Shih-Hsi Tong Graduate Student.

Daniel C. H. Yang Professor.

Mectianical and Aerospace Engineering Department,

School of Engineering and Applied Science, University of California, Los Angeles,

Los Angeles, CA 90095

Generation of Identical Noncircular Pitch Curves In this paper the theory and algorithm for generating pairs of identical noncircular conjugate pitch curves are presented. A set of criteria for two identical rigid bodies engaging in conjugate rolling motion is established. Based on these criteria, an algorithm is developed, which can be used to design this type of rolling pairs with almost unlimited profile varieties and any number of lobes. Geometrical conditions for having C' (or slope) continuity at the intersections of tips and roots of lobe profiles are also established. A family of similar profiles can be designed at the same time by using a single parameter, the pitch noncircularity, via a dimensionless analy-sis. Results from this investigation should have applications to the design of noncircu-lar gears and lobe pumps.

1 Introduction

The basic working theory for noncircular gears is that the two centrodes of the two engaging bodies roll on each other and each centrode has its own fixed rotation center. These cen-trodes are called pitch curves. If two pitch curves are used as the profiles of the two rigid bodies, these two bodies should have conjugate rolling contact with respective fixed rotation centers. This kinematic relationship is the basis for noncircular gearing. Tooth profiles of noncircular gears should be generated along these pitch curves. In 1910 Dunkerley described elliptical gears and lobe-gear pumps in his book [1]. Later Gobler used speedgraf method to design rollcurve gears in 1939 [2]. Most publications concerning noncircular gears concentrate on the design and application of elliptical gears [3, 4 ] . The pitch curves of this type of gears are two ellipses with foci as their rotation centers. Other than elliptical gears, the theory for syn-thesizing pitch curves of noncircular, non-identical gears was investigated by Bloomfield and Cunningham [5, 6] . Bloom-field's report was more general, in which methods for pitch curve generation were developed based on either one desired pitch curve or the function of desired gear ratio [5] . Cunning-ham [6] discussed the design and application of gear profiles with different functions including sinusoidal, logarithmic spiral, logarithmic, and reciprocal functions. Benford [7] discussed eccentric gears by mounting standard circular gears eccentri-cally. Bernard & Freudenstein [8] designed internally meshing noncircular planetary gears which were applied to slider mecha-nisms. The manufacturing method of noncircular gears using the roulette method was introduced by Horiuchi [9] .

To date, as for the investigation of identical pitch curves, ellipse and logarithmic spiral are the two only known geome-tries that have been used [10, 11]. There is no systematic way to design identical pitch curves with general profiles. This paper represents a research effort to this concern. Kinematic theory is firstly developed for generating pairs of identical conjugate noncircular pitch curves. A simple algorithm is then established which enables the design. Several examples are used for illustra-tion.

2 Background on Pitch Curve Generation

Referring to Fig. 1, let ri{di) and ^2(^2) be the pitch curves of the right and left rotors, respectively. The center distance of the two rotors is /, and the two angular positions of the two

Contributed by the Mechani.sms Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 1996. Associate Technical Edi-tor: K. Kazerounian.

rotors are 6i and 6*2, respectively. If rx{Bi) and / are given, the profile of the rotor 2, r^idx), to have pure rolling conjugate motion should be

r^e,) = I - r,{9,) (1)

The angular position of rotor 2, as a function of ^ ] , can be calculated by

u de. - ri (2) In the case, in which / and 62 = f(di) are given, the pitch curves of the two rotors can be designed by the following method. To have pure rolling, the instant velocities at the contact point for two rotors are the same, therefore

d0^

dt ^2

d92

dt

/-, = ^2 de.

(3)

By substituting/' for dO^ldOi and (/ - n ) for r2, the two rotor profiles can be derived as

r,(e,) = f'l 1 + / '

and

r2(^ , ) 1 + / '

(4)

(5)

For a pair of conjugate rotors having recyclable continuous motion without oscillation, the two rotor profiles have to be simple and closed. Assuming ri is simple and closed, for r2 being also simple and closed, the following condition needs to be satisfied:

62 = In-K when

where n is a rational number.

