Meck. Mack. T&,ary Vol. 28, No. 2, pp. 181-192, 1993 0094-I 14X/93 $5.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ~ 1993 Perpmon Preu Ltd

F E A S I B L E D E S I G N S P A C E S O F T H E P R E S S U R E A N G L E

F O R S P U R G E A R S

H. L. CHANG? and Y. C. TSAI Department of Mechanical Engineering. National Sun Yat-Sen University, Kaohsiung.

Taiwan, R.O.C.

(Received 5 March 1992; received for publication 22 April 1992)

Abetract--Various types of spur gears have been designed for different specific purposes, but the pressure angle plays an important role. In this paper, the feasible design spaces of the pregure angle for spur gears are presented by using polar angular functions. Convex curves, straight lines and concave curves are the three basic elements of tooth geometries which have proprietary design spaces existing in driving gears, driven gears and racks. The design space of mating gears are constructed by the intersections of their individual proprietary design spaces. The possibility of cusp occurring in each design space is also investigated.

INTRODUCTION

Various types of spur gears have been designed to meet different specific purposes. Involute gears are widely used today since they are favorable to manufacturing and assembly. Cycloid gears, which possess a constant specific sliding and avoid the undercutting problem, have been applied to clockwork and gear pumps. The concave-convex of noninvolute meshing gears [1, 2] are developed for increasing the load capacity of gears. By varying parameters of basic rack, Mitome and Seki [3], and Costopoulos et aL [4] have developed the gear pumps with optimum performance. It is implied that the peculiar tooth geometry in each type of gears intrinsically affects the performance and meshing properties of gears.

Practically, tooth profiles of gears are mostly generated by rack cutters (hobs) or pinion cutters. The straight cutting edge of a rack cutter produces the convex profile of an involute gear. Varying the ratio of the pitch radius to the radius of rolling circle, the convex side of cycloidal rack cutter will generate three types of hypocycloidal profiles of gears such as convex profiles, a straight line and concave profiles [5]. However, epicycloidal profiles of gears are always convex curves generated by the concave side of cycloidal rack cutters. Pinion cutters may also generate the same tooth profiles as the rack cutters do. Similar to the rack cutter, the convex cutting teeth of pinion cutters may not always generate convex tooth profiles. Therefore, it is not easy to select the proper tooth profile of rack (pinion) for a particular designed gear geometry.

In gearing kinematics, pinions (driver), gears (driven) and the rack are conjugate with one other. The mating configurations of these conjugate tooth profiles can possibly be classified in four meshing categories: convex to convex, convex to concave, concave to convex and concave to concave. However, the characteristics in each mating configuration are not clarified. In order to provide a concise concept for gear design, it is necessary to explore the mating configurations.

In this paper, the general mathematical model of parametric gears proposed by Chang and Tsai [6] will be used for investigating the mating configurations. Specifying the pressure angle as some arbitrary parametric functions of the rotation displacement of gears, a large amount of parametric tooth profiles in the dedendum part and addendum part of gears and rack can be logically developed and analyzed by the model respectively. For example, involute gears are classified as the constant parametric profiles, cycloid gears and circular-arc gear belong to linear parametric profiles [6]. By differential geometry and gearing kinematics, three types of tooth elements, the convex curve, the straight line and the concave curve, can be discriminated by the polar angular functions. The feasible design space in each part of gears is constructed by the

tPresent address: Department of Mold and Die Engineering, National Kaohsiung Institute of Technology Kaohsiung, Taiwan. R.O.C.

181

182 H.L. C ~ S G and Y. C. TSAZ

limitations and the first order derivative of polar angular functions. By the intersections of the feasible design spaces of gears and the rack, the mating configurations for pinion/gear as well as the rack/pinion/gear are established. The constraint equations corresponding to the mating configurations of gears are also presented.

The mating configurations indicate that the meshing pair of concave to concave profiles fails in tooth conjugation. The cusp of tooth profiles, which may result the undercutting problem in profile generation, are also described in each feasible design space of mating configuration. With atlas and tabulated constraint equations of mating configurations, the analysis and synthesis of gears will become more concise and efficient.

