Determination of Surface Singularities of a Cycloidal Gear Drive With Inner Meshing

Mathematical and Computer Modelling 45 (2007) 340354www.elsevier.com/locate/mcm

Determination of surface singularities of a cycloidal gear drive withinner meshing

Yii-Wen Hwang, Chiu-Fan HsiehDepartment of Mechanical Engineering, National Chung-Cheng University, 168 San-Hsing, Ming-Hsiung, Chia-Yi 621, Taiwan, ROC

Received 10 January 2006; received in revised form 17 May 2006; accepted 23 May 2006

Abstract

The cycloidal gear drive is widely used in industrial applications, such as gerotor pump, speed reducer, transmission apparatusand so on. In this paper, the profile of inner rotor is with equidistance to an epitrochoidal (or extended epicycloid) curve, andthe mathematical model of the internal cycloidal gear with tooth difference is created by the theory of gearing. The proposedmathematical model can simulate not only gerotor pump but also cycloidal speed reducer. The design of outer rotor depends ondifferent applications. Being applied to the speed reducer, the outer rotor will be a pin wheel (outer rotor arc teeth). Besides, fora better design of the gerotor pump, the mathematical model of the generated shape between outer rotor arc teeth will also beproposed. Lastly, a simpler dimensionless equation of undercutting will be derived from the proposed mathematical model. And amore explicit procedure to determine the feasible design region without undercutting on the tooth profile or interference betweenthe pins will be developed and demonstrated by some numerical examples.c 2006 Elsevier Ltd. All rights reserved.Keywords: Cycloidal gear drive; Speed reducer; Gerotor pump; Undercutting; Pump efficiency

1. Introduction

The cycloidal gear drives can be applied to gerotors. The gerotor is a positive displacement pump mechanism,which delivers a known, predetermined quantity of fluid in proportion to speed. It can be applied to both fluid powersand fluid transmissions. Because the inner gerotor advances only one tooth space per revolution, gerotor elementsrevolve in the same direction at low relative speed. Thus, the gerotor can be made to pump in the same directionregardless of rotation direction. Additionally, it provides a relatively pulseless flow, high volumetric and mechanicalefficiency, and balanced and quiet operation.

Besides, the cycloidal gear drives can also be applied to many other fields such as cycloid speed reducers andair motors [1,2]. Cycloid reducers use rolling elements and a multilobate cam to transmit torque and provide speedreduction. These reducers provide high reduction ratios with low backlash, high accuracy, and high stiffness. Theycontribute much to the renovation and replacement of the directional products of the mine hoist, cement equipment,rod and bar mill train, lifting machines, sugarcane extraction mills, crushers and power generation equipment.

Corresponding author. Tel.: +886 5 272 0411x33308.E-mail addresses: [email protected] (Y.-W. Hwang), [email protected] (C.-F. Hsieh).

0895-7177/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2006.05.010

Y.-W. Hwang, C.-F. Hsieh / Mathematical and Computer Modelling 45 (2007) 340354 341

Nomenclature

c center distance between the outer rotor and inner rotorR pin radius of the outer rotorr center distance between the circular-arc pin and the outer rotorm the tooth numbers difference between the outer rotor and the inner rotorN the tooth number of the outer rotorSi coordinate system i where i = 1, 2, fOi origin of the coordinate system Si , i = 1, 2, fi , i rotation angle, i = 1, 2 profile parameter of the circular-arc pin of the outer rotorMi j coordinate transformation matrix from system j to system ii pitch radius of the outer rotor when i = 1, pitch radius of the inner rotor when i = 2ri position vector represented in Si , i = 1, 2, i = a, bP the eccentric ratio parameterQ the pin radius ratio parameterQu the minimum value Q of undercutting1min the meshing point is located at the tooth bottom of the inner rotor1max the meshing point is located at the tooth top of the inner rotorQc the maximum value of the pin radius ratio

