Biernacki - Analysis of Stress and Deformation

Analysis of stress and deformation in plasticgears used in gerotor pumpsK Biernacki* and J Stryczek

Faculty of Mechanical Engineering, Wroclaw University of Technology, Poland

The manuscript was received on 21 December 2009 and was accepted after revision for publication on 12 March 2010.

DOI: 10.1243/03093247JSA630

Abstract: The mechanism of induction of stresses and deformations in plastic cycloidal gearsused in gerotor pumps has been analysed using the finite element method and the ABAQUSprogram. It has been found that the gear system remains under the influence of the mechanicalload resulting from the torque on the pump shaft and the hydraulic load resulting from theactivity of pressure in the intertooth displacement chambers. It has also been discovered thatthe intertooth forces and stresses are formed only in the part of the gear that can be referred toas ‘active’. In the other part of the gear, forces and stresses do not occur, and the part can bereferred to as ‘passive’. Another finding of the research is that gear deformations occur suchthat the teeth of the external gear are deformed and moved in the direction of the active part ofthe mesh, and the teeth of the internal gear in the direction of the passive part. Thus, radial andaxial intertooth clearances are formed, which result in internal leakages in the pump, as well asin lower working pressure and efficiency of the machine. A way of determining the load rangefor plastic gears used in gerotor pumps has been specified. It has also been observed that theloading of the pump should not result in reduced stresses sz higher than compressive stressessc equal to the plasticity limit Re of the plastic, that is, sz ( sc 5 Re. At the same time, the radial

clearance hr and the axial clearance ha should not exceed the extreme values hr max and ha max

assumed for a particular size of gerotor pump, that is, hr , hr max and ha , ha max.

Keywords: gerotor pumps, cycloidal gears, plastics, finite element method, stress, deformation

1 INTRODUCTION

Gear pumps are basic energy generators in hydraulic

drive systems. There are three generations of gear

pumps:

(a) external toothing gear pumps;

(b) internal toothing gear pumps;

(c) gerotor pumps.

A fundamental design feature of gear pumps is the

internal cycloidal gear with the tooth difference

z2 2 z1 5 1. This feature gives the gerotor machines a

considerable advantage over internal and external

gear pumps, namely a very compact structure, a

smaller size, and lower weight. The intertooth

displacement chambers in the gerotor pumps have

a high volume, which enables a high efficiency of the

relatively small and lightweight units to be achieved.

The pumps work with a lower delivery pulse and

pressure than gear pumps, which results in a

reduction in noise while in operation. At the same

time, gerotor pumps are reliable and durable.

The history of gerotor pump development is long.

According to Pippenger and Dong [1], gerotors were

discussed back in the eighteenth and nineteenth

centuries. The first work on the subject was written

by Myron Hill, who in 1926 published a book entitled

Kinematics of Gerotors.

An important step in gerotor development was the

start-up of mass production of gerotor pumps by

Henry Nichols in the 1930s in the United States.

Another stage in gerotor machine evolution, in the

1950s, was the application of gerotor elements in

orbital motors by Lynn Charlson.

Research on gerotor elements at that time focused

on their geometry and kinematics, as well as on their

*Corresponding author: Faculty of Mechanical Engineering,

Wroclaw University of Technology, Lukasiewicza 7/9, Wroclaw

50-371, Poland.

email: [email protected]

465

JSA630 J. Strain Analysis Vol. 45

possible applications in selected fluid power ma-

chines. Publications on the subject were rather rare

and unthorough, and the specialist knowledge was

protected by the manufacturers. An original ap-

proach to the subject was presented in 1948 by

Sannikow [2], who described the geometry and

kinematics of various gear types working in gerotor

machines. Another distinguished study of the pro-

blem of gerotors was published by Colbourne [3],

who presented possible applications of trochoids

(cycloids) of various types in the manufacture of

rotodynamic pumps. A different approach to the

geometry and kinematics of gerotors was presented

in studies by Stryczek [4–7]. Gerotors were treated as

gears with corrected and modified cycloidal profiles,

and their geometry and kinematics were described

by a uniform system of parameters: z was the tooth

number, m was the modulus, l was the tooth depth

ratio, and v was the correction coefficient.

In the 1970s and 1980s, new gerotor hydraulic

machines were produced in Germany, namely the

ring rotor pump presented by Scholler [8] and the MZ

planetary motor described by Wusthof and Schneider

[9]. The machines feature a higher working pressure

as well as volumetric and total efficiency, and they are

also quieter than the Nichols and Charlson machines.

Further development of gerotors was dictated by

the development of the theoretical fundamentals of

this machine group, and particularly on the develop-

ment of hydraulic models. Bednarczyk and Stryczek

[10] formulated rules for designing the axial clearance

compensation in gerotor pumps, whereas Antoniak

and Stryczek [11] provided a model of flow in the

channels and internal clearances of gerotor pumps.

As a result of research, a gerotor pump featuring axial

clearance compensation, a high working pressure,

and volumetric efficiency, as well as a low-speed,

high-torque orbital motor with a double-gear gerotor

set of very high displacement (10 000 cm3/rev), was

made and described by Stryczek [12]. Italian en-

gineers built a miniature gerotor pump featuring axial

and radial clearance compensation [13].