PARAMETRIC TOOTH PROFILES

The model proposed by Chang and Tsai [6] divides the conjugate tooth profiles into two groups. The first group ofconjugate tooth profiles includes the dedendum part of gear I, ~,j, the addendum part of gear 2, Y~, and the addendum part of rack, Y~,. The second group of conjugate tooth profiles includes the addendum part of gear I, 2~,, the dedendum part of gear 2, Y~d, and the dedcndum part of rack, 57,d. In Figs I and 2, the contact points of the first group P~ and the second group P; which are initially set at the pitch point P can be described by the following [6]:

Pl = P + An, (I)

e l = e + )-'n', (2)

where :. = IP, PI, :.'= IP;PI, and the common normals n and n' are separately defined as

n = cos ~i + sin ~j, (3)

n' = cos ~'i + sin co'j, (4)

z2,

Fig. I. Coordinates of meshing gears.

Pressure angle for spur gears 183

!

R l

Pitch Circle

n •

X~ d

! b .J

Pitch Line

u

, t

# • #

Z la

- u N

n'

Fig. 2. Coordinales of gear I and the rack.

and

--- n/2 + pressure angle =f(qb, ), (5)

a ' = it/2 - pressure angle = f ' ( ~ , ), (6)

where f ( ~ t ) and f ' ( ~ , ) are arbitrary parametric functions of rotation angle of gear I, ~,. Equations (1) and (2) describe the common normal of a pair of meshing teeth at any contact

position always passes through the pitch point P such that the law of gearing can be satisfied. Based on these equations, the geometries and geometric properties of gear i, gear 2 and the rack in the first group and second group are expressed in terms of the three kinematic parameters (a, 3., ~, ) or (,,', 3.', ~, ) respectively. Additionally, these kinematic parameters are strictly correlated by the first order derivative equations [6],

d~. d~, = R~ sin ~, (7)

d3.' d~l -- R~ sin ~', (8)

where Rt is the pitch radius of gear 1. In the first group, while gear I rotates counterclockwise with displacement ~ , gear 2 will

rotate clockwise with displacement ~2, and the rack will translate along the positive F, axis with a displacement u which is defined as

U = S l ( ~ i = R 2 ( ~ 2 , ( 9 )

where R2 denotes the radius of pitch circles of gear 2. In the second group, the moving senses of the displacements @,, ~2 and u are contrary to that

in the first group.

184 H.L. C'UANG and Y. C. Ts~J

The initial values of these kinematic parameters in both groups are set at the pitch point P and defined as

=rip ,

~ = 0 , 0[e t ~--- 0(p,

~i' = 0,

~, =0,

~2 = O,

u = O, ( 1 0 )

where ~r and • ~. are the polar angular values of the first group and second group at the pitch point P, respectively.

T H E E L E M E N T S O F T O O T H P R O F I L E S

The tooth profiles of gear, pinion and rack might be composed of three types of elemental tooth profiles, the convex curve, the straight line and the concave curve. In Fig. 3(a), the three types of profiles in the first group are initially located on the pitch point. The normal of convex profile n,x at an arbitrary position has the profile angle % less than the profile angle ~e at pitch point.

~ . = ~p - ~p . , (I I)

where ¢~. is some positive value. The normal of straight line n,L always keep the same direction along the line, thus the profile

angle ~L iS a constant value and can be defined as

• ,L = "P- (I 2)

The direction of the normal of concave profile n,. at an arbitrary point is described as

aw = cce+ ~o~,, (13)

where ~o~ is some positive value.

% j'~ %" ~'tL % " %

~ Pitch Circle

(a)

Y i ~ 0 x "

(b)

Fig. 3. Profile angles of three types of tooth profiles. (a) The first group, (b) the second group.

Pressure angle for spur gears 185

The three types of tooth profiles in the second group, shown in Fig. 3(b), are also located at the pitch point initially. Based on the normal vector n~, at the pitch point, the directions of a~, u~ and n~ of these three tooth profiles at an arbitrary position can be individually defined as following,

and

a~ = ' ' (14)

=,'L = cz~., (15)

• • • 06) 0t~ ---- Gt/, + fp~,

where ct~, ate' L and at~ denote the profile angles of convex profile, straight-line profile and concave profile in the second group respectively, as well as (p~ and ¢~, are positive values.

THE LIMITATIONS OF TOOTH PROFILES

The geometries and geometric properties (radius of tooth curvature, relative curvature, specific sliding, etc.) of pinion, gear and rack described in parametric model [6], are characterized by functions of ~, d~t/d~t and a', da'/d¢~,. Therefore, the limitations of these conjugate tooth profiles should be evaluated by the polar angular functions and their first order derivatives.

