Involute Profile Of Non Circular Gears

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Involute Profile of Non-Circular Gears

Involute Profile of Non-Circular Gears

Dr. Bálint Laczik Associate Professor

Institute of Mechanical Engineering, College of Dunaújváros, 2401 Dunaújváros, Hungary

e-mail: laczik@goliat.eik.bme.hu

Abstract: This paper presents an approach to calculate

the parameters of a family of noncircular gears. The

calculations of the general elliptical rolling curves are

based on complex algebraic methods. Two methods was

developed to calculate the contour curves of involute

profile. The first based on the virtual manufacturing in a

CAD system, while the second by determines the contour

curves analytically. These gears were manufactured by

wire EDM technology.

Keywords: non-circular, involute, elliptical gears,

Maple, EDM

1 Introduction

The earlier version of spatial non-circular gears

shown on a sketch made by Leonardo da Vinchi. In

the 17-18th century makes has more practical

applications of the non-circular gears in the

clockworks, musical instruments, mechanical

theatres and another automatic toys.

There are some well-known non-circular gear

models made by F. Reuleaux to study the

kinematics in the technical education in the

beginning years of 20th century. (Fig.1).

Figure 1

In the middle of 20th century the non-circular

gearing was used in electromechanical systems to

control and drive non-linear potentiometers. In spite

of the improvement of digital techniques, the

importance of non-circular gears are not decreased.

The profound summary and list of earlier references

of this discipline is found in book [1]. For a huge

lifework of Mr. F. L. Litvin [2] is a little-known

part. In this work all basic questions of this special

machinery elements are discussed exhaustively, but

the applied calculation, construction and

manufacturing methods are now anachronistic. In

the corresponding chapters of new books [3-4] of

the author summarise the most important

theoretical results of [2].

The articles [5], [6] reflect some issues of the new

constructions and manufacturing techniques. The

[7] demonstrate the practical basics of the

metrology of non-circular gears.

In this article a method is presented to calculate the

basic parameters of general elliptical gears. The

involute profile of teeth obtained by i.) virtual

manufacturing in a CAD system and ii.) complex

analytical method.

To formal derivation and numerical calculation we

carried on in the symbolical mathematical program

Maple® V. 7R.

2 General elliptical pitch curves

The polar equation of general elliptical curve is

(1)

where R is a set of real numbers1.

The curve ρk hase k ρkmax maximal values and k

ρkmin minimal values of its radius. The pitch curve

have 2k congruent part between ρkmax and ρkmin

values, that can be transformed to each other by

rotation and/or mirroring. The involute profile of

teeth’s can be obtained by well-known generating

(envelope) method.

The profiles of teeth curves located in the same

positions of the congruent parts of pitch curve are

identical, if the symmetric line of teeth of the basic

rack coincide with the polar vector ρk

corresponding φ = 0, and number of teeth is

zk = 2k.( ξ + ½) (ξ=…, 4, 5, …) (2)

The number of teeth in one congruent part of the

pitch curve is ξ + ½ (not integer!). If k1 = k2 the

always the same teeth’s of driven and driver wheel

1 In case k = 1 the pitch curve is a classical ellipse. The

rotation shaft coincides with one of its foci.

,...2,1=

:= ρk

− 1e

()cos kφR∈

kk e

are connected, while in case of k1 ≠ k2 the teeth

connecting with periodically identical position2. In

The mean transmission ratio of gears is ηmean =

k1/k2, and the ratio of actual relative angular

velocities is η = ρk1/ρk2.

In practice to calculate parameters of pitch curves

the origins k1, k2, ηmax and the distance of axes

„a” are given. The system of basic equations

(3)

is linearly depend on the formal parameter „A”.

