INFLUENCE OF A NON-STANDARD GEOMETRY ... - Dunarea de Jos

THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
11
INFLUENCE OF A NON-STANDARD GEOMETRY
OF PLASTIC GEAR ON SLIDING VELOCITIES
Laurenţia ANDREI 1 , Douglas WALTON 2 ,
Gabriel ANDREI 1 , Elena MEREUŢĂ 1
1 The University “Dunarea de Jos” of Galati, 2 The University of Birmingham, UK
laurencia.andrei@ugal.ro
ABSTRACT
In this paper the sliding velocities of plastic non-conventional spur gears is
analysed. There are two peculiarities to be taken into account when metal gear
practice is applied: the variable tooth height along the gear face width and the
special meshing conditions of plastic gears. The study is based on the curved face
width spur gear solid modelling which enable gear tooth geometry to be produced.
A primary analysis shows the effect of the suggested spur gear design on the sliding
velocity. Different curvatures and heights of the non-standard gear teeth show the
influence of the tooth geometry on sliding velocity variation, a particular criteria
for a further study on the optimisation of the gear geometry.
KEYWORDS: non-standard curved face width spur gear, plastic gear, sliding
velocity
1. INTRODUCTION
Curved face width spur gears, with variable
tooth height along the gear face width [1] are
especially designed for plastic gears in order to
increase their transmissible power level. The
advantages of these gears, compared to standard
spur gears are:
- higher contact ratio for a given size of gear and
number of teeth;
- lower bending and contact stresses;
- better meshing in plane misalignment conditions;
- no axial forces as are inherent in helical gears and
- enhanced lubrication under operating conditions.
Against these advantages there are limitations:
- the difficulty in gear design due to the complex
tooth geometry compared to conventional designs;
- the difficulty in gear train mounting;
- the sensitivity to center distance variations;
- the manufacture is limited to cutting processes, as
designing a moulding die is almost impossible.
Experimental tests carried out by the authors on
the running curved face width spur gears, with
modified geometry, pointed out the peculiar thermal
behaviour of the non-standard gears [2]. With these
non-standard gears, running at high loads where
conventional spur gears would fail, the gear surface
temperature increased at an extremely high rate,
recommending lubrication. It was obvious that the
manufacturing process, leading to a rough flank
surface, was the main cause for the high induced
temperatures. Other influences on the friction forces
generated should also be considered.
It is well known that wear is the predominant
mode of failure for plastic gears, generally running
with no lubrication. Over the traditional wear caused
by the nonconformal tooth contact with both rolling
and sliding components, plastic gears exhibit wear as
a result of high friction and associated high
temperatures [8]. Laws of Sliding Friction,
considering the dependence of friction force on the
normal load and its independence on both the
apparent area of contact and sliding velocity, are not
obeyed by plastic materials. It was shown [12] that,
once sliding is established in polymer/polymer
contacts, the dynamic coefficient of friction is
dependent on sliding velocity and it is higher than the
static coefficient of friction.
This paper analyses the influence of the
modified tooth geometry of the curved face width
spur gears on the sliding velocity, as one of the main
causes of frictional losses that increase the gear
temperature above that of standard spur gears. The
analysis also considers the special meshing
conditions of plastic gears: the low value of the
material’s Young’s modulus results in high deflection
of the meshing gear teeth and influences the sliding
velocity due to changes in the point of contact.

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THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
2. CURVED FACE WIDTH
SPUR GEARS WITH MODIFIED
GEOMETRY
The curved face width gear geometry is produced
using both the traditional conjugate surface generation
theory [3] and solid modelling techniques [4]. The
simulation of the non-standard gear generation, using
solid modelling, was based on the following
assumptions:
- the virtual gear blank is a cylindrical primitive that
is rotated about its axis and is provided with
translation motion, tangential to the base circle of the
gear, in order to get the rolling motion required for the
involute tooth form generation;
- the virtual tools, for the gear tooth concave and
convex flank generations, are conical solid primitives
that substitute the rotational motion of the imaginary
rack-cutter flank, with zero pressure angle, about its
inclined axis. The modified gear tooth geometry is
defined by the tool axis inclination β, and by the
radius of the “generating circle”.
Fig. 1. Tri-dimensional representation of curved face
width spur gear teeth
Figure 1 shows the representation of three
generated teeth as solid model. The tooth geometry,
numerically identified in several sections along the
gear face width, is used for further investigations.
