Geometric Design for a Gerotor Pump With High Area Efficiency

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Chiu-Fan Hsiehe-mail: [email protected]

Yii-Wen Hwange-mail: [email protected]

Department of Mechanical Engineering,National Chung-Cheng University,

168 San-Hsing, Ming-Hsiung,Chia-Yi 621, Taiwan, R.O.C

Geometric Design for a GerotorPump With High Area EfficiencyThis paper proposes a gerotor profile design based on the theory of gearing. Presentedfirst is the curve of the outer rotor, whose conjugate profile is the inner rotor. Next, theanalysis assesses the feasibility of three different design curves: an epitrochoid curve, ahypotrochoidal curve, and a curve made by continuously connecting the epicycloid andhypocycloid. The undercutting equation derived from the mathematical model—which ispresented in parametric form—facilitates identification of the design parameter limits,while the sealing property and nonundercutting on the profile are estimated using curva-ture analyses. Finally, the offset concept is applied to the gerotor design. First, twoconjugate curves are obtained, whose offset curves are then found. Pump performance—including area efficiency and sealing—is also compared for these designs, which includeboth offset and nonoffset rotor profiles. �DOI: 10.1115/1.2779887�

IntroductionGerotor pumps are widely used in the automotive industry for

uel lift, engine oil, and transmission systems. However, volumet-ic efficiency and cavitation damage are causes for concern inerotor pumps with high output flow. Therefore, to optimize pumperformance and reduce cavitation damage, it is essential to un-erstand the fluid dynamics inside the pump.

The gerotor design offers a lower cost alternative to other fluidower and fluid transfer mechanisms over a broad range of appli-ations. Designers of engines, compressors, machine tools, trac-ors, and other equipments requiring hydraulic systems can use anntegrally designed gerotor pump for a more compact, lowereight unit. In most cases, it will also run more quietly.Much relevant research exists on gerotor �cycloidal� pumps.

or example, Tsay and Yu �1� proposed an analytical method forerotors with outer rotor arc teeth and inner rotor trochoid teethnd compared its design variables with those of the traditionalesign. Beard et al. �2� derived relationships that show the influ-nce of the trochoid ratio, the pin size ratio, and the radius of theenerating pin on the curvature of the epitrochoidal gerotor.hung and Pennock �3� presented a unified and compact equationescribing the geometry and geometric properties of the differentrochoid types and the geometric properties of a conjugate enve-ope. Litvin and Feng �4�, who developed computer programs forenerating the planar cycloidal gearings and rotors of screw Rootslowers, presented an improved design that eliminated profile andurface singularities. Chiu �5� presented a rotor profile design andanufacturing method consisting of a hypocycloid and an epicy-

loid. Kang and Hsieh �6� compared the area efficiency for theddendum tooth profile using an extended cycloid, cycloid andingle circular arc, respectively. Litvin et al. �7� investigated thenvelope’s relation to surface family by considering envelopesormed by several branches for cycloidal pumps and conventionalorm gear drives.More recently, Paffoni �8� presented a vector analysis that pre-

isely describes the geometry of a hydrostatic gear �gerotor� pumprom which he deduced parametric equations. Subsequently, Paf-oni et al. �9� proposed a teeth clearance influence on the toothumber contact in a hydrostatic �gerotor� pump using arc circularrofiles. Specifically, they found that a severe reduction in theumber of point contacts and amount of diminution in wide sur-ace contact of the pinion could be predicted based on tooth clear-

Contributed by the Power Transmission and Gearing Committee of ASME forublication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 24,006; final manuscript received December 10, 2006. Review conducted by Philippe
elex.
ournal of Mechanical Design Copyright © 20

om: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 06/22/201

ance, following which the pressure contact and film thicknesscould be deduced. In addition, the authors found that, despiteappearances, the benefit of high geometrical conformity, whichnormally leads to low pressure and a thick film, was stronglyaltered.

Hwang and Hsieh �10� proposed that the mathematical modelcan be applied to simulate not only the gerotor pump but also thecycloidal speed reducer. The carryover phenomenon had been im-proved from the traditional design. Here, the carryover phenom-enon means a larger fluid or gas that may be carried from high-pressure port back to low-pressure port. Thus, the pumpperformance may be reduced. Besides, the equation of nonunder-cutting had been developed, and some numerical examples hadbeen presented to design and prove the proposed feasible designregion, which is without undercutting on the tooth profile or in-terference between the adjacent pins.