In order to take another step in gerotor machine

development, the gerotor elements had to be

strengthened. Research enabled the maximum gear

load to be found. Mechanical and hydraulic loads

induce contact stresses between the mating teeth,

causing their wear and deformation and generating

intertooth clearances. This results in internal lea-

kages in the machine, a decrease in working

pressure, and a lowering of volumetric efficiency.

The first serious strength analyses of gerotor

elements were carried out by Colburne [14], who

examined the influence of gear profile on intertooth

contact stress. Strength analysis of trochoidal gears

in pumps, carried out by the finite element method

(FEM) by Gomez-Montero et al. [15] showed that the

maximum stresses occur in the tooth pair moving

around the central point of the toothing. FEM

research into the strength of epitrochoids working

with orbital motion was carried out by Nag et al.

[16]. They proved that the deformations of particular

teeth assume various values and enable flow through

the intertooth channels of the machine.

The research outlined above has not exhausted the

topic, however. In particular, the question as to the

source of gear deformations, and consequently

clearances in the pump, remains unanswered.

Knowledge of the profile and size of clearances can

be used for improvement of the hydraulic pumps

mentioned above [10, 11], and ultimately for their

technological development. It is particularly impor-

tant in the case of plastic gears, which are less

durable than steel gears [17]. The application of

plastics is advantageous for design reasons, because

gerotor pumps can work with ecological fluids and

generate a lot less noise. It is also advantageous for

technological and economic reasons, because plastic

gears can be manufactured by means of the simple

and cheap injection moulding method.

Thus, the following specific aims were formulated:

(a) determination, by means of FEM, of the

mechanism by which stresses and deformations

occur in the plastic cycloidal gear set used in

gerotor pumps;

(b) determination of the load range for the plastic

cycloidal gears, depending on the stress and

acceptable deformations.

For these purposes it was necessary to:

(a) analyse the design and operating principle of the

gerotor pump, as well as of the gear set working

within it;

(b) develop the fundamentals of the geometry and

kinematics, and specify the load distribution for

the gears;

(c) develop a model of the gear set;

(d) carry out a strength analysis of the model by

means of FEM.

2 DESIGN AND OPERATING PRINCIPLE OF THEGEROTOR PUMP

The design of the gerotor pump according to

Stryczek [4, 12] is shown in Fig. 1. The main body

466 K Biernacki and J Stryczek

J. Strain Analysis Vol. 45 JSA630

of the pump consists of three plates: the front plate

1, the central plate 2, and the back plate 3. In the

central plate 2 there is a gear set, 4 and 5, which is

driven by a shaft 7 located in bearings 6. Between the

internal toothing cycloids with the tooth difference

z2 2 z1 5 1, intertooth displacement chambers are

formed (TCR). In the back plate on the input side (I),

a cylindrical input channel (ICL) and a kidney-

shaped input chamber (ICR) are formed, whereas on

the output side (O), an output channel (OCL) and an

output chamber (OCR) of the same shape are

formed. The input chamber is separated from the

output chamber by bridges (LHB and HLB).

The operating principle of the pump is as follows.

The working fluid enters the pump via the input port

(ICL) and fills the input chambers (ICR). Simulta-

neously, the intertooth displacement chambers

(TCR) move, increasing their volume and sucking

in the working fluid. Next, the intertooth displace-

ment chambers (TCR) move on to the bridge (LHB)

where the suction process ends and the delivery

process begins. Successive intertooth displacement

chambers (TCR) move along the output chamber

(OCR), decreasing their volume and delivering the

working fluid. The working fluid flows out of the

pump through the output chamber (OCR) and the

output channel (OCL). Finally, the intertooth cham-

bers move on to the bridge (HLB) where the delivery

process ends and the suction process begins.

The torque essential for revolution of the gears

and for realization of the delivery process is

determined, according to Stryczek [4], by

M~q

2pp ð1Þ

In the process of delivery, the working fluid

remains at pressure p. It also flows on to the front

area within axial clearance ha.

As shown in Fig. 1, it is possible to distinguish four

zones in the pump, through which the displacement

chambers (TCR) move:

(a) the low-pressure zone (LPZ) determined by the

input chamber (ICR), with input pressure pi;

(b) the zone of pressure increase (LHPZ) deter-

mined by bridge LHB, where the pressure rises

from pi to po;

(c) the high-pressure zone (HPZ) determined by

the output zone (OCR), with pressure po;

(d) the zone of pressure decrease (HLPZ) deter-

mined by bridge HLB, where the pressure falls

from po to pi.

The conclusion that can be drawn from the above

description is that the gear set in the gerotor pump

remains under combined mechanical and hydraulic

load. The mechanical load is generated by torque M

driving the gear set. The hydraulic load is caused by

the activity of the pressure p of the working fluid in the

intertooth displacement chambers, and also by the

activity of pressure p on the front surfaces of the

mating gears. These combined loads cause gear defor-

Fig. 1 Design and operating principle of the gerotor pump: 1, 2, 3 – body plates; 4, 5 – cycloidalgears; 6 – bearing; 7 – shaft; TCR – internal tooth chamber; ICL – input channel; ICR –input chamber; OCL – output channel; OCR – output chamber; LPZ – low-pressure zone;HPZ – high-pressure zone; LHB – low–high bridge; HLB – high–low bridge; LHPZ – low–high pressure zone; HLPZ – high–low pressure zone

Analysis of stress and deformation in plastic gears 467


mation, as well as an increase in intertooth clearances,

namely the radial clearance hr and the axial clearance

ha between the gear front and the pump body.