In gearing kinematics, the contact path in the first group is always constrained inside the pitch circle of gear 1, thus the polar angle a must be limited in the range,

~ < ~ ~<n. (17)

The contact path in the second group is constrained inside the pitch circle of gear 2, therefore, the limitation of the polar angle ~t' should be written as

08)

Notice that the limitations of polar angles represented in equations (17) and (18) are valid for all types of tooth profiles of gear I, gear 2 and the rack simultaneously.

The tooth profile of gear I In Figs 4(a) and (b), the initial position of the three types of profiles in Ztd will move from

position P to position Ps if gear 1 rotates counterclockwise with a displacement 0,. At this

/ - Pitch Circle ; - , ( ,M

I! / k--" _v .., ::-'

( a )

~ " - - ~ : - - Pitch Circle !"J° d-

N 'vl I1 'M

(b)

Fig. 4. Polar angles of three types of tooth profiles of gears. (a) Convex profile, (b) concave profile.

186 H.L. CItANG and Y. C. Ts^l

moment, the normal of tooth profile at the contact position should pass through the pitch point P. The polar angles of these three types, therefore, can be described as

~,t=~,s+dp, ( j = x , L , r ) , (19)

where ,,~, aL, and a,., denote the polar angles of n~, nil and n,. t for convex profile, straight-line profile, and concave profile of I:,a respectively.

The derivative of each type polar angle becomes

- - = I - - - -

d~l d~l •

dat, - - = I ,

da,q d~o w - - = l + - - d~, d~ , "

(20)

Considering the line of P s P in Figs 4(a) and (b), the angle as, is a limitation of the polar angles of the tooth profiles, since the contact position of a pair of conjugate teeth is impossible to stay at the position Ps when the tooth profiles move away from the pitch point. As a result, the polar angles in the three types should be greater than the limit angle as,.

aj, > as, ( j = x, t'). (21)

From the triangle O, PPs, the limit angle as, is defined as

n ~b, as, = 5 + "2-" (22)

The intersection of straight-line profiles, aL,, with the limit line, a = n, will obtain the maximum rotation displacement ~,a,,,,, of the straight-line profile.

~b,L, m,~ = n -- ae. (23)

Similarly, the characteristics of polar angles a~,, a~., and ~,,'.,, as well as the limit angle ast and the maximum rotation displacement ~tL, m,~ of the addendum profile Y.,~ can be investigated as that of the dedendum profile Eid.

Comparing the convex and concave profiles with the straight-line profiles, the limitations of the dedendum profile Y,d and the addendum profile E,~ of gear I are listed in Table I.

The tooth profile o f gear 2

The tooth profile of gear 2 (driven) is geometrically similar to that of gear I (driver) and have the same gearing kinematic. However, the kinematic parameters defined in gear 2 are contrary to that in gear 1. For example, the kinematic parameters of a and 2, characterize the dedendum part of tooth profile of gear 1 and the addendum part of tooth profile of gear 2; the kinematic parameters of a ' and 2', characterize the addendum part of tooth profile gear ! and the dedendum part of tooth profile of gear 2.

Theoretically, the previous investigations for gear I can be immediately applied to that of gear 2 if the kinematic parameters defined in two parts of tooth profiles of gear I are alternately changed by the following equations:

a ' = n - ~. (24)

Therefore, the polar angles a,2, ~',.2 and ~'t2 of three types of addendum profile ~2,, and a~2, a~.2 and a h of three types of dedendum profile Y:d are described as

c t s s = ~ e - q b , = ~ e - m ~ , ( j = x , L , t , ) ,

a;~ = ~'e -- ~P2 = a'e -- m~p, ( j = x, L, c), (25)

where m is the gear ratio.

Pressure angle for spur gears

Table I. The comtrainU of tooth profiles

187

Profile type Gear ! ~,, Gear 2 Y,. Rack ~,, Gear ! ,r,. Gear 2 ~-a Rack

Convex %, <"Ls =~= > =,, '% > =L, "~, < =1.1 =~ > 'Q , =~.> a~. profile d=,, d~',,

• ,, >=~ ~.2 > ~s~ d¢---~ > 0 =;,<=~, ='.z< =~, ~-T>O d"x, d~,z > d"~, d,v,~, > - - < 1 - -m - - < 1 - -m dO, d'~1 d~i d~m

Straight a~, = a , + ~ , a ~ = a e - m ~ ~ az.,=ae a~., = a~.+ ¢, a ~ = a ' e - m d ~ a~, = a~, fine dau I data da~, da~., da~ '

. . . . . m - - = 0 - - = I - - = - m dat" .. 0 dO, dOl dO, dO~ dO, dO,

Concave a~, >a~, a ,z<a ~ a,, < a~, a~, > "~.~ a~ < a~z a~, < a~, profile da,, < 0 a~, < a~, a~ < a~ da~,

a,, > a s, ~ > as~ d~, d~"-'~ < 0

d~,, > I da'z d~;, > d=;z - - < - m 1 - - < - m d~, d~, d¢~ d~b~

Note: Note:

~ = M s ~ n ( M = x . L . v ; N = l , 2 . r). O ~ r ' ~ s ~ 2 ( M = x . L , v ; N = l . 2 . r).

