x-coordinate belongs to

X(2,1). This process is repeated until

Y= Yc - rc (with the yc of circle 1), filling

the appropriate Y(I) and X(I,1).

d. The angle ' is now decremented by A4, as

indicated by Figure C-3. J = 2 for this

second position of the fillet circle. Again,

the coordinates xc and Yc of point C2 are

found, together with the y-coordinate of the

effective rack point of circle 2. The later

is identical to one of the y-coordinates of

circle 1. In order to find the value of I

which corresponds to this Y(I) of circle 1,

all previously computed values of Y(I) are

compared to the present one. (This sorting

process is made possible by multiplying the

values of all the y-coordinates stored in Y(I)

C-11-85-

by one thousand and then truncating them

to make a four digit integer out of them.

(See Program description in Section 2 ).

Once the specific value of I = f has been

found, it is assigned to X(f,2) for storing

the x-coordinate of the effective point of

circle 2. The process of decrementing the

y-coordinates of the second circle by .001

inch each time and computing the associated

x-coordinates, until the lower limit of this

circle has been reached, follows. It is

identical to the one described for circle 1

in part (c) above. All values of these

coordinates are assigned to the appropriate

V(I) and X(I,2).

(The above operations are repeated in the

program many times. Tablc C-1 ) which shows

a typical vector Y(I) and its associated matrix

X(I,J) will be discussed below).

e. Figure C-3 shows two circle points on the

horizontal line through Y(I). These are X(II)

on circle I and X(I,2) on circle 2. Since

X(I,2) is closest to the tooth center line, it

becomes the accepted value for the x-coordinate

C-12-86-

Ii

Fcol. 1 col. 2 col. 3 col. 4 col. 5 col. 6 col. 7 col. 8

I Y(I) X(I,1) X(I,2) X(I,3) X(I,4) X(I,5) X(I,6)

J=1 J=2 J=3 J=4 J=5 J=6

00 -3 0 P -6 0 -9 0 = 1 2 0 -150

1 3.628 - .981

19 3.610 -. 977

29 3.600 -. 976

49 3.580 -.974

66 3.563 -.974 -.968

104 3.525 -.978 -.963

122 3.507 -.982 -.963 -.1963

169 3.460 -1.001 - -.969 -.960 - .966

206 3.423 -1.027 -.982 -.965 -965 -.976

231 3.398 -1.051 -.995 -.972 - .968 - .976 - .992

289 3.340 -1.045 -1.001 -. 986 -.987 -.999

342 3.287 -1.053 -1.020 -1.013 -1.021

387 13.242 -1.073 -1.051 -1.054

421 3.208 -1.101 -1.094

446 3.183 -1.143

459 3.170 -____-____ ___ _________-.0

Table C-i

Typical Vector Y(I) and Matrix X(I,J)

(Diagonal lines indicate largest value i~n any

row. All values in inches).

C-13-87-

of the tooth profile associated with Y(I).

(In the actual program there are many X(I,J)

values for a given Y(I). Again, that

x-coordinate which is closest to the tooth

center line becomes the desired point on the

root profile).

Table C-i Illustrates the relationship between the

vector Y(I) and.the matrix X(I,J) with selected numbers from an

actual computer run. Column 1 contains the counter I which

identifies the y-coordinates on the fillet circle segments.

While I takes values from 1 to 459, in steps of one, in the

actual run, the table shows only a limited number of y-coordi-

nates. Column 2 contains these coordinates. Each one is common

to all circle points on the same horizontal line. Columns 3 to 8

list the x-coordinates of these circle points. With t in 0

fin= 15° and AO = -3 °, the counter J, which identifies the

circle number, ranges from 1 to 6. The first number in each column

represents the x-coordinate of the effective point of the rack in

the given position. (This was the basis of the choice of the I's

in the table).

Once such a "table" has been completed by the computer,

the x-coordinates of the root profile are determined by the largest

C-14

-88-

(i.e. least negative) number in any one row of the matrix

X(I,J). For example for I = 231, where y = 3.398 in., the

profile coordinate x = -.968 in.

3. Analytical Method of Obtain Root Profile Coordinates

Figure C- 4 shows a rack with a fillet radius r c which

is generating the indicated root profile. Since the Euler-Savary

Curvature Theory (**) shows that, the contact point between a gener-

ating curve and its envelope lies on a line connecting the instant

center P and the center of curvature C of the generating curve,

point S represents a point on the root profile. To obtain the

coordinates of this envelope point the components of the vector

-S= + rq = b- + r n (ClO)c Cs

must be determined. The unit vector ncs has the direction of

line R, as indicated above.

The components of the vector UC were given by equ's.

(C3) and (C4) or by equ's. (C) and (C6) for undercut and non-

undercut teeth, respectively.

In order to obtain a single set of expressions, certain

angles, found in the above equations, and which were first introduced

N. Rosenauer and A. H. Willis: Kinematics of Mechanisms, pp. 60-62,

Dover Publications, Inc., No. S1796, New York, 1967.

C-15

-89-

ii . .. .. - "kq W,' W*

Rack Pitch Line

Rack m

p 2 Fillet Radius C = rc

~Tooth Centerline

b 2 Root Profile

V Figure C-4

Determ~ination of Root

Profile Coordinates by

Curvature Theory

0 X(i)

C-16-90-

in Appendix B, are now redefined as follows (see Figure

C-4):

For undercut teeth, let

eb=- + 0 (CIlI)

[See also unit vectors nop and npr for undercut teeth, as given

by equ's. (B5) and (B6)].

For non-undercut teeth , let

b beffT = n = T-- + 0 + Rbsine - tan (C12)

[Compare the above to the angles of the unit vectors of equ's.

(B13) and (B14)].

With the above, equ's. (C3) to (C6) can be generalized to the

following form.

For the x-component of vector M-:

C-17

-91-

: " -- • . . . . . . i i i il i i i i - ... .... . . . .. .

-C(R p b 2 sin (T+*p)-I(R p + a2COS (T + )

For the y-component of vector UC:

YC=( - b 2 ) COS (T + p) + (R p4 + a 2) sin (T + ~

(C14)

To derive an expression for the unit vector nc'it is first

necessary to express the unit vector nip and inpr, alluded to

earlier, in terms of the angle T. Thus,

n op -sin (T + q)i 4- COS (T + p)j (C15)

and

npr =O co(T + P)i + sin (T + P)j (C16)

Then, as Figure C-4 shows:

C- 18

-92-

=- (PR) Epr - (RC)ii Op(C 7Cs I~ /(?R) 2 + (RC )2

or, with the appropriate substitution:

TCS

(R i + a2) [cos(Tc + 01) + sin( T + 4)j b 2 1-sin(T + 'I + COS (T + lp)j]

/(R~ i a) +b

(C18)

or

nCs

[(R~i + a2)cos(T+ )+b 2sin(T + +p] I [(Re p + a2)sin(T + )-b2cos (T+ )]3

/(Rip*+ a2 ) 2 +b2

(C19)

Finally, the root envelope coordinates xenv and y env may be

determined according to equ. (C1O).

C- 19-93-

x x i + y c + r cn Cs (C20)

with the help of equ's. (M1), (C14) and (C19), one obtains:

X env (R p -b2) sin (T +W) + (R p +a 2) COS ( P

(R P + a 2) COS (T + ) + b 2 sin (T +WI)

C p' + a2 )2 + b~ 2C1

and

'ev=(R - b) COS (T +i + (R + a ) sin (T +)

+ (RP + a 2) sin Cr+ b - 2 COS (T + )

'(R'h + a ) 2 + b 22 2 (C22)

C- 20

-94-

APPENDIX D

INITIAL AND FINAL ANGLES FOR TROCHOID AND INVOLUTE GENERATION

When computing gear tooth profile coordinates it is

necessary to know the initial and final values of the trochoid

generating angle p (see Appendix B) as well as of the involute

generating angle a (called roll angle in Appendix A).

The following shows how the relationship of the trochoidal

path of the effective point of the rack cutter with respect to

the rest of the gear tooth profile may be utilized to determine

these angles for undercut and non-undercut teeth. It must be kept

in mind that, while the effective --int of the cutter tooth always

determir s the magnitude of the inner form radius and with that

the initial trochoid generating angle 'in, it does not furnish

the angle associated with the lowest point of the gear root

whenever there is a fillet radius on the cutter. This final

trochoid angle 'fin is defined as that angle for which the effective

point of the cutter coincides with the centerline of its tro-

choidal path (see sections 4 and 5 below).

The initial involute generating angle cin is a function

of the inner form radius. The final involute generating angle

"fin depends on the outside radius of the gear tooth.

Sections 1 and 2, on the trochoid angle I and on the

determination of the inner form radius for undercut teeth

respectively, are taken with minor nomenclature changes from the

D-1-95-

work of R.A. Shaffer: An Analysis of the Undercut Problem in

Involute Spur Gearing, Report No.R-1606,0MS5530.11.557, DA

Project 505-01-003, Frankford Arsenal, Pitman-Dunn Laboratories,

May 1962.

1. The Trochoid Angle T

Figure D-1 shows the trochoid described by the effective

point of the indicated rack tooth. The line S'U', of length beff

coincides with the center line SU of the trochoid when the

generating angle ', as used for this purpose, is zero together

with the length QS , along the rack pitch line. The trochoid

angle T is measured from the trochoid center line and it defines

the position of the radius R (used later to get the form radius

Rf) from the orgin 0 to the instantaneous location of the effective

rack point at U'. Then,

S= o - ' (Dl)

where

a = ) TO' = tan - l - tan - 'R2 - (Rf - bpff)2 (D2)TO Rp - beff

and

, arc? QS> = U - - (Rp - ff)2

p p p R (D3)p

D-2

-96-

Position of PitchLine for i > 0

Hob or Rack Tootl

Position of Pitch

Line for i = 0Q S

Effective Point ofI" Rack Tooth

' --RFIGURE D-1i

Trochoid Anple 1

D-3

-97-

AD-All? 593 CITY COLL RESEARCH EOUNOATION NEW YORK F/S 19/1

AUTOMATED STRENGTH DETERMINATION FOR INVOLUTE FUZE BEARS. (U)FEB 82 6 6 LOWEN, C KAPLAN DAAKIO79-C-0251

UNCLASSIFIED ARLCD-CR-81060 NL

H~~ ~ 0. .0 j~

MICROCOPY RI SOLUTION TISI CiIARItqA!I ?NA. LIIA [ HOP,

1-

Substitution of equ's (D2) and (D3) into equ. (Dl) furnishes:

T = tan-I /R 2 - (Rp - beff) 2 / R 2 _ (Rp - beff) 2

RP - bf f Rp (D4)

2. Determination of Inner Form Radius for Undercut Teeth

Figure D-2 shows the side of a basic rack of pressure

angle 0 in line with the base radius OV = Rb, of the indicated

involute surface. Point V represents the theoretical origin of

this involute. The effective point of the rack cutter, shown at

point U', is positioned such that the involute profile will be

undercut, i.e. beff > R psin 2e. The trochoid of this effective

point intersects the involute at point P and the length OP

represents the inner form radius Rf. Note that the center line

of the trochoid is represented by line OU, which forms angle 6

with the side of the rack tooth.

To obtain an expression for Rf one sets the following

angles equal to each other:

T = 6 + 6 (D5)

where

= obtained from equ. (D4), with R = Rf

Further,

= . - ' ,(D6)

D-4-98-

i|

Trochoid .._Basic Rack

so Involute Surface

beff p

T U -U 1

Deterination of Inner For

TRadius for Undercut Tooth

D-5-99-

where i' may now be expressed in terms of the rack pressure

angle 0. [ See also equ. (D3) ].

= (Rp- beff)tane (D7)

The angle 6 has its origin in involutometry. If the angle a

represents the involute generating angle with respect to base

radius OV, and if one defines the angle POS = 4, then:

= INV ¢ = taniF - R

bb _tan-i/R f-b (D8)

Substitution of equ's. (D4), (D6), (D7) and (D8) into eou. (D5)

furnishes:

-ltn -(M - beff)' - vR 2 - (Rp - bffYtan R f- ef

Rp Rb R

(R -beff)tane + / 2 b ___

=0- p -- + f ta-1 f R b (D9)

This expression may be solved for Rf either by a

computer adaptation of the Newton - Raphson method or by a

trial and error method.

D-6-100-

3. Determinatinn of Inner From Radius for Non-undercut Teeth

Figure D-3 shows the effective point of a rack cutter

making initial contact with the prospective involute surface

above the origin point V of the involute. Since now beff <

R sin2e, the tooth will not be undercut.P

As in Figure B-4 of Appendix B, the contact point Q

and with that the effective point lie on the line of action

UP. The inner form radius Rf may be described as follows:

Rf = (U) 2 + 2 (DlO)

where

=PU - Q = RbtanO - beffsine (Dll)

with the above, equ. (DlO) becomes:

Rf = !RbtanO - beff )2 + R2 (D12)

For the special case when beff = R psin 20, equ. (D12) becomes:

D-7

-101-

Rack Side

Involute Surface

Line of Action

_bef f -

Rack Pitch Line

Rf R

FIGURE D-3

Determination of Inner Form

Radius of Non-undercut Tooth

D-8-102-

_Trochoid Center Line

in* xit Positior ofFffective Pack Point

Base Circle

S , Staneard Position

of Effective llacl-

Point

b effS fin Trochoid Center

Line Position of

in Fffective Pack Point

FT(rURF D-4

Initial and Vinal Trochoid

Cenerating Angles for Undercut Teeth

* D- 9

-10 3-

Rf = Rb (Dl3)

as is to be expected.