3. THE THEORETICAL SLIDING
VELOCITY
The theory of gearing gives the following
relation for the sliding velocity in gear pairs, with the
standard involute tooth flank geometry (fig. 2):
vs
= ( ω1
+ ω2
) ⋅(
PM )
(1)
where ω 1 and ω 2 are the angular velocities for pinion
and gear respectively and (PM) is the distance
between the pitch point P and the contact point M.
If equation (1) is re-expressed as a function of
θ 1 , the pinion roll angle - the independent variable for
the computer simulation, it becomes:
R1
⋅ sin θ1
vs
= ( ω1
+ ω2
) ⋅
(2)
cos( α w + θ1
)
where R 1 is the pinion pitch circle radius and α w - the
pressure angle.
pinion
Rb2
ω 2
gear
P M
v s
ω 1
θ 1f
θ
1
θ 1a
R b1
Fig. 2. Sliding velocity for a pair of teeth with the
standard involute for tooth flank
Due to the modified geometry of the nonstandard
gears, the sliding velocity variation is
modified along the gear face width. To analyse its
variation it is assumed that the curved face width gear
is made up by a stack of elementary standard spur
gear sections. The calculation is developed in the
particular case of a gear train with modulus 2 mm, 30
teeth and 24 mm facewidth. Here two identical gears
are meshed together giving a gear ratio equal to one.
The gear geometry is defined by a tool axis
inclination β = 5° and a radius R g = 24 mm. Each
elementary gear section, with 2 mm facewidth,
positioned by H i - the distance from the centre of the
face width, has a particular geometry (table 1) whose
parameters are identified using the gear solid model.
Also, θ 1 varies, in every section H i , from θ 1a to θ 1f ,
given by:
2 2
A sin α wi − Ra
− Rbi
θ1a
= α wi − arctg
Rbi
Rbi
θ1 f = arccos − α wi
Ra
(3)
where A is the center distance and R a – the radius of
the addendum circle, constant for all the elementary
spur gear sections.
Section H i
[mm]
R f
Table 1. Gear section geometry
R b α w
[mm] [°]
[mm]
0 27.5 28.19 20
2 27.51 28.20 19.95
4 27.53 28.22 19.84
6 27.56 28.27 19.55
8 27.64 28.34 19.15
10 27.68 28.46 18.44
12 27.77 28.65 17.25
Figures 3 – 6 show the variation of the sliding
velocity for several sections along the gear face
width. Section 0 (fig. 3) has the standard involute for
the gear tooth flank geometry leading to maximum

THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
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sliding velocity, as known, at the first and last points
of contact. It may be seen that in other sections (fig.
4-6) the maximum velocity is gradually increased,
reflecting the increase of the length of the line of
action due to the increase in gear base circle radius.
Section 3, corresponding to the end of the gear
face width, with a base circle radius with 1.6% higher
than the standard value of the considered geometry,
has an increase of 10% for the maximum sliding
velocity. The sliding velocity is calculated for a speed
of 1000 revs/min.
in section 2 (H 2 = 8 mm)
Fig. 6. The sliding velocity variation
in section 3 (H 3 = 12 mm)
Fig. 3. The sliding velocity variation
in section 0 (H 0 = 0 mm)
Fig. 4. The sliding velocity variation
in section 1 (H 1 = 4 mm)
Fig. 5. The sliding velocity variation
Block [5] was the first who determined the
influence of relative sliding velocity on the flash
temperature between sliding gear tooth surfaces:
0.5
ϕ f = F(
k, ρ,c,w,b,dH ) ⋅v (4)
s
where F expresses the dependence of flash
temperature (ϕ f ) on the thermal conductivity (k),
density (ρ), specific heat (c) of the surfaces, Hertzian
contact length (w), gear facewidth (b) and the
instantaneous energy loss due to friction (dH).
The analysis on curve face width gear geometry
related to the sliding velocity variation, for the above
mentioned dimensional parameters, shows that:
- as the gear tooth height gradually decreases; the
maximum reduction is about 6%;
- the maximum sliding velocity gradually increases;
at the end of the gear face width its value is higher
with 10% than the standard value recorded for the
gear centre section;
- the increase in sliding velocity leads to an increase
of 4.8% for the gear’s flash temperature, one of the
components of the maximum gear surface
temperature [8].