Figure 1 presents a cycloidal pump schematic in which rotors 1and 2 perform the rotational motion about axes O1 and O2, respec-tively. The rotor tooth profiles with trochoidal curves form closedchambers as the space and all rotor teeth are in mesh simulta-neously. In a gerotor pump, the fluid is sucked into the inlet portand then shifted to the outlet port. Because of the rotor clearances,flow leakage occurs between the high-pressure and low-pressuresides of the pump. Therefore, to limit pressure, excess fluid isrecirculated to the inlet port through a pressure relief valve. Theflow through the rotor clearances creates high fluid velocity andlocalized low-pressure areas, which produce air and vaporbubbles, thereby causing cavitation damage and noise. To addressthis issue, this paper presents three methods for designing gerotorprofile and discusses their feasibility, sealing, and area efficiency.

2 Geometric Design of a Cycloidal PumpThe tooth numbers of rotors 1 and 2, N1 and N2, respectively,

are specified in the design as N2=N1−1. However, the profile ofrotor 1 �see Fig. 1� can be designed by using trochoidal curvesthat may be epicycloid, hypocycloid, or equidistant. Here, the epi-cycloids and hypocycloids are first designed to form the adden-dum and dedendum of the outer rotor; however, a profile is alsogiven for the outer rotor with equidistance to the trochoid. Theprofile of the inner rotor can then be generated using the profile ofthe outer rotor. The following sections present the derived math-ematical model in parameter form and examples that illustrate thedesign results.

2.1 Mathematical Model. As shown in Fig. 2, curves �1�1�

and �1�2� are epicycloid and hypocycloid, respectively, and each

has a common half tangent O1E. Therefore, to generate the inner

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otor, coordinate systems are created, as shown in Fig. 3, whereoordinate systems S1�x1 ,y1�, S2�x2 ,y2�, and Sf�xf ,yf� are rigidlyttached to the outer rotor, inner rotor, and frame, respectively.he gear ratio or angular velocity ratio can then be determined by

m21 =�2

�1=

�2

�1=

N1

N2�1�

ere, �1 and �2 represent the rotation angles of outer and innerotors, respectively. First, the position vectors of the addendum

1�1� and dedendum �1

�2� are represented in coordinate S1 as fol-ows:

r1�1��1� = �r1x

�1�

r1y�1�

1� = � − re1 sin��1 + �1� + �rb1 + rp�sin �1

− re1 cos��1 + �1� + �rb1 + rp�cos �1

1� �2�

Fig. 1 Schematic of the cycloidal pump

Fig. 2 Design of the outer rotor profile

Fig. 3 Coordinate systems applied for the gerotor pump

270 / Vol. 129, DECEMBER 2007


r1�2��2� = �r1x

�2�

r1y�2�

1� = � − re2 sin��2 − �2� + �rb2 − rp�sin �2

re2 cos��2 − �2� + �− rb2 + rp�cos �2

1� �3�

where rb1 and rb2 are the radii of the rolling circle, �1 and �2 arethe angles of the base circle, �1 and �2 are the angles of therolling circle for curves �1

�1� and �1�2�, respectively. rp is the pitch

radius of the outer rotor, and re1 and re2 are defined as

re1 = rb1�1 �4�

re2 = rb2�2 �5�

where �1 and �2 represent the trochoid ratio of the epitrochoid andhypotrochoid, respectively. In this particular case, �1=�2=1;therefore, the epicycloids and hypocycloids can be formed basedon the profile of the outer rotor. However, because obtaining acomplete profile of the outer rotor only requires consideration ofthe symmetric curve from point F to point G �see Fig. 2�, thefollowing equations must be taken into account:

�1 =rp

rb1�1 �6�

�2 =rp

rb2�2 �7�

c = re1 + re2 �8�

rp = �rb1 + rb2�N1 �9�

� =�rb1

rp�10�

=�rb2

rp�11�

=�

N1= � + �12�

The following coordinate transformation then yields the equa-tion of the inner rotor �11�:

r2�i��i,�1� = M2f��1�M f1��1�r1

�i��i� = M21��1�r1�i��i� i = 1,2

�13�

where

M21��1� = �cos��1 − �2� − sin��1 − �2� − c sin �2

sin��1 − �2� cos��1 − �2� − c cos �2

0 0 1�

Operating Eq. �13� then yields the following conjugate equations:

