Design of the conjugated straight-line internal gear pairs for ...

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Original Article

Design of the conjugated straight-lineinternal gear pairs for fluid powergear machines

Wei Song1, Hua Zhou1 and Yonggang Zhao2

Abstract

The conjugated straight-line internal gear pair, including a pinion with straight-line profile and an internal gear with profile

conjugated to the pinion profile, is the key components to the conjugated straight-line internal gear pumps. Truninger

[Truninger gear pump. Patent 3491698, USA, 1970] pointed that this kind of gear pairs leads to a very small trapped

volume, which is 10 times smaller than that of the conventional involute internal gear pumps. Therefore, it can yield

significant reduction in noise levels of the pumps. However, to the authors’ knowledge, there is no research on the design

of these gear pairs until now. First, our study describes the straight-line profile of the pinion mathematically with

definitions of some geometric parameters. Second, according to the theory of gearing, the mathematical models of

the rack-cutter and the conjugated internal gear are derived. Then, the condition to avoid overcutting is obtained for the

generation of the pinion by a rack-cutter. Finally, the condition to avoid interference in the internal gear pair is derived as

a non-linear equation system and an algorithm is developed to find a numerical solution. Two examples are presented to

demonstrate how to determine tooth numbers of the internal gears for avoiding interference. It is hoped that the study

in this article should be helpful to the designers of the conjugated straight-line internal gear pairs. Moreover, it could

provide a prepared knowledge for the researchers to investigate the performances of the fluid power gear machines,

pumps, and motors, with the conjugated straight-line internal gear pumps.

Keywords

Straight-line profile, internal gear, overcutting, interference

Date received: 13 August 2012; accepted: 9 October 2012

Introduction

Gears are widely used in applications of motion trans-mission and power transmission, which are an import-ant element in most types of machinery.1 With theadvantages of interchangeability, insensibility to vari-ation in centre distance, and low cost of manufacture,symmetric involute gears are usually the favorite ofthe designers.2,3 However, in some occasions, asym-metric gears or non-involute gears, are required withthe purpose of obtaining higher strength and effi-ciency, less vibration and noise, etc.4–6 In thesecases, some problems may occur in the design andmanufacture of these gears. For example, the conven-tional cutting tools for involute gears are not suitableanymore, which indicates that work on cutting tooldesign should be made to facilitate these new gears,and the conditions to avoid overcutting or interfer-ence for involute gears are not established for thesegears either.

Researchers generally utilize the theory of gearingto solve the problems that occur in the design and

manufacture of gears. Although involute gears havebeen studied for a long time, more efforts are recentlymade to help people get a clearer understanding ofthem. Litvin7 has presented the detailed mathematicaldescriptions of the generating processes of involuteexternal gears and involute internal gears and derivedthe conditions of non-overcutting for external gearsand of non-interference for internal gears. With a newconcept of virtual involute, Su and Houser8 havedemonstrated an approach to determine the rootfillet shapes in the generation of involute externalgears and presented a better and simpler proof ofovercutting. Meanwhile, researchers adopt the similar

1State Key Laboratory of Fluid Power Transmission and Control,

Zhejiang University, P.R. China2707 Institute, China Shipbuilding Industry Corporation, China

Corresponding author:

Hua Zhou, State Key Laboratory of Fluid Power Transmission and

Control, Zhejiang University, No. 38 Zheda Road, Hangzhou,

P.R. China.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

0(0) 1–15

! IMechE 2012

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DOI: 10.1177/0954406212466479

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approaches to study the design and manufacture ofnon-conventional gears. Kapelevich9 has developedthe geometry and design of asymmetric involutegears. Litvin et al.10 have proposed the modifiedasymmetric involute gears as a combination of aninvolute gear and a double-crowned pinion, whichcan reduce the transmission errors and localize thebearing contact, and designed the rack-cutter basedon the generating principle. Yang11 has reshaped theprofile of the conventional rack-cutter by the devi-ation function to generate the driving gear of internalgear pairs. Results show that this approach can com-pensate the elastic deformation and reduce the contactstress between the gears. Yang12,13 also has investi-gated the generating processes of asymmetric involutehelical gears, including external and internal gears, inthe three-dimensional analysis and obtained the math-ematical models of the corresponding cutters.Costopoulos and Spitas14 have presented externalgears with asymmetric half-involute tooth, whichhas a decrease of maximum bending stresses up to28% compared to the standard gears, and investi-gated the generating process of gears by rack-cutter.Sankar and Nataraj15 have introduced circular rootfillet instead of standard trochoidal root fillet in thegear to increase the tooth strength, which is more aptfor the gears with less number of teeth.