In order to achieve the aim of this paper, that is, to

specify the mechanism by which deformations occur

and to determine the load range for the gear set, it is

necessary first to specify precisely its geometry,

kinematics, and load distribution.

3 GEOMETRY, KINEMATICS, AND LOADDISTRIBUTION OF THE GEAR SET IN THEGEROTOR PUMP

3.1 Geometry

Figure 2 illustrates the method for designing the

epicycloidal gear set used in gerotor pumps accord-

ing to Stryczek [4–7].

Fig. 2 Designing the epicycloidal gear set: (a) designing the epicycloidal tooth; (b) designing theepicycloidal gear set; (c) correcting and modifying the epicycloidal gear set



The first step in the design process is to assume

the tooth profile. As shown in Fig. 2(a), the profile is

determined by the arc of the shortened epicycloid,

which is made by point M connected to the circle of

radius r rolling slidelessly on the outer side of the

fixed circle of radius rb.

The second step is to make the internal gear

element. It is made by the successive arc/tooth

number z1, formed in a closed shortened epicycloid

(Fig. 2(b), shown in red). On the basis of Stryczek [4],

the following equations are adopted for the shor-

tened epicycloid forming the tooth profile

xe1~m

2z1z1ð Þ cos g{l cos z1z1ð Þg½ �

ye1~m

2z1z1ð Þ sin g{l sin z1z1ð Þg½ �

ð2Þ

Using the external toothing gear and tooth

number z1, the internal toothing gear and tooth

number z2 5 z1 + 1 is designed (Fig. 2(b), shown in

blue). In this arrangement, the external toothing gear

with centre O1 turns around fixed centre O2, making

a family of shortened epicycloids. To this family, the

external envelope is drawn, which makes the tooth-

ing of the internal gear.

According to Stryczek [4], the equations used for

determining the profiles of the successive shortened

epicycloid making the family have the following

form

xef2~m

2l cos cz z1z1ð Þ cos

z1g{c

z1

�

{l cosz1z1ð Þz1g{c

z1

�

yef2~m

2l sin cð z z1z1ð Þ cos

z1g{c

z1

�

{l sinz1z1ð Þz1g{c

z1

�

ð3Þ

while the equation of the envelope for the shortened

epicycloid family according to Stryczek [4] is

sin z1g{l sin z1gzz1 g{cð Þ{c

z1

� �z sin

z1 g{cð Þ{c

z1

~0 ð4Þ

As can be seen in Fig. 2(b), equations (3) are used

to design particular shortened epicycloids that make

a family. Equation (4), considering the position of

the particular epicycloids (angle c) is used to

determine the point that is an element of the

envelope in a given epicycloid (angle g). System of

equations (3), (4) has an implicit form, but it can be

solved by means of computer.

Figure 2(b) shows that both the external and the

internal gears feature disproportionate gearing.

Meshing of the convex teeth of the external gear

with the sharply ended teeth of the internal gear is

not good owing to considerable intertooth thrust and

wear. The teeth of both gears are also difficult to

manufacture.

Thus, in the third design step it was necessary to

introduce corrections and modifications of the tooth

profile. As depicted in Fig. 2(c), the correction of the

teeth is made without a change in distance of the

O1O2 axis. The correction is made by shifting the

profile of both gears towards the inside of the gear

set in section g. As a consequence, the external

toothing gear has a profile made by the internal

equidistance to the shortened epicycloid. On the

basis of Stryczek [4], the profile (shown in red) is

derived by means of the following equations (in the

equations, the correction coefficient v 5 g=r is

introduced)

xeke~m

2z1z1ð Þ cos g{l cos z1z1ð Þg

{ncos g{l cos z1z1ð Þgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1zl2{2l cos z1g

q375

yeke~m

2z1z1ð Þ sin g{l sin z1z1ð Þg

{nsin g{l sin z1z1ð Þgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1zl2{2l cos z1g

q375

ð5Þ

The internal toothing gear, however, should have a

profile made by the equidistance to the envelope.

Nevertheless, the complex profile is replaced with

r 5 g arcs drawn from the vertices of the envelope.

The arcs are connected by arcs of the circle drawn

from the centre of circle O2. The modified profile of

the circle is shown in blue.

The epicycloidal gear set finally achieved, featur-

ing the corrected and modified toothing (Fig. 2(c)), is

the most frequently applied system in hydraulic

machines such as gerotor pumps, orbital motors,

and control systems.

375

375



3.2 Kinematics

Figure 3 illustrates the kinematic analysis of the

cycloidal gear set used in the gerotor pump.

