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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
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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)
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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*
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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-
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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
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APPENDIX G
PROGRAM LEWIS CIRCLES
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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