With regard to the researches concerning the cycloidal gear drives, Tsay and Yu [3] proposed an analyticalmethod for gerotors with the outer rotor arc teeth and inner rotor trochoid teeth. The relations among design variableof the traditional design method and the proposed analytical method had also been studied and compared. Yangand Blanche [4] investigated the characteristics of the backlash and torque ripple and analyzed the relationshipsamong the machining tolerance, drive parameters (namely, gear ratio, pitch diameter and normalized tooth heights)and drive performance indices (namely, backlash and torque ripple). They found that, as a gerotor cannot beadjusted to compensate for wear, the contact forces should be kept to a minimum and the curvature of the lobesdecreased to reduce the wear rate. Subsequently, Beard et al. [5] derived relationships that show the influence ofthe trochoid ratio, the pin size ratio and the radius of the generating pin on the curvature of the epitrochoidalgerotor.

Drawing on the theory of the envelope, Shung and Pennock [6] presented a unified and compact equation describingthe geometry, the geometric properties of the different types of trochoid and the geometric properties of a conjugateenvelope. Mimmi and Pennacchi [7] dealt with a general method showing the analytical condition for avoidingundercutting using the concept of the limit curve. From the same perspective, Vecchiato et al. [8] applied the rotorprofiles of a cycloidal pump as a circular arc and a conjugated epicycloidal curve. They investigated the formation ofan envelope by branches and discussed the determination of singularities and computerized pump design with rotorprofiles free of singularities. Litvin et al. [9] investigated the envelopes relation to surface family by consideringenvelopes formed by several branches for cycloidal pumps and conventional worm gear drives. Demenego et al. [10]developed a tooth contact analysis (TCA) computer program and discussed avoidance of tooth interference and rapidwearing through modification of the rotor profile geometry of a cycloidal pump whose one pair of teeth is in mesh atevery instant.

Of particular interest for this paper, Fong and Tsay [11] proposed a dimensionless equation of non-undercuttingto study the feasible design region without undercutting for the internal cycloidal gear with a small tooth difference.They solved the undercutting curve using a fixed eccentric ratio parameter and showed the impact of tooth differenceon the feasible design region. Specifically, when the tooth number difference is equal to 1, the feasible design regionfalls only under the lowest point of the undercutting curve, but when the tooth number difference is equal to or greaterthan 2, the feasible design region is determined by considering the undercutting curve and max curve simultaneously.However, not only is their dimensionless equation of non-undercutting extremely complicated, but the authors did nottake into consideration interference between the pins. Therefore, the result for the speed reducer may not point clearlyto a feasible design region.

342 Y.-W. Hwang, C.-F. Hsieh / Mathematical and Computer Modelling 45 (2007) 340354

Fig. 1. Traditional design of gerotors.

Therefore, in this paper, the mathematical model with tooth difference is derived by the theorem of gearing and theoperation of the coordinate transformation for the profile of the internal gear. The proposed mathematical model canbe applied to simulate not only the gerotor pump but also the cycloidal speed reducer. A better geometry design is alsopresented in gerotor pump design. According to the proposed mathematical model, a simpler dimensionless equationof non-undercutting will be derived and a design region of non-undercutting will be presented in the followingsections.

2. Geometric design

As shown in Fig. 1, the traditional design of gerotors has carryover phenomenon between inner and outer rotors.That is, when the outer and inner rotors rotate at the location shown in Fig. 1, the inability to completely discharge theliquid reduces pump efficiency. In this paper, the profile of inner rotor will be designed first and then the outer rotorgenerated according to the inner rotor, so that the carryover may thus be improved. The mathematical model of rotorswill be created in the following.

2.1. Mathematical model

One method for generating the internal cycloidal gear is shown in Fig. 2. Circles 1 and 2 are in internal tangency,and their radius are radii 1 and 2 separately. Point I is the instantaneous center of rotation. When circle 1 rotatescounterclockwise around the circumference of circle 2 in a pure rolling motion, the eccentric throw r is the distancebetween point p and the center of circle 1, and point p generates an extended epicycloid path. This path could bethe center of rollers and then the profile of inner cycloidal rotor (or cycloidal wheel) would be generated by the innerenvelope methods.