This article presents a special kind of gear pair, theconjugated straight-line internal gear pair, which iscomprised of a pinion with straight-line profile andan internal gear with profile conjugated to that ofthe pinion (Figure 1). This type of gear pair is thekey component of the conjugated straight-line inter-nal gear pumps which are also called Truninger gearpumps and have a crescent to isolate the dischargechamber from the suction chamber. As mentioned inTruninger,16 this kind of gear pair leads to a verysmall trapped volume and the change in trappedvolume, with respect to the pinion rotation angle,lies within a range of only 0.7% of the delivery

volume per tooth, which is about 10 times smallerthan that of the conventional involute gear pumps.As change of trapped volume induces large pressureripples and high noise level to hydrostatic systems,this kind of gear pair can yield a significant reductionin noise levels of the pumps. What is more, thesestraight-line profiles have smaller curvature comparedto the conventional involute profiles, and therefore,high hertz contact stress during meshing process canbe prevented, which can increase the service life ofgears. As mentioned above, obviously, the conjugatedstraight-line internal gears have good influences onthe performances of gear pumps. As manyresearches17–21 have proved that gear pairs have adirect impact on the performances of gear pumps, itis necessary to have a good understanding of the char-acteristics of gear pairs in the design of gear pumpswith high performances. Although the conjugatedstraight-line internal gears can improve the perform-ances of gear pumps in noise level and service life, tothe authors’ best knowledge, there is no literatureabout the design of the conjugated straight-line inter-nal gear pairs until now, which is the basic in thedesign process of fluid power gear machines, includinggear pumps and gear motors.

This article demonstrates the design method of theconjugated straight-line internal gear pairs andderives the mathematical models of the gear pairs.Based on these mathematical models, geometricmodels of the gear pairs and their cutting tools areobtained and the problems of overcutting and inter-ference are investigated. It is expected that this studyshould be helpful to the design and manufacture ofthe conjugated straight-line internal gear pairs andalso provide a prepared knowledge for investigationsinto the performances of fluid power gear machineswith the conjugated straight-line internal gear pairs.

Mathematical model of the pinion withstraight-line profile

Figure 2 shows the tooth shape of the pinion withstraight-line profile, which consists of an arc oa, aline ab, and an arc bc. Here, one tooth of the pinionhas a central angle of �t, the value of which is 2�/z1and a profile angle of �. Also, it is defined that thetooth thickness, s, equals the width of space on thepitch circle. Therefore

s ¼1

2�

2�r1z1¼�r1z1

ð1Þ

where r1 is the pitch radius of the pinion and z1 thetooth number of the pinion.

Equation (1) yields

r1 ¼1

2�

2s

�� z1 ¼

1

2mlz1 ð2Þ

Figure 1. Cross section of the conjugated straight-line

internal gear pumps.

2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)




Here, referring to the tooth parameters of involutegear,7 parameter ml is defined as the module for thegears with straight-line profile, which is represented as

ml ¼2s

�ð3Þ

With the definition of module ml, another two par-ameters are determined

ra1 ¼ r1 þmlh�a ð4Þ

rd1 ¼ r1 �mlh�d ð5Þ

where h�a is the addendum coefficient, h�d the deden-dum coefficient, ra1 the addendum radius of thepinion, and rd1 the dedendum radius of the pinion.

With the parameters mentioned above, coordinatesystem S1(O1, x1, y1), which is rigidly connected to thepinion, is built to make a mathematical description ofthe profile, as demonstrated in Figure 2. Here, theorigin of S1 coincides with the pinion centre and they1 axis with the midline of the tooth. Then, the threeparts of the profile, arc oa, line ab, and arc bc, can berepresented in S1 as follows

Roa1 ¼

xoa1

yoa1

zoa1

264

375 ¼

x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2a1 � x21

q1

2664

3775, 04x14xa ð6Þ

Rab1 ¼

xab1

yab1

zab1

264

375 ¼

x1

kx1 þ b

1

264

375, xa 5 x14xb ð7Þ

Rbc1 ¼

xbc1

ybc1

zbc1

264

375 ¼

x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2d1 � x21

q1

2664

3775, xb 5 x14xc ð8Þ

where b is the intercept of line ab, k the slope of lineab, x1 the x-coordinate of the point on the profile ofthe pinion, and xa, xb, and xc the x-coordinates ofpoints a, b, and c, respectively.

Here

k ¼ � cot� ð9Þ

b ¼ r1 cos�

2þ r1 sin

�

2cot� ð10Þ

xa ¼�kb�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ k2Þr2a1 � b2

q1þ k2

ð11Þ

xb ¼�kb�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ k2Þr2d1 � b2

q1þ k2

ð12Þ

xc ¼ rd1 sin �t ð13Þ

where � is the corresponding central angle of the tooththickness s, as represented by equation (14).

� ¼s

r1ð14Þ

Therefore, the mathematical model of the pinionprofile can be illustrated by

Rp1ðx1, s,�, z1Þ ¼ Roa

1 Rab1 Rbc

1

� �0ð15Þ

Mathematical model of the rack-cutterfor the pinion with straight-line profile

This article assumes rack-cutter as the cutting tool forgenerating the pinion with straight-line profile.According to the generating principle, the pitch lineof the rack-cutter is always tangent to the pitch circleof the pinion in the cutting process,1 which means thatthe profile of the rack-cutter is conjugated to that ofthe pinion. Therefore, the profile of the rack-cuttercan be regarded as an envelope to the family of thepinion surfaces. In what follows, the profile of thepinion is assumed as the generating surface whilethe profile of the rack-cutter as the generated surface.

Figure 3(a) shows the generating process of thepinion by a rack-cutter. Coordinate systems Sc(Oc,xc, yc) and Sg(Og, xg, yg) are built, which are rigidlyconnected to the rack-cutter and the ground, respect-ively. Here, the xc axis of Sc coincides with the pitchline of the rack-cutter and the origin of Sg with thepinion centre. The pitch line of the rack-cutter is tan-gent to the pitch circle of the pinion at point Pc.Figure 2. Tooth shape of the pinion with straight-line profile.