It can be seen in Fig. 3(a) that all teeth, 19, 29, …,

69, of the external (active) toothing gear (shown in

red) mesh with teeth 10, 20, …, 70 of the internal

(passive) toothing gear (shown in blue), making

contact points A1, A2, A3, …, A7, which are arranged

in the contact line of the conchoidal profile (shown

in green). The contact lines can be divided by

vertical symmetry axis O1O2 into the active part

(the green continuous line) and the passive part (the

green dashed line). In the active part, the teeth of the

external (active) gear transmit the torque on to the

teeth of the internal (passive) gear, whereas in the

passive part the teeth slide on one another without

transmitting the torque. It is then necessary to

determine the contact ratio e, which, according to

the definition given by Stryczek [4], equals the

relation of the arc of action to the circular pitch. As

shown in Fig. 3, the arc of action is arc BD made on

the pitch circle with radius rw1. The pitch circle rw1

must turn by this arc while one pair of teeth mesh

(19–10), beginning from the engagement at point A1

to the disengagement at point A1k. Also, on the pitch

circle rw1 the gear circular pitch pw 5 BE is marked.

Hence, the contact ratio has the form

e~BD

pwð6Þ

where BD~ prw1ð Þ=z1zprw1 is the arc of the mesh

and pw~ 2prw1ð Þ=z1 is the circular pitch.

The result of the above is the formula for the

contact ratio

e~z1z1

2ð7Þ

which establishes the fact that the value of the

contact ratio is greater than 1 and is far greater than

the value of the contact ratio for involute gearing.

Kinematic analysis has been carried out for the

rotation of the external gear by one pitch, namely by

angle a 5 2p=z1.

Three characteristic positions of the external gear

have been specified:

(a) the initial position (Fig. 3(a)), when tooth 19 of

the external gear engages and is turned in

relation to gearing axis O1O2 by angle

a 5 {p=z1;

(b) the central position (Fig. 3(b)), when tooth 19 of

the external gear is in gearing axis O1O2, namely

a 5 {p=z1;

(c) the final position (Fig. 3(c)), when tooth 19 of

the external gear remains turned in relation to

gearing axis O1O2 by angle a 5 zp=z1.

In the initial position, tooth pair 19–10 engages at

point A1. At the same time, in the gearing on the

active part of the contact line, tooth pairs 29–20, 39–

30, and 49–40 mate at points A1, A2, A3, and A4. On the

Fig. 3 Kinematics of the cycloidal gear set: (a) initialposition of the external gear a 5 {p=z1; (b)central position of the external gear a 5 0; (c)final position of the external gear a 5 zp=z1



passive part of the contact line, tooth pairs 59–50, 69–

60, and 19–70 slide at points A5, A6, and A7.

In the central position, tooth pairs 19–10, 29–20, and

39–30 at points A1, A2, and A3 shift on the active part

of the contact line. The tooth pair 49–40, however,

disengages at point A4 and goes onto the passive part

of the contact line. On the passive part there remain

the tooth pairs 59–50, 69–60, and 19–70 which come

into contact at points A5, A5, and A7.

In the final position, another tooth pair 69–70

engages, and contact point A7 replaces point A1.

3.3 Load distribution of the gear set

What stems from the geometric and kinematic

analysis is that, while rotating, the teeth of both

gears remain in permanent contact, providing tight-

ness of the displacement chambers. At the same

time, the chambers initially decrease and then

increase their volume, which results in the pumping

process. The intertooth displacement chambers

change their position in relation to the characteristic

zones of the pump defined in Fig. 1, which results in

a change in the mechanical and hydraulic load

distribution.

The mechanical and hydraulic load analysis,

similarly to the kinematic analysis, was carried out

for three characteristic positions of the internal and

external toothing gears with symmetry axis O1O2,,

namely the initial, the central, and the final positions

(Fig. 4). In the initial position (a 5 {p=z1) shown in

Fig. 4(a), the gear set is worked on by torque M

which generates the intertooth forces P’2{P’’2,

P’3{P’’3, and P’4{P’’4 in tooth pairs 29–20, 39–30, and

49–40 located in the active part of the contact line.

In pair 19–10 there are no intertooth forces because

the teeth engage and the profiles slide against each

other. Neither do intertooth forces occur in the other

tooth pairs located on the passive part of the contact

line. At the same time, the intertooth displacement

chambers are worked on by the pressure of the

working fluid. The chambers located in the low-

working-pressure zone (LPZ) are worked on by the

input pressure pi, whereas the chambers located in

the high-pressure zone (HPZ) are worked on by the

output pressure po. The chamber located in the zone

of pressure increase (LHPZ) is also under the

influence of pressure, the value of which is equal

to the value of the output pressure po because it is

assumed that the chamber is connected by the axial

clearance ha and radial clearances hr to the high-

pressure zone (HPZ).

Similarly, the chamber that remains in the zone of

pressure decrease (HLPZ) is influenced by high

pressure po. Simultaneously, the working fluid at

high pressure po works on the front surface of the

gears. Finally, the broken line C, O1, O2, 49–40

Fig. 4 Load distribution of the cycloidal gear set: (a)initial position of the external gear a 5 {p=z1;(b) central position of the external gear a 5 0;(c) final position of the external gear a 5 zp=z1



separates the intertooth displacement chambers and

the front surfaces of the gears remaining under the

influence of input (low) pressure pi and output

(high) pressure po.