Similar to gear I, the limit angles ~s~ and a ~ , and the maximum rotation displacements ~).a,.~ and 4)w2m.. construct the constraints for the three types of profiles of gear 2 which are also listed in Table I.

The tooth profile of the rack The rack is translating along the pitch line, consequently, the orientation of tooth profile keeps

constant, and the polar angle should be identical to the profile angle. In Figs 5(a) and (b), the common normals ns, and n;, ( j = x, L, v) are characterized by the polar

angles, i • as,=a,s ( j=x,L,v) , c~s,=a,s ( j=x,L,v) , (26)

/ / - - - - Pitch

n..~

Y f - -

f

e ~ n 'far

n'..u

Line [ / - - - Pitch Line

avt l o

-2 i (a) (b)

Fig. 5. Polar angles of three types of tooth profiles of the rack. (a) Convex profile. (b) concave profile.

188 H.L. CRANG and Y. C. TSAI

where the polar angles a,,, ~h,, %, as well as a~,, aL, a~, designate Y~,, and Y~ as convex profiles, straight-line profiles and concave profiles respectively.

The constraint equations of 2:~, and ~,d of the rack are described in Table I. However, it should be noted that those constraint equations of the rack are considered as the basic rack (or rack cutter) with respect to gear 1.

Considering the three conjugate tooth profiles of gear 1, gear 2 and the rack simultaneously, the convex-side profile of the rack meshing with gear 1 can be deemed as the concave-side profile of the rack meshing with gear 2, and vice versa. Therefore, the constraint equations of Y~,, and Y~,d listed in Table I for coneex (or concare) profiles are adapted to that for the concave (or convex) profiles of the basic rack with respect to gear 2.

THE FEASIBLE DESIGN SPACES OF MATING CONFIGURATIONS

The mating configuration of gears The feasible design spaces of the mating configurations of gears are defined by the common

design spaces existing in the appropriate bounded regions of gear 1 and gear 2. Since the straight-line profile is characterized by a particular linear parametric function, thus there are no bounded region existing in ~b, -a and ~ l - a ' coordinates. Geometrically, convex and concave profiles are distinguished by the straight line and have their particular appropriate bounded regions which are described by the limitations listed in Table I. Consequently, the mating configurations of convex and concave profiles of pinion/gear are divided into four categories: (1) convex to convex, (2) convex to concave, (3) concave to convex and (4) concave to concave.

Intersecting the bounded regions of convex and concave profiles of gear ! and gear 2, the feasible design spaces in both groups are described by the listing constraint equations in Table 2 and depicted in Figs 6(a) and (b). It is indicated that the concave to concave mating profiles do not possess feasible design space. Therefore, this mating configuration fails in the law of gearing. On the other mating configurations, the domains of the feasible design spaces are characterized not only by the mating category but also by the gear ratio m. Further- more, each mating configuration is restricted by the first order derivative of the polar angular function.

The mating configuration of the conjugate profiles of the rack, pinion and gear

Considering the three conjugate tooth profiles of the rack (basic rack of gear i), pinion (gear I) and gear (gear 2) simultaneously, the mating configurations of rack/pinion/gear have six categories (see Table 3). However, only four mating configurations have feasible design spaces which are shown in Figs 7(a) and (b). The domains fl 3, i'll, t~ 2 and fl~ of pinion/gear are individually identical to the domains fl,3, fl~3, fl,2 and fl~2 of rack/pinion/gear. However, the set of the feasible design spaces f~,,I and t~,t2 (or fl~, and fl~2) are equivalent to that of fl~ (or i'll ). It implies that the design space for the pinion cutter with convex profiles is greater than that for the rack cutter with convex profiles.

THE CUSP OF TOOTH PROFILES

The cusp will exist in a curve when the radius of curvature of the curve equals zero, i.e.