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r2�1��1,�1� = �r2x

�1�

r2x�1�

1� = � �rb1 + rp�sin��1 − �1 + �2� − re1 sin��1 − �1 + �2 + �1� − c sin �2

�rb1 + rp�cos��1 − �1 + �2� − re1 cos��1 − �1 + �2 + �1� − c cos �2

1� �14�

r2�2��2,�1� = �r2x

�2�

r2x�2�

1� = � �rb2 − rp�sin��2 + �1 − �2� − re2 sin��2 + �1 − �2 − �2� − c sin �2

�− rb2 + rp�cos��2 + �1 − �2� + re2 cos��2 + �1 − �2 − �2� − c cos �2

1� �15�

owever, solving the conjugate curve requires consideration ofhe equation of meshing, which can be represented as follows:

f �i��i,�1� = � �r2�i�

��i� k� ·

�r2�i�

��1= 0 i = 1,2 �16�

here k is the unit vector in the z direction. Substituting Eqs. �14�nd �15� into Eq. �16� then yields the following:

f �1� = cm21�rb1 sin��1 − �1� − re1 sin��1 − �1 + �1��

+ re1rp�m21 − 1�sin �1 �17�

f �2� = cm21�rb2 sin��2 + �1� − re2 sin��2 + �1 − �2��

− re2rp�m21 − 1�sin �2 �18�

2.2 Mathematical Model of an Offset Curve. In this paper,he offset curve method is used to generate new profiles for theuter and inner rotors without interference between the rotors be-ause using this method can improve pumping efficiency. Offseturves, also called parallel curves, are defined as the locus of theoints at constant distant d along the normal from the generativeurves. One basic property is that an offset and its generator havecommon normal and thus a common direction of local tangent

ectors. Hence, the offset curve of profile �1 can be representedy

R1�i��i� = �R1x

�i� R1y�i� 1�T = r1

�i��i� ± n1�i��i�d i = 1,2 �19�

ere, when i=1, the curve is an epitrochoid, the � sign implieshe inward offset, and the sign implies the outward offset. How-

ented as

ournal of Mechanical Design


ever, when i=2, the curve is a hypotrochoid, the � sign impliesthe outward offset, and the sign implies the inward offset.

The unit normal vector can then be derived by

n1�i��i� = �n1x

�i� n1y�i� 0�T =

��r1�i�/��i� � k

�r1�i�/��i

i = 1,2 �20�

where

�r1�i�

��i= �Tix

�i� Tiy�i� 0�T �r1

�i�

��i = ��Tix

�i��2 + �Tiy�i��2 i = 1,2

�r1�1�

��1= ��rb1 + rp�cos �1 − re1�1 +

rp

rb1�cos��1 + �1�

− �rb1 + rp�sin �1 + re1�1 +rp

rb1�sin��1 + �1� 0 T

�21�

�r1�2�

��2= ��rb2 − rp�cos �2 − re2�1 −

rp

rb2�cos��2 − �2�

��rb2 − rp�sin �2 − re2�1 −rp

rb2�sin��2 − �2� 0 T

�22�

Above, �Tix�i� Tiy

�i� 0�T-means the tangent vector. Operating Eq. �20�yields the following:

n1�1��1� = �− rb1 sin �1 + re1 sin��1 + �1�

�rb12 − 2rb1re1 cos �1 + re1

2

− rb1 cos �1 + re1 cos��1 + �1��rb1

2 − 2rb1re1 cos �1 + re12

0 T

�23�

n1�2��2� = � rb2 sin �2 − re2 sin��2 − �2�

�rb22 − 2rb2re2 cos �2 + re2

2

− rb2 cos �2 + re2 cos��2 − �2��rb2

2 − 2rb2re2 cos �2 + re22

0 T

�24�

fter which the unit normal vector can be represented in coordi-ate system S2 by

n2�i��i,�1� = L21��1�n1

�i��i� f �i��i,�1� = 0 i = 1,2 �25�

here

L21��1� = �cos��1 − �2� − sin��1 − �2� 0

sin��1 − �2� cos��1 − �2� 0

0 0 1�

imilarly, the offset curve of conjugate curve �2 can be repre-

R2�i��i,�1� = �R2x

�i� R2y�i� 1�T = r2

�i��i,�1� ± n2�i��i,�1�d f �i��i,�1�

= 0 i = 1,2 �26�

in which the definitions of signs � and are as above.