In the central position (a 5 0) shown in Fig. 4(b),

intertooth forces P’1{P’’1 occur in tooth pair 19–10,

which engage, as well as forces P’2{P’’2 and P’3{P’’3between tooth pairs 29–20 and 39–30, which are

already in the mesh. In tooth pair 49–40, the forces

vanish, because the teeth go out of the active contact

line. Similarly, the intertooth forces do not occur in

the other tooth pairs located in the passive part of

the contact line. In the central position, the load

distribution generated by the pressure of the work-

ing fluid changes. The chambers located in the low-

pressure zone (LPZ) and on the border with the

pressure increase zone (LHPZ) and the pressure

decrease zone (HLPZ) are influenced by low pressure

pi, while the chambers in the high-pressure zone

(HPZ) and on the border with the pressure increase

zone (LHPZ) and the pressure decrease zone (LPZ)

are influenced by high pressure po. At the same time,

the working fluid at output pressure po works on the

front surface of the gears. The vertical line C, O1, O2,

49-40 separates the intertooth displacement cham-

bers and the front surface, which remain under the

influence of input (low) pressure pi and output

(high) pressure po (shown in red).

In the final position (a 5 zp=z1) shown in

Fig. 4(c), another tooth pair 69–70 engages and

replaces tooth pair 19–10. The intertooth forces and

load distribution that are generated by the pressure

of the working fluid are the same as those in the

initial position. The only difference is the change in

the position of the key through which torque M is

transmitted.

From the analysis carried out it can be concluded

that the mechanical and hydraulic load distribution

of the gear set in the gerotor pump changes

according to the angle of rotation. Therefore, a

complex state of stresses and deformations can be

anticipated, and the best method of studying the

problem is the finite element method (FEM).

4 COMPUTABLE MODEL OF THE CYCLOIDALGEAR SET

4.1 Geometric model, loads, and fixings

On the basis of the diagram of the gerotor pump

shown in Fig. 1, a geometric model of the cycloidal

gear set was prepared (Fig. 5). The parameter values

of the epicycloid listed in Table 1 were adopted.

The toothing profile of the external gear was

determined from equation (5). The toothing of the

internal gear was shaped in the form of arcs of a

circle drawn from the vertices of the envelope

circumscribed by equations (3) and (4). The geo-

metric model is influenced by the mechanical load

resulting from the transmission of torque M by the

driver shaft of diameter d1 5 25 mm on to the key

situated in the external toothing gear and on to the

internal toothing gear. The hydraulic load of the

model stems from the impact of pressure p of the

working fluid located in the intertooth displacement

chambers of the gear set. The gear set remains in a

state of balance because, in accordance with formula

(1), torque M on the shaft will be equal to the torque

generated by pressure p.

The external gear toothing profile is derived from

equation (5). The internal gear toothing profile is

made according to the arc of the circle drawn from

the vertices of the envelope described by equations

(3) and (4).

The model must be fixed, and the way of fixing

also stems from the operating principle of the

gerotor pump shown in Fig. 1. It can be seen in the

figure that the external toothing gear is set on the

shaft 7 and can only revolve around axis O1. In the

model presented in Fig. 5(a), the gear is fixed radially

on the shaft of diameter d1 5 25 mm and can revolve

around axis Z. The internal toothing gear depicted in

Fig. 1 is located in the port of the central body 2 and

can revolve around axis O2. In the model presented

in Fig. 5(a), the gear is fixed radially on diameter

d2 5 75 mm, but it can revolve around axis Z.

As shown in Fig. 1, the gear set is also worked on

by pressure p of the working fluid in axial clearance

ha made between elements 2 and 3 of the body. This

presses the gear set to the body 1. On account of this,

in the model shown in Fig. 5(a), the end face of the

gear set is loaded with pressure p, while the opposite

face is fixed.

4.2 Finite element grid

The numerical model of the cycloid gear set was

made by means of the ABAQUS system version 6.7-5.

The license number 05UWROCLAW of the program

was leased by Wrocławskie Centrum Sieciowo-

Superkomputerowe at Wroclaw University of Tech-

nology, Poland.

The finite element grid shown in Fig. 5(b) was

made by means of eight-node HEXA solid elements,

typically used for creating solid models. The finite

element grid for the gear set was made from ca



500 000 elements. Those elements made the profile

of the gear set on plane XY. In order to model the

thickness of the gear set, 15 layers of HEXA elements

were built. The layers were arranged ca every 0.7 mm

along axis OZ until a thickness of the gear set of

b 5 10.4 mm was achieved (Fig. 5(a)). The HEXA

element is an eight-node cube. The number of

nodes indicates that it is the first-order element.

Each node of the HEXA element has three degrees of

freedom, and they are shifted in relation to axes X, Y,

and Z. In the anticipated loci, grid refinements were

made, which can be clearly seen in Fig. 5(b). The loci

Fig. 5 Model of the cycloidal gear set: (a) load distribution; (b) finite element grid

Table 1 Values of parameters for the epicycloidal gear

Number Parameter Symbol Value

1 Number of teeth in the external toothing gear z1 62 Number of teeth in the internal toothing gear z2 5 z1 + 1 z2 73 Pitch circle radius r 4.8754 Modulus m m 9.714 mm5 Tooth depth coefficient l 0.7776 Tooth profile shift g 11.1125 mm7 Tooth profile shift coefficient, correction coefficient v 5 g=r v 2.2888 Radius of the arc of the internal gear teeth r 5 g r 11.1125 mm9 Face width b 10.4 mm



of the refinements are tooth contacts of the mating

gears on the suction side (the left side of the model

in Fig. 5(b)) and the area of the splineway in the

active gear. Smaller refinements were made at the

corners of the tooth bearing of the passive gear. The

grid of the external toothing active gear consists of a

greater number of elements because the authors

believe that it is more strenuous.