4. Determination of Initial and Final Trochoid Generating Angles

for Undercut Teeth

Figure D-4 shows the outline of the rack cutter in its

standard position, as defined in Appendix B. The effective point

of the rack makes contact with the undercut trochoid, generated by

it, at point S. The flank of the rack is in line with base radius

UV = Rb, where V represents the origin of the involute curve. The

associated line O- = R to the pitch point is at the pressure angle

O of the cutter with respect to line "9. The angle i is zero.

Also, the distance 7--, along the pitch line and normal to beff ,

eouals a, as given by equ. (B2) in Appendix B.

In order for the effective point to make contact with

point Sfin at the trochoid center line, the rack pitch line must

roll clockwise through the angle qfin until points Pfin and Rfin

coincide. The distance rolled through equals a = Mand therefore:

fin - a (D14)

D-10-104-

(Note that the above angle is negative because of the zero

position of p.)

When the effective point of the rack cutter is located

at point Sin' the intersection of the trochoid and the involute

surface, * = Pin . To determine this angle, one must first find

the total roll angle ytot of the pitch line, associated with the

change of the effective point from Sin to Sfin ' Ytot depends on

the arc length PinPfin = PinR.n and it equals the angle between

the pitch radii 6Pin and 0Pfin . With the inner form radius Rf

known, one obtains

Pinfin in R - (R - beff)Z (D15)

Rf must be computed with the help of equ. (D9).

Then:

arc Pin Pfin - - (Rp - beff)2 (D16)

R Rp p

Finally, as figure D-4 indicates:

*in = Ytot " Vfin (D17)

D-11

-105-

or

SR2 (Rp be f 2in - a (D18)

in R RRp

5. Initial and Final Trochoid Generating Angles for Non-Undercut

Teeth

Figure D-5 shows the effective point of the rack cutter

making initial contact with the involute profile at point S. Since

this point lies on the line of action IMp, located above the origin

V of the involute, there will be no undercutting. (Note that this

figure describes the same relationship between rack and involute

as Figure B-4 of Appendix B ). The distance P-R along the rack pitch

line equals a, as defined by equ. (BIO). The generating angle *

equals zero in this initial position, while the angle between the

pitch radius 'M = R (with point 0 not shown) and the base radiusP

OU = Rb equals 0, the pressure angle of the rack.

In order for the effective point of the cutter to make

contact with point Sfi n at the trochoid center line, the rack pitch

line must, as in the last section, roll clockwise through the angle

*fin until points Pfin and Rfin concide. The distance rolled

through is again given by PR - a. (It is important to remember

that the length a is now given by equ. (BlO), rather than by equ.

(B2) of Appendix B). Therefore,

D-12-106-

Trochoid Center Line

I Y(T)

Lieof Action R Pitch Circle

RpLS f d Base Circle

Pf in-

Rb

Dis ance R.S. =b

1 0. f

Figure D-5INITIAL AND FINAL TROCHOIDGENERATING ANGLES FOR NON-UNDERCUT TEETH

D-1 3-10?-

afin - (D19)

It can be shown that, between points S and Si.n the

trochoid profile moves away from the theoretical involute profile SV. In

addition, point S also respresents the last point of tangency

between the trochoid and the involute, as angle i increases. To

make this clear, figure D-5 shows that point S', which is

associated with the arbitrary positive angle p', is at a consider-

able distance from the involute profile. For the above reason

'in = 0 (D20)

6. Initial and Final Involute Generating Angles for Undercut and

Non-Undercut Teeth

Figure D-6 indicates that, once the inner form radius

is known, the initial involute generating angle ain may be computed,

regardless of whether the tooth is undercut or not, with the help

of the following:

aV U T inU in (D21)in Rb Rb

D-14-108-

T f in

Tin

R Rf

Point V is U.origin of involute i

at base circle i.

Rfn

.. fi n

Figure D-6DETERMINATION OF INITIALAND FINAL INVOLUTEGENERATING ANGLES

0

D- 15-109-

Since

in in = (D22)

the above becomes:

ain f (D23)Rb

Similarly, with the outer form radius Rfo known, the

final involute generating angle afin is given by

afin = 0 Rb R (D24)Rb

D-16-110-

APPENDIX E

GEAR STRENGTH CALCULATIONS. DETERMINATION OF LEWIS AND AGMA FACTORS

This appendix first gives a review of the origins of the

Lewis gear strength formula and the associated Lewis and AGMA Geome-

try Factors. + Subsequently, the essentials of a computer determina-

tion of the Lewis Factor for any involute gear profile are developed.

+For more detail see: AGMA Information Sheet. Geometry Factors for

Determining the Strength of the Spur, Helical,Herringborne and BevelGear Teeth. AGMA 226.01, American Gear Manuf. Assoc., Aug. 1970.

E-1~-iii-

1. Maximum Bending Stress in a Parabolically Shaped Beam

Figure E-1 shows a cantilever beam with a parabolic

cross section. The parabola originates at point 0 and is described

by

2y=Bx (El)

where B represents a constant. The beam of thickness b experiences

the indicated endload F . Using the usual bending stress formula,

the maximum bending stress max at the abitrary section f-f becomes:

0 Mc _ 1.5 s (E2)

'Tiax T b x 2

where for the present case:

IT = Fsy , the moment at section f-f

c = x , the distance to the extreme fiber of the beam

I = b(2x)- the moment of inertia of the arbitrary cross section12

of thickness b.

If one now substitutes the y of equ. (El) into equ. (E2),

the following expression for the maximum bending stress at any point

along the beam results:

amax = Fs .5B (E3)

E-2-112-

x

if FS

f

Beam Thickness -b

Figure E-1

Cantilever Beam with Parabolic

Cross Section

E-3

-1.13-

The above shows that, the maximum bending stress in a

parabolically shaped beam is constant along the beam.

2. Derivation of Lewis Gear Strength Formula

Figure E-2 represents a typical gear tooth which is acted

on by a contact load F. The line of action of this force passes

through point L, the earliest possible contact with a matinig

tooth, at the end of the involute profile. If there is no load

sharing with a second set of mating teeth, i.e. the contact ratio

equals unity, this condition also produces the largest possible

bending stresses at any cross section below point V, where the line

of action intersects the tooth centerline.

The load angle B is defined by the line of action of force

F and the normal to the tooth centerline. The bending load F is5

obtained from

F = F cosa, (E4)5

and it acts normal to the centerline at point V.

Now, let point V become the origin of a parabola which

is tangent to the tooth profile at some section e-e, as indicated

in Figure E-2. Since, as has been shown in the previous section,

the bending stress along a parabolically shaped beam is constant,

section e-e must represent the location of the maximum bending

E-4-114-

K Tooth Centerline

L L Tooth Profile

(I % Inscribed Parabola with

Origin at point Vh

Figure E-2

Background to Lewis Formula

(Line of action of load F, as

shown, corresponds to unity contact ratio).

E-5-115-

stress of the gear tooth. All other cross sections have portions

outside the parabola and will therefore experience lower bending

stresses due to the load Fs '

Thus, if one can determine the point of tangency

between the parabola and the tooth profile the maximum bending

stress may be found with the help of the nomenclature introduced

in Figure E-2:

(Fsh)(2) 6tF ho - 6 s (E5)max bt 3

tb t2

where

h = VA , the distance from point V to the tangency section e-e

t = 2(A-), the tooth thickness at section e-e

The moment arm h may be expresed in terms of the distance

x = A , which generally is found by graphical means. (See the afore-

mentioned AGMA Standard 226.01). Thus, with the right triangle

VCB and the resulting similar triangles AVC and ACB, one obtains the

relationship:

x. t/2 (E6)

t/2 h

E-6

-116-

and consequently,

t2

h - (E7)4x

Equ. (E7) is now substituted into equ. (E5). This results in the

following expression for the maximum bending stress in the gear

tooth:

s 3(E8)

max b 2x

In order to make equ. (E8) as general as possible, one

assumes first that the dimension x was determined for a tooth of

arbitrary diametral pitch Pd' The magnitude x, for a tooth of

unity diametral pitch will be given by:

x = x Pd (E9)

The maximum bending stress in a tooth of unity diametral pitch

then becomes according to the above:

E-7

-117-

Fa maxl b s (ElO)max1 bY

where the Lewis factor Y is given by:

2 x1 2 x Pdy- = (Eli)3 3

This factor may either be directly determined from a unity

diametral pitch tooth, or by the indicated multiplication by

Pd if x originates from a smaller tooth.

Naturally, for a given load Fs and tooth thichness b,

the bending stress will be higher for teeth of larger diametral

pitch. Therefore, the general form of the Lewis Gear Strength

Formula becomes:

a sd (E12)max b Y

E-8-118-

3. Determination of Origin V of Lewis Parabola

Before the computer procedure for the determination

of the point of tangency between the Lewis parabola and the tooth

profile, as well as of the distance x , can be given, it is first

necesary to find the distance Rv = V from the center of the gear

to point V, the origin of the Lewis parabola.

Figure E-3 shows the line of action of force F passing

through the arbitrary point WR of the involute profile. (Since the

force F is always normal to the involute profile, its line of action

coincides with line 1-FW ). The radius R = - is associated withR R R

point WR *

The load angle 8R of this line of action with the normal to

the tooth centerline is identical to the angle between the tooth

centerline and the indicated base radius 5UR = Rb This later angle

may be defined as follows:

aR= OR - OR (E13)

where

R= cos - Rb (E14)R

E-9-119-

Line ofDeteroninftioncof

Load Anl Q aR

______R M_______________ R-X

E- 1

-120-

and

R = WRSR (El5)R

The above arc length is equal to one half of the circular

tooth thickness TR at radius R. To determine this tooth thickness,

one makes use of the following relationship from involutometry

which allows the determination of a circular tooth thickness T2 at

a radius R2 if the tooth thickness TI at radius RI is known:

R22 =2R 2 ( INVc - INV 1 )

(E16)

where

INV = tan i - i (E17)

To adapt the above to the present case, it is necessary

to know the actual circular tooth thickness Tcs at the standard

pitch radius R of the gear. (The magnitude of Tcs depends onp c

whether the gear is standard or modified). The associated angle

0 corresponds always to the pressure angle e of the rack cutter.

E-11-121-

II - ... . II IIIII

Equ. (El5) for eR may now be expressed with the help

of equ. (El6) in the following manner:

U TR = I [T - 2R (INV R - INV 0)]R - T - T R c s P ( E l )

or

T= 2R - INV pR + INV e (E19)

Finally, Eau. (E13) for load angle 6R becomes:

Cos-l Rb Tcs + INV INV e (E20)R R 2RpR

For the case when the contact ratio of the mesh is unity

and therefore the crucial contact is made at the end of the involute,

one lets R = Ro , the outside radius of the gear, in eau. (E20). This

gives the load angle 6 for the Lewis parabola:

E-12-122-

cos ( )- p + INV - + INV (E21)

where

= I - b (E22)

To determine the position of point VR where the line

of action intersects the tooth centerline,one considers the following:

(See Figure E-3)

RVR = 5V = OMR - RV (E23)

Since the line WIF is normal to the tooth centerline,R R

= RsinOR , (E24)

while

O- = RcosOR (E25)

E-13

-123-

L i*'

and

MRVR =Rsin OR 1-an R (E26)

Substitution of equ's. (E25) and (E26) into equ. (E23)

leads to the position of point VR with respect to center 0 of the

gear:

RV R ( cosu,, - sineRtan6R) (E27)

For the case when R = R 1the above expression becomes:

R v= R 0(cosO R - sinO R tans) (E28)

0V

where 3 is given by equ. (E21) and OR is obtained with the help

of equ. (E19) ,i.e.:

Te R CS- INV + INV 0 (E,-9)

0 2RP0

again, R Rbs (E30)0 0

E- 14

-124-

4. Computer Determination of Lewis and AGMA Geometry Factors

The computer determination of the Lewis and the AGMA

Geometry factors requires that the coordinates of the gear tooth

profile are fully known and stored in the associated vectors

X2(K) and Y2(K). Subsequent to this, the following operations

are performed (See Figure E-4):

a. The distance R= V, of the origin V of the

Lewis parabola, is found with the help of equ.

(E28).

b. To accommodate the specific situation, the

equation of this parabola becomes:

2R - Y2(K) = B X , (E31)v par

where Xpar is the x-coordinate of the parabola,

associated with the coordinate Y2(K). Again, as

in equ. (El), B represents the constant of a

specific parabola. Since X is required, eou.par

(E31) becomes:

Rv Y2 (K)

par =2B )(E32)

E-15

-125-

Ad

F71

y Gear Tooth Contour

Trial Parabola _______

P v

Xpar

X2(K)Y2(K)

Figure E-4

Computer Determination

of Tanpent Parabola + s

0

E-1.6

-126-

c. In order to find the tangent parabola and with

that the value of the parameter x of the Lewis

formula, the appropriate constant B must be

determined. To this end, a comparatively large

B is first chosen to make sure that the associated

parabola does not contact the tooth profile. The

inner parabola of Figure E-4, with the constant

B1 , represents such a case. As the magnitude of

B is progressively decreased, the resulting para-

bolae approach the tooth profile more closely.