4. MESHING CONDITIONS OF
LOADED PLASTIC GEARS
Polymer gear teeth, with a relatively low
Young’s modulus, will deflect elastically under load,
the deflections being larger than those experienced by
metal gears. As a result of the applied load, a
premature engagement takes place, as well as a
delayed disengagement. Hence, the contact point for
a plastic gear train is differently positioned compared
to the ideal rigid gear pair and is influenced by the
amount of tooth deflection.
Gear tooth deflections were first calculated by
Timoshenko and Bread [11] and the analysis has been
refined since the original work by many workers.
There are three kinds of deflection to be considered:
deflection due to the local surface Hertzian
compression and bending and shear deflections.

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THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
4.1. Influence of Hertzian deflection on
sliding velocity
The Hertzian compression, as an effect of
tooth’s load, modifies the flank geometry, changing
the standard involute into a straight line [10].
Following Weber’s analysis [13], the real contact
point extends down to the pressure line by an amount
2δ H :
M t
2δ H =
(5)
2 ⋅ Rb
⋅ k H
where M t is the transmitted torque and k H – the
stiffness of Hertzian contact. According to Hertzian
law and assuming that the gear tooth surfaces are two
cylinder of diameters d 1 ’ and d 2 ’ respectively, the
stiffness is given by:
π⋅ E ⋅b
k H =
(6)
' '
2 2 2d1
2d 2
4(1 − ν ) ⋅(
+ ln + ln )
3 s s
where E and ν are the Young’s modulus and
Poisson’s ratio, s – the width of rectangular contact
area. From the results derived by Yang [14] the
stiffness k H is practically a constant along the entire
line of action and it can be calculated by the relation:
π⋅ E ⋅b
k H =
2
(6’)
4(1 − ν )
So, the deflection due to Hertzian contact is
constant and independent to the position of contact
point. It increases the sliding velocity by an amount:
H
∆ vs = ( ω1
+ ω2
) ⋅ 2δ
(7)
H
4.2. Influence of bending deflection on
sliding velocity
Bending deflection is calculated assuming that
the tooth is a beam based on a rigid foundation. For
the standard involute flank geometry, the contact
force, exerted during gear meshing, is along the line
of action. In the case of plastic gears, the variation of
tooth profile determines contacts in points that do not
belong to the theoretical line of action and, therefore,
the actions of the contact force are in unknown
directions. But, since the contact points are in the
vicinity of pressure line and, furthermore, they have
the same sliding velocity as the point being on the
same flank and on the line of action, the study on
bending deflection can be developed, with small
errors, considering standard conditions.
The displacement of the point where the load is
applied has a bending and a shear component. Using
Walton’s assumptions and calculation [9], the
deflection due to both bending and shear is expressed
as follow:
- bending deflection:
32M t 3
b =
⋅ h
3 3 j
π ⋅b
⋅ E ⋅ m ⋅ Rb
δ (8)
where h j is the current tooth height, measured from
the deddendum circle to the contact point. The
increase in sliding velocity, due to bending
deflection, varies along the gear tooth flank with:
b
∆ vs = ( ω1
+ ω2
) ⋅δb
(9)
- shear deflection:
2M t
δ sh =
π⋅b
⋅G
⋅ m ⋅ R
⋅ h j
(10)
where G is the shear modulus. The increase in sliding
velocity, due to shear deflection, varies along the gear
tooth flank with:
sh
∆ v = ( ω + ω ⋅δ
(11)
s 1 2 )
5. THE REAL SLIDING VELOCITY
Taking into account the total deflection of the
plastic gear tooth, the real sliding velocity should be
calculated as:
r
H b sh
vs
= vs
+ ∆vs
+ ∆vs
+ ∆v
(12)
s
To analyse both the influence of the nonstandard
geometry and the influence of the low
stiffness of the plastic gear tooth on the sliding
velocity, the curved face width gear with the
geometrical parameters mentioned in paragraph 3, is
treated as a series of n standard spur gears. It is
assumed that the transmitted torque is equally
distributed on the elementary spur gears, with
specific geometry. The gear is made by ERTALON
66SA with E = 3450MPa, G = 1320 MPa and ν = 0.3.