Song et al. 3




During the generation, Sc moves with the rack-cutterwhile S1 rotates with the pinion. When Sc has a dis-placement of d, S1 rotates over an angle of ’1, both ofwhich are related by equation (16)

d ¼ r1’1 ð16Þ

The profile of the rack-cutter can be determined byapplying coordinate transformation from S1 to Sc, asrepresented by equation (17).

Rccð’1, x1, s,�, z1Þ ¼Mc1ð’1, r1ÞR

p1ðx1, s,�, z1Þ ð17Þ

where Rccð’1, x1, s,�, z1Þ is the envelope to the family

of the pinion surfaces, Rp1ðx1, s,�, z1Þ the profile of the

pinion as illustrated by equation (15), and Mc1ð’1, r1Þthe matrix for coordinate transformation from S1 toSc, as represented by equation (18).

Mc1ð’1, r1Þ ¼

cos ’1 � sin’1 r1’1

sin’1 cos ’1 �r1

0 0 1

264

375 ð18Þ

Considering the necessary condition of existence ofRc

1, equation (19), which is called as equation of mesh-ing, should be satisfied.7

Np1 � v

ð pcÞ1 ¼ 0 ð19Þ

where Np1 is the normal to the pinion surface and v

ð pcÞ1

the relative velocity between the pinion and the rack-cutter. Both are represented in S1.

Obviously, Np1 and v

ð pcÞ1 can be represented as

follows

Np1 ¼ T

p1 � k1 ¼

dRp1

dx1� k1 ð20Þ

vð pcÞ1 ¼ v

ð pÞ1 � v

ðcÞ1 ð21Þ

where Tp1 is the tangent to the pinion surface, v

ðcÞ1 the

velocity of the rack-cutter, and vð pÞ1 the velocity of the

pinion. They are represented in S1.It is easy to obtain two relations as follows

vð pÞ1 ¼ x

ð pÞ1 � R

p1 ð22Þ

vðcÞ1 ¼M1cð’1, r1Þv

ðcÞc ð23Þ

where xð pÞ1 is the angular velocity of the pinion as

represented in S1, M1cð’1, r1Þ the matrix for coordin-ate transformation from Sc to S1, and vðcÞc the velocityof the rack-cutter as represented in Sc .

Here

M1cð’1,r1Þ¼ �

cos’1 sin’1 �r1ð’1 cos’1� sin’1Þ

sin’1 cos’1 r1ð’1 sin’1þ cos’1Þ

0 0 1

264

375ð24Þ

vðcÞc ¼ �r1!1ic ð25Þ

where !1 is the value of the angular velocity of thepinion.

Equations (23) to (25) yield

vðcÞ1 ¼ �r1!1 cos ’1i1 þ r1!1 sin ’1j1 ð26Þ

Meanwhile, Rp1 can be represented by

Rp1 ¼ x1i1 þ y1j1 ð27Þ

where (x1, y1) is the coordinate of the point on theprofile of the pinion.

Substituting equation (27) into equation (22), thus

vð pÞ1 ¼ �!1y1i1 þ !1x1j1 ð28Þ

Equations (21), (26), and (28) yield

vð pcÞ1 ¼ v

ð pÞ1 � v

ðcÞ1 ¼ ð�!1y1 þ r1!1 cos ’1Þi1

þ ð!1x1 � r1!1 sin ’1Þj1 ð29Þ

Figure 3. Generating processes of the conjugated straight-line internal gear pair: (a) pinion and (b) internal gear.





Equations (20) and (27) yield

Np1 ¼

dRp1

dx1� k1 ¼

dy1dx1

i1 �dx1dx1

j1 ð30Þ

Combining equations (19), (29), and (30), the equa-tion of meshing can be represented as follows

ð’1, x1Þ ¼dy1dx1ðr1 cos ’1 � y1Þ

�dx1dx1ðx1 � r1 sin ’1Þ ¼ 0 ð31Þ

Equation (31) provides a relation between the twoparameters of the rack-cutter profile, x1 and ’1.Then, based on the given pinion profile, the mathem-atical model of the rack-cutter can be obtained byequation (17).

Mathematical model of the internal gearwith conjugated profile

As mentioned previously, the profile of the internalgear is conjugated to that of the pinion. Therefore,the profile of the internal gear can be regarded as anenvelope to the family of the pinion surfaces. Then,the pinion is assumed as a shaper-cutter to generatethe internal gear and the mathematical model of thepinion is regarded as the generating surface.

Coordinate system S2(O2, x2, y2) is built to demon-strate the relationship between the pinion and theinternal gear during the generation (Figure 3(b)).Here, S2 is rigidly connected to the internal gear andthe origin of S2 coincides with the gear centre. Thepitch circle of the internal gear, which has a radius r2,is tangent to that of the pinion at point P. The rota-tion angles of S1 and S2, ’1 and ’2, are related byequation (32).

r1’1 ¼ r2’2 ð32Þ

Here

r2 ¼1

2mlz2 ð33Þ

where z2 is the tooth number of the internal gear.Then, according to the theory of gearing, the pro-

file of internal gear can be obtained by equations (34)and (35).