(The constant B2 of the outer parabola of Figure

E-4 is smaller than the constant B of the inner

one). liventually, the tangent parabola will have

the correct value of B.

d. The above process is accomplished by computing

for a given value of B all X as functions of thepar

full range of the Y2(K). Each Xpa r is compared with

the corresponding value of X2(K) of the tooth profile.

Whenever

I X2(K)I - I X I 0 (E33)par

the correct value of B was used and the associated

values of X2(Ktan ) and Y2(Ktan ) lie on the tangency

E-17

-127-

cross section of the gear tooth. The tooth

thickness t as defined by Figure E-2 then

becomes:

t = 2 [j X2(K tan ) I] (E34)

e. Finally, the Lewis distance x of equ. (E6) may

be found by:

2 X2(K tan ) ]2x=[t/2] ta

h RV - Y2 (Ktan)

(E35)

The AGMA Geometry factor YAGM is obtained from

the substitution of the above into

Y AGMA=cos6 (1.5 tanB

cos- x 2[IX2(Ktan)II

(E36)

E-18

-128-

APPENDIX F

PROGRAM MATRIX

The present appendix describes and lists program MATRIX

which is used for the determination of the sizes of the various

arrays in programs LEWIS CIRCLES and LEWIS ENVELOPE. A sample of

the program, together with two output pages is given in section 5

below.

1. Determination of Length of Vectors Xl and Yl

(Used in programs LEWIS CIRCLES and LEWIS ENVFLOPF.)

The vectors Xl and Yl are used for the initial storage

of the involute coordinates of the tooth profile. The common length

of these vectors is determined with help of equ. (1) i.e.

Yl-rows = Sfin a in (F1)Act

The explanation for the above nomenclature follows equ. (1) in

section 2.

2. Determination of Sizes of Vector Y and Array X

(Used mostly in program LEWIS CIRCLES.)

The number of rows, both of the vector Y and the array

X, depends on the number of one one-thousandth of an inch contained

in the vertical distance between the initial point of the first fillet

F-1

-129-

L L - .. ... I . . . . . . . . i l ... Il i .. . ]II il ... . . . . . . .

circle and the final point of the last one.

The y-coordinate Yin of the first point of the first

circle is that of its effective point. It is determined for =in

according to equ. (B9) or equ. (B25):

Yin = (Rp - beff)cos(t + 'in) + (R pin + a)sin(T + in) (F2)

where, for an undercut tooth

S= = 0 , according to equ. (B9) (F3)

in initial roll angle of rack pitch line according to

equ. (D18)

For a non-undercut tooth:

T = - + 0 + beff - tanO, according to equ. (B2S) (F4)2 Rp sinO

in = 0, according to equ. (D20)

The final point Yfin of the last circle is obtained

according to equ. (3) or (5), i.e.:

F-2

-130-

Yfin Ycfin rc (F5)

Ycfin represents the center of the fillet circle for =fin'Then, according to equ's.(C4) or (C6):

Ycfin = (Rp - b 2 )cos(T + 'fin ) + (RP~fin + a2)sin(T + fin)

(F6)

When the tooth is undercut the angle T is used

according to equ. (F3), otherwise equ. (F4) must be used. The

angle l fin for undercut and non-undercut teeth is the same. It

is given by equ. (D14) or equ. (D19).

Thus, the number of rows of X and Y are found from:

Y-rows 1000(Yin - Yfin ) [in formn of an integer] (F7)

The number of columns of the array X depends on the

number of circles involved. This in turn is a function of the

angular increment AW as well as in and 'fin

Therefore, as in equ. (10):

X-columns = in [also made into an (F8)integer]

3. Hand Cbmputation of Sizes of Vectors X2 and Y2

(Used in programs LI'WI, CIRCLES and LI:1"IS E1NV1IOIV)

F-3

-131-

The size of vectors X2 and Y2, which contain

the final tooth profile coordinates, are not computed in

program MATRIX, but may be obtained by adding the results of

equ's. (Fl) and (F7):

Total rows of X2 and Y2 = YI-rows + Y-rows (F9)

The above will give an exact result for use in

program LFWTS CIPCLFS and a somewhat larger than needed number

for program LEWIS ENVELOPE (The y-coordinates of the root are not

generally computed every one one-thousandth of an inch by this routine.)

4. Description of Computer Program MATPTX (See section 5 of this

Appendix)

a. Input Parameters

The following input data are required:

N = number of teeth in gear under consideration

BLFF = beff for unity diametral pitch (See Appendix B)

PD Pd = 1.000, the diametral pitch

TIIETAD = 0, the pressure angle of the hob

CAFrCS Tcs, the circular tooth thickness of the gear at

F-4

-132-

the standard pitch radius Rp terms of Pd 1. 000

RC rc. the rack fillet radius for unity diametral pitch

KADD Addendum constant of gear, with the magnitude of the

addendum defined by KADD/PD, in terms of Pd = 1.000

DEiLPSI = A , the increment of the roll angle of the rack

pitchline on the pitch circle of the gear

DELAL Ac, the increment of the involute generating angle

b. Computations

The initial computational sequence in program MATRIX

is in many ways identical to that of the early parts of programs

LEWIS (See section 2). Following the test which establishes

whether the tooth under consideration is undercut or not, the

appropriate parameters for equ's.(Fl) to (F8) are computed.

The evaluation of these expressions starts in line 61

in the following sequence :

YIN = Yin, according to equ. (F2)

YCFN = Ycfin' according to equ. (F6)

YFIN = Yfin' according to equ. (F5)

F-5

-133-

The integral number of rows of X and Y is found

in lines 64 to 66, according to equ. (F7). The integral number

of columns of array X is determined according to equ. (F8) in

lines 67 an 68. The number of rows of vectors Xl and Yl, based

on equ. (Fl), are found in a similar manner in lines 69 and 70.

c. Output Parame-ers

(All lengths are given in inches and all angles aregiven in degrees)

The output of the program first states the numerical

values of the input parameters N, BEFF, PD, THETAD, CAPTCS, RC,

KADD, DELPSI, and DELAL.

Subsequently, it is noted whether the tooth is undercut or pot.

In addition, the numerical values of the following computed

quantities are given:

RF - Rf

AL PHEN = a.in

ALPHFLN = 'fin

PSIIN =in

PSIFIN = fin

YN = Yin

YCFIN = Ycfin

F-6

-134-

!4

YFIN = Yfin

Finally, the number of rows of the vectors Xl and Yl,

together with the size of the tooth root array X, are printed out.

S. Sample Program and Sample Output

The following pages contain a listing of program MATRIX

as well as two sample outputs.

The programs were run for the same teeth as found in

programs 1I01TS CIRCIES and LFWIS ENVELI.OPIE in section 3 of this

report.

The input data for the first run on p. F-12 are as follows:

N = 36,

BEFF = 1.215 in.,

PD 1.000,

THETAD = 200,

CAPTCS = 1.8460 in.,

RC = 0.040 in.,

KADD = 1.425 in.,

DELPSI = 0.125 degrees

F-7

-135-

DELAL = 0.25 degrees

The program found that the tooth is not undercut. The

various output parameters from RV toYFIN may be used for checking.

Finally the array sizes are given as:

Length of Vectors Xl and Yl 94

Length of Vector Y . 370

Size of Array X : 370 x 85

The lengths of vectors X2 and Y2 are obtained by addition

of the lengths of Yl and Y, according to equ. (F9), i.e.,

94 + 370 = 464

The second run of the program on page F-13 concerns itself

with an undercut 12 tooth pinion.

F-8

-136-

I -

SC FC

C

C

I I vPL ICITK F AL*q(4-1-,C-7)2 CCNNrN lP,P,RP,THETh

2 lEAL NvKACC'1 IEC (9,51) %,99FF,P0,THFTAD,CAPV .S,R-5 14EAC(,,%S) KACr6 QE6Cl5.3?) CFLDSI#DEL4L1 51 FORYAT16~F9.5)

11 1 FClPAT('l','N =',F9.5,5XvBEFr- =',F9.9,5X,'P1 =',F9.5)12 ',ITE (6,?)TlHET (,CAPTCi,PC1 3 3 F C R 14,8T TI- E TA ' 'F 15, 9X , IC AP IC S 'F 9 .5 99 X ,RC 'F 9. 5

15 ~2 FfM'Qv.T(//' ','KAC(C =',FQ.5,'XICELPSt =',F9.59,'DEt4L =,F91.5//)0= 3.1415S

ii 7 =PI/lEc.18THET4 THETAC*Z

19 4P= NI(?.*PC)c QB = _P*r:CCSiTHFTA)

23 7HETAB CAPYCS flCOC(THE~TA) /RR42.*( DVAN(TH4ETt )-THETA)24 FPS =THETfU!/2.!CO + THETA

?5 E1Th F/(PP*flSTN(THET4)) - OTAN'(THETA)217 PLLC0 RP*CSIN(THETA)*CS!\(THETA)

21 IF(EPFF.CT.PALLCA) GC TO 412E TF(FBFFF.LE.EALLCW) CC TC 42

CC ** TCT- IS UNDERCUTC *4** ***~*~******

21; 41 A (PP)*CTAN(T-ETA)3 c TAL D31 PT R!2 C Q33 RF=ECEK!KERPT,C, 1J-lA, 1.C-lP,100)34 hLPI-IN CS4~ASR*FR*R)R

3 ALP -F = (T.CCCS(RP/RM)36CA?'T -(CSZQT(I-bS(RF*RF-(RP-BEFF)**2)))/RP

31 PSTIN -A/'QP +GAMI3EPSIFIN -A/L4P

WRII E (6 9)4C 65 FORMAY'(1/' ***t* TCOTH IS UNDERCUT *****0//)

41 CC Ta 41Cc e************ *******

C ** TOOTH IS NOT UNDERCUT ***

C** ***************;**

c4; 42 A P/CTAN(T-ETA)

43 TAU EPS + CELTA44 RF = SQRT(DOS((R*D3TANTHETA).-B/DSTN(THETA))**2+R5*RB)

4! ALP14114 = DSQRT(OA8S(RF*RF-RP.*RB)l/RBACIF (8LPHIN.LE.O.CUII AIPHIN = 0

F- 9-137-

4-1 -L~tIF CTqNCACS(RR/RC))4$. FS I IN C.c

cSIFT% -qRC 'ARITEC(,'z6)

!1 66 FCR~FkT(//# 9*t** TCOTiH IS NOT UNnERrUT 4*'I

c 4 62 FJRI6Tdh:l//I 1',''1F =I9F-1.9//)

C *101"* rr4 ALL TYPES 'r- TEETH **

rtPS[FC =PSIF IN/Zri- 3LINC = oD I/

E I~~LF IN(' = L

5 (12= F :~C*CSIN(T'ffTA)6c 2 A - QC*CCS(THFTA)

'tI\ ($0-P)*fCCS(TAU+PSIIN)+IRP*PSIINA)*tSLN(TAu.PSI IN)Y CF IN RPe (P~) *fCCS (TA4 PS IF IN + RP*PSIFTN+ f2 *f'S IN TAU+PS IF 111

U''FIN =YCPIN - R

6, ~IRCV%= RCe7 ' 0.L (PS [INO - PSFF)/DELPSI

E IC1t. = CCLic Q( %IN (4LPIF-ALPFI%)I(Z*CELAL)

7C ~ IqINVi jjp INV

-12 5 F'RMAT(//t #,'ALPI-!N =',F9.5,,5XPIALPHFIN =19F9.5)

1 IT C f . c6 ) PSEINCPSIFC36 Fb PCJV.T(/ 9,'PSIIN =',F9.5,5X,'PSIFIN =',F9.5)

75 ARITF(C-,92) YIN,YCFIN,VFIN

7t 4RTTF( 9 5) !R!NVP5 rCRV3T(/////* ',SX'LENGTH CF ROWS FO~l INVCLLTE VECTI7RS XI AND YI'

iTTE(6,87)R 7 F(:MAT(!/@ lv5XONU-9ER CF ROWS AND COLUMNS FOR TOOTH RCCT ARRAYS*

Fc RtTE16,@3) IRCIhdCCLEl P 2 aChWT(//'OY[\=',Fll.5,5X,'YCFIN=IF1.5,5X,'VFI'N=',Fll.5)

V2 3 FCfRPhl//' lo' RCN'tv16,5Xvl CDOL',L6)STCP

P COUiLE PIECISICN FUNCTIO~N DEKKER[B9C,EPSvTCL9LIMIT)C

CC TITLE- CEKKERIS ALGCRITHMC,C PURPCSE- TC FINC A 'RCCT CF A NON-LINEAR ALGEBRAIC EQUATIN F(X)0O.C fESCRIPTtCN CF PARtPETERSC, P z THE LATEST ITERATE AND CLOSEST APPROXIM~ATION TO THE ROOT.C A :27E PREV!>,uS ITERATE.C C z P-E FREVICUS CR AN CIDER ITERATE.C COPPENTSC, AT ALL TIFPE; e ANC C BRACKET THE ROOT.C IP41TIALLN A IS SET E'CUAL TO C.