2.3 Equation of Undercutting. According to the theory ofgearing �11�, undercutting occurs when a singular point exists onthe generated tooth profile. The mathematical definition of toothprofile singularity is represented by the following equation:

Vr2�i� = Vr1

�i� + V12�i� = 0 i = 1,2 �27�

where Vr1�i� and Vr2

�i� represent the velocities of the contact point�1� �2�
displacement along the tooth shape ��1 and �1 � in coordinate
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ystem S1 and the generated shape ��2�1� and �2

�2�� in coordinateystem S2, respectively. V12

�i� is the sliding velocity. Equation �27�mplies that the following two determinants are equal to zero:

��R1x

�i�

��i

− V12,1x�i�

f�i

�i� − f�1

�i� d�1

dt� = �

�R1y�i�

��i

− V12,1y�i�

f�i

�i� − f�1

�i� d�1

dt� = 0 i = 1,2 �28�

Equation �28� is the equation of undercutting, while the slidingelocity is represented as

V12,1�i� = V12,1x

�i� + V12,1y�i� = ��1,1

�i� − �2,1�i� � � R1

�i�� − �E � �2,1�i� �

i = 1,2 �29�here

�1,1�i� = k� �30�

�2,1�i� = m21k� �31�

E = c sin �1i� + c cos �1j� �32�

ubstituting Eqs. �19� and �30�–�32� into Eq. �29� yields V12,1�i� as

V12,1�i� = �− R1y

�i� + m21�R1y�i� − c cos �1��i� + �R1x

�i�

+ m21�− R1x�i� + c sin �1��j� i = 1,2 �33�

he following equations can then help solve Eq. �28�:

�R1�i�

��i=

�R1x�i�

��ii⇀+

�R1y�i�

��ij⇀= r1

�i� ± n1�i�d i = 1,2 �34�

f�i

�i� =�f �i�

��if�1

�i� =�f �i�

��1i = 1,2 �35�

here

r1�i� =

�r1�i�

��in1

�i� =�n1

�i�

��ii = 1,2

2.4 Equation of Curvature. Because the gerotor profile maye an offset curve that is equidistant to the trochoidal curve, it isossible to analyze the profile’s curvature by first considering thatf the trochoidal curve. According to gearing theory �11�, thequation of curvature is as follows:

�1�i��i� =

r1�i�i

�i� · n1�i�

r1�i

�i� i = 1,2 �36�

here

r1�1�1

�1� =�2r1

�1�

��12 = �− �rb1 + rp�sin �1 + re1�1 +

rp

rb1�2

sin��1 + �1�

− �rb1 + rp�cos �1 + re1�1 +rp

rb1�2

cos��1 + �1� 0 T

�37�

r1�1�1

�2� =�2r1

�2�

��22 = �− �rb2 − rp�sin �2 + re2�1 −

rp

rb2�2

sin��2 + �2�

��rb2 − rp�cos �2 − re2�1 −rp

rb2�2

cos��2 + �2� 0 T

�38�ubstituting Eqs. �21�–�24�, �37�, and �38� into Eq. �36� thenields the following:

�1�1� =

rb13 + re1

2 �rb1 + rp� − rb1re1�2rb1 + rp�cos �1

�rb1 + rp��r2 − 2rb1re1 cos �1 + r2 �1.5 �39�
b1 e1
272 / Vol. 129, DECEMBER 2007


�1�2� =

− rb23 − re2

2 �rb2 − rp� + rb2re2�2rb2 − rp�cos �2

�rb2 − rp��rb22 − 2rb2re2 cos �2 + re2

2 �1.5 �40�

The corresponding profile �concave or convex� can be deter-mined through a curvature analysis, from which the contact stressmay also be estimated to show the contact situation between theouter and inner rotors. The curvature of the generated profile canthen be represented as follows:

�2�i� = −

nr2�i�

vr2�i� = −

nr2x�i�

vr2x�i� = −

nr2y�i�

vr2y�i� f �i� = 0 i = 1,2 �41�

vr2�i� = vr2x

�i� i� + vr2y�i� j� = vr1

�i� + v12�i� =

�r1�i�

��1+ ��1,1

�i� − �2,1�i� � � r1

�i��

− �E � �2,1�i� � i = 1,2 �42�

nr2�i� = nr2x

�i� i� + nr2y�i� j� = n1

�i� + ��1,1�i� − �2,1

�i� � � n1�i� i = 1,2 �43�

where

vr2�i� = �Tix

�i� + r1y�i� + m21�r1y

�i� − c cos �1��i� + �Tiy�i� + r1x

�i�

+ m21�− r1x�i� + c sin �1��j� i = 1,2 �44�

nr2�i� = �nix

�i� − n1y�i� + m21n1y

�i��i� + �niy�i� + n1x

�i� − m21n1x�i��j� i = 1,2

�45�Substituting Eqs. �44� and �45� into Eq. �41� then yields the cur-vature of the generated profile.