I 0, (27) K

where g is the curvature of tooth profile. If a tooth profile has a substantial cusp, the generated profile may confront an undercutting

problem which diminishes the strength of tooth and contact ratio. To avoid this problem, involute gears are generally constrained by tooth number. For instance, the minimum tooth number for 14.5 ° involute gears is 32, and that for 20 ° involute gears in 18 [7].

Gea

r I

ft,

Gea

r 2

~h

Tab

le 2

. T

he c

onst

rain

ts e

quat

ions

of

feas

ible

des

ign

spac

es f

or p

inio

n/ge

ar m

eshi

ng t

ooth

pro

file

s

Con

stra

int

equa

tion

s D

omai

n G

ear

I Yr

. G

ear

2 ~E

2~

Con

stra

int

equa

tion

s D

omai

n

Con

vex

prof

ile

Co

nca

ve

prof

ile

Con

vex

prof

ile

Con

cave

pr

ofil

e

Con

vex

prof

ile

Co

nca

ve

prof

ile

=>

=#

-m¢

~,;

-<

=r+

¢~

,;

:~x

; tl

, C

onve

x C

onve

x ~

t'>

0t~

.-m

~.

~t'<

~t~.

+qb

,. 0

<~

i<~

t;

~,

da

prof

ile

prof

ile

. ~

m

da'

=>

+-~'

-; --

m<~

--~<

l;

O<~

,<x;

ot

<~--

~t;

--m

<~<l

; ot

;~O;

= ¢

~=

p--

m~

,;

~II

< -

-m;

tie

Cou

cave

~t

' <~t

~,--

mq~

,; O

< q

~t <

--

; pr

ofil

e m

x

~,

2u,,-

-x.

d='

o <

~,, <

_--

--_-

-. --

<-m

; .."

;. O

; d¢

~,

dog'

">

=e+

¢,;

=

~x

; ti~

C

onca

ve

Con

vex

='>

..~

.+¢~

,;

TT

>I;

pr

ofil

e pr

ofil

e d=

;

0<

¢~

t<

2+

m

;

No

feas

ible

des

ign

spac

e --

C

onca

ve

No

feas

ible

des

ign

spac

e pr

ofil

e

ti;

ti;

o;

Rack

E,,

Tab

le 3

. T

he c

onst

rain

ts e

quat

ions

of

feas

ible

des

ign

spac

es f

or r

ack/

pini

on/B

ear

mes

hing

too

th p

rofi

les

Con

stra

int

equa

tion

s D

omai

n R

ack

];,e

Y

,./'Z

.~

Con

stra

int

equa

tion

s D

omai

n

Con

vex

prof

ile

Con

cave

pr

ofile

Con

vex/

conv

ex

Con

vex/

conc

ave

Con

cave

/con

vex

Con

vex/

conv

ex

Con

vex/

conc

ave

Con

cave

/con

vex

=<=,

+¢b,

: O

<~-~

,<l;

O

<¢~,

<x;

0<~<

I;

='<

~-m

¢,;

No

feas

ible

des

ign

spac

e --

C

onve

x/co

ncav

e N

o fe

asib

le d

esig

n sp

ace

Iden

tica

l to

f~

ti,

2 C

onca

ve/c

onve

x Id

enti

cal

to f

~

da

g a

<at

; -m

<-

~T-~

<0

; ~,

,z

Con

cave

C

onve

x/co

nvex

=

'<=

~.;

=

'~0

; =

'<~

-mcb

,;

prof

ile

L

da"

¢~J"

='

>~.-m

Ch;

-m

<~<

0;

0<¢~

,<x;

=

>=

r-ra

P,;

0

<¢

~,<

2=

r-~

; =

>~

+~

-.

Iden

tica

l to

ti3

ti,

3 C

onve

x/co

ncav

e Id

enti

cal

to t

i~

No

feas

ible

des

ign

spac

e --

C

onca

ve/c

onve

x N

o fe

asib

le d

esig

n sp

ace

ti~,z

190 H.L . ~ and Y. C. T s ~

With the conditions for predicting the cusps in ~d, Z,. and £,~ can be respectively described as

the equations of curvature radii of parametric tooth profiles in the first group [6],

d.__aa _ 1 R, cos a d¢~l /~ '

da R~cosa ( da R, cosa) d ~ + l = - - - ~ - - - o r ~ - ~ + m - - ,l '

d-u = -T- or d~, ,~ " (2S)

(') aJ .au a~

I

I l l . 2

Ct~L20tL2 ( re<l )

( re>l) (m--I)

0 I , | i

0 ~. x ~t

(b) oc Cru

X ~ . . . . . .