F-10-138-

C LSE EPS=I.C-7 AND TCL=l.C-1O.

C SUPQCUTINES aNn/CR FUNCTION SUBPROGRAMS REQUIRED- FIX)

fE tIMPLICIT REAL*8 (h-1",O-Zf

!s KCLNT=-1FCA=F (A )F OB= F(F

I FCC=FIC)1;2 IF((FCP*FCC).GT.o.no)GO TO 6Q3 R KtLNT=gOLT4I.1;4 IF(DAV51FCP).LE.0ABSCFCC))GC TO 7r! T=F

91 CEKKER=BCE C=T

c T=FCPIc r FCP=FrCIf'2 FOC=71413 FOA=Fcc1C4 7 IFIFCP.EC.0.rlCRETURNI C! TCIL1:2.CC*PC*CIBS CB)+O.5D.0*TCLIIIIFCDIEI (F-C).LE.2.flC*TCLIRETURN

in, IF(CtP5 CFP-FCA).LT.l.C-7O)PETURNICE IFI(LlNrT.CE.LImIT)GC TO 12

C~ s nI=A-C (B-A)*F IU/(FCR-FOA)

III IFC.N\CT. C C(C.LE.OI ).INC.(CC!.LE.P) ).OR.C(B.LE.DI).ANO.(rI.*LE.C) )))

112 1Ff Cf C.LF.CL).ANC.fnl.LE.RFl.OR.f (B.LE.D1).h'Jt.(OI.LE.C)))1 bNC.(DblPS(EI-P).LT.TCI-I))PP=B+nlS]GN(TDLI, CC-R)

I I -_ I F I C R (1 -C.L T. CAP S ( -Ch,4. COPP= fk+C 12. rc

in CEIKKER=fl

I l~F(CF"P*FCC).GT.O.00)C=A12c IF( (FCC!FCC).C-T.0.0OIFOlC=FOAIZI Co TC e

1:; 6 IPRITEfI*15)1223- ST CP121 12 1,ITE(6oll)

17 c c TCP11-6 15 F~l'I.-'Tt-'.' lr: IN INPUT CSTA. FfB) HAS TIE SA"E SIGN AS',

1?pi 13 FO z~T(9-l,'NUNPEQ rF ITFR'kTI.CNJS HAS EXCrFCFE L1~'1T. S!TQP CrC.PuTA

1 .1 fENC

17S, rCLEtF PFRECISICN FUNCTIrN FM~

I ?2 F =rA %K (CSC.''T( fX**2- (RP-R ) **?) ('P-R ) )-DSCR fX*'*?- RP-F ) **2) /;P

P3QETLV\ F1

114 ENC-139-

2. 2

-j U

C.)

a,

a' c

at

LU U'

u or

w7 w' Q0JU

r. r i4 or

LL w.

orUte u)

_0. aLIw . j 9.'

to i > - tr 9- C 0

C3 w

u ' I..

N~~~. 7 U. I l

Li A

4,0 0 w 09U. L.L of u Q 0U

L. I 9-'FU12

* C .0 -140a

cr Acc

00 L fl-

a* Ic D

C)o

w 4.I> U. 0

f. x -

c cy

C -

wu -,

o to

- 4o ,- LL

o " Li L

*~4 cca-4 ~-~- orU C

14 11 f

APPENDIX G

PROGRAM LEWIS CIRCLES

G-1

-143-

0..

1,3

cc U,

M zDU0. 0o

G-2J

U., I*~ I144 -

CN

41

.0 -

VI S

-- 0. . : .t

VI A5 Q

C.3 U -a

_j 'n M

C47 Z *m I'-1 -1 :t - 1

Q ~ ~ I~S L) ai 0 . 3-30l

- - 3

4 - C ccc - - - . .-145-c

C)L

C3

.6

0 0

4-4

-4G-4

416

CL'

a..

WN

(3l

4x 4

.0 - --C a. L

-M.IL

0 0- .4U I U.cc2 -

a U. .4

J, -2-

ah -

Uo w3 I

Li -D-- 5 -0 5 0 ooow .4

LU . a. G-5

- N 4 U.147-

* z

ac

If.

!LI I aj -1 .

14 - 4

If IP fIfI.

^4 r, . a N

= - .qoo -v o ,

oru u z~~u uo

- '~ ~~-6 U LI - - L

4 ~ -48-.J~ L

'Ij

wc

4L

4.L

w :, x -Z m

41 WI Q Li

W Ix a - -- - 1J. IA. * -

P- zJ 0=*

x i ex m -2-& -0. 1 * Z -j" --

- w 1 a ). - IJ .eW e-*11 -

u -1 e l -. = L SI *

U .Wu~ -V i 2.5w -J-

~~001. 0V NJ- . - I

--. o

0

0.J

o;-

u.

- 2-:

-10

CcUC

oI

i 7

- _ DIQ IN

'Aw~I-

II

- CCL

UL

00

:3 S3

4x5.

Cii-

Ci N

ID%

'- 11, it

N .

01

- d ..152-

o .-. I INII

" C i, O. S ,.,,

0 5- *-10-152-*

Ci IC '5 Ci

.... . ... . . .. . . -... .. ..... ... .l % J u j ., . N r .... an .r ? -iJ' - , t.I- ' -

. . . . . . ...

I111 1 IlI!I Il IIl ill lIi 11 9 1 1 h u l 1 1 II i

IN i ll"""1 flu I I I II aI I a. N II IIN II l I I* II iN 1 1

II l II II 111 II I I II *g , , Iii I I I"- i * , , " , . . .

w K x x x x a x x x x x x x xa K x C Ic w K x x x K x A n x x K X w < x

N Q

.. . . . . .. . . .. . . ... .. : . . . . . . ... . . . . . . . . .

NN # # NN #Y

-11-15 3-

- , -. . . . ... .. . .. . . .. . . . . .. .. . . ... . . .. .- - -

1 1 0 0 0 I. .1 H I If N I t I 1 1 I If 1# .0 1 IiI I 0 1 1 - I

0000 OONI e I, 01 .I IO 00 0 go",' I N go gUNI I NH I gogoogI H N N

ot II I 4

I I

th

I go1 goo go 01000I

0 'gII

09 0 0 0 04 OII I 0'

n

""

I I I I

N~~~ N ' N -N NNN '. , y Ny N - z Z0 N Nz NZ N N z ;z Nz zo z z z z Z, 7 0A o( xi w~ w( -CN Cf CN C C KC xC fi x xC A f.x wC xl x C x flx NC xf XC NC xN NC wC NC KC x C KC wC N( xC C

~7~. d N. . .M ....M . . . . . . . . . .'. .. .e . .

Z N -T tf ZO M t 45 ID' ZN'''C~ 0 ~ " ' 'n -P W, '~S N .0 cc 'ot -3 -' N "~'t .~0 ~-N '' f

G-12-154 -

-- 4 -- - - - - -- - - - - -- - -

I I i i 1 j I I I I I I I i Z I I I I II I

I I 1 II II 1 Sr IIrII I r ' " II rIg II 5 II 5, ii II

rI 5 II Ir II r!

rr I, s, rg ii 5I II I II5 rI It I r 4 .I

ii i I . . .'.i 4 . . . ,

a. a. a a . . a. a a a . . a. a . . . . . . a. . a. . . . a. . . . . . . . . . a. . O a. O O . P O I-a . a ~ a a a z -

G -13-155-

I ,, ,, , . . ' . ' _ - ....

"lii IIII i Iii II i IIl I# I II I I iI ii i gi li i g 99 ll l iu "N 99 *I

i PH NW NN W # # I I go in" n i e P I I ii P ii, I, i

.... ....~~~~~. .. ..... ...-. .. ..: .. ....

, , • • , . 2 , . . . . . . . • • . . • . , a • • • • a a • • • • • • a • a a a a,.

~----------------

. .. .. . ....... . ... .. .... .. . .... . . .....i0 3 0 0 0 0 .. ... . ..... J ) L i J 0Un il ll a ii w .. . . ... .. . . . ... .. ..

IA- Pd4 0 41 Pd IA4 ,4,4P 0 .0 -Y ; .r a 10 J4 dP , I, N4 "dP 40 4 , ,P , 1 P d dp430 .4Pd4 4 C 0 .. d

ii ~ ~ ~ ~ ~ . o z. Ni N Ni i i i gi 0 i i i I i I i I II I I I i N i i N 9 i I i I I I N " N Ng i P I W 0 0 P i I t I i i

Va INI yf l Y I

------------------------------------------------------------------------------------------

G- 14-156-

.444' gs~~ .4 .S .4 .. .' .~ .~ . . .P . . . .4 . W . ' .~ . U' . . . 4 e P~~* 'U 4N "

4. .NU 0 o .g-4. 0 "'41 UN4 -ee. .N 4% 0 1' eQ-U 0V 4 0' s3 N 0' 0 N 9 R. 1-' ' " PS15 ~ S - "C-, ' P 1 PS PS S p P.PSPS 4 4 4 ' 4 ~ U ' ' ' 'V4-44-44-' -4 - -7'V-7 -4- 4I -. -. -*-

*~~~~ z zI -Z;Vz zie ZZ zuh z'h 1(11 zee ICI;lIIC z zz11X11 z , z~11

X X X N X C XXNXX X CX C A X t XX VtX X K X X VCX X X CNXP K X VC t - x CeX XN""xI XX

a.~~~~~ W. u. Va. w %.a .a - . .a .a . . a a (j a a. c. cj tK C7 u. W. w. 4.a .a .a .a . p 10 a. v. w. L.a .a .a .a .NNNN N.4 NNN n.N NNN NNN N4. NNN NNNN NNNsIN NNN N'~N~S '' CIw\-

a. ~~ ~ ~ 0 VW 00' 00 a.a .a .a .a .aa .M .a .a . .a 0.K .a .a . .a .a Ma . a.a .a . .a 0 ., a .a .a

V! . . . . ..z .

--------------- -- -

4~~~~~~~~~~~~~~~~~~~ W.4 1.0m 1. I''1~4 #U PQO0 NP V45'N~7 '0-3~0- N 2,9'V4 Is". 0'0 NIX 1 . 4 0 PLW 7' - JP

-157-

f I i . i I I I i 11111I ! I I J I I I I i t | I III Ig l i i i I I ! !Ili J ! II I I I gill 1 11 I I I

)lII) i ! SIII H III II I 111 . 11111111i II S I N NIH Il II II U N e I II I, I N II N l I I IIII 1Il l I * 1 II 1 I I f1 II CU Ut h I t,

•~~~ ~ . . . . . . . . . . .

1 IN II Nf NI NI it I i N N4 Nl I') N N N t Nf * Ni N N N N | IN t N1 4 N IN NI N N N NI I NI Nt N II N IU NI Ni N N iN '4 N€ i N ' CI ' N I N *uJ C~.0 LD u ' U /U u) uL U.) w. .4.'

.3u L)44)J

*. . . . . . . . . .

u. I.N .N .N .N .N .N .N.N .N .N .N .N .N .. I.N I. f. .. . N. . NN. . .it N. N. A . 2f N. at N . Ulc . cN e

G-16

IoN N IN : 4: u' 1 0 N - w 0 .0 NO: 400 0 P4 0' a0 N - I' .0 0S ON I: - -0 - I5 OjNCSnSf 4M N" -. *.

-- IV V IN .. IV V'

is - Is N 0 K a 'NNN 'off "mN 11NON-NNN.',JN NNNNNNNN N n NNN ION -p""' 00- N '.. .,N

v;; za. ZZ 7. a.z z. zzz a.; z. z. a. z. a. a. a.z.a a . .a.a a a . a a . .a.a a a . .a.a a . .a.a.a a . .a.a a z . .a.a a .

ox x x xcc ~ xx x x K x0 Vi 00 00 00C 00 0 PC X000 x~ W x0 A0 K0 0 000 PC 000 000 0 0 X I00 I0 X A0P

000 00 0 '~ 00 00 00 0000 0 0 0000 000 000 00 0 00 0000 0000 0 0IN0*N .W 4S' Ny ocl SI-, -WiC e S4S' N- w - .0r Wn Will5g' Wel m. & f n4P - U 4 CS S O~N ~4P

N ~ ~ if 0 N pI N Nf Nt N N N N NW S0~ gNN NN NN NN N Nf IN H N 0 a IsN W N . 5

ly IV4 I4N N, N NN N m" IV -V Ny N N IN IV- IN Nyr N4 N 1. N Ny NV N4 1.-M N N 11 '4 IN IN Ny "N N N N NY Ny~ z z z N z44"5.N '"

.0004~N Nf. 6 0 -S0 0' - P : v Wa 3Sm7.O00 0 0 00 ;N N NMS' E. S ON f" SIS S M MMn M f~M'Si S M% on m' MM m m. W W4 W W4 4 4 4P lr W4 W W V

G-17-159-

It

It.... . ......... I ! 1 1

t II I II I If I I I f 1141499 I | I I I I | ! ! | II" | | I . !

I " I N 'J I N I N 'I NI II N! NI NI I N NI N! NI N I N I N N I I N ' I oN N t II I N II IN N

3, a * x 9 x a • , w at 2c w 3& A If %C mc W. •*•*•

G- 18-160-

' - III .. . . -II) ... . .4 40 40 .4 I ° I . .0 . . ' . 0 ) .0. . ....