The coordinate systems are created as shown in Fig. 3, where the coordinate systems S1, S2 and S f are rigidlyattached to the outer rotor, inner rotor and frame, respectively. The ratio of the inner rotors rotation angle to that ofthe outer rotor is inversely proportional to the ratio of the tooth number. Therefore, the relationship between rotationangle 2 and 1 is represented as follows:

2 = NN m 1 (1)where N is the tooth number of the outer rotor, m is the tooth number difference between the inner rotor and the outerrotor. Thus, the tooth number of the inner rotor is N m.

If the circular arc pin tooth of the outer rotor is represented in S1 as

r1() =x1()y1()

1

= R sinr R cos

1

. (2)The equation of the inner rotor can be determined by the following coordinate transformation:

r2(, 1) =M21(1)r1() (3)


Fig. 2. Generation of the extended epicycloidal curve.

Fig. 3. Applied coordinate system.

where

M21(1) =cos(1 2) sin(1 2) c sin2sin(1 2) cos(1 2) c cos2

0 0 1

.The mathematical equation of the inner rotor can be yielded by Eq. (3) as follows:

r2(, 1) =x2(, 1)y2(, 1)

1

=r sin(1 2)+ R sin( + 1 2) c sin2r cos(1 2) R cos( + 1 2) c cos2

1

. (4)The equation of meshing is represented as [12,13]:

f1(, 1) =(r2

k) r21

= 0 (5)

where k is the unit vector in the z direction. Substituting Eq. (4) into Eq. (5) yields

f1(, 1) = mr sin + cN sin( + 1) = 0. (6)Eqs. (4) and (6) considered simultaneously determine the generated tooth profile of the inner rotor.


Fig. 4. The basic ideal of generated shape.

Fig. 5. Coordinate system for generating outer rotor.

The above mathematical model is applicable for a cycloidal speed reducer. However, if the application is a gerotorpump, the carryover shown in Fig. 1 can be improved by generating the outer rotor profile using the envelope of theinner rotor profile. This basic concept is illustrated in Fig. 4.

For the coordinate system shown in Fig. 5, the equation of the inner rotor (Eq. (4)) in Sa is written as ra(, 1), therelationship between the rotation angle 2 and 1 is represented as:

2 = N mN 1. (7)Operating the coordinate transformation as in Eq. (3) then yields

rb(, 1, 1) =Mba(1)ra(, 1)

=r sin(1 2 + 1 2)+ R sin( + 1 2 + 1 2) c[sin 2 sin(1 2 2)]r cos(1 2 + 1 2) R cos( + 1 2 + 1 2) c[cos 2 + cos(1 2 2)]

1

(8)where

Mba(1) =cos(1 2) sin(1 2) c sin 2sin(1 2) cos(1 2) c cos 2

0 0 1

in which the inner rotor tooth profile is obtained by considering simultaneously the equation of meshing:

f2(, 1, 1) =(rb1

k) rb1

= 0. (9)


Substituting Eqs. (1) and (7) into Eq. (8) and operating Eq. (9) yields

f2(, 1, 1) = mr[sin1 + sin

(1 mN m 1

)]+ mR

[sin( + 1)+ sin

( + 1 mN m 1

)]+ cN sin

(1 + NN m 1

)= 0. (10)

Simultaneous consideration of Eqs. (6), (8) and (10) then allows determination of the generated shape between theouter rotor arc teeth.