0 r///llg//Ik==-~---~ -- - . ,d'...

b' o ,, ' ; . ~ - - \ ' -. ~-.~ - ,, \ \ - _ OCS: Z OCL2 OCL2 OCL2 OCS2 (m-- ! )

(re>l) (m~l) (re<l) (re>l)

Fig. 6. The feasible design spaces of pinion/gear. (a) The first group, (b) the second group.

Pressure anlgle for spur lears 191

Since the parameter ,1. is a positive value and cos • is a negative value, therefore the possibility of the cusp existing in these profiles should pose the polar angular functions in equation (28) to satisfy the following conditions:

d= <1,

dO"--; < - I or ~ < - m ,

d"'u < 0 or ~ < 0 . (29)

(a) a a u

aSl

x d

a

X ~

G L2 GL2 (re<l) (re>l) (re=l)

0 , I I x x t: 0 ~-

(b)

I ~LI

x_r s. 2

a ' ~ °fLr c~b. 2

0 "-r3" "d ' , -

\ ors2 f L 2 °'1.2 a'L2 a'S2 (m= I )

(re>l) (re=l) (re<l) (re>l)

Fill. 7. The feasible design spaoes of rack/pinionll~.ar. (a) The tint group. (b) the second group.

192 H.L. CH~O and Y. C. Ts~

Examining the above equations, it is found that the feasible design spaces of f~ and f2,2 in the first group never confront the cusp.

Similar to the first group, it is easily proven that the cusp does not occur in the feasible design spaces of f~ and f~3 for the second group.

C O N C L U S I O N

The atlas and constraint equations for the feasible design spaces of pressure angle for spur gears have been presented. By differential geometry and gearing kinematics, the proprietary design spaces of convex profiles, straight-line profiles, concave profiles of gears and the rack are characterized by polar angular functions. The feasible design spaces of the mating configurations for pinion/gear, and the rack/pinion/gear are established by intersecting the proprietary design spaces. It is indicated that the concave to concave mating profiles fails in law of gearing and the feasible design spaces are substantially affected by the gear ratio. The cusp never exists in the feasible design spaces o f the mating configurations of t~ 2, f~,2, fY3 and f~3. It is believed that the design of parametric gears will become concise and efficient by using the design spaces proposed in this paper.

Acknowledgement--The Authors are grateful for research support from the National Science Council of R.O.C. through Contract No. NSC-80.0401-E110-05.

R E F E R E N C E S

I. A. O. Lebeck and E. !. Radzimovsky, Trans. ASME JI Engng Ind. pp. 543-553 (1970). 2. G. W. Eggeman, Trans. ASME JI Mech. Transm. Automtn Des. 109, 419-423 (1987). 3. K. Mitome and K. Seki. Trans. ASME JI Mech. Transm. Automm Des. 105, 736-741 (1983). 4. Th. Costopoulm, A. Kanarachos and E. Pantazis, Mech. Mach. Theory 23, 141-146 (1988). 5. V. L. Doughtie and W. H. James, Elements of Mechanisms. Wiley. New York (1960). 6. H. L. Chang and Y. C. Tsai, Trans. ASME JI Mech. Des. 114, 8-16 (1992). 7. H. H. Mabie and F. W. Ocvirk, Mechanisms and Dynamics of Machinery. Wiley, New York (1978).

DIE G E E | G N E T E N ENTWICKELTEN RAUME DF~ EINGRIFFSWINKELS DER S T I R N Z A H N R ~ D E R

Zmammeafmsm~--Viele Bauarten der Stirnrider sind f6r verschiedene speziellen Anwendungszwecke entwickelt. Abet hier spielt der Zahn-Eingriffswinkel eine wichtige Rolle. Im Rahme dieser Arbeit werden die gceigneten entwickelten Riume ffir den Zahneingriffswinkel der Stirnrider mit einer Polar-Winkei-Funktion dargestellt. Die konvexen Kurven, die geraden Linien und die konkaven Kurven sind die drei Hauptfaktoren der Verzahnungsgeometrie, die einen gceigteten entwickelten Entwuffsraum for die treibenen Zahnrider, die getriebenen Zahnrider und die Zahnstange vorhanden ist, Entwurfsraum der Zahnrider wird dutch ihrcn induviduellen gceigneten Entwuffsraum konstruiert. Die m6glichen Phenomen der Verzahnung im jeden Entwurfraum werden auch bier untersucht.