3 Geometric Design and DiscussionThis paper presents three methods for designing a gerotor pro-

file, the first using an epitrochoid and its equidistant curve �offsetcurve�, the second using a hypotrochoid and its equidistant curve�offset curve�, and the third using both the epicycloid and hypo-cycloid and their respective equidistant curves �offset curves�. Thefollowing sections provide a clear discussion of the design processand each method’s feasibility.

3.1 Gerotor Design Using the Epitrochoidal Curve. Thefirst design uses the tooth profile of the outer rotor as the epitro-choidal curve, whose formula is given as Eq. �2�, and the innerrotor as the conjugate tooth profile, given in Eq. �14�. The profilesof the inner and outer rotors can be obtained by simultaneouslyconsidering the equation of meshing.

This method is illustrated using some examples with the fol-lowing parameters:

N1 = 5 N2 = 4 rb1 = 6 rb2 = 0 �2 = 0

Equations �4�–�12� give c=re1, rp=30, =�=0.2�, and =0, andEqs. �4� and �5� give re1=6�1 and re2=0. The design results ofthese examples are shown in Fig. 4, while �1 equals 0.5, 0.7, and0.9. If the definition of curvature difference is �1

�12�=�1�1�−�1

�2�, theanalytical result of the curvature difference is as shown in Fig. 4.When the trochoid ratio �1 is larger, the curvature of the epitro-choidal curve and its generated curve are closer to the larger anglerange. In other words, as shown in Fig. 4, the larger the trochoidratio �1 is, the more the curvature difference between inner andouter rotors will be approximate to the horizontal line within thelarger angle range, which indicates that the profiles of inner andouter rotors are closer to this larger angle range, thereby produc-ing a better sealing with less carryover phenomenon.

Nevertheless, according to the so-called offset method, the ac-tual profile of the gerotor is the offset curve of the trochoidalcurve. Thus, the profiles of the inner and outer rotors can beobtained by finding the two conjugate curves and making them aninward offset. However, because offset distance d will affect thepotential for undercutting, an undercutting problem must be
avoided. As a result, two different aspects must be considered: the


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inimum radius of curvature in an outer rotor and the minimumistance of the offset with nonundercutting on the inner rotor pro-le, which should be derived based on the undercutting analysis.omparing these two values and taking the smaller of the tworoduces the extreme value of d. Thus, we can ensure that nondercutting on the inner and outer rotors will exist.

First, the radius of curvature of the epitrochoidal curve is de-ned as follows:

�1�1� =

1

�1�1� �46�

here �1�1� is as given in Eq. �39�. If the curve has an extreme

alue, then the following condition should be satisfied:

��1�1�

��1=

PQ − PQ

P2 = 0 �47�

here

P = rb13 + re1

2 �rb1 + rp� − rb1re1�2rb1 + rp�cos �1

Q = �rb1 + rp��rb12 − 2rb1re1 cos �1 + re1

2 �1.5

P = rpre1�2rb1 + rp�sin �1

Q = 3rpre1�rb1 + rp�sin �1�rb12 − 2rb1re1 cos �1 + re1

2 �2

Substituting the solved value from Eq. �47� into Eq. �46� thenives the extreme value of the curvature’s radius, whose analyticalesult is shown in Fig. 5. The corresponding undercutting analysis

Fig. 4 Curvature difference between outer and inner profiles

Fig. 5 Radius of curvature for outer profile



is given in Fig. 6. As shown in Fig. 6, Ud means a value of d thatmakes the undercutting on the tooth profile of the inner rotor. If�d min is the minimum value of the curvature’s radius and Ud min isthe minimum undercutting value, the result is as shown in Table 1.Thus, it becomes possible to determine the design range of theoffset distance d. Indeed, using this method, the value d can beobtained randomly within any feasible design range. To illustrate,Fig. 7�a� shows the results of the nonoffset case and Fig. 7�b�shows the results when d is given as 17, 14, and 8 �Cases 1, 2, and3�, respectively.