Ri2ð’1, ’2,x1, s,�, z1Þ ¼M21ð’1, ’2ÞR

p1ðx1, s,�, z1Þ

ð34Þ

Np1 � v

ð piÞ1 ¼ 0 ð35Þ

where Ri2ð’1,’2, x1, s,�, z1Þ is the envelope to the

family of the pinion surfaces, M21ð’1, ’2Þ thematrix for coordinate transformation from S1 to S2,

as represented by equation (36), and vð piÞ1 the relative

velocity between the pinion and the internal gear, asrepresented in S1 by equation (37).

M21ð’1,’2Þ¼

cosð’1�’2Þ �sinð’1�’2Þ ðr2�r1Þsin’2

sinð’1�’2Þ cosð’1�’2Þ ðr2�r1Þcos’2

0 0 1

2664

3775

ð36Þ

vð piÞ1 ¼ v

ð pÞ1 � v

ðiÞ1 ð37Þ

where vðiÞ1 is the velocity of the internal gear, as repre-

sented in S1.Here7

vðiÞ1 ¼ x

ðiÞ1 � R

p1 þ E1 � x

ðiÞ1 ð38Þ

where xðiÞ1 is the angular velocity of the internal gear

and E1 the position vector drawn from O1 to O2. Bothare represented in S1.

Two relations can be illustrated as follows

r1xð pÞ1 ¼ r2x

ðiÞ1 ð39Þ

E1 ¼M1gð’1ÞEg ð40Þ

where M1g(’1) is the matrix for coordinate transform-ation from S1 to Sg, as represented by equation (41),and Eg the position vector drawn from O1 to O2, asrepresented in Sg by equation (42).

M1gð’1Þ ¼ �

cos ’1 sin ’1 0sin ’1 cos ’1 00 0 1

24

35 ð41Þ

Eg ¼ ~O1O2 ¼ ðr1 � r2Þjg ð42Þ

Equations (37) to (42) yield

vð piÞ1 ¼

ðr2 � r1Þ!1

r2

� ½ðr1 cos ’1 � y1Þi1 þ ðx1 � r1 sin’1Þj1� ð43Þ

Combining equations (35) and (43), the equation ofmeshing in the process of generating the internal gearscan be represented as follows

gð’1, x1Þ ¼dy1dx1ðr1 cos ’1 � y1Þ

�dx1dx1ðx1 � r1 sin’1Þ ¼ 0 ð44Þ

Therefore, the mathematical model of the internalgear profile can be obtained by considering equations(34) and (44) simultaneously.

Song et al. 5




Geometric models of the rack-cutter andthe internal gear

With the mathematical models obtained in the previ-ous sections, the geometries of the rack-cutter and theinternal gear can be determined with the given pinion.A program written by the Matlab code is developed toplot the profiles of the rack-cutter and the internalgear based on these obtained mathematical models.With the parameters listed in Table 1, the profiles of

the rack-cutter and the internal gear are plotted inFigures 4(a) and 5(a), respectively.

As shown in Figure 4(a), the obtained profile ofthe rack-cutter includes three parts, line1, curve 2,and line 3, just corresponding to the three parts ofthe pinion profile. Obviously, it is not a continuouscurve. Here, line 1 is not connected with curve 2 andcurve 2 intersects line 3. To obtain a continuous pro-file, two treatments are carried out: (a) adding a ver-tical line from the upper end of curve 2 to connectwith the added extension of line 1 and (b) removingthe parts of curve 2 and line 3 outside the intersectpoint, which lie inside the Space zone. The modifiedrack-cutter profile is demonstrated in Figure 4(b).Here, the caption Tooth refers the tooth of therack-cutter and the caption Space the tooth spaceof the rack-cutter. Similarly, see Figure 5(a), theobtained profile of the internal gear is also discon-tinuous and comprised of three parts: arc 1, curve 2,and arc 3. Using the same method, the modifiedprofile of the internal gear is demonstrated inFigure 5(b). To get a brief description, the term

Figure 4. Profile of the rack-cutter before (a) and after (b) modification.

Figure 5. Profile of the internal gear before (a) and after (b) modification.

Table 1. Parameters of the internal gear pair.

Parameter Value

z1 28

z2 38

s 5 mm

� 25�

h�a 1

h�d 1





profilemeans the modified profile in the following partof this article.

As mentioned above, shapes of the tooth spaces ofthe rack-cutter and the internal gear can be obtained,as shown in Figure 6(a) and (b), respectively. WithFigures 2 and 6, the basic geometric dimensions of

the rack-cutter, the pinion, and the internal gear canbe calculated by the parameters of z1, z2, s, �, h

�a, and

h�d, as listed in Table 2.Based on the kinematic relationships shown in

Figure 3, Figure 7 demonstrates the generating pro-cesses of the pinion and the internal gear by their

Figure 7. Generating processes by the cutting tools: (a) pinion and (b) internal gear.

Figure 6. Tooth shapes (tooth space) of: (a) rack-cutter and (b) internal gear.

Table 2. Calculation of the basic geometric dimensions.

Name Notation

Calculation formula

Rack-cutter Pinion Internal gear

Module ml ml ¼ 2�/s

Pitch radius r – r1 ¼ ml z1/2 r2 ¼ ml z2/2

Addendum ha mlh�a

Dedendum hd mlh�d

Addendum radius ra – ra1¼ r1þ ha ra2¼ r2� ha

Dedendum radius rd – rd1¼ r1� hd rd2¼ r2þ hd

Song et al. 7




corresponding cutting tool. Here, the rack-cutter islocated outside the pinion, while the shape-cutter islocated inside the internal gear. Obviously, the gener-ated profile is the envelope of the families of the cuttersurfaces. Moreover, the geometric models can be builtwith the help of the computer-aided softwareSolidWorks, as shown in Figure 8. These geometricmodels are helpful for researchers in the design ofthese gears, such as, finite element analysis, kine-matics and dynamics analysis, assembly and interfer-ence analysis, etc.