= 27r (6)

Journal of Mechanical Design Copyright 1998 by ASME JUNE 1998, Vol. 120 / 337

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userHighlightneed to get : ***r2 in terms of theta2***or theta2 varying linearly

Fig. 1 Pitch curves of conjugate roiling pairs

Geometrically, n represents the ratio of the numbers of lobes of rotor 1 and rotor 2, or

number of lobes of rotor 1

number of lobes of rotor 2 (7)

3 Theory of Identical Two-lobe Pitch Curves

Equations (1) to (5) can be used to design conjugate pitch curves. However, there is no existing method for generating identical conjugate pitch curves. In this investigation we want to present a systematic and easy method for this purpose. Refer-ring to Fig. 2, assume rotors 1 and 2 are two identical rolling pairs. Let the initial contact points. A, and Dj, be at the root and the tip (or vice versa) of each rotor profile. The profile segment A,Bi on rotor 1 conjugates with the profile segment D2C2 on rotor 2. The corresponding angular spans of the two arc segments are 61 and 62 on rotors 1 and 2, respectively. Since rotors 1 and 2 are identical, the profile portion DiCi on rotor 1 is the same as D2C2 on rotor 2. Let the profile segment A,Bi be represented by the function/(6,), which is monotonically increasing and with C' continuity. The reason to require a monotonically increasing/(#,) is to avoid the emergence of undesired nonidentical lobes. Because / is monotonically in-creasing, when / = 1/2 (i.e. r, = rj = 1/2), the points B, and Ci shall meet. Meanwhile the sum of the span angles ^1 and

Let T = a9u if we substitute r/a for 9^ in Eq. (15), we have, dOi = dr/a, q{adi) = qir) and the upper bound of the integration changes from t^i/a to 0 , . Consequently,

Jo I

/

I-fie,) d6

a Jo / -I . 1 aT 7r

dr = = (16) qir) a 1 2

This proves the first part of Eq. (13). Therefore, function/(^i) which equals ^(a^i ) could be used

as pitch curves for these two identical two-lobe rotors.

(fl)0,=O

5 Examples: Design of Two-lobe Pitch Curves

Example 1: A Linear Function.

Let a = \, b = 2, and the initial design is chosen as ^(^i) = 1 + 6x12. (Please note that the condition ^(0) = a is satis-fied.) In this case, / = 3. For having qi4>i) = 112 = 1.5, we find (pi = 1. By substituting q{9\), 4>u and / into Eq. (12), we obtain

Jo I q Jo 3 - (1 -I- 9i/2) d9, = 1.7261

Let 1.7261 = aTr/2, therefore, a = 1.0989. The final profile, f(9i), resulting from this initial guess, q{9i), will be

f(9,) = qiadt) = 1 + 0.54955,

The two conjugate rotors based on function / are drawn in Fig. 3. For illustration, the relative positions of the rotors in three angular positions, 9i = 0, 40, 70, are shown in Figs. 3a, 3b, 3c, respectively.

6 Design of iV-lobe Pitch Curves The algorithm developed in section 4 can be easily extended

to design identical pitch curves with A lobes, where N is an

(6) (9, = 25

(c)e,=45

Fig. 4 A pair of identical 3-lobe parabolic pitch curves for / f{0,) = 2.4298flf + 3.117601 + 4

14 and

(o)^,=0

(6)^, =40

(c)

(a)e,=0

(b) 0, = 20

( c ) e , = 3 5

Fig. 5 A pair of identical 4-iobe parabolic pitch curves for / = 14 and f{e,) = 4.31978? + 4.1568ei + 4

Example 3: N = 4.

By the same procedure, we obtain a = 2.0784. The final profile off{6\) will be

/ ( ^ , ) = q{ae,) = 3 + (2.07846li + 1) '

= 4.31970? + 4.15686li + 4

The rotors generated are drawn in Fig. 5, with Figs. So, 5b, and 5 c corresponding to angular displacements 0, = 0, 20, 35, respectively.

7 Design of Smooth Profiles It is sometimes desirable that the entire profile of the rotor

is smooth. Referring to Figs. 3 -5 , these rotor profiles are not smooth at the tips and roots of their lobes. The reason is that C' continuity is not designed at these tips and roots, the intersec-tions between segments of profiles. Referring to Fig. 2, in order to have C' continuity at these intersections of profile segments, the function of profile needs to satisfy the two conditions

ddi = 0 at 6*1 = 0 and r,(0) = a (17)

The equivalent conditions of Eq. (17) in our design procedure are

dqi9:)

d0x = 0 at 0 and q(Q) = a (18)

In other words, in the range of interest, the initial design func-tion ^(^i) should have a local minimum, and the initial point, ^ (0) , should start at this local minimum. This is in addition to the previously stated conditions of monotonically increasing and C' continuity in the same range of interest. One example is given here to illustrate this process of designing smoothness.