The system enabled modelling of the surface

contact by means of a special algorithm character-

istic of the ABAQUS system. The contact was located

in the area of the tooth contact on the suction side of

the gear set (the left side of Figs 5(a) and (b)). The

friction factor m was selected considering the plastic

type and the conditions present in the area of

contact (e.g. high pressure). The value m 5 0.4 was

determined on the basis of research results [18].

The numerical model allowed very precise deter-

mination of stresses and deformations for the gear

set, and the discrete action error level does not

exceed 0.02 per cent for the deformations, and ca 1.3

per cent for the stresses.

4.3 Selection of the plastic

As the plastic for the matched gears, pure polyox-

ymethylene (POM) was chosen. It is characterized by

relatively high mechanical strength and low strain.

The material has low water absorption and good

processability. Compression strength testing of POM

was conducted, and the stress–strain diagram

sc 5 f eð Þ was obtained (Fig. 6). The figure shows

that, for deformations e 5 0–0.15, stress sc increases

in an approximately linear manner, and the range

can be described as elastic. For deformations

e . 0.15, however, the stress sc changes according

to a parabolic law, and this is described as the plastic

stress range. In this case, the stress sc 5 80 MPa

corresponding to the deformation e 5 0.15, which

separates the elastic range from the plastic range, is

assumed to be the plasticity limit Re for POM.

For further stress analysis it was decided that the

compressive stress sc 5 Re 5 80 MPa would stand for

the maximum acceptable stress for the gear set, and

the reduced stress sz specified in FEM would be

compared with it.

4.4 Research programme

Research on the stresses and deformations in the

plastic cycloidal gear set was carried out for two

characteristic load states resulting from the operat-

ing principle of the machine (Fig. 1), as well as from

the assumed model (Fig. 5(a)). These are:

(a) the state of total mechanical and hydraulic load,

where torque M works on the mesh and

pressure p works on the intertooth displace-

ment chambers;

(b) the state of partial hydraulic load, where

pressure p works on the face end of the gear set.

In both load states, research on the stresses and

deformations was conducted for three characteristic

positions of the gears, for which a prior theoretical

analysis of kinematics (Fig. 3) and load distribution

(Fig. 4) had been carried out.

Fig. 6 Stress–strain diagram for polyoxymethylene (POM)



In the research process, the model was loaded

successively with torque M 5 7.16, 14.32, 17.9, and

21.48 N m, derived from formula (1), and pressure

p 5 4, 8, 10, and 12 MPa. The assumed displacement

of the gerotor pump was q 5 11.24 cm3/rev.

5 RESULTS AND DISCUSSION

Figure 7 depicts example results on the model of the

cycloidal gear set loaded with the total load resulting

from the activity of torque M 5 7.16 N m and pres-

Fig. 7 The state of stresses and deformations in plane XY of the model of cycloidal gears for itsdifferent positions. The pressure load of the model p54MPa and the torque load M57.16Nm: (a), (b) initial position of the external gear a 5 {p=z1; (c), (d) central position of theexternal gear a 5 0; (e), (f) final position of the external gear a 5 zp=z1



sure p 5 4 MPa. The distribution of stresses and

deformations for the gear set in three characteristic

positions is presented. In order better to compare

the deformations, the deformed profile (in colour) is

shown against the background of the undeformed

gear profile (continuous black line).

The state of stresses and deformations for the

initial position of the external toothing gear

(a 5 {p=z1) is presented in Figs 7(a) and (b) respec-

tively. Figure 7(a) shows that, in successive tooth

pairs 19–10, 29–20, and 39-30, stresses (intertooth

forces) occur, whereas in pairs 49–40, 59–50, 69–60,

and 19–70 no stresses (intertooth forces) occur.

The highest stress can be observed at the bottom

of tooth 19, where the activity of the splineway can

also be seen. In this area, the so-called ‘stress bridge’

is formed, and this bridge is the most strenuous spot

in the whole gear set. The reduced stress equals

sz 5 33.17 MPa and is lower than the maximum

acceptable stress sc 5 80 MPa assumed in section

4.3.

The state of the stresses can be explained by

analysing the state of deformations shown in

Fig. 7(b). In the figure it can be seen that the external

toothing gear deforms towards the active part of the

mesh, whereas the internal toothing gear deforms

towards the passive part of the mesh.

This causes mutual pressing of teeth 19–10, 29–20,

and 39–30 and the formation of stresses (forces)

between them. At the same time, teeth in pairs 49–40,

59–50, 69–60, and 19–70 disengage, making radial

clearances hr. The greatest clearances are formed

between teeth 49–40 and 59–50. The maximum value

of the clearance recorded in pair 49–40 in this

position equals hr 5 0.065 mm.

The state of stresses and deformations for the

central position of the external gear (a 5 0) is

presented in Figs 7(c) and (d) respectively. Fig-

ure 7(c) shows how, in the successive tooth pairs

19–10, 29–20, and 39–30, gradually diminishing stresses

(intertooth forces) occur, while between teeth 49–40,

59–50, 69–60, and 19–70 stresses are not induced. The

maximum stress does not occur on the contact of the

teeth, as previously, but on the left side of the

splineway at the point where torque M is imposed.