Overcutting analysis

Overcutting is a phenomenon where the gear profile isdamaged by the cutting tool in the generating process,which usually occurs to gears with a small toothnumber, and this decreases the tooth strength.According to some literatures,7,22 overcutting can be

determined by the appearance of singular points onthe generated surfaces. In this approach, the gener-ated gear surfaces are usually derived by the givensurfaces of the cutting tool. However, our study pre-sents a contrary case, in which the surfaces of therack-cutter are obtained by the given surfaces of thepinion, as mentioned in the ‘Mathematical model ofthe rack-cutter for the pinion with straight-line pro-file’ section. Considering the kinematic relationshipshown in Figure 3(a), the pinion can be assumed asa cutting tool for the rack-cutter. Meanwhile, thepinion is regarded as the shaper-cutter for the internalgear, as shown in Figure 3(b). Then, based on theseassumptions, the profiles of the rack-cutter and theinternal gear can be obtained, as demonstrated inFigure 9. Here, the pinion has a tooth number of17 and other parameters are listed in Table 1.Obviously, singular points have appeared in the pro-files of the rack-cutter and the internal gear. In

Figure 9. Singular points on the profiles: (a) rack-cutter and (b) internal gear.

Figure 8. Geometric models: (a) rack-cutter and pinion and (b) shaper-cutter (pinion) and internal gear.





addition, with envelopes shown in Figure 10, it is clearthat overcutting has occurred to the profiles in thepositions inside the circles.

To determine the condition of non-overcutting, astudy is carried out, which refers to the approachproposed by Litvin.7 Considering the generating pro-cess of the pinion by a rack-cutter, the straight line ofthe pinion profile satisfies the equation of meshing.Then, equations (7) and (31) yield

f ð’1, x1Þ ¼ kr1 cos ’1 þ r1 sin ’1

� kb� ðk2 þ 1Þx1 ¼ 0 ð45Þ

The equation of overcutting yields7

@x1@x1

vð pcÞx1

@f

@x1

@f

@’1

d’1dt

��

�� ¼@y1@x1

vð pcÞy1

@f

@x1

@f

@’1

d’1dt

��

�� ¼ 0 ð46Þ

where vð pcÞx1 is the x-component of v

ð pcÞ1 and v

ð pcÞy1 the

y-component of vð pcÞ1 :

Equations (31), (45), and (46) yield

r1ðk2 þ 2Þ cos ’1 � kr1 sin’1 � ðk

2 þ 1Þb

� kðk2 þ 1Þx1 ¼ 0 ð47Þ

kr1 cos’1 � r1ð2k2 þ 1Þ sin’1 þ ðk

2 þ 1Þx1 ¼ 0

ð48Þ

From equation (48)

xlim ¼ x1 ¼r1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2k2 þ 1Þ2 þ k2

qk2 þ 1

sinð’1 ��’Þ ð49Þ

Here, xlim is the limiting value of x1 to avoid over-cutting and

�’ ¼ arctank

2k2 þ 1

� �ð50Þ


�k sin ’1 þ cos ’1 ¼b

2r1ð51Þ

Considering equations (9) and (10)

sinð’1 þ �Þ ¼1

2sin

�

2þ �

� �ð52Þ

Therefore

’1 ¼ arcsin1

2sin

�

2þ �

� �� ð53Þ

Substituting equation (53) into equation (49), thevalue of xlim can be solved.

The condition to avoid overcutting can be illu-strated by equation (54).

xlim 5 xa ð54Þ

Figure 11 gives two examples to illustrate the con-dition represented by equation (54). Figure 11(a)shows the condition of the case demonstrated inFigure 10(a). This case has an xlim of 1.305 and anxa of 0.966, which does not satisfy equation (54), andovercutting has occurred, while Figure 11(b) presentsthe condition of the case demonstrated in Figure 7(a),which has an xlim of 0.795 and an xa of 0.986. Thiscase satisfies equation (54) and there is no overcutting.These two examples prove that the condition illu-strated by equation (54) can be used to determinethe parameters of the conjugated straight-line internalgear pairs for avoiding overcutting.

Figure 10. Overcutting on the profiles during the generation: (a) pinion and (b) internal gear.

Song et al. 9




Interference analysis

Compared to external gear pairs, internal gear pairshave smaller relative velocity between the meshingtooth surfaces since the gears have the same directionof rotation. Also, the reduction of the tooth numberdifference between the gears can decrease the relativevelocity to a much smaller value, which is good forimproving the performances of internal gear pumps.According to the work of Zhou and Song21 on theconjugated involute internal gear pumps, decreasingtooth number difference, especially by increasing thetooth number of the pinion, is much effective fordecrease of pressure ripple and increase of deliverycapacity. However, a small tooth number differencewill increase the risk of interference between thepinion and the internal gear, which affects thenormal operation of gear pairs. Mimmi andPennacchi18 have pointed out that, with respect tothe involute internal gear pairs, the tooth number ofthe internal gear is commonly at least 19 or 20 and thetooth number difference needs meet the conditionz2� z15 4, to avoid interference during the gener-ation and operation. Recently, Slodczyk andStryczek23 have developed a systematic method ofdesigning involute internal gear pumps. With thedeveloped computer program, an internal gear pairwith a smaller tooth number difference is obtained:z1¼ 13 and z2¼ 16.23 As mentioned above, it is mean-ingful to design internal gear pairs with a small toothnumber difference for improving operation perform-ances and reducing pump size, on the condition thatno interference has occurred.