Example 4: A 4th-order Polynomial with Smooth Profile.

We choose the initial design ^(^i) = 1 + 0\l\(>, and this function satisfies both Eqs. (8) and (18). Let N = 2 and / = 4, and after following the design steps given in Section 4, we obtain a = 1.8383. The rotors generated are drawn in Fig. 6. As shown in the figure, the lobe profiles at both tips and roots are smooth.

8 Design of Families of Profiles by Using a Dimen-sionless Parameter

We can use a dimensionless parameter bla to design a family of similar profiles. Denoted by k, i.e., k = bla, this parameter, geometrically, can be considered as a measurement of the non-circularity of a pitch curve, or, kinematically, the maximum speed ratio. Without sacrificing the generality, we assign I = a + fo = 1 for simplicity. To satisfy the condition ^(0) = a stated in Design Step 2, we need to express parameter a in terms of k, and the result is a = 1/(1 + k). One example is given here to illustrate this idea.

Example 5: A Cosine Function,

In this example we want to design a family of two-lobe rotors with cosine function profiles. The initial design of the profile is chosen as

?(^i) = 1 1

cos 6,. 2 2(k + 1)

To satisfy Eq. (12) given in Step 4, we need

4{k+ 1) , _, 1

TrVfc tan" fk-

Therefore, the family of rotor profiles parameterized in k can be obtained by using

f(9,) = q{ae,)

^}__ _k

~ 2 2(k+ 1)

1 A{k + 1) tan -pi

Fig. 6 Identical pitch curves having smooth profiles with / = = 1 + 0.7137et

4 and f(e^)

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A = 2

it =2.5

A: = 3 Fig. 7 Pairs of identical 2-iobe cosine pitch curves witli f(0i) - 1/2 {k - 1)/(2(k + 1)) cos (4(fr + 1)/(iTi^) (tan^' 1/Vk)e,)

A family of rotors of this design with three different k values, k = 2, 2.5, and 3, are generated and drawn in Fig. 7. It shows

that these three pairs of pitch rotors have very similar shapes, and a rotor increases its slenderness with the increase of its pitch noncircularity k.

9 Conclusion

A complete procedure for generating identical pitch curves is presented. A set of criteria for two identical rigid bodies engaging in conjugate rolling motions is established. The final design rotor profiles are similar to the given desired functions. Based on these criteria, an algorithm is developed, which can be used for the design of this type of rolling pairs with almost unlimited profile varieties and any number of lobes. All exam-ples presented are simulated by interactive graphic software. Geometric conditions for having C' continuity at the intersec-tions of tips and roots of lobe profiles are also established. Families of similar profiles can be designed at the same time by varying a dimensionless parameter k, the pitch noncircularity. Results from this investigation should have applications to the design of noncircular gears and lobe pumps.

10 Acknowledgment The authors would like to express their sincere appreciation

to Professors Joseph K. Davidson at Arizona State University and Bernard Roth at Stanford University for providing us with very useful reference materials and valuable opinions.

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1960, pp. 59-66. 6 Cunningham, F. W., and Cunninghain, D. S., "Rediscovering the Noncir-

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9 Horiuchi, Y., "On the Gear Theory Suggested by Leibnits," Bulletin of Japan Society of Precision Engineers, Vol. 23, No. 2, June 1989, pp. 144-162.

10 Artobolevsky, 1.1., Mechanisms in Modern Engineering Design, Mir Pub-lishers, Moscow, 1977, pp. 21-31.

11 MacConochie, A. F., Kinematics of Machines, Pitman Publishing Corpora-tion, 1948, pp, 190-193.

12 Litvin, F. L., Theory of Gearing, NASA Publication, Washington, DC, 1989.

13 Cunningham, F. W., "Noncircular Gears," Machine Design, Feb. 19, 1959, pp. 161-164.

14 Morrison, R. A., "Rolling Surface Mechanisms," Machine Design, Dec. 11, 1958, pp. 119-123.

15 Huckert, J., Analytical Kinematics of Plane Motion Mechanisms, The Mac-millan Company New York, 1958, pp. 122-126.

16 Dooner, D. B., and Seireg, A. A., The Kinematic Geometry of Gearing, John Wiley & Sons, Inc., 1995, pp. 56-63.

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