LUC

u0i

P 0

Li L

i

004 0D

fm

o1t

Q -i

o N v

NJ cc 4 ODb

*i 0

190

Nrl,.0.t0NI4I0U'N00NNO..O0~.N~tU'UNI..~NU'NNI.........U' ...-. NUNN

In 4o a .0 ey NI 0 M -D N, $0 C ON .00N I 0 0 I 4 M' 4WD 0 ft UNI WI 0 0 - M 0- n VY 4 4 W0. 0 W P- CCN 'U'U 0-

I 1I1IlI1I I I I I I I I I 1 I I l 1 I1 1 I1I I I I I11 1 11 1 1 I I I

I0 Nf 1, N I t . 'O I0N 0NN 1 0 N -4 '. If If 0. If If N 0 NItN 00 00 04 I0 0

4444~~~~~~~~~~~~~~~~; 2' N 2. z zI N Nz 0N. IN I40N 4N II4N 0U 4U ~N0-. ~.4 I40N '0U .NC

ZI ZZZ; zI -~ zNI 7z ZZI 4Z~ z.4 U'; -; ZZ0 N ZNI zI z. zU' -;Z z ;I 0 IO I0N4 zNI ~ ~ Z~Z U' N 0Z z U' NONU I-0 NU IN00 N. I

w w x xx w vN w w ) w x x w x x K w x w b. I w x Nxvc w x x w w o w x w N w w w x w0x Nw w w wx N0x

M. Vf0 A0~ 0 -N IN N IN w 1 000.... 'U 'U or V' f4 W' 04 4 4d (4 It 0 cN a.4)I I WWM0N NN N N NNNf-M0 W co Q01M6 I. . . . . . .. . WI. . . . . . . . . . . . . . . . . .,, - : 1w

.... 0.0.0~~~ ~~ .00..000..000..000..04 .0000000..............000................

69~~~~W 61 69 691 1 I 6 9 6 I U U 9 U 6 6 Z6 9 11 U P 6 1 M I UO M 6 9 M II 6 I 1 I 1 I 1 6 M 6 1 U U 9 I * 6

--- -- --- -- --- -- -- -- -- -- -- -

NN N ~ NN XNISNN It It . . " . If "on "N No NN"N f

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

I, IN .1 : ! ,Z1 W 0 -w &P 99; 2 ,wq r qp4 V WW

G- 20-162-

I I I o l I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 III I

a * P a aa 0. pa a 0 pMNa a a a a a 5 5 8-0 5 a * 0 ' a N 0 0a ft a a. v 0 a p p a .0 a 0 N p 4 9 1 p N p

4 (N N 4 P N -a*000, JCr ON%%0U96..496%9'E%0%O%0 m w mw w 649 NUS-.6UU0ONEN-.06 ft 09% IV. 44 4 4 4 990 0 0 Nm 9% 449 m 0 4 N9 N 9% 04 fn .I N 40 EU W% NN44 4 N. Nl W

W1.0 4 .0 4Q4) w 9.0.w.0 409% v, v, 9 W 9% in% 9m~,Wv % W49 n9,%W 09% N 00 0 in% in% 000N,0000in0w9v, 0' 0 '

* p a SP a Ppp pap N pUsp p 0 0 a U 6 N P a . . . . a a a P . .. . P .a a . . .P .P p P N

G-21-163-

.iKH .~l .H rl.R ." .".' . .KS .. . -. .

gt0 00 00000 coc c 0 000? Nic o 00 0000 0 ' 0000 K0N~;;;08;CI, C 'C;tcO'~~cc'Cor-.

oooooooo0;! ooocooooooo oooooooooSoo 0 O 0 o c00 c 5!0

2 o ?Qc CC

1 1 1 1 1 A I 1 1 1 6 g i l 1 1 6 1 31 6 11 I 1 6 6 1 1 1 1 1 I I

H ~ ~ ~ ~ ~ , W - : W _:WHH H H H II H H H H H H A H H H H H H A S I H H H I H H A H IH H H AlH H I

N~~~~~~~~~~~1 N; N: N: N: .N . . .N .N.

K~~~ ~~~~~~~~ ~~~~~~~~~~~ . .. . . . . . . . . .H . .. . .. . .HKKKKKK K Kf KKKKKKKKKH

0U'NCz zin ZinO Z;ZN T.Z0';Z~~~ K'V'OCC ZZ' ZNZZ0 0 00 0 0 0 0 00 0 0 0 0 OOOCzOCzCzCOC

.PCKOONC'N ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ v~; N.NH'H00 K O O O 0 0 0 O 0 O O C O 0 O 0 O 0 0 C 0 ~ C

K. .tK . . . . . . . . . . . . . . . . . . . . . .'0 ~ ~ ~ ~ ~ W % f ri 000H . - ' W, e" Z, "f W" W' le 19 be NNNNNNH N' . . .0' .'. . .0i 'I i l

'C ~ ~ ~ ~ ~ ~ ~ ~ ~ W I2 w00 0 . . 00 0 0 0 . 00 . 0Art' n f 'ArH 'inA ninnicg'A nArif'UtU nAl lvr 'U l'A rU'e nine 'Ari'e i t

inr A A 'eni s'i'eni e l I'i itv i l A nU' Kn .5 5 .'i' nA I t .,. .'i v ninA'Al sArA n nA ii e nine n nA i t A i i22 i I

H l Hi HH H H lH i' il t I H H H H H H H H Ht it H H I ii H' HH H HH16H4 - I I iI

V 0% - 4AN P- W0 0% a NC 0.0~ 4. Cw A10 0 9.. I % OW - C61N N N N GJ 0% 490CN %0 160%0 0 N 0

0% " O6 0 0 40 N 40 w 10 0% N 61. v%-0 N O 0 0 v It N 90C .0 N -1 0 N .t NO6 0 p - 1. 0 %-6 . % 6 %-

N Nl N I N N N N N f WI -f N N N N , N M N N A N N N N N N WN NNN N N N0 00 Nvm o-. N INNN INN

0% N ~0 % 61 N 0% C 0 N - 0 % 61 N 00% CNewN"-".Nl'l1 Nme.* PCN." 00 N .0 WN .%0N 6 OWN .61N 11 .0 0% N- 0 0' X1 .. "%. ' N U 0%

0%0%%C EC C .N C 0% %0 0 %

%0 0% 0% NN NN NNOO OO OO O0 0% %0 009 2316

0%0 0 %0%0 % %0%0 % 0 0%0 0 0 N %00 ~ %0 0 %0%0% %0% %0% %0%0 % 0 % 0 % %0% %065-~t . . . . .

NNN 00000 e 0 m00. U'P '..4.4 0.Np.N 4 0Il0-000N0 on mP-0P FN

. IF I. P . t. . . f.. IF I. I.I . I.P. . . . p . P . .P.P . P..P.P . . . P.P. P . . . .. ' I.. I.. . IF P P. .. .P P.P .p . " IF IF .. p . . .. IF p. t .

Com w 0' N000,4, 'ON N NNNN NNNN NNNN NNNN NNNN LN Nr N Nt N N Nr N N N NZ Ni Ny N NNN N FNN

1. . IF 1 11 IF .1 11 11 of IF It 11 I t if .1 I IF if . OF M. it M M M 1 i t M M 1M .M IF IF IF IF IF 1. 1 M M IF "C M M . 1. M, 1, If IF IF 4I I

.0 ~ ~ ~ ~ ~ - >'- .N- 0 0 4 . U' 4. N 3.~ : t. > P. . r .7 N : - 0 >- *' P . >- >- .7 P. N .- O P.0U . PP- N- >.. 0 > >- .. >' . - > )0 ~ p 0U

ry4 w 'o' .4.74N, w7 .7 w 7. .%P 4. .u7 .7; ", .7 w7. l'. v7 u7 w 7. C7 V7 4. L 7.7 w 7747 w7 .7 N 77.. .7 4

U ~ ~ qj a 1- UPU 'U s- 'U ' 'U ' -U'pp U U ' i'U'U m m U M 'I'i'U 'U U 'U 'U 'PPU 'U i 'U 'U t 'U 'U 'U 'U 'U 'U 'U 'U 'P

"y 14 PP ", 1. IN, 1%0 'N.,P i PP i i i i i IN, 4 'N 14 t%, '4 1.4 ii P i i i P I i P P i i i P P P i i i i i i ii P i i i P i P

M M M M M M I' l I t I I f I F 1 I . 0 to 0 I 0 IF I , 01 1 o 1 o , I o " I 1M Ml H l , I I O lMl 4 .

N ~ N N o N f N N f w N 3, N N iN t NI N N N, N N, N e N N N c N IF N N No N N~ N 9 x N e Nu Nd U, NF N NNN Ne N N , N N N N N N N o N N N, Ne w

G

-166-

0 wee e00-0% s o- 0 %U- 0 % 0~~ s ~P-0%em as e 0' t- 0- e. P N N m-0 0 I0N 0"-E 00 -4 -N- 1 0

0 cp a 00-Clp . t-4---ta > -

31-3..3. X. 31-3. 3. 3-- 1'. 3'. P- I-) - 1 V-31 lN3-3- XNNN1-2 . 3- .N03- NN- NNN3NNNNNPNNN

mfae OF, "MIAeo ao02 Xo ae o o ej m'oo eo c oo ooVo oc o oOO~oaooaoaooAoooa AooAoaooacaA 1zA Ag Z Toaoooocoo

N 0 oe e0 e0'0'0.00'0'Ue~ m~ e ee- -0p--~44 44 44G4'125 1E E EE E EE . . E EE%0%0 0 %0%0 ~ 5% %0 0 %0%0 0 %0%00 00 00 % %%0%0 0 %0167-%%0%%%0%00

00 00 CC rO Z NoO eC eo cc 0 N O 4 e 4D 0 40 fo. ooo0.ooer oc Woo ao eWeoW91 1 1 11 1 1 11 9 1 19 1 1 11 1 1 91 1 1 &1 1 1 911 1 1 g l

W N W N N N W N p .P W N W N N W W nZ.0p. g p . N p . N gt ~ N .

CC~~~~~~~~~~~~~~ 00 C NC L 00 0 C COO OC O O O O 0 0 0 0 0 0 0 0 0 0 0 0C O C KC

K~~~~~~~~~~~~~~~~~~ - -N - -f 0KKN. - N - - 1N-

KKKKKKKK.... .... .... KK ..................OC OCU KK ~ aO K

r~~~~~~~~~ - K 2 K K K K K K , % fml oN J N N N

NIn 44 N0E94~~o ......4tO - 0 e pN 4 0 . EE 0 N N~ 0~~U N N4 4- 4 4 4 4 0~P ~ ~ . A , N A N N N N N N N N N~ 2 C O 2Gee e~co oec e 4z

a 6N a0 0w 00 0a av 6a aa 0a 1 aa4 0a a0 a4 0 p0 v

N NNN N-N NNN NN NN NN NNN NNN NNN NN NN NN NNN NNN NNN00 00 00 00 00 00 00 0 000 000 00 000 000 00 000 000 00 000 0000N

IIIIO6I~T .~eia~ .1i iii ,I ige1eg .g~ gg g u u. gg g g .um . s

S m a a U a I m Wmauaaewa in 1"91 *W w in ww wa Wav% g, aaaaw aaaaaaaa

N 40 40 N 0 0 Np 40 0 0 0 v a0 0m N 4 a N a00N N 0 0 a a a4 0 0 . . . £N40

3. 3. 31- 3- )1- 3. 3- 3- P- 3- 31- 3- 3-- 1-- )- 3N N N N N

00 0 00 0 00 0 00 0 00 0 00 0 0 00 00 0 00 0 00 0 0 00 0 0 or If, W

awaaaaaaphaaaaaawoahaagaaaaaaaa qaa1.aaaaaa.4 .aaawaaaaa:a.0w..

00 0 0 0 000 0 000 0 0000000000000e0000etc00000000Ce 000 000 000lic *ei ii ii 11 1 1 1 .0 01 1 45el 0e 0g , gi 4 ge0gees0 0 4 0z 0.01 111 11 1

ccc0000Oc o000000 a co000000000000000a0000000 00000000'00000DC)0 00000oc.c0 Lc 2 00 0 0 0 0 c 0 0 20 CC 000000000000 0 00 0 0 C O OO 0 cc cc CC) 0 0 020 00 0c 0cc 00 EC 0 0 a cc 0 0 c~c 0 00

NNNNN NNNN NN N N N N

N~~ . . ..N .MI . . I. . W ~ , . s . ~ a e I a a a , . I I p a e . C ' P I

N ~ , N N;, N N N N N N N NNNNNNNNNNNNNNNNNNN 2

Zft zJ ZZ zPt z z zV z, 74; zJ z b z z z~ zt ft z z1 zt zt zI z z zf zt TVC N Z zi zI z z zf ft zt zi Z. 0z N. zN zt z' zU N z 7t. zN~~~~~~~ N N ttfff f . )ft ). f . )-. ft ft ). F ). ft )-t *t >. ft 0 - 1- ft 01 ).- :0- )ft ft ft ft >.4 4.1 )1 )1. 3. >c4 14 tf tf tf ,

G - 2 8-170-

.0-10,4: 0100 'a'loolococlo o4 -DIIO "6" 10444b,44 'D 04 'D0 'O ,z 0 c

NNo P N 44 4 N 0* c;,; 0 Ic - p 4 N 444k P 0 U 0 N0P- 4c N N NN N 0 NP - # O P 4 N N 4 P N .t4 P Nr P

* Pe~ O-- N N * 04 N @ P 0 N N * 0 4 N P P0- N E 0f N 4 a 0 t04 C-N8

P 5 UU U UP * UU U UP UUU PU UU U UUUU u UUU P*a pU * s. a aU a U a p m N a a w a p a 4 0 0 0 f. p

z z z N;Z.Z Z7;;;zz;~; Z.;zzzz; .;;~ Zz;~Z

0, NNN 0 NNN , NNNw NNN NNN NNN NNN NNN NNN NNN NNN NNN NNN NN"I

a, I" in VN 0. W, WN V 10 W w wo w ov w -u a Q a vW rZ 1. 10 w a N w .0N ND N0 4. a N Ma Nl N0 .0 41 *0 N, 1 NVal NNNNNNNN

ooooaoecooe oaoec ooe oooeooeo aoo oeeeO O~o OOG-e CC29C

-mc Niin ON'"N 'N'-n

4 f , M 0 o '

£CCCC CCCC CCCC CCNNN N NN N N N NN N N NN N N N N N PNN m N N . p. .