3. Equation of undercutting and design constraints

For the generated rotor profile, if a singular point exists on the tooth profile, then undercutting will occur and therelative velocity definition can be represented by the theory of gearing [12,13]:

V(2)r = V(1)r + V(12) = 0 (11)where, V(1)r and V

(2)r represent the velocities of the contact point displacement along the tooth shape in coordinate

system S1 and the generated shape in coordinate system S2, respectively. V(12) is the sliding velocity. Eq. (11) impliesthat the following two determinants are equal to zero [12,13]:

x1

V(12)x1f f1

d1dt

=y1

V(12)y1f f1

d1dt

= 0. (12)Eq. (12) is the equation of undercutting. And the sliding velocity is represented as:

V(12)1 = V(12)x1 + V(12)y1 =[(

(1)1 (2)1 ) r1

] (R1 (2)1 ) (13)

where

(1)1 =

k (14)

(2)1 =

N

N m

k (15)

R1 = c sin1

i + c cos1

j . (16)

Substituting Eqs. (2) and (14)(16) into Eq. (13) and transforming it into a dimensionless equation by dividing r yieldsV(12)x1 =

1N m (m mQ cos PN cos1)

V(12)y1 =1

N m (mQ sin + PN sin1)(17)

where P = cr denotes the eccentric ratio parameter; Q = Rr denotes the pin radius ratio parameter.Letting

1 =

x1

V(12)x1f f1

d1dt

, 2 =y1

V(12)y1f f1

d1dt

(18)and setting the angular velocity of the outer rotor to unit d1dt = 1 = 1, a dimensionless equation can also be obtainedfrom Eq. (6) by dividing r :

f = m sin + PN sin( + 1) = 0. (19)


Operating Eq. (18) yields the following dimensionless equation:

1 = 1N m [m cos(m + mQ cos + N P cos1)+ N P cos( + 1)(m NQ cos N P cos1)] (20)

2 = 1N m [m cos(mQ sin N P sin1)+ N2P cos( + 1)(Q sin + P sin1)] (21)

where

r1

= x1

i +y1

j = Q(cosi + sin

j )

f = f

= m cos + PN cos( + 1)

f1 = f

1= PN cos( + 1).

According to the theory of gearing, the singularity exists when both determinants are equal to zero,1 = 2 = 0,the coordinates of the singular point can be solved as follows:

21 +22 = 0. (22)The rotation angle 1 of the outer rotor at the singular point is denoted as u and can be solved by simultaneous

consideration of Eqs. (19) and (22). If the undercutting value u is in the range of1max u 1max, undercuttingwill occur on the tooth profile. However, we solve the values (Q) by fixed eccentric ratio parameter P . If the minimumvalue of the undercutting is denoted as Qu , the following equations are helpful to obtain the minimum value of theundercutting as:(

11

)2+(21

)2= 0 (23)

where

11

= 1N m {N P[m sin( + 1)+ cos(m sin1 + NQ sin( + 1))+ N P sin( + 21)]}

21

= 1N m {N P[m cos cos1 + N P cos( + 21)+ NQ sin sin( + 1)]}.

Eq. (19) can yield the following:

= tan1(

PN sin1m + PN cos1

). (24)

Substituting Eq. (24) into Eqs. (22) and (23) then produces the minimum value of undercutting Qu .When the lower bound of the eccentric ratio parameter P is equal to zero, the upper boundary values of eccentric

ratio parameter P may be derived from the differentiation of Eq. (19) and written as follows:

d f ((1), 1)d1

= f

dd1

+ f1

= 0. (25)

Thus, the transformation yields:

dd1

= f1f

= N P(N P + m cos1)m2 + (N P)2 2mN P cos1 . (26)

In Eq. (26), when f1

= 0 then dd1 = 0, the function (1) will reach its extreme value. Therefore, to obtain thecondition of extreme value of parameter P , the numerator of Eq. (26) must be zero, and then we can yield as follows:

1 N Pm

1. (27)


Fig. 6. The tooth top and tooth bottom radii of the outer rotor for m = 1.

It should be noted that the denominator of Eq. (26) must not be zero. Finally, the design constraint of P can bedetermined as follows:

0 < P

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Determination of Surface Singularities of a Cycloidal Gear Drive With Inner Meshing