This section proposes a method to determine thesmallest tooth number difference of the conjugatedstraight-line internal gear pair, which seems to bethe limiting condition to avoid interference. It isassumed that the tooth number of the pinion isgiven. Then, the problem here becomes a solution

for the smallest tooth number of the internal gear,which insures that there is no interference in thegear pair.

Figure 12 presents a limiting case to make the inter-ference analysis, in which the tip point of the piniontooth, A, just touches the tip point of the internal geartooth, K. Obviously, the trajectory of point A in S2

passes through point K, which is the intersect pointbetween the work profile and the addendum circle ofthe internal gear (Figure 13).

The trajectory of point A can be obtained fromequation (34)

xtra¼ cosð’1�’2Þxa� sinð’1�’2Þyaþðr2� r1Þsin’2ytra¼ sinð’1�’2Þxaþ cosð’1�’2Þyaþðr2� r1Þcos’2

ð55Þ

Therefore

xk¼ cosð’1�’2Þxa� sinð’1�’2Þyaþðr2� r1Þsin’2yk¼ sinð’1�’2Þxaþ cosð’1�’2Þyaþðr2� r1Þcos’2

ð56Þ

where (xk, yk) is the coordinate of point K in S2.

Figure 11. Conditions in the determination of overcutting: (a) z1¼ 17, �¼ 25�, h�a ¼ 1, and h�d ¼ 1 and (b) z1¼ 28, �¼ 25�, h�a ¼ 1,

and h�d ¼ 1.

Figure 12. Interference in the internal gear pair.





Since point K is the intersect point between thework profile and the addendum circle, the equationof the work profile should be derived first. By substi-tuting equation (7) into equation (34), the equation ofthe work profile can be obtained as follows

x2¼cosð’1�’2Þx1�sinð’1�’2Þðkx1þbÞþðr2�r1Þsin’2

y2¼sinð’1�’2Þx1þcosð’1�’2Þðkx1þbÞþðr2�r1Þcos’2

(

ð57Þ

Equations (9) and (57) yield another form of thework profile

y2 ¼ � cotð’1 � ’2 þ �Þx2

þb sin �þ ðr2 � r1Þ sinð’1 þ �Þ

sinð’1 � ’2 þ �Þð58Þ

Here

� cotð’1 � ’2 þ �Þ ¼ �1� tanð’1 � ’2Þ tan�

tanð’1 � ’2Þ þ tan�

ð59Þ

’1 � ’2 ¼z2 � z1

z2’1 ð60Þ

Equation (58) shows that the work profile of theinternal gear has a similar form to that of the straightline. However, its slope and intercept are not con-stant, as illustrated by equations (59) and (60). Sincethe tooth number difference is very small compared tothe tooth number of the internal gear, an approxima-tion can be obtained

� cotð’1 � ’2 þ �Þ � � cot� ¼ k ð61Þ

Then, we replace the work profile by an approxi-mate line, as represented by equation (62)

y2 ¼ kx2 þ b0 ð62Þ

As shown in Figure 13, the approximate line inter-sects the addendum circle at point K0, which is used toreplace point K in the solution process. Then, we getan approximate profile, as shown in Figure 14.According to the definition of the gear with straight-line profile, the approximate line passes through pointC and a relation can be obtained

xc ¼ r2 sin �cyc ¼ r2 cos �c

ð63Þ

where (xc, yc) is the coordinate of point C in S2.Here

�c ¼s

2r2ð64Þ

Equations (9) and (62) to (64) yield

b0 ¼r2

sin �sin

s

2r2þ �

� �ð65Þ

The coordinate of point K0 can be solved by equa-tion (66)

yk0 ¼ kxk0 þ b0

x2k0þ y2

k0¼ r2

a2

ð66Þ

where (xk0 , yk0 ) is the coordinate of point K0 in S2.Then, the solution of equation (66) is

xk0 ¼�kb0 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ k2Þr2

a2� b02

q1þ k2

yk0 ¼�k2b0 � k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ k2Þr2

a2� b02

q1þ k2

þ b0

8>>>>><>>>>>:

ð67Þ

Figure 14. Approximate geometry for determination of

interference.

Figure 13. Geometry for determination of interference.

Song et al. 11





xk0 ¼ cosð’1�’2Þxa� sinð’1�’2Þyaþðr2� r1Þ sin’2yk0 ¼ sinð’1�’2Þxaþ cosð’1�’2Þyaþðr2� r1Þcos’2

ð68Þ

Meanwhile, the coordinate of point a can be rep-resented as follows

xa ¼ ra1 sin �aya ¼ ra1 cos �a

ð69Þ

where �a is the central angle of point a, as shownin Figure 2.

