Gears

1. 149
ADVANCED ENGINEERING
3(2009)2, ISSN 1846-5900
INFLUENCE OF TIP RELIEF PROFILE MODIFICATION
OF SPUR INVOLUTE GEARS ON STRESSES
Buljanović, K. & Obsieger, B.
Abstract: In this paper the linear tip relief profile modification has been observed. The amount
of tip relief profile modification depends on elastic gear tooth deflection that needs to be
compensated. The standard gear model without linear tip relief profile modification and also
modified one have been developed and analyzed using FEM analysis to compare gear tooth root
stress, influenced by mentioned profile modification.
Keywords: Spur involute gears, profile modification, tip relief, FEM
1 INTRODUCTION
During the meshing of gear pair, there appears so-called contact shock due to the
contact of two new teeth. This impact produces noise and amplifies inaccuracies in the
pitch and cause deformation of the teeth under load. In order to reduce the impact
influence, the involute in the tip region is modified through a relief curve. This process
is called profile modification at the tip and depends on elastic gear tooth deflection that
needs to be compensated [1].
2 LINEAR TIP RELIEF PROFILE MODIFICATION
Tip relief profile modification can be designed in few different ways. In this paper
linear tip relief modification has been considered. This type of correction is shown in
Fig. 1.
Fig. 1. Linear tip relief profile modification
Ca
da
dk
d
df
Δs(d)
Δs(d)
Linear tip relief:

2. 150
Tip relief profile modification is defined as thickness Δs(d) of the material
removed along the tooth flank with reference to the nominal involute profile. To define
changes in tooth thickness Δs(d), tooth tip diameter da, profile relief at tooth tip Ca and
diameter at the beginning of correction dk have to be calculated by (1).
k
a
a k
( )
d d
s d C
d d
−
Δ =
−
. (1)
3 TIP RELIEF CALCULATION FOR NOMINAL LOAD
Profile relief at tooth tip Ca has been obtained as a sum of elastic deflection of the spur
gear caused by distributed load and Hertz contact deformation. Diameter at the
beginning of correction dk has been found at characteristic point B of the tooth flank.
3.1 Elastic tooth deflection of the spur gear
Elastic tooth deflection caused by nominal transverse load in plane of action is shown
in Fig. 2.
Fig. 2. Elastic tooth deflection of the spur gear
Elastic tooth deflection caused by nominal transverse load has been calculated
using simplified expressions [2, 3]:
( )
( )P
2
bti
b1,2
1 C yF
A Be D
b E
ν
δ ⋅
−
= + + . (2)
Where:
( )1,75 1,68,1
1,05 153 xx
A e z− −−
= − + , (3)
( ) 1
0,63 7,35 0,924B x z−
= + − , (4)
( ) 1
1,28 2,88 3,68C x z−
= − + , (5)
( )n
1,06 0,638lnD m z= − + , (6)
δb
Fbti
αFY
yP
df
y
x

3. 151
( )p b b fp
P
n n
cosr ry
y
m m
α ω− −
= =
⎡ ⎤⎣ ⎦ , (7)
b
b
arctan
r
ρ
α =
⎛ ⎞
⎜ ⎟
⎝ ⎠
, (8)
b
p
b
cos
r
r
α
= , (9)
( )b
1,25
2
mz
r m x= − − , (10)
b
b
r
ρ
ω φ= − , (11)
( )n
n
4 tan
inv
2
x
z
π α
φ α
+
= + , (12)
n
f n
cos
2
m z
r α= , (13)
n n n
inv tanα α α= − . (14)
3.2 Deformation caused by Hertzian contact stress
Hertzian contact stress refers to the localized stresses that develop as two curved
surfaces come in contact and deform slightly under the imposed loads. This
deformation is dependent on the elasticity of the material in contact. Deformation
caused by Hertzian contact stress is shown in Fig. 3.
Fig. 3. Deformation caused by Hertzian contact stress
The expression that has been used for calculation of deformation caused by
Hertzian contact stress is:
2
bti n
H1,2
H
2 (1 )
1,27 0,781ln
F m
b E b
ν
δ
π
−
= +
⎛ ⎞
⎜ ⎟
⎝ ⎠
. (15)
ρ2
T2
ρ1
bH
bH
T1
2bH
δH2
δH1
δH1,2

4. 152
3.3 Relief at tooth tip
Profile modification should be calculated for each tooth flank of the mating gears. The
maximum values of the profile relief at the tooth tip of each gear are equal to the sum
of elastic tooth deflection and deformation caused by Hertzian contact stress [1, 2, 3],
so stands:
1,2 b1,2 H1,2a1,2
1
2
C δ δ δ== + . (16)
4 GEAR TOOTH ROOT STRESS
The mashed gears teeth are subjected to bending, compression and shear. One side of
the tooth root is strained by tension and the other side by compression. The stresses are
analyzed on the tooth side loaded in tension because the first cracks are expected to
appear there. ISO 6336 standard [4] specifies the fundamental formulae for bending
stress calculations for spur gears.
The critical cross-section of the tooth is determined by defining tangents on the
profile root fillet under the 30° angle to its axis symmetry, as shown in Fig. 4. Bending
stress σbn has been calculated depending on tangential force component Ft.
Fig. 4. Critical cross-section of tooth
Nominal tooth root stress for the ith
point of contact on tooth flank can be
determined by B-method [4].
t
F0-B F S β
n
F
Y Y Y
bm
σ = . (17)
The helix angle factor Yβ equals 1 for spur gears. Although the tooth form factor YF
and stress correction factor YS are defined in [4] at the critical cross-section, the MAX
method presumes the calculation of maximum stresses in tooth root. That means
critical position for the stress analysis is for the (YFYS)max.
5 GEAR PAIR MODEL
Gear pair with following geometrical parameters has been analyzed:
- number of teeth z1,2 = 58/67,
- profile shift correction x1,2 = 0 mm,
Fbti
αFY
yP
Fr
Ft
sFn
ρF
30° 30°