The stress is sz 5 26 MPa and is also lower than the

acceptable stress sc 5 80 MPa.

It can be concluded from analysis of Fig. 7(d) that,

under the influence of the pressure, both gears are

moved in opposite directions and deformed, the

effect of which is that, between teeth 49–40, 59–50, 69–

60, and 19–70, radial clearances hr are formed. The

greatest clearances are made between teeth 49–40

and 59–50. The value of clearance hr, generated as a

consequence of deformations, equals hr 5 0.082 mm

and is the highest at this position.

Finally, the state of stresses and deformations for

the final position of the external toothing gear

(a 5 zp=4) is presented in Figs 7(e) and (f) respec-

tively. In terms of quality, the obtained states of

stresses are similar to those obtained for the initial

position of the external gear (Figs 7(a) and (b)). In

tooth pairs 69–70, 19–10, and 29–20 gradually decreas-

ing stresses (forces) are induced, while between the

teeth 39–30, 49–40, 59–50, and 69–60 radial clearances hr

are formed.

The highest stress is induced on the left side of the

splineway, at the spot where torque M is imposed,

and amounts to 25 MPa. The maximum value of the

radial clearance in tooth pair 39–30 amounts to

hr 5 0.037 mm.

The stress and deformations of the cycloidal gear

set put under partial load, i.e. pressure p 5 4MPa

affecting the end face of the gear set, were also

studied (Fig. 5(a)). The achieved stress was

sz 5 4 MPa, which is of course lower than the

maximum acceptable stress sc 5 80 MPa. The max-

imum axial clearance generated by the moving gear

set is ha 5 0.018 mm, which is far lower than the

maximum radial clearance hr 5 0.082 mm.

Next, strength analysis of the gear set for the

successive load values determined by torque M and

pressure p assumed in the research was carried out.

It was observed that, for loads M 5 17.82 N m and

p 5 10 MPa, the reduced stress sz induced in the

gears shows values close to the acceptable for POM,

namely sc 5 80 MPa. Higher loads might cause

plastic deformation of the gears.

Deformation analysis was also conducted, and

characteristics of radial clearance hr and axial

clearance ha were developed, depending on the

pressure p working on the gear set. Figure 8 shows

that the clearance increase is directly proportional to

the increase in pressure p of the working fluid

working on the gear set.

For the boundary pressure working on the gear set,

p 5 10 MPa, the clearances are hr 5 0.0204 mm and

ha 5 0.046 mm. These values are higher than the

boundary values hr max 5 0.1 mm and ha max 5

0.03 mm assumed for the gerotor pumps in earlier

studies [10, 11]. Ultimately, on the basis of the

characteristics presented in Fig. 8, it can be stated

that the plastic gear set should not be loaded with a

pressure higher than 5 MPa.

Figure 9 compares theoretical analyses carried out

on the cycloidal gear set made of undeformable



material with computer analyses conducted by

means of FEM for the gear set made of deformable

POM.

Analysing the geometry of the gear set made of

undeformable material (Fig. 9(a)), it can be ascer-

tained that the gears keep their theoretical shape

based on equations (5), and the teeth of the external

gear contact the teeth of the internal gear.

The gear set made up of deformable plastic POM

(Fig. 9(b)) undergoes deformation. The external gear

teeth undergo deformation and are moved in the

direction of the active part of the meshing, while the

internal gear teeth are moved in the direction of the

passive part. This is illustrated in Fig. 9(b) by brown

arrows. As a result of deformation in the area of the

lower bridge on the passive side, intertooth radial

clearances hr are formed. The higher the pressure of

the working medium p, the greater is the strain and

the larger are the radial clearances hr (Fig. 8(a)). By

considering the shape of the gear set of deformable

material POM in the plane perpendicular to the end

face, it is ascertained that it is pressed and its width

decreases in relation to its initial value.

Analysing the kinematics of the gear set made of

undeformable material (Fig. 9(a)), it can be ascer-

tained that the contact line has the shape of a

conchoid that is divided by the axis of symmetry

O1O2 into the active part (continuous line) and the

passive part (broken line). Because of this, the tooth

pair engages at point C and disengages at point A1k.

In the case of gears made of POM (Fig. 9b), the

contact line is also a conchoid, but, as a result of

Fig. 8 Dependency diagram between the clearances in the cycloidal gear set and the workingpressure p imposing a load on the gears: (a) change diagram of the radial clearances(maximal and minimal) hr 5 f pð Þ; (b) change diagram of the axial clearances ha 5 f pð Þ

Fig. 9 Comparison of the geometry, kinematics, and force (of stresses) in the cycloidal gear set:(a) undeformable material; (b) deformable material POM



tooth deformation and the formation of intertooth

clearances hr, the pair engages at point C9 before

point C and disengages at point D9, which is situated

ca 1 pitch before point A1k.

The arc of the active part of the contact line and

the contact ratio e decrease. Using equation (6), the

dependency on the actual contact ratio is deter-

mined

er~z1{1

2ð8Þ

The values of the actual contact ratio er evaluated

from equation (8) are smaller than the theoretical values

of the contact ratio e evaluated from equation (7).