Substituting equation (69) into equation (68),a relation is obtained

cosð�a � ’1Þ ¼r2a2� r2

a1þ ðr2 � r1Þ

2

2ðr2 � r1Þra1ð70Þ

where ra2 is represented by equation (71)

ra2 ¼ r2 �mlh�a ð71Þ

The solution of the non-linear equation system forz2, comprised of equations (68) and (70), determinesthe minimum tooth number of the internal gear toavoid interference. Here, it is assumed that thedesign parameters, m, s, �, h�a, and h�d, are alreadygiven. Considering the discussions above, the non-

linear equation system contains three equations withthree independent variables of z1, z2, and ’1, whichindicates that they are solvable. Since it is hard to getan analytical solution, an algorithm is proposed toobtain a numerical solution, as shown in Figure 15.

The procedures of the algorithm are demonstratedas follows:

1. A value of z1 is given as the input, which insuresthat there is no singularity on the profile.

2. The initial value of z2 is obtained by adding z1 by astep �z.

3. With the values of z1 and z2, the value of ’1 can besolved from equation (70).

4. Considering the characteristics of cosine function,the obtained value of ’1 should make the absolutevalue of equation (70) no greater than 1. If not, z2is continuously increased by another step �z untilthe value of ’1 makes equation (70) meaningful.

5. Two objective functions, f1 and f2, are derivedfrom equation (68), as represented by equation(72), and their values can be calculated with theobtained z1, z2, and ’1.

6. With two neighboring groups of z1, z2, and ’1, thedifference between the values of f1 can be obtainedand so is f2. If the absolute values of these twodifferences are not greater than the given error,e, simultaneously, the latter group of z1, z2, and’1 is considered as the solution of the non-linearequation system. If not, z2 is continuously

Figure 15. Flow diagram of the algorithm.





increased by a step �z and the procedures from 3to 6 are repeated until i exceeds the given limitvalue. Here, we assume that the initial values f1and f2 are 0 and are represented by f

ð0Þ1 and f

ð0Þ2 :

f1¼cosð’1�’2Þxa�sinð’1�’2Þyaþðr2�r1Þsin’2�xk0

f2¼sinð’1�’2Þxaþcosð’1�’2Þyaþðr2�r1Þcos’2�yk0

(

ð72Þ

Then, based on the algorithm, a program edited byMatlab codes is developed to find the minimum toothnumber of the internal gear. In what follows, twoexamples are presented to illustrate this approach.Here, e¼ 1e� 5, �z¼ 1e� 5, and i¼ 50.

Example I. With the parameters of the pinion listedin Table 1, the numerical solution of z2 is 31.4303. It iseasy to understand that the minimum value of z2should be 32, the smallest integer larger than the solu-tion. However, Figure 16(a) shows that the toothnumber of 32 still has the occurrence of interference.While, the larger value 33 has no interference with thepinion (Figure 16(b)). Therefore, in this case, the min-imum value of z2 is 33.

Example II. In this case, we choose a smaller pinion,the parameters of which are listed in Table 3 andinsure that there is no singularity on the profile.Since the obtained value of z2 is 13.6476, similarly,we first try the internal gear with a tooth number

of 14. As shown in Figure 16(c), there is interferencein the gear pair. Then, the internal gear with a toothnumber of 22 is tried and has no interference with thepinion (Figure 16(d)). Therefore, the minimum valueof z2 is 15.These two examples demonstrate how to determinethe minimum value of z2 for avoiding interference.Since approximations are used in the analysis, whichmakes the solution process much feasible, there isinevitably some deviation from the exact solution.Nevertheless, the minimum value of z2 can still befound from the solution very quickly whether thetooth number of the pinion is small or large,which may need one or two trials with the help ofcomputer-aided software, such as SolidWorks. Ingeneral, the approach presented in this article isstill effective to determine the minimum toothnumber of the internal gear for avoidinginterference.

Figure 16. Internal gear pairs with different parameters: (a) z1¼ 28, z2¼ 32, �¼ 25�, h�a ¼ 1, and h�d ¼ 1; (b) z1¼ 28, z2¼ 33,

�¼ 25�, h�a ¼ 1, and h�d ¼ 1; (c) z1¼ 12, z2¼ 14, �¼ 25�, h�a ¼ 0:6, and h�d ¼ 0:8; and (d) z1¼ 12, z2¼ 15, �¼ 25�, h�a ¼ 0:6,

and h�d ¼ 0:8.

Table 3. Parameters of the pinion

in Example II.

Parameter Value

z1 12

s 3 mm

� 25�

h�a 0.6

h�d 0.8

Song et al. 13




Conclusion

This article has carried out an investigation into theconjugated straight-line internal gear pair, including apinion with straight-line profile and an internal gearwith profile conjugated to that of the pinion. The fol-lowing conclusions can be drawn from this study:

1. The tooth shape of the pinion with straight-lineprofile is determined by four parameters of s, z1,�, h�a, and h�d:

2. Based on the theory of gearing, mathematicalmodels of the rack-cutter and the internal gearare derived.

3. Geometric models of the rack-cutter and the inter-nal gear are obtained. Also, formulas of basic geo-metric dimensions for the rack-cutter, the pinion,and the internal gear are presented.

4. Overcutting analysis in the generation of thepinion by a rack-cutter is carried out and the con-dition for avoiding overcutting is obtained.

5. Interference analysis for internal gear pairs is car-ried out and an algorithm is developed to find anumerical solution. Two examples are presentedto illustrate how to determine the minimumtooth number difference for avoiding interference.