N N N N N N N N N N N N N- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -INN N N N

oc ~ ~ ~ ~ ~ ~ ~ ~ ~ . 3oo e o o o o o eo o o o o o o o o oo o o o e o o o o 000. 3.1-CC

* NW w .N W vW N N u: ri w KNI NW N .NW W QW Nw Nw N m il w W W N~ IW ;f w NW N o NQ N KNN

fi w Q i fi fi zi fi wi wi t i ~ f if

meNN 9 w w" bNNNtNxNN3"e'N NN wN vNN vNNupNNwNNN NNNNw"NNNNNNNN

- tn iK ilf t nfinn cnn in , nn n n nn,. nlN i -ffi nnfnnynGin t i30flti

0000 00 00 00 00 00000 '000 00 00 00 00 0 00 00 00 00 00 00C200 0

N O. -K 0K ON K. . N U'N ~NKN N N U' N a -9 .Of Of IN N of O N U' 0 - 0 N 0 a $I a 0 H .0 1. N 1- h 00 N C

00c0000000000000004000000000oo000000000000 0 0 0 0 0 0 0 0 ccc ccCCC

c, 47 (7 CO OD OD C II III IIIII IN 4W W' II WI II a -4 I M o 'f WC CC

. . . . . . . . .I .IN I I H N N N N N N N N N N N N I N N N N N I I N N II I N a l, I

0 ~ ~ ~ ~ ~ ~ ~ Pl 000 00 00 0 00000 000 0 000 00 N 00 0 0 0 0 0 000 C0 0 LU C 0 INC0

U'~~~~ zK z z 0z ;N;,;z ; Nz0;; ;. U. t 0Oz K . ' -N 0 KN.0U'K N- KK . U Nz :K A KH'N;NNNNN- - - - - - -OO zO 1Z ;;: Y0000 ;z zK zzK z K Z Z z Z zKK ; zz N zz N zz zz lzOz0zZ ZU ;z Z

0 L U 0 0 0 0 0 O 0 0 0 0 0 0 U 0 0 0 0 C 0 K K K K KKa.('

v' 31 U' 29 29 V, U' U'be vx' v' %e Ne if U' e' x' 4' bc u' 3rU,'U 'U IKKKKKKK K

00 000 0000000000000000oo0oc000 00000000 00 00000000000000000cI II I 1111 li i ~I I~ III i l ll I I g ggg gugg g gmf1 g1 11I g

00 0 0 0 0 00 0 000 r-r rr 00 0 0 1.00 0 00 00 000 CC. 00 0.00

I 0 c: 0: c 00 C .0 00 0 0000000000 00 0 00 0 . '0 00 0: . 00 0 0 0 00 0 00 . .C 0 . . . 0c-r4 w O rr wwwwrwww1 T- Tww4- . fr '

4~ 4444444 P 1 11IN P . N N N N N OO O C00. 11 If $

Of~~~~~~~ ; :10 ;1 Of :'; 01 Of ;fO fO O fO 110:fO f fOffO O zO f ;fO f fO fO O fO f fO ;O O fO f fO fO O fO f 1C'1 fZfC

0. w woo a v ;:

If. ut 11 .K. K 1 K 1 31 I If 9 u t 1 J, uI wI ft 3, N e w 3 l 3K g at 39 3C Nt 3( Kif K c p 1 w If! 39t K t K t W 1 P f P K 9 P 1K KIKP N It Ng P t P PCK

G, - 312-174-

I' ~ ~ ~ ~ ~ ' 4 N~ 0 0-i'0CO . NP. IV P. N P.Nm - 0 Il 0 0 - - m 0 N P". C qC y W, OD W, & 0 0 N (P 4 4 ft 0 .0 N -P19%0 N0V , NP

N NN NN NN NN NN NN N N % l .aaa, , , , , , , C C C C fP , C CC CitC CCC t C~ CC C * ' I ~~a c Cy e cc N C4 w~ C : C , C C CCCCCCCCCCCCCCCCCCC0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 4 0 0 00 00000 00000 4 00000000000C0C000C CC11111111111~~~~~44 44 gi 44 g ~ l l l l l il ~ l l l l i

a a a a a a a a ii a a a a a a a a a p a a a a a t a a a a a a a a a a a a a it a it it. i. II.

NCC N NC 0 IL0

N NNNNNN NNNNNN NNNNNN NNNNNN NNNNNN... NNNNNN

0 00C 000 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0W 00 0 00.0 0 0 0 CD WC W C k e O'W' ODC C C CCC U'CC CCP. . . . . . P.P.. C. .0CCC~~CV~~~~CVC . . C . C . C. , ~ i*'~~~~~~ WC w w 'C 4 Cr 'r ;, 4 C C w 't CF 'r C C. W. w C C wCr w CS C C E

C CC C CC C CC CC CC CC CC C-PC CC CC CC C CC C CC C CC CC CC CC C; CC2 CZ CCz

paz an a fnaa a,,. aaa aagzaa aa fl za ZZZa naa aa za italatf anna atli flt., a,,,,z Z.z . Z;;;zz

i t C C C C ~ K V i t ~ i C M m -i i i ~ t t ~ t t t ~ ~ t i i C C C V x C ~ t x x C pwCPVwC CwC

N 9wNN NN NNN NNMw NNMugNN NNN NN NN NN NNN NN NN NN NNNM NN w NNw

t-C- CC-CC-CCC175-C--CC>C-CC-CCCCCCCCCC P-CC >

4 C4 4U' N C4 -0 44 N4 i' .44 .0 -0 4 0 w m 4 w 4d w Ow xi 04 w m &4. a. N C e 4 L- e L. 04.C

4444 ~ ~ ~ c 444444 4444w w m444 44 4 N~~w Nw c N ww w .NNN-NNN4 4 4 4 44 4444444444444U'4 atw

4 4 44

4 44 444 444 44 444 44 444 444 44

4 44

4 C. 4444444444.4.

0CC~~~~~~~~~ 0.00 0 00 0 0 0 .OO O O O .0C . .OO.O.O.CO.C.O.C.C.O. 00C

111111 ......111 ii.W.lilww ... itli ... xxiii. - ...111 1 1 1 41 1 1

It ~ ~ ~ ~ ~ 0 0 0 I S S Iot I I H H ImSH I t H H I t I H H H H H H H H H H H H H H H H H H H H H H H I H IS H H It H It IS. H H H H H It.I. H

4444 4444 4444 4444 4444 444 4444~ 44 4444 4444 444

N r N N .N -N -N. Nr N. N N NP N U. N N N N- N N1 N. N1 N N N N. & N. N7 N N- cr N N. N, Nc N NNNNNNNNNNN

it 4. 4" W" W" b" 40 41* 4 I" it 1 No Z 4e W" .0 up HC 1"-- - W" w444 .C .4 .44 4 4 44 4 4 w44I,4"2 " "W

000CC~~~~~~~~~~~~~~~ -o~~o~C C O C.14C000 OCCOaOCOCO000COC

(. i L CCCC CCC CC0 000 000 0C00 000 000C 000176-0C0000

Wf m NA N N4

N~~~~ mm Nf ON 0 P. 01 -ri ,4 lfNM V E - - - -l -t -~a II E OOE O' E E 'M t. 0N 'f

e0'N C, 0

"""Own NnM NNN NMN NNN NNNN NNNoN N N NM M N NI MM NI" INN N M,* lI, S

C 1: W LP M N 0 0 1 0 k 4 M N

N~~~ N. N N N N N.

Zft zft Z Zft 7.; zt ft zt 7-ZZ;~ ftZ ZZ 7t ZZ ZZ zt zt ZZftft zf f z f MZt t f t -- ftMCf f f f t t t t t t f f t t tztftft Iiftf ff

000 00 00 00 00 00 00 0::::0=00 00 00 000w 000 0 0 0 Co.) C C 0000t, Cr Q0 pW M

MN ~~le ::M MMww iN 55M UN MN UNN up .111M IV MUMPS MUM 29 If PMN M M

N NNN NNN NNN NNN NNN NNN NNN NNN NNN NNN GNNN NNN

~ ~ I->.177-

o~ 'Ve 't i- ' -i'0NU N 0 4HpNN e '-N P.- MP M aP 0 a' .- I 0 N - m HP ' 0 HP U 0 NO P 0 U- r. o4 -4 4D 't t- f 0 r 4 E I

0 .It t0 11 1 1,1. 0 N I I 1 I t 1-i 0 0 1 1 0 011 0 1. . 1.ll t I 1It 111 1 1 1I It

z N N p-NNNNN NN N N N- N -N -N -N N CN 4 N- N N N N N N- Q N N N- N N N N N N N N -N N -N N -N N N N v NN N- Z NZ

MIT& 4 ~ w NI 0 I_ HPw 4U 4'N-C PNCU40N-00H .0p P9N 00HP 4'0 N , - 4 PP 4 504 N 4,-,-0N4 '00 0 .'o 5WH PH P PH 44 4 4 4 . ' 'U 'JU U' w' 'a :,v 44 444 44 4 0 P 90

444 4 4.0 44 4.0.04.004444 444 4 4.0 44 44 444 44 444 444 44044 44

4**~~~~~~~~~~~~~'~~ >44 44 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 44 . 4 4 4 4 4 4

liP ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ r w 1 IH P~ i, ~l" P P P I IPlPHH H P H P HP IP HP1lP PtI

HPHPHPHP w -a4'PPPPPHHHHHHHHPPPPPPPPHVPPPPPHPPPPPPHN N N v N N No N -N N N N- S

~H ' ~ N H ' N P H P H ~ H P ) ) H P > ) ) > HP3)N3>

4U' HP P00 P, U' HP0 4F W W 04H 00 P9 V4~ 0N 4V178-N H'

cc m. 'D a, N 0 -N

to 00 01 0 00-- - N WN

0 000002---

glllllllll I l

It Nf ft 0 If N N N 0 Nt N N N

.0 °°0.0°°°.0. . o.000...

GO "4 NW In 4 444 t 4 4444

G-37/l-179N/#9-

APPENDIX H

PROGRAM LEWIS ENVELOPE

-18 1-

a,00

4j4

I-I.

- 01

40 40. *

".. 0-,u

0 W 0- 4 .6 . wo

I3 Q w 0 4 3 A0-1 41. t;; -

cc *0 3 4w 0 0OO

:WO -? Z 4A 41 4a 0 rtH w44 4b %2 _ a6Q 0 44% V% V% 4.~ : ~ v I-I n0 I aW

a D- il %n K t

a * *4 41* j

K1 0 0 0% Wa.I w0 W *43

II.

4.) a -ft444 2 42a .44.3.0 0..J-4- 0 34493 u

A ~ ~ ~ ~ ~ ~ ~ Ztll 0M O-U 4 4.4 4W"1 d3 0 00 ~ ~ ~ ~ ~ ~ ~ ~ ~ o U4 * Lb 0in 4 - 4 0 0 41 J 4-

4 ~ 3 3 4- 0~.- w s-w . ~ 0' 0-444 .1444 wwu4

3d 4 .J4 b. ll * WI4 3d 41. 0~~ 4.%- 04 f4~ CJ. 4 ~ -** & H* * *' 4-4410 B 43d3 4 ~~ 44-Z La W ~ 0. 4-e~ls 2.c *I3000 4 2 B.

'a. 4- 4 * 3d- 0a ~~ a 4440 '00' .. a . 0 *4& 4--W0fr3d * .

22~ ~ ~ ~~~~~~~~f 2.H N * 40 d 4

%~% W C4 d0 ~ * *

uj

CY0

N44

- . a:

0 * 4L

I.