The design outcomes mentioned above are those that resultwhen the eccentric throw c=re1, a compact construction. In suchcases, when two rotors are operating, the chamber between thetwo rotors can keep a sealed space for inhaling, compressing, andexhaling the fluid. For instance, as shown in Fig. 8�a�, in case 2 ofTable 1, when the design is c�re1 and the corresponding param-eter is c=4�4.2, the tooth profile of the rotor would have under-cutting and interference between the outer and inner rotors. There-fore, it could not be used as a gerotor pump. Again in Case 2,when c�re1 and the corresponding parameter is c=5�4.2, asshown in Fig. 8�b�, the tooth profile of the rotor would have no

Fig. 6 Undercutting analysis for three examples

Table 1 Design range of d

Fig. 7 Generation of gerotor profile

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ndercutting phenomenon but would have an apparent clearanceetween the outer and inner rotors, which might lead to a leakageroblem. Therefore, it is not good to make a gerotor pump either.

3.2 Gerotor Design Using the Hypotrochoidal Curve. Inhis section, various examples using the hypotrochoidal curve onhe outer rotors are tested to assess whether the resulting designs

ake suitable gerotor pumps. All examples are based on the fol-owing parameters:

N1 = 5 N2 = 4 rb1 = 0 rb2 = 6 �1 = 0

Based on these measures, Eqs. �4�–�12� give c=re2, rp=30, =0.2�, and �=0, and Eqs. �4� and �5� give re1=0 and re26�2. Figure 9 shows the results for these examples, in which �2

s 0.5, 0.7, and 0.9 and, once the inward and outward offsets are inlace, any value of d �=5�. These results do not indicate that theooth profile under the offset would be a gerotor pump.

3.3 Gerotor Design Using the Epicycloid and Hypocycloid.ccording to Refs. �5,6�, when an epicycloid and a hypocycloid

re made into a tooth profile, their connection point is continuous,ot only achieving position continuity but also satisfying the ra-ius of curvature continuity. Thus, even though the radius of cur-ature on that point is zero, it can connect smoothly. That is, if inhis unique situation, only an epicycloid or a hypocycloid is useds the tooth profile, the connection point is singular �a cusp�. Inhat case, it would not be suitable as the design index. However,ecause the mathematical model in this paper is in parametricorm, it holds for any design value. For example, the parametersf the epicycloid and hypocycloid may be given as

�1 = 1 �2 = 1 N1 = 5 N2 = 4 rp = 40

The parameters calculated are shown in Table 2. From Eq. �17�,1=�1, and from Eq. �18�, �1=−�2. The results also prove thathen c� �rb1+rb2�, the tooth profile has no continuity and there-

ore suffers from an undercutting phenomenon. For example, if

Fig. 8 Design constraint of tooth profile

he value of c in case 1 of Table 2 is changed to 7 ��rb1+rb2� or

274 / Vol. 129, DECEMBER 2007


9 ��rb1+rb2�, the curvature radius of the inner profiles is notcontinuous �see Figs. 10�a� and 10�b��. Rather, only when c= �rb1+rb2� is the design feasible.

3.4 Comparison of the Area Efficiency. The followinganalysis compares the area efficiency for the examples presented

Fig. 9 Gerotor designed with hypotrochoidal curve and theiroffset profiles

above in Secs. 3.1 and 3.3. For simplicity, the span angle is de-



fitu�

df

wA

Ibs

F�

J

Downloaded Fr

ned as �span�outer� �see Fig. 11�. Applying trigonometry to the inflec-

ion point obtained in Sec. 3.1 yields a span angle of �span�outer�, and

sing a similar operation on the span angle in Sec. 3.3 gives

span�outer�=2. The area efficiency can then be calculated for the twoesigns having the same �span

�outer� using the following efficiencyormula:

� =Aouter − Ainner

Aouter� 100% �48�

here Aouter stands for the cross-section area of the outer rotor andinner stands for the cross-section area of the inner rotor.The rotor’s area can be calculated by using the Green’s theory.

n the view of the close curve on a plane, when going on theoundary of the area, the formula for the area keeping on the leftide is

A =1

2��1

�2

�yx − xy�d� �49�

or example, the area from point E to G can be calculated by Eq.49� in Fig. 2. In the same way, the area on the right side is

Table 2 Parameters calculated

Fig. 10 Tooth profile analysis by the radius of curvature



A =1

2��1

�2

�xy − yx�d� �50�

Similarly, the area from point E to F can be calculated by Eq.�50� in Fig. 2. The angle ranges of �1, �2 and �1, �2 depend on thetooth profile design. The rotor’s area can be obtained from eitherEq. �49� or Eq. �50�, whose value is then substituted into Eq. �48�.Thus, the area efficiency can be solved.