Analysis of the system of forces and loads of the

gear set made of undeformable material (Fig. 9(a))

established that the intermediate forces P’2{P’’2,

P’3{P’’3, and P’4{P’’4 have an influence only in the

active part of the meshing CA1k in its entire range.

The directions of action and the values of the

intermediate forces change, and therefore variable

stresses in these teeth should be expected. The

forces and stresses are difficult to determine analy-

tically.

In the gear set made of undeformable plastic

(Fig. 9(b)), the intertooth forces P’1{P’’1, P’2{P’’2, and

P’3{P’’3 are also active in the active part of the

meshing C9D9, which is, after all, smaller than part

CA1k because of the deformation. The directions of

the forces change in the course of the gear revolution.

The values of the forces (stresses) present in indivi-

dual tooth pairs 19–10, 29–20, and 39–30 decrease. It can

be stated that the ratio of the force values in the

following tooth pairs is P1 : P2 : P3 5 1:2/3:1/3.

6 CONCLUSIONS

The aims formulated at the beginning of the work

have been achieved.

Using the finite element method and the ABAQUS

package, the mechanism of generation of stresses

and deformations in the plastic cycloidal gear set of

gerotor pumps has been described. It has been

observed that there are two parts in the mesh: the

active part, where intertooth forces are induced

between the teeth of the internal and the external

gear, and the passive part, where intertooth forces do

not occur. The division line runs approximately

along axis O1O2 which connects the centres of gear

revolution. The intertooth forces and stresses have

various values. The most important forces and

stresses are induced in the first tooth pair, which

engage in the area of the mesh C pole. In successive

mating tooth pairs, the forces and stresses gradually

decrease (Fig. 7).

A characteristic of the deformations in the gear set

is that the external gear teeth are deformed and

moved towards the active part of the mesh, whereas

the internal gear teeth are deformed and moved

towards the passive part. Thus, intertooth radial

clearances hr are made, and their highest values are

achieved between the tooth pairs located in the zone

of pressure increase and the high-pressure zone

(Fig. 9).

A way of determining the loadability of the plastic

cycloidal gear set used in gerotor pumps has been

specified. It has been concluded that stress and

deformation analysis must be carried out together.

It must be ensured that reduced stresses sz in the

gears are not higher than the acceptable stresses,

which in this case are compressive stresses sc equal

to the plasticity limit Re of POM, namely sc 5 Re.

This is formulated as

sz¡sc~Re

At the same time, it has to be made sure that the

deformations of the teeth and gears, as well as the

radial clearances hr and axial clearances ha resulting

from them, are smaller than the extreme clearances

hr max and ha max assumed for the particular size of

gerotor pump. This is formulated as

hrvhr max, havha max

From the viewpoint of hydraulics, the condition of

deformations is more important than the condition

of stresses. This is so because, by keeping the

stresses lower than the acceptable (sz , sc), it is at

the same time possible to achieve deformations and

clearances greater than the extreme (hr . hr max,

ha . ha max), which causes flaws in the pumping

process, as well as lowering of the working pressure

and pump efficiency. In general, the gear deforma-

tions can be reduced in two ways:

(a) by applying a higher-strength plastic, i.e. a

plastic with higher acceptable stresses sc 5 Re;

(b) by modifying the cycloidal gear set design.

Finally, knowledge of the deformation mechanism

in plastic cycloidal gears, and resultant knowledge of

the profile and size of the internal clearances in the

pump, can be used for designing high-precision

hydraulic models of the pump, and for improving its



design and at the same time its technical para-

meters.

F Authors 2010

REFERENCES

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2 Sannikow, V. Planetarnye reduktory s vnecentroid-nym cevocnym zacepleniem. Masgiz, 1948 (Moscow).

3 Colbourne, J. R. The geometry of trochoid envel-opes and their application in rotary pumps.Mechanism and Mach. Theory J., 1974, 9, 421–435.

4 Stryczek, J. Koła zebate maszyn hydraulicznych.Oficyna Wydawnicza Politechniki Wrocławskiej,Wrocław, 2007, available from http://www.dbc.wroc.pl/dlibra/docmetadata?id520045&from5

&dirids51.5 Stryczek, J. Cycloidal gears in design of gear

pumps and motors. Arch. Mach. Des., 1990, (1–2),57–88.

6 Stryczek, J. Principles of the design of cycloidal gearsin hydraulic machinery. Arch. Mach. Des., 1990, (3),201–218.

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APPENDIX

Notation

g equidistance shift

h whole tooth depth

m modulus

M theoretical torque on the pump shaft

p 5 po 2 pi difference in output pressure po andinput pressure pi

pw circular pitch

q pump displacement per shaft

revolution

r radius of the arc of the internal gear

teeth

rb base circle radius

rw1, rw2 pitch circle radius

v 5 g=r correction coefficient (Fig. 3(c))

xe, ye epicycloid coordinates

xef , yef epicycloid family coordinates

xeke, yeke epicycloid equidistance coordinates

z1, z2 tooth number of the active gear andpassive gear respectively

a rotation angle of the internal gear

c eccentricity rotation angle

e contact ratio

g epicycloid angle

l tooth depth ratio

r generating circle radius



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Biernacki - Analysis of Stress and Deformation