6LI

- - Wi 5

0- dc I4 z L4 0W* I. cc 49 '. 0-414 IL

*I

0. 0 55- H

SM 0 4 9 1SmI ID0: * H#Iz. :4.1 0 me I *0 04 .41 0 *0 44 a *. ~

- 044 4 - * T:7 40

-MZ C 04 0 . 1 at W -S 0

N~9= 04 SM -4j* 40 4 L) 04 0 a ID a U

S. v 0 4q 4 In IL S. Wi L W.it .. I I I 2 .S

0 4 SM 0 X~ -A 40 IL P. XSOC- 54. IOU 44.4 : . - . xM 040 0Nz b

*Z44 1- : SM0 401

-W 4 0l P. a4 Ol0 " F 0 *4 M lip 4.44 04 P. W a..0;:.ftW0 w 0 4 0r . 91.0S1 i 0 * M 4 -0 S .Z

a 0zo wl on4- 0 4 wow W"I H 0 a,,S.K-S 0 9) ?1. SM a04.wwuiEWi w 0 4 Z wC 4'OOZ4 HS.4 u04 wgw * 0 4 V544 wEw

Wi 0 14 % 4 4 5545 0 E4% 4. 6P4.Wi-183-4 %'

.j I

I-I

IL bW- +

A, W

4J0 4 4

x, .U, Q

U , C V 4

0 * It f., IL 4 Z1Z O-u 4p 4*0 0& 'La

ChC aUL %m -- 4- 0 -1w W *U,- : 0 CL W

LU~I I0. 4 LA .* tl z ; t OW ft 0. of c

4~~ 01 IIL4 4 - f

W* 0 ILwt - U L I- 0, C 0 tAL .Wt 40 0 C L

OfC t CO4 LAO19

ILa aIA Ii of I*WWz oW

.j - . w IUM em -- - 1 aL &f- w 00 00 0IL,..I j~N LA IC OL)UU~A4t~I.

-. pwL* V W MI I al v a odea b.0 - N:; -C

U, W 10,0,

LU Z2.IIL LLLLLIL IL

k0

LizPu6 1- jt juWt

fvm bUU0P af

%P ~ ~ I 4-I W

-184-

vi

W)a. z

0 w u -9- a

0- ILAIIA 0. .6 .

- imI 0. 0 in - * N ;t.C 42 4 e IL j

* Ni- - . .a f lIAP 4pI .im 0. C m

4~1 * . w

iU % 0 0- VU. z

CLV a IL ~ 3 .xv U. 0 4w -. &.*O

OIL U' ow a Ko * W RzL)Z *2z, Eg W- 94-.U -

OW v LN 4wz o 04 K.~L wNC

4~Z- 0. . I-N % 0 0.

K W - Z a FaJ6 -3 1-4 -asi a 40L M a .WOm - *-a

62..~ ~ ~ ~ ~ ~~U-W 1.. 1 gU. &Z0Z XJtoOwc 00

tv f" IF r-t mWp. :0.0VW:tv tv "I- m" USp:: OW0 6, C, NON 0,42 .

z: . -Z WZW .1 n.

w'e ow MU VC CE KKo %I

*% ** U - 0 £ ~1 85-N ~ O .*% a. * W*

0j

00

wq4A 0 b

IDv0. w P

"Ilozi-

N16

4L

P.

- . PC

v 0 * 0 9

u a N 4c to: -1 4 Ca

Is a 01 at'

- , Is - 0 j or S* - , C2, t, O . 0 o- a t i 4j U 00 I-~ g.O 9 4

a 6. of 0- 4 aar = 9z 00 9 0 K a* -

9-~~ 0 - ~w 6. 1 0 U. -oom a i oo

W6 'a x W C) 0 J - 00* -a Se 5. 1 9- ofM 40C0c

U4 a 0 0 Ce

0 *. a. '64 6*- M &L-* 0 4 1 .. aC 1 1; 9:

I. C wi CA Z; 61 fm *% 1 33

a * 1a * W V%9-4 ::l~

a- 16U. - 3-* - U. =U. 09-.- U.9-

OL : O O.- Of C, 00 lo-at* 09-Mi- -A-M& O wano4 wlsq;- - 0 0

j z 0 a4-0 094 le0 ' 0 *~ -Ua ( .3w' u xu a0was 0 Ilg 4In -9-0 's a'.04 M D.

4 ~ ~ 1 0 : M.Jaa "14 :OU 0 .4-4 uu -

a. C Mi a-i 09- v f" IV Wa -n, "I.0 *

* N UE~~~~~i ~ ~1W wA I Ou U G w 1 G G K - U A U 0 4 4

Mi * 9I.ZZ .~aW* U-- U m a UU aU~- US 187-- U

04

w4

-0

0z .

w ~

39 IL

z 0

VP IL W "

I" I

Li 4,-L imA

- 188

IL

40

at

0 4D* l

or

me

z ..

_ 00

z :4

Eow

W

aI" 4

"0ww 4

£09

- 189

AL9

0 0 0 01

U a.

14 on I

o 9

o 9; 0

4H 0

* 1904

1. OF .:V .m ~09000000000 0c,, Z4;44clo4c ~o;0 8000 cc0 0000 0 0'c 0 cc00000000000 00

1 41 8 0 4 1 9 a a a a a a aN a a I a a a a a 1 v a Ni * a * a I a a 0 1 9 q a a a a a a a a

n -- --- nf-------------------- ;--------fl-------------------

xK K XK xM M X bM K K K KKIMOC K K K KIM MMc KKP K13Mixc m KMUCI Mlxxx K I X K MX mMM 1mX 2 K M WI K KM t

* .-*0P. NO ft4 adpmm~mmP m D 0 0£ . p. Puem a 1 d 0m- .- *mPtP .um

C y646~66 p46 in ft o- 4 .0 q c * 0 * c. 00 : 4 0 pp, *r a * .** * . WI * 11 4 . . r 0" . o- * **

Q.- -- - ---- - -- -- - - - - - -- -- --- -- - -- - - - - -- - - -- -- - - --- -- - - -- - - - -

o az;; z 141 ge1414 1414 a4f4 a fury fyfgrgfuNNV N44NNgNeVNN agz weue z aaa Zgague ZZ4 aza.I.P .a.a aba3a.a a3aa a a a- -- -1- V-3-loP -3- 9 - 1 1 X -P 9 -3 X l 3 l - -P 3 -P-3aA P -J

Kt K K Kn K K 0 V. K K K l K K in K Of K W K %; K Kn K0 KW 0' W Kt PM Kg KI K K K KP W KV K JKKKPKKMKKKW Pa~~d~d~~d~d ad 41.0 .01s N me mld agodd WJ Od~dd~N n a to m on 4" 4" 4r r 4r qr 0' W V% ON um~d

-li

6NN d. OCOO O O0.f ID m 01"0 P M

000 -- ---- - -- -- -- -- -- - - ----- -o -C--b

.,eecooceeaceoaO ~ aoC ~ oO4P,~

ow. ~ ~ ~ ~ ~ ~ P etew ICOI 0ol tas , i se o. go g- e-f *lowtg a~

03343U ~ ~ ~ ~ ~ ~ -- t.*:UW II tN U ----------OU USU UU tU t U Un

- .~ 0

P 1-3. 3.1, P P NP ~N N N f t 3. 3 . - -V-3. . .3Jb-f 3- .3.

oqpWX aw v a w

*P"'EN.N 29 stmagaWtNU ~ V1 W % ~ 0

H-12.

pU~ggUg 5UU S5 SU UU UUSSU US USU US U U PU192-S UU

-~~~~~~ !: w m. AD f P .- me o-.6em m .0.mt."": P. - P. @@ NO -P.4 N .. 0 lP. 4S 'm P-O N a 5 14;g 004. No N ewf4 P. mOP. P.4 . P. ftNb%.16 P. : 0 N 040 2* m l .P .PP4P 0P ~NP .0 .. 0NP .. 0-.. 4 0NP

NP0 mP1 Osl T P-0 2 P AE44 0M .PP. .4 4.P. P. P. P% mmop N000O ~0 0 .~ .P 4 P. Ip, m. .04

--------- --- -------- -- -------- --- ---

0: v~ N ralm0a0G m N f4 P. P. N ; 0.-aW F

.0 P. PP.0 0.0 P.v 4 m ..0 0 - P. N P: 4 p. P. P.. . . P.OP P.O P 0 P0 . .*P p. mm I-W.0 m lu W mmP1 .09 mP .. 0 0..P .P 0 0 0v 0 .. 0 P. IV P. 444 v.P 4 * .mP .5'NP..P.-

-- ------------ -- ------- ----- ------ --- ------

3- gg gf 3. 3-p gg g p3p SS PISS SSS3.SU SP PU WP S

MMMNMNMMMNNM MNMNMMMM MMMNMMNMMMMMMMMMM M N

NP.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~Zu NP.cm P. NPWP"P'PWN.N.N.NN P P .NP P P P.NN.NNNP N." NP. N.P. W, NP.O NP NP. N NP3..

H-13-193-

IL

Ir aO a V. - F M

N 0

bNN

4ug 0

4 -4

oy 0

-1 . R I'3- N

416 W,.

a w

ft 0 10 0b

* 0

o 4L

o 0 IL pm 0

a 195-

AD-All? 593 CITY COLL RESEARCH FOUNDATION NEW YORK F/6 19/1AUTOMATED STRENGTH DETERMINATION FOR I NVOLUTE FUZE GEARS.(U)FEB 82 6 6 LOWEN. C KAPLAN DAAK1G -79-C-0251

UNCLASSIFIED ARLCD-CR81060 NL33ffffffffffff

I LmhE

L 72

MICROCOPY RESOLUTION TEST CHART

N~ATIONAL KIRMI of TANIARD, 196i A

if . .4 00%OA% 4 0 . 0N00,Dow%*4 U4 @ 4 NMW W. I', a%. m 'D4.0

0000000000000000 000 cc000000000000000

- -- --- - - - -- .0 -. 0 n-- - -- - - - -

U0 V. P. ' 33-3 U--3 3-% 40 - 0 - 1- V. P. V% P- PN P' P 31 -U - 10 W 4% 0 N 3' 0' N- b- 3- 3-% 3- $- XN 3- N3- *% 3N 3 3- 3. 0

4%W 4%4%CWS~m U NN N NN @ 40... 4 two on 4" 49 1? 4p @ 4fM 'r oM 0 In n N%

-196

:PGll IN ID 0 N ""I ..'C9 " 404fmE..0 10.. I-499 n fn 949 on' #n 4 me m W IN 9 0 t l a 94A M 0 ftu -r4 if' 4 WDaN~ifCaA 40 09 10 49 4 4 9 0 09 a N R m -p 4 N ... w 9i~9NO094,14 o 1N4 0 a 99 No 14n a43,a0 invot 0fm4 D Ip0 'ON9m 0N9 0 a 9NMA0NO 4q99499 ~ ~ ~ f ow- e A 4" 1" W r VAm W%499 4 4 :0 N 1- P000 0 9 99 9 0 000 0- &a,--

P . . P. P. * -P P. f* * . I. '. P* f- 4D a * a a a

4. us sl V-0g I At A ~ 9 11s41IL80 a c 8 a 0 a aIf 8 0 a 0 4 Ia 1 a a I a 1 0 0 a a 1 * a a a 0 8 Is a a a v a a

nan- aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

* wwwwwwwwwwwwwwwwwwwwwwwwwwwww------------------------------------------------------ ------

09 9 49 N m Npj 4 a 0 Nm Iw No 4w NO 494 am 99 N , Alm, UO ~ 4 N-I W9 NOul0u44 N

9~~~~~~~~~~~~~ N179 0 0 4 4 4 44 N0 9 aN N9 6 0NO NOa4N0 094m~a e049 m~aO9 N494 .N -196N 9.7a e. N m.-ao . N . 9 .

a P.* * .a. * a.... .. .... a.......a .. a... a.....*. *a.. ...... a

oft

-198-

W, .1 O 4pP @ h a F- a N ew lp 0 v 'd OP V% W% W

..... ~~~~~ NO 10 0. C! 94 1O P: 0 .11111 N0P *

N ~ ~ ~ ~ ~ ~ ~ ~ a oi N" N P. . . . . 00 - a maN

000n0 a 00 0 00 0 in inww4-WW -WW V-- I.0Oro r w W -f' - I! * W- -

a a s a 1 a a a a a a 8 0 9 a 0 0 4 a a 0 68 a I a ts a a a a

g~ g, , o, :P llgP

H - 19 -NNNNN NNN NNN NN*JNNN'NNNN199 N

.4 vo

U, IL

N

z 3..

ca w

U' C

U I

0 sobe

a.

Iw N

do

oI-

93 aGa K

H-20200-

p~ ~~ Af v.. cm m~ P9 N -0 .0 P0 lo% * me - m~ pa 0*m a- me N Pd 4 & -p...

~~4 4 : ::4 eP0 P0 ~ ~ e00~ .0 . Ka .a

m+mmmmmmmm.*..*..;.epmmmmmmpmmmmm.0.00.0..0.00.0..0.00.0.o...C. cm- .. .. ... a. .a. a . n c Im.ntt l 't l ' 1. .1.m a .~... e . .

oocoo, oCJcoo 0coeo 0 Jo oooJo o Jooo o

IeIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

IL

.... . cu u 'u u, 5U.m .. , u s mm sc gm c

P. 3- Pd Pd J. 30 1d ad 3- 3- PdP P P Pd Pd V. 3. 3 p P P P P P P X- P- Pd 3d 3d X- 3d 3- 1 3P d 3- 3- V- b- 30- V- So, 0- 3- Pd 3- 3- 3d 3-P P d

ow CAp w me baw m de on w .0m w oin -me vs wom m Pe Z Z. mX 0.C ~ ~ ~ ~ ~ ~ 4 t f Mq Wep.e~pPpe~.e Wm.Uem in Ne0em-P..d.em .%-

*mmmmeemC0Pepmpmm...0~ 0pm~~a~J621O~m-?-md

ch c --tftfmo~ 0 V It in n .D 4.0 -P F- .aa aa 0 ---j. . . .. . .