The results for area efficiency in a nonundercutting situation areshown in Fig. 12. Clearly, when the tooth profile of the gerotor iscomposed of both an epicycloid and a hypocycloid, given thesame span angle, the area efficiency is better than in any otherdesign. Moreover, when the gerotor tooth profile is an epitro-choidal curve with the same span angle, the area efficiency mayincrease as trochoid ratio increases. However, increasing the spanangle will lead to the rotor having an undercutting phenomenon,which may limit design flexibility.

Based on the results above, in the case of a gerotor tooth profilecomposed of an epicycloid and a hypocycloid, the tooth profilewill have an undercutting phenomenon after being an inward or anoutward offset. Nevertheless, the following section, besides dis-cussing the merits and weaknesses of this type of design, showsthat a gerotor pump having an undercutting situation may still befeasible.

As shown above, the offset profile for an epicycloid and a hy-pocycloid combination—for example, many screw compressordesigns �12,13�—fails to achieve a single continuity at the con-nection point. Rather, in such designs, only the tooth profilereaches a continuity position at the connection point that can also

Fig. 11 Definition of span angle as �span„outer…

extend to pump performance �14�. However, such continuity can

DECEMBER 2007, Vol. 129 / 1275


bif

naor

cdf

Fn

1

Downloaded Fr

e achieved if the offset is given the value d and a commonntersection point is found for the two curves using the followingormula:

Rjx�1��1� − Rjx

�2��2� = 0

Rjy�1��1� − Rjy

�2��2� = 0j = 1,2 �51�

As shown in Fig. 13�a�, the design results of gerotors are theonoffset ones. The results of setting the distance at d=5 andpplying the three cases from Sec. 3.3 to the inward and outwardffsets are shown in Figs. 13�b� and 13�c�. So, the variations ofotor profiles can be observed clearly from Fig. 13.

In addition, the maximum distance of the inward offset, whichan make the outer rotor tip a cusp, can be solved by denoting

max�outer�, taking into account Eq. �51� �j=1� and considering theollowing equation:

ig. 12 Comparison of the area efficiency underonundercutting

Fig. 13 Design results of gerotor

276 / Vol. 129, DECEMBER 2007


�span�outer��2,d� −

�

N1= 0 �52�

Similarly, the maximum distance of the inward offset, whichcan make the inner rotor tip a cusp, can be solved by denotingdmax

�inner�, taking into account Eq. �51� �j=2� and considering thefollowing equation:

�span�inner��2,d� −

�

N2= 0 �53�

Because dmax�inner�

�dmax�outer�, the offset’s extreme value is usually

set at dmax�inner�. In addition, because the cusp may lower both gerotor

pump efficiency and tooth profile strength, designing the gerotorrequires a consideration of the cusp caused by the offset’s extremevalue. In other words, if the value d is larger, it may reduce thesealing property in a gerotor pump, which would increase leakage.However, an appropriate d can increase area efficiency.

To illustrate, given seven initial span angles �span�outer� �nonoffset�,

24 deg, 30 deg, 36 deg, 42 deg, 48 deg, 54 deg, and 60 deg, oncethe inward and outward offsets have been found, a new span angle�span

�outer� and area efficiency can be calculated from the initial spanangle and the offset value d. The final result is as shown in Fig.14, where the thick black line is the nonoffset area efficiency, theupper area is the area efficiency after calculating the inward offset,and the area below is the area efficiency after calculating theoutward offset. It is apparent that no matter which initial spanangle �span

�outer� is set for the outer rotor, the inward offset will havea higher area efficiency, while the outward offset will produce alower area efficiency. Moreover, in computing the gerotor’s areaefficiency and span angle, this process also enables the derivationof the design parameter, which enhances design flexibility andgives the designer an important reference.

4 ConclusionThis paper proposes an application of the offset or offset con-

cept for gerotor design, as well as a parametric mathematicalmodel. In line with the conjugate theory, firstly, two conjugatecurves are obtained and then their offset curves are applied to thegerotor based on the design requirement. The designer can set anyprofile by using an undercutting and a curvature analysis to pro-duce an outcome that benefits the design limitations.

Specifically, the discussion outlines three different gerotor de-signs. The first, based on the epitrochoid curve of the outer rotor,produces a conjugate inner rotor. Calculating the inward and out-

Fig. 14 Area efficiency of offset rotors

ward offsets then yields two new conjugate tooth profiles, while



aps

aBcf

ittb

atrsbled

R

J

Downloaded Fr

n undercutting analysis gives the offsets’ extreme values. Alsoroposed is an analysis of curvature difference to improve theealing property as the trochoid ratio increases.