Using equations 5 -11 it was found that:
- the Hertzian deflection is not significant and it can
be ignored. Along the gear face width, the ideal
sliding velocity is increased by 0.12 ÷ 0.1%; the
maximum influence is recorded at the gear’s centre,
where the gear tooth flank has the standard involute,
and is decreasing towards the gear face width ends
due to the increase in base circle radius;
- the shear deflection is also not important on the
sliding velocity. The maximum increase of 0.86% is
recorded at the gear centre and is decreasing along
the gear face width to a minimum value of 0.73% at
the gear end sections;
- the bending deflection is one of the components of
gear tooth deflection that also decreases along the
gear face width, a consequence of the higher gear
tooth rigidity. The sliding velocity varies by 2.5÷
1.8% from its ideal value due to bending deflection.
Figure 7 shows the variation of the maximum
sliding velocity (at speed of 1000 revs/min) at points
the gears come out of mesh, where the bending
deflection is maximum. It can be seen that the real
sliding velocity, compared to its ideal value, varies
from 3.5% at the gear half face width to 2.67% at the
gear ends.
b
sh

THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
15
of action length. At 16 mm radius of the tooth
curvature, the increase of the sliding velocity is about
26% from its standard value, decreasing to a 7%
variation for R g = 28 mm. A further increase in
generating circle radius gets the non-standard gear
close to the standard spur gear where the maximum
sliding velocity tends to its standard value.
Fig. 7. The real sliding velocity variation
Table 2
Geometry of the non-standard gears (β = 5°)
R g [mm] 16 20 24 28
R b [mm] 29.17 28.82 28.65 28.55
R f [mm] 27.94 27.83 27.77 27.7
α w [°] 13.5 16.12 17.25 17.9
Table 3
Geometry of the non-standard gears (R b = 28mm)
β [°] 2 5 8
R b [mm] 28.41 28.55 28.69
R f [mm] 27.65 27.7 27.78
α w [°] 18.74 17.9 17
A further study is developed considering the
influence of the curved face width gear geometry on
the real sliding velocity. The non-standard gear
geometry is determined by two independent parameters:
the radius of the generating circle (R g ) and the
tool axis inclination (β). It should be specified that
the variation of these parameters is limited by both
the gear face width and by the variation of the tooth
height which may reduce the gear backlash. The solid
modelling enabled the gear geometries to be
produced and sections along the gear face width to be
numerically described (tables 2-3).
As a first step, the tool axis inclination is
defined by β = 5° and the radius of the generating
radius is incrementally increased from 16 to 28 mm.
Table 1 presents the gear geometry at the end of the
gear face width where the gear generation provides
the maximum modification of the gear standard
parameters.
Figure 8 shows the variations of both ideal and
real sliding velocities, at a speed of 1000 revs/min,
along half of the gear face width, for several nonstandard
gears, with different tooth curvature along
the gear face width.
It can be seen that the smaller the generating
circle radius is, the higher is the increase in the ideal
sliding velocity at the gear face width ends. This is
explained by the low value of the pressure angle and
the increase in base circle radius, modifying the line
Fig. 8. Sliding velocity versus tooth curvature
(β = 5°)
Fig. 9. Sliding velocity versus tool axis inclination
(R g = 28 mm)
As regards to the real sliding velocity, this is
also decreasing with an increase in the generating
circle radius, with a lower rate than the ideal velocity.
The maximum variation for the real sliding velocity,
recorded at the end of the gear face width ends is
about 24% from the value calculated at the gear
centre, and decreases to around 6.5% for R g = 28 mm.
If the generating circle radius is increased, the gear
tooth is more flexible and leads to an increase in
velocity variation. Compared to its ideal counterpart,
the real sliding velocity is higher by 2% for R g = 16
mm and by 3% for R g = 28 mm, respectively. This

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THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
means that the tooth deflection has a reduced
influence on gear sliding velocity.
Table 3 presents the curved face width gear
geometry, at the gear face width end, defined by R g =
28 mm and several tool axis inclinations. Figure 9
shows the variations of the ideal and real sliding
velocity along half of the gear face width. It can be
seen that the ideal velocity varies, along the gear face
width, by 4.2% (β = 2°) to 10.7% (β = 8°). The real
sliding velocity variation is about 3.8% (β = 2°) and
9.8% (β = 8°).
The data show that the plastic non-standard gear
sliding velocity is affected by the independent
parameters that control gear geometry as follows:
- an increase in generating circle radius reduces both
the ideal and the real sliding velocity variation;
- an increase in tool axis inclination increases the
sliding velocity variation.