Solution of system (3) is

= pk1

2aAηmax

+ + Aηmax 2ηmax A = pk2

+ + Aηmax 2 (4)

= ek1

− + Aηmax

+ + Aηmax 2ηmax A = ek2

− + Aηmax

+ + Aηmax 2

Substitute the solution (4) in to equation (1) can be

the value of common modul of gears can be

obtained (5)

(5)

where k = k1 or k = k2. (If the gears are

manufactured with wire EDM technology, m can be

an arbitrary value different from the standardised

ones.)

The curvature radius of pitch curve is

:= K

+ ρ2æ

çö

∂

φρ

çö

+ − ρ22æ

çö

∂

φρ

ρæ

∂

∂2

φ2ρ

(6)

In the practice usually K > 0 and the pitch curve is

convex3.

3 Virtual manufacturing in CAD system

The basic rack cutter consist of two straight lines

that form a pressure angle α with respect to the real

2 This is in working practice unfavourable, but this is a

basic property of non-circular gears.

3 This is one advantage for generating technology with

CAD, but with the method presented in chapter 4 the case

K < 0 can be equally treated.

axes of complex co-ordinate system. The co-

ordinates of corner points of first teeth in the basic

rack are

))(.

(( 110

tanhIhmQ +−= (7.a)

))(.

(( 221

tanhIhmQ −−−= (7.b)

))(.

(( 222

tanhIhmQ −+−= (7.c)

))(.

(( 213

tanhIhmQ ++= (7.d)

where I2 = -1, see Fig. 3. The pitch line of the cutter

is coinciding with the complex axe. The fillet

radiuses of standard tool are in this case neglected.

Complex co-ordinates of sth corner point in the rack

are

Qs := Qs-4 + I.m.π(8)

s = 4, 5, …, 4ζ – 1, where ζ = ξ + 1 is the total

number of teeth of basic rack.

The pitch line of the cutter rolls on the pitch curve

without slipping. Let the angular position of pitch

curve (1) φ = ψ. The pitch line of the tool is

actually the tangent of pitch curve (1). The complex

co-ordinates of the sth corner point of the tool are:

:= w +

− QsId

+ ρ2æ

çö

∂

φρ

φe()Iµρe()Iψ

(9)

and the slope angle of the tangent line is

:= µ + + ψ

2θ (10)

The angle between radius vector and tangent line of

(1) is

(11)

By evaluate the equations (1), (7-11) the complex

co-ordinates of corner points of tool rack can be

calculated. The first part of generated gear in case

of m = 0.4973, P = 8.56996, e = 0.231692, k =2

shown on Fig. 2.

Figure 2

:=θ

arctan

∂

pk1 () − 1ek2

() + 1ek1 pk2

k1 () + 1ek2

() − 1ek1 pk2

ηmax

= +

pk1

− 1ek1

pk2

+ 1ek2

a = +

+ 1ek1

− 1ek2

:= m

+ ρk

2æ

çö

∂

φρk

ξπ

In the CAD systems standard manipulation

commands like MOVE, ROTATE, SUBTRACT,

and MIRROR can be used to virtually manufacture

the gear. The movement of the tool object is

determined by equations (9-11). In every step the

tool object is subtracted from the workpiece object

which results the final contour curve of the gear.

Apply the appropriate mirroring and rotating

functions the complete contour curve can be

produced.

By manufacturing the gears with wire EDM cutting

technology, the offset value ∆ is very important an

depend on the used technology, look Fig. 3.

Figure 3

4 The basic curve and the involute

profile

The basic involute profile curves of the teeth are

determined by the enveloped figures of the profile

normal of the teeth of the rack. The normal lines

PT1 and PT2 coincide with the tangent line at the

contact point P of the pitch curve (Willis-Kennedy

Theorem).