The second uses the hypotrochoidal curve, with the inner rotors the conjugate curve, to produce a design for the outer rotor.ecause of the offset, the profile of these two rotors will not beontinuous and will suffer an interference phenomenon. There-ore, this design pattern cannot be used for a gerotor.

The third design, again for the outer rotor, is made by connect-ng the epicycloid and hypocycloid continuously. Not only doeshis design produce a better sealing property than that yielded byhe curvature difference in the first design, but it also leads to aetter area efficiency in a nonundercutting situation.

In the compressor rotor design, all profile curves need not havesingle continuity on the connection point if pump performance is

he only concern. Therefore, this paper uses the offset concept toesearch yet another design pattern. In this model, given the initialpan angle �span

�outer�, the area efficiency after inward offsetting wille higher and the area efficiency after outward offsetting will beower than that for the nonoffset. Not only does this result indicatenhanced design flexibility, but these design steps also enable theevelopment of several different gerotors.

eferences�1� Tsay, C. B., and Yu, C. Y., 1990, “The Mathematical Model of Gerotor Pump

Applicable to Its Characteristic Study,” J. Chin. Soc. Mech. Eng., 11�4�, pp.
385–391.


�2� Beard, J. E., Yannitell, D. W., and Pennock, G. R., 1992, “The Effect of theGenerating Pin Size and Placement on the Curvature and Displacement ofEpitrochoidal Gerotors,” Mech. Mach. Theory, 27�4�, pp. 373–389.

�3� Shung, J. B., and Pennock, G. R., 1994, “Geometry for Trochoidal-Type Ma-chines With Conjugate Envelopes,” Mech. Mach. Theory, 29�1�, pp. 25–42.

�4� Litvin, F. L., and Feng, P. H., 1996, “Computerized Design and Generation ofCycloidal Gearings,” Mech. Mach. Theory, 31�7�, pp. 891–911.

�5� Chiu, H. C., 1994, “The Mathematical Model and Computer Aided Manufac-turing of Root’s Blower Gerotor Profile,” J. Technology, Taiwan, R.O.C., 9�1�,pp. 3–19.

�6� Kang, Y. H., and Hsieh, C. F., 2000, “Study on CAD/CAM of Root’s RotorProfile,” Proceedings of the 2000 Manufacture Technology Conference at theNational Tsing-Hua University, Taiwan, R.O.C., pp. 357–363.

�7� Litvin, F. L., Demenego, A., and Vecchiato, D., 2001, “Formation by Branchesof Envelope to Parametric Families of Surfaces and Curves,” Comput. Meth-ods Appl. Mech. Eng., 190, pp. 4587–4608.

�8� Paffoni, B., 2003, “Pressure and Film Thickness in a Trochoidal HydrostaticGear Pump,” Proc. Inst. Mech. Eng., Part G: J. Aerosp. Eng., 217�4�, pp.179–187.

�9� Paffoni, B., Progri, R., and Gras, R., 2004, “Teeth Clearance Effects UponPressure and Film Thickness in a Trochoidal Hydrostatic Gear Pump,” Proc.Inst. Mech. Eng., Part G: J. Aerosp. Eng., 218�4�, pp. 247–256.

�10� Hwang, Y. W., and Hsieh, C. F., 2006, “Geometric Design Using Hypotrochoidand Non-Undercutting Conditions for an Internal Cycloidal Gear,” ASME J.Mech. Des., 129, pp. 413–420.

�11� Litvin, F. L., 1989, Theory of Gearing, NASA, Washington, D.C.�12� Litvin, F. L., and Feng, P. H., 1997, “Computerized Design, Generation, and

Simulation of Meshing of Rotors of Screw Compressor,” Mech. Mach.Theory, 32�2�, pp. 137–160.

�13� Stosic, N., Smith, I. K., and Kovacevic, A., 2003, “Optimization of ScrewCompressors,” Appl. Therm. Eng., 23, pp. 1177–1195.

�14� Chang, Y. J., Kim, J. H., Oh, S. J., Kim, C., and Jung, S. Y., “Development ofan Integrated System for the Automated Design of Gerotor Oil Pump,” ASME
J. Mech. Des., submitted.
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Geometric Design for a Gerotor Pump With High Area Efficiency

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