The variable tooth stiffness reduces the variation
of the real sliding velocity compared to the ideal
velocity, due to the decrease in tooth deflection from
the gear centre to its end sections.
6. CONCLUSIONS
An analysis on sliding velocity is developed
for the case of a plastic curved face width gear train,
with modified geometry, as the sliding velocity has a
direct influence on the gear flash temperature
generated. Experimental investigations on the thermal
behaviour of non-standard curved face width gears
showed that high temperatures were induced.
There are two peculiarities of these gears,
which required investigation as they influence the
variation in sliding velocity:
1. the gear geometry that is dependent on the gear
generating process, by the radius of the tool
“generating point”, that is provided with rotational
motion, as well as by the tool inclination axis;
2. the lower material stiffness that affects gear
meshing conditions;
The ideal sliding velocity is calculated using the
traditional practice for metal gears and considers the
influence of the gear geometry on its variation. As a
consequence of the base circle radius and pressure
angle variations, along the gear face width, the line of
action has an increased length and the maximum
sliding velocity is higher towards the gear face width
end sections.
The real sliding velocity considers the influence
of both the gear geometry and tooth deflection. It was
found out that the sliding velocity is mainly
influenced by the gear geometry. The gear tooth
deflections do not have a significant influence on the
sliding velocity - they reduced its variation compared
to the ideal calculation.
REFERENCES
1. Andrei L., 2002, Study of plastic curved face-width spur gear
generation and behaviour, PhD Thesis, University “Dunarea de
Jos” of Galati.
2. Andrei L., Epureanu Al., Andrei G., Walton D., 2004,
Investigation of the Thermal Behaviour of Non-metallic Curved-
Face-Width Spur Gears, Tribotest Journal 10-4, Leaf Coppin, p.
299-310.
3. Andrei L., Andrei G., Epureanu Al., Oancea N., Walton D.,
2002, Numerical simulation and generation of curved face width
gears, Int. J. Machine Tools Manufact., 42, p. 1-6.
4. Andrei L., Andrei G., Mereuta E., 2002, Simulation of
curved-face-width spur gear generation and mesh using the solid
modelling method, Proc. 10 th Int. Conf. on Geometry and
Graphics, Kiev, Ucraine, Vol.1, p. 245-9.
5. Block H., 1933, Les Temperatures de Surface dans les
Conditions de fraissage sous Pressions Extreme, Congr.
Mondialopetr. Le 2 Congr., Paris, 47—486.
6. Davoli P., Gorla C., 1996, Meshing Conditions of Loaded
Plastic Gears: Numerical Analysis and Experimental tests, Berichte
no 1230, p. 383-396.
7. Goodwin A. J., 1994, The wear of dry running steel gears,
Mphil thesis, The University of Birmingham, UK.
8. Hooke C.J., Mao K., et al, 1992, Measurement and Prediction
of the Surface Temperature in Polymer Gears and Its Relationship
to Gear Wear, J. Tribology (trans. ASME)-12, p. 1-6.
9. Meuleman P. K., Walton D., Weale D. J., Driessen I., 2000,
Minimisation of transmission errors in highly loaded plastic gears.
Research done under the European Industrial and Materials
Programme (Brite EuRam III), Mid-term report, Project No BE97-
4752.
10. Nikas G.K., 1996, Load Sharing and Profile Modification of
Spur Gear Teeth in the General Case of any Flank Geometry,
Berichte no 1230, p. 923-935.
11. Timoshenko S., Braud R.V., 1926, Strength of gear teeth is
greatly affected by fillet radius, Automotive Industries, p. 138-142.
12. Walton D., Cropper A.B. Weale D.J., Meuleman P. K.,
2002, The efficiency and friction of plastic cylindrical gears: Part 1
– influence of materials, Proc. IMechE, Part J: Journal of
Engineering Tribology, vol. 216, p. 75-92.
13. Weber C., 1949, The deformation of loaded gears and the
effect of their load carrying capacity, British Scientific and
Industrial Research, Sponsored Research (Germany), Report No 3.
14. Yang D.C., Lin I.Y., 1987, Hertzian Damping, Tooth Friction
and Bending Elasticity in Gear Impact Dynamics, J. Mechanical
Transmission and Automation in Design, vol. 109.