This study should be helpful in the design andmanufacture of the conjugated straight-line internalgear pairs. Moreover, it will provide a prepared know-ledge for the researchers to investigate the perform-ance of fluid power gear machines with the conjugatedstraight-line internal gear pairs. In addition, furtherresearches on the influence of gear parameters on theperformances of fluid power gear machines will becarried out in the next future.

Funding

This research received funding from the National ScienceFoundation of China (grant no. 51175453).

References

1. Dudley DW. Handbook of practical gear design. Boca

Raton, FL: CRC Press, 1994.2. Colbourne JR. The geometry of involute gears. Berlin:

Springer-Verlag, 1987.3. Jain S and Amin SS. Gear metrology. Charlotte, NC: The

University of North Carolina, 2004.4. DiFrancesco G and Marini S. Structural analysis of

asymmetrical teeth: reduction of size and weight. Gear

Technol 1997; 14(5): 47–51.5. DiFrancesco G and Marini S. Structural analysis of teeth

with asymmetrical proEle. Gear Technol 1997; 14(4):

16–22.6. Crocker MJ. Handbook of noise and vibration control.

Hoboken, NJ: John Wiley & Sons, Inc, 2007.7. Litvin FL. Gear geometry and applied theory. Englewood

Cliffs, NJ: Prentice Hall, 1994.

8. Su X and Houser DR. Characteristics of trochoids andtheir application to determining gear teeth fillet shapes.Mech Mach Theory 2000; 35(2): 291–304.

9. Kapelevich AL. Geometry and design of involute spurgears with asymmetric teeth. Mech Mach Theory 2000;35: 117–130.

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lysis. Comput Method Appl Mech Eng 2000; 188(1–3):363–390.

11. Yang SC. A study on using deviation function methodto reshape a rack cutter. Int J Adv Manuf Technol 2006;

A30(5–6): 385–394.12. Yang SC. Study on an internal gear with asymmetric

involute teeth.Mech Mach Theory 2007; 42(8): 977–994.

13. Yang SC. Mathematical model of a helical gear withasymmetric involute teeth and its analysis. Int J AdvManuf Technol 2005; 26: 448–456.

14. Costopoulos Th and Spitas V. Reduction of gear filletstresses by using one-sided involute asymmetric teeth.Mech Mach Theory 2009; 44(8): 1524–1534.

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16. Truninger P. Truninger gear pump. Patent 3491698,

USA, 1970.17. Colbourne JR. Gear shape and theoretical flow rate in

internal gear pumps. Trans CSME 1975; 3(4): 215–223.

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Appendix

Notation

b intercept of line ab represented in S1

b0 intercept of the approximate linerepresented in S2

d displacement of the rack-cutter in thegeneration of the pinion

e error used in the solution of non-linearequation system

E1 position vector drawn from O1 to O2

represented in S1





Eg position vector drawn from O1 to O2

represented in Sg

f1,2 objective functions used in the solutionof non-linear equation system

fð0Þ1,2 initial values of f1,2h�a addendum coefficienth�d dedendum coefficienti loop count used in the solution of non-

linear equation systemk slope of line abml module for the gears with straight-line

profileMij coordinate transformation matrix from

coordinate system Si to coordinatesystem Sj

Np1 normal to the pinion surface repre-

sented in S1

r1,2 pitch radii of the pinion and the internalgear

ra1,2 addendum radii of the pinion and theinternal gear

Rab1 vector of line ab of the pinion profile

represented in S1

Rbc1 vector of arc bc of the pinion profile

represented in S1

Roa1 vector of arc oa of the pinion profile

represented in S1

Rp1 pinion profile represented in S1

Ri2 envelope to the family of the pinion

surfaces represented in S2

s tooth thicknessS1,2,c,g coordinate systems rigidly connected to

the pinion, the internal gear, the rack-cutter, and the ground, respectively

Tp1 tangent to the pinion surface repre-

sented in S1

vðcÞ1 rack-cutter velocity represented in S1

vðiÞ1 internal gear velocity represented in S1

vð pÞ1 pinion velocity represented in S1

vð pcÞ1 relative velocity between the pinion and

the rack-cutter represented in S1

vð piÞ1 relative velocity between the pinion and

the internal gear represented in S1

vðcÞc rack-cutter velocity represented in Sc

vð pcÞx1 x-component of v

ð pcÞ1

vð pcÞy1 y-component of v

ð pcÞ1

xa,b,c x-coordinates of the points a, b, and cxlim limiting x1 value to avoid overcutting(x1,y1) coordinate of the points on the pinion

profile represented in S1

(xc,yc) coordinate of point C represented in S2

(xk,yk) coordinate of point K represented in S2

(xk0 , yk0) coordinate of point K0 represented in S2

(xtra ,ytra) coordinate of trajectory of point Arepresented in S2

z1,2 tooth numbers of the pinion and theinternal gear

� profile angle�z step used in the solution of non-linear

equation system� central angle of the tooth thickness on

the pinion pitch circle�a central angle of point a�t central angle of one pinion tooth’1,2 angles of the pinion and the internal

gear during generation processes!1 value of the angular velocity of the

pinionxðiÞ1 angular velocity of the internal gear

represented in S1

xð pÞ1 angular velocity of the pinion repre-

sented in S1

Song et al. 15



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