* a a* . .. a a ... . . . . a. . m a . a m .. * 9 * ~ S * * *oo eo o oo~a ~ ee ao o eo oa 0 00 00 0 00 00 0 00 00 0 00 0* ~ ~ ~ ~ ~ ~ ~ WWl *,," n~wo .1 .01 in~gg .0c ge ,, ,, Oz*s e e m g ,, , , e . . se

wem in Inmm ina aor umua ua g a am mmmmam apm m mm

KKKKKKKKKKKKKK KKKKKKKKKKKKKKUUKKKKKKKKKK KKKKKKKKVKKKKKKK KKKK

11-- - -- -----xf" ;.,w ; *n p;It !

N NN NN NN NN NN NNN NNN NNN NNN NNN NN NN INNf- NN NN2MWWWWWKWHW KKWUWWKWNWWKKKNWKUWWHWHKK 20 X

03 V .00 .GMEN. N4% MEd~O~U@

-- 0-0 0 s01 .v N v n - -%- - w-- In In ---- v%

.. .. ... .. 0 00 90.6.9 n

~~ -a a0 aa 00 00 a0 aa a00 aa 6a 0a

ALzzzzzzzzzzzzZZXZZZZ K memwZinutZemwZsZZqZZ

Z Z ZS 9OS %' mlE W~S 14p n REUmUmv ON 0i~ 0ggg swes. ..

H -23-203-

I~ PIP.-..

Ad~ m Pe dPA .O 0PP .4AOUA .A .@ dP CP- a.P 4P 4 EP.P A P* A PP 0 A -PP A W PP Ad ot PW ft a Pb PUA A dA dA.PU. 04 A.PU e4 i o. dpP

* o f % 5 a a e f e E @ e , , , , 8 s o *i i y o 4 1 1 1 4 a e ~ # C C I . 1 1 C l lS g o s o w g e1ge a 6 9 0 , 1i

4.J. P .PS O 3. m. A P. O P. P. P -P I- o. PPU -3 .. P.4 0A P P A. P.3,PP - 10-A-P 3-m 10 Xb P - PA. SO P PP Ad Ad Ad P3-P

0# A P PA.- P A.- P ~- 04 A A P . d P04PPUPA.Ad . 4 PPOpU VPO

A. -P4 -oe aU P0 . Ad0 . A PA.4 d A H- 4 4 4 0 0 0

P P P ~ 000 PP PP UU UA. . A A.44 P- ~ P dA A d--000P

*UzzmU4UUUP.UU44UUUUP.Uep.UzUUU4.0PzaU4@4UU-zEPzeUP;l.U0PzdU-U

p. p. m- 3~-P . .p .p -p .p. p. I- S. I- A. w- P. P. w. P. P p. P. V. 3. P. P. P- .p .P . P. P. So P. P.3 P . P. P. P. P. P. P. P. 3. P. A- P.- W. P .

arn.4"w~ ..u.. ... a1a ** ***ua.* m.. ....00000000000000 ;00 NNoo~ 0000 Vo o oo o o o 00 0 0 00 0 0

- - ----- an - ----

4000D a' 010.. .. . 0. 0 4 . .01 0 0 me 0 4 In 0 0 me 4 0 90 N . 0 N C0 40IDD 0 a a% 0 D IC 000000 0000 0000 00 0 0000 00- --- - N N NN N m 6 mm C P. P. P. P. P.6 4

000 000 m 00 00p N% 1 0000 0 0 000 00 00000 00O 00 0m 0 0 000 0 00otm-b

MMNNM M!!-No app M:VN NU N N~g~VN MgN~N N N N N NN NOcvv &NNttf N N f"MN O Z S 2 --- N- - - - -N NNNNNNNNNw NNNNNNNNNNNNNN

N ~ ~ ~ ~ ~ i in a MN M n NNNNNNNNNNNNN

21. O 9. W C P. N- U P C P. N. V. P 0 e, P. P N 3. me . P. w N. 3- 3. 3- 4. N P0 I. N- 91 N. . N. PS-b - - X' 1- 3- P' mm m- XmPm3.3-- P

-~~~~m Nowi C04 6%9 9 4N9 4m aN 0%N 460N 4mO emo mm m m m

me 0 N 0 C N CNONmm 0 4 C 9 N NO 26%m m4C 9NC O m.14CN9N mm4C N6NmN NN 0000.9-mOON NN 444m m6 CC C~m mm NN-2060099

'04 ~ ~ ~ ~ ~ i m; O3 3 N-. %3 0 0 V0 fm v ; f &,a vovNftv 003 4 % , Vft 4 0 O..4N: a4 3% N - -Nmv 10 3% UW40NmU 4 -4N. U 3% ~ I NC4 I, P. =% 0: W NNNNNN Nt N N- N % : 400 N Ca :0 N , 4-Nw C

N ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ e N .0 00 ~ NNNN I44 %3 04NNNUU0 0 - N o "I- % 0NN U0 0 W, p 3

a0 44 4 4 N a N N N N N Nl N N v v N N N N a N N a p N N N Ne 0 v0 N 1 a v IN a 0 a ON 0 O0 0f e

ow 0w 00000o 0 :0Z0000 00m0-Q: 00000 0 0 0 0 0 00 00 00 0 00 00 0e t , g e e ,~ ~ ~ ~C g g ! see e e g&f fiiW W g em m t s , e g . e e i e g o g e Ie t e

U 40 3' 4 N- S' N 3 2 U- 1-' N 30, N- 3%NI, N - - N 3- P' N- 0' U 4P 3- N ' 4 -N 3- 3 0 3 0N4

N ~ ~ ~ ~ ~ ~ ~ r 0. U. V m z N 0U4 %.. %N0 4 N5 4 %m.0 5 4 N 0N3

on fm0 o0n on on On Un Un Uo U Ur Nn Nn N N N' A4

000 L 00 0 00 0 00 0 0 @00 0 00 0 00 0 0 I00 ' '0'0 0 ' . 0 H-27'W0'0 0* I P I P * * I I I * * * I I I I I * I P * P * II I * 207 * * I I

P oo UPP8 0 NO 0 N , N, 0 -P -N - N - -4 P wwwp0 - p4N w NN4P UP-N 4NUCca, p P P P p00 0 0 0 00 - -- -cc -- NNNNNNNNNNN 4 " 44 4P

0 . -w 0o Owr"1ao. 0 0 t". * on in v% I" on 00 f.a on .a . . .9.. .* .* * * .* . . . . . . . . .0 Im00000000 0000 - -8-0 00000800000000f.000 QQ000 w%:::wwwwwm0A

Of 11a a INv 0Ova 0" a 11 a at at af a a Of a a a a1 a wa 0 gal It~ ga1me0 a v 0 a a a a a vu SOa m 01 a v a a a

-------------------------------- zzz zzz zzz z;; zzzj~.v . .2 .*x .v P. 3. 30 wwM 3. wM v 3- 3- 3- 3-M 2-3 MM V -P MMV 0 ,X -o b3 -- *D3 ,b M- 3- D-M PM PV PM P- V -3 X V -M .) M vb j.

- - - - - - - - --; "- - nn nn nn nn nnn nn nn nn nn nn nnn nn nn nn nn nnn nn nn nn nn nnN ~ ~ ~ ~ ~ ~ ~ f N Ni Nr N N N" N% N Na Nopez NNNNNNNNNNN NNNNNNNNNNN

rMM X M M M M M N M M M M V WM M M M M M V M W X M V MMM We4 MMWwT o I:l, rl0"

P *4 U -U44N U 4 Pp @ pa- PNP U 4U p4U C -NP-U O

N 0P 4 4N0P N 4N~ p N4 4N aP N4 P4 NN UOU20844 PP 4

-N d)4 4 2 N M 4 4 a @0- 4 PI. 4@, NYM 4 " P 10 4 aNM ~4Ua~iM Ce iNSCN4* ~~in 9% 1 : 0 0440 mm9NmmmmFminmI.-m mm P4.ppP4P:p0wpPap44O

0:00:00:008000000 0 00000a0 0 0 0a a a wa -a. 0- a~a CaeOaaC.elm;;

NNWW X MN W"XNNNXNNWNWNNNNNWNW WWN N WNW WNMNNWNK WW M X NWW N W

I wi N N N N-N -w - - w - a aw * aW aV Mr 4 4 w wQ N NW NV Nw N o 4r *4 Mi -r M or 0 -w4 -4 W I I# w -V Nr Nr Nw NW N N

0- U 3- V- mm. Pm U 10 - W N N N N- 1- N- N N N- N N- N- N- N- N- 3- N- P- N N0 * N- N N N N . p N- N. N. N N 3- N. N V-3,b P- N0 V- 3- N-

on 0 a% am *-, *r w w z z q SSSSSSS*SSSP.410 a III me 44 41. 4 4 4 4 4 44 4 4 4 4 4 44 4 4 4 4 4 r"

mupugsr 0 S pm mugs' mmr~er srqw su~ m puu Vu p rnVuu insu in~S m%:Miniu I n1

NNN WNNN NN NN NN WNNW NN NW NN WW WW NNNN WNH-29WNNW

NNN NN NN NN NN NN NN NN NN NN NN NN NN NN NN NNN209NN NN

wo lo .0 Do. OPla.4 P P0 a Wo *WWf* a PPN~ NP N I" on W N a S o 00 P. P. 9 or

PU'N Pc 4P.00N44 ON# am PP 0 440P :NPNPN4v 0 4 W!0P40.m :Ol a

ceococod ooo oWooevoo a cooO-----C------ -

ft PR of N 0z OR pmN P0 ON WPP P 4 0 40 0 0P.O4 4PN

. . P * P.*V. ft 0 i a . . * *a 0am a* 4 4 4 E44 E4444 4 4..4......... 4 44 44* 4 4* 44 * 4* 4

PasaPa'mIPP ~ ~ ~ ~ ~ ~ 0 P. PP P. P. P awe.PP .PPPPPUPPUPPPPSPPPPP

-210-

41~~~~~~~~~~~. of, ftP4-4P3P Oi4Pd4f-dPd1@W,@441dO94U 441dN9~

3.**~~~~~~~~~ ..3 3..... .. n*** * -I** * 3* * * ** ** * *

----------------- tft --- ----- 00--00 0- ---- 0-. 00---- Va,

* . * , 0 8 p a a * 4 g 0 0 0 9 a 0 0 " 0 0 0 4 . 0 a 9 s 6 6 a e

P* 3- 3- 30 3 P- 3- 34 W 3- 3--3-V ua 3 4 3-1 34 43b3 4 03433 4 P-g I-'-P -P -3 -3-:-V-I-X -P - -V -I-3 - A 3. J. 3. 31

waaaa aaaaa aaaa aaaaa aaaag~g ~~wg4 4g ~M~ In

PdP dP dP dP dP d dP dP dP dP dP Pd li f l ow Pd PdP dP dP dP dP dP dP dP dnd n dP dP dP dP dP dP P2 Z

0. U a- v Ob NW a' NW U -0

3. *. P . * P. P.. P. P 3.

-------- % op- --

.e##.eeee14e32

*U *,h~U~gU -212O

DISTRIBUTION LIST

CommanderU.S. Army Armament Research and

Development CommandATTN: DRDAR-LCN, H. Grundler

G. DemitrackA. Nash

R. BrennanW. Dunn

C. Janow

F. Tepper (5)L. Wisse

DRDAR-TSS (5)Dover, NJ 07801

AdministratorDefense Technical Information CenterATTN: Accessions Division (12)Cameron Station

Alexandria, VA 22314

Director

U.S. Army Materiel Systems Analysis ActivityATTN: DRXSY-MPAberdeen Proving Ground, MD 21005

Commander/DirectorChemical Systems LaboratoryU.S. Army Armament Research and

Development CommandATTN: DRDAR-CLJ-L

DRDAR-CLB-PAAPG, Edgewood Area, MD 21010

DirectorBallistics Research Laboratory

U.S. Army Armament Research andDevelopment Command

ATTN: DRDAR-TSB-SAberdeen Proving Ground, MD 21005

ChiefBenet Weapons Laboratory, LCWSLU.S. Army Armament Research and

Development Command

ATTN: DRDAR-LCB-TL

Watervliet, NY 12189

-213-

CommanderU.S. Army Armament Materiel

Readiness CommandATTN: DRSAR-LEP-LRock Island, IL 61299

DirectorU.S. Army TRADOC Systems

Analysis ActivityATTN: ATAA-SLWhite Sands Missile Range, NN4 88002

CommanderHarry Diamond LaboratoriesATTN: Library

DRXDO-DAB, D. OvermanWashington, DC 20418

City College of the City University of New YorkMechanical Engineering DepartmentATTN: G. Lowen (10)137 St. and Convent AvenueNew York, NY 10031

-2 14-

DATILMEIO