The basic curves of the left and right profiles of the

teeth are different. The complex equation of normal

line parameterised by λ is

W = ρ.ei.φ + λ.ei.(φ + θ ± α)(12)

The differential equation of the enveloped curve

derived by

:= ()tan µ =

çö

ℑ∂

∂

φW

çö

ℜ∂

∂

φW

çö

ℑ∂

∂

λW

çö

ℜ∂

∂

λW(13)

where I and R symbolise the real and imaginary

part of partial derivatives of W. Solution of

equation (13) is

:= λ

çö

∂

φρ

ρ2

çö

()sin α

+ − 2æ

çö

∂

φρ

ρ2()ρφ æ

∂

∂2

φ2()ρφ

(14)

Considering formula (6) it is evident that K =

r.sin(α).

The continuous function of the rack profile curve is

Q(u) see Fig, 4:

≤u

ua ,

−≤xa,

Q(u) =

≤

−u

ua ,

).(

(15)

≤− xb,

)2.( −ua

otherwise

where

a = h1 b = h2

)(

tan

)(

πγ

tan

+= (16)

expanded in to odd type of trigonometric series

)2cos())12sin((

)( 212

0juBujA

uQ j

å+−+= −(17)

The Fourier-coefficients are

).sin().(

1duukuQAk

).cos().(

1duukuQBk

(18)

Figure 4.

Based on equation (9) the point of rack profiles

curve at the angular position ψ and profile

parameter u determined by equation (19)

(19)

The involute points of teeth E1 and E2 are

intersections of the tool curve (19) and the normal

lines of teeth. The equation of normal lines is

)

±±

Λ+= I

Ieek (20)

Solving the equations (19) and (20) Λ and u

parameters can be calculated. Substitute Λ or u into

equation (19) or (20) the co-ordinates of involute

point can be determined.

5 The undercutting line of the teeth

In case of the conventional involute gears the teeth

are undercutted if radius of pitch circle is smaller

than

min )sin(

r=(21)

In case of non-circular gears, the teeth are

undercutted if K < rmin. The approximate

undercutting curve can be calculated by the

sweeping path of corner point Q4s+1 and Q4s+2. The

limit point of undercutting is the intersection of

exact involute profiles and the sweeping path

curves of Q

4s+1 and Q4s+2 points The part of

undercutting line is shown in Figure 5.

Figure 5

6 Applications

From the practice of author some products are

shown in Fig. 6 which ware calculated with the

presented method. Usually the gears in the

industrial applications (flow measuring

equipment’s, printing machines and robots) have

modul cca. 0.4 – 3.5 mm

Figures 6

7 Summary

This paper present an approach that calculates the

parameters of a set of noncircular gears by general

elliptical rolling curve applying complex algebraic

methods. The complex formulas easy to use for

calculating the undercutting limit. To formal

derivation and numerical calculation ware carried in

the symbolical mathematical program Maple® V.

7R. To manufacture the gears by wire EDM

technology the final tool paths were generated in a

CAD system.

ψµ

ρϕρ

II eedIuQw .).)((

∂

+−= ò

References

1 Olsson, U.: Noncircular cylindrical gears,

Acta Polytechnica, Mech. Eng. Series

Stockholm (1953) X. 1-216

2 Ф.Л.Литвин: Некруглые зубчатые

колеса, изд. МАШГИЗ, Москва-

Ленинград, 1-203 (1956)

3 Ф.Л.Литвин: Теория зубчатых

зацеплений, изд. НАУКА, Москва-

Ленинград, 1-203 (1967)

4 F.L. Litvin: Gear Geometry and Applied

Theory, Prentice Hall, New York (1994)

5 Shinn-Liang Chang, Chung-Biau Tsay,

Long-Iong Wu: Mathematical model and

undercutting analysis of elliptical gears

generated by rack cutters, Mech. Mach.

Theory, Vol. 31. No. 7, pp. 879-891,

Elsevier Science Ltd., (1996)

6 W.C.Smith: The Math of Noncircular

Gearing, Gear Technology, pp. 18-21,

(2000. July/August)

7 B.Laczik - Z. Szaniszló: Measuring of

Gears with General (Non-Circular) Pitch

Curve, Proc. INES 2001, Helsinki (373-

376)

Involute Profile Of Non Circular Gears

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