5. 153
- normal module mn = 12 mm,
- normal pressure angle αn = 20°,
- gear facewidth b1,2 = 330 mm,
- tool addendum factors ha
*
01,2 = 1,25mn,
- bottom clearance factors ca
*
1,2 = 0,25mn,
- tool tip radius factors ρa
*
01,2 = 0,25mn,
- transverse contact ratio εα = 1,79.
Material assigned to both gears has been steel with following material parameters:
- Modulus of elasticity E = 210000 N/mm2
,
- Poisson’s ratio ν = 0,3.
According to theoretical background for tip relief profile modification Ca and dk
have been calculated:
- relief at tooth tip Ca1,2= 0,061/0,061 mm,
- diameter at the beginning of correction dk1,2= 349,223/403,358 mm.
6 NUMERICAL ANALYSIS
Finite element nonlinear contact analysis was chosen for modelling and simulation of
gear pair in mesh.
The analysis has been carried out by using software package ANSYS 10.0. [5].
Newton-Rapson’s method [6] has been used for the convergence of the results for this
non-linear analysis.
The load has been applied by putting in contact pinions’ and wheels’ teeth and
applying the torsion moment on the pinion.
The gear models have been discretized by 2D finite elements that are adequate for
the contact analysis. The stress state has been considered to be a plane stress and the
friction has been neglected.
6.1 Geometrical model of gears
Modelling of entire gears in mesh would significantly increase the complexity and size
of geometric and numerical model which would, in turn, result in prolonged calculation
time. Thus, already in modelling phase certain simplifications have been made. Only
parts of the rims of the wheel and the pinion have been modelled (Fig. 5.), both with
two whole teeth and two teeth segments [7].
Rim thickness has been set to 100 mm that is approximately 5mn in order to avoid
the influence of too thin rim on the results.
6.2 Meshing of gear model
Three types of finite elements have been used for meshing of gear models.
Gear models have been divided in areas and they have been mashed with elements
PLANE183 [5]. These elements are defined by 8 nodes, having two degrees of
freedom at each node: translations in the nodal x and y directions and are well suited to
modelling irregular meshes. These elements may be used as plane elements (plane
stress, plane strain and generalized plane strain) or as axisymmetric elements. These
elements have plasticity, hyperelasticity, creep, stress stiffening, large deflection, and
large strain capabilities (Fig. 6.).

6. 154
Fig. 5. Geometry of gears in mesh
Fig. 6. PLANE183 finite element
Due to contact problem analysis the contact elements usage have been necessary.
Parts of teeth flanks in contact have been meshed with contact elements TARGE169
and CONTA172 [5]. These parabolic elements (Fig. 7.) with two nodes on end and one
midside node each with two degrees of freedom (translations in the nodal x and y
directions) are very suitable for analysis of problems with states of plane stress and
plane strain. As they can’t be used as standalone elements, they must be overlaid over
existing 2D solid elements – in this case PLANE183 Contact occurs when the element
surface (CONTA172) penetrates one of the target segment elements (TARGE169) on a
specified target surface.
Fig. 7. TARGE169 and CONTA172 contact finite elements
x
y
TARGE169
CONTA172
x
y
PINION
WHEEL

7. 155
In order to further decrease calculation time, finite element mesh has been adapted
as well. Areas around contacting surfaces have been meshed with larger density of
finite elements mesh because these areas are crucial for results accuracy. Coarser finite
elements have been used in areas of less significance such as gear rim and parts of gear
teeth that are not in the contact.
Meshed gear model is shown in Fig. 8.
Fig. 8. Meshed gear model
6.3 Boundary conditions
The gears have been loaded by positioning mating teeth i.e. their flanks into contact
due to inadequacy of other loading models [8]. Namely, concentrated force couldn’t be
applied due to high local deformation of the material which takes place near point of
force action and significant influence on the results.
After positioning the mating teeth in desired position the boundary conditions have
been applied.
The wheels’ nodes placed on inner rim radius and on the ends of rim have been
constrained in global Cartesian coordinate system (x, y) in all directions i.e. the
movements in directions of both axis have been disabled (Δx=0, Δy=0). The pinions’
nodes placed on inner rim radius have been constrained in global cylindrical coordinate
system (r, φ) in a way: Δr=0. Centre of both mentioned coordinate systems have been
centre of rotation of the pinion.
Rotation of the pinions’ nodes placed on inner rim radius around the centre of the
global cylindrical coordinate system has been enabled. Angle of rotation Δφ of these
nodes has been increasing in stepwise fashion until it resulted with momentum which
has been higher then nominal torque at the pinion. Final value Δφ has been determined
from two closest rotation steps by the interpolation method.
Target surface
(TARGE169)
Contact surface
(CONTA172)
PLANE183
PLANE183

8. 156
7 RESULTS
Gear tooth root stresses along the path of contact in standard model have been
calculated and then compared to the stresses in modified one to present the influence of
determined profile modification on gear tooth root stresses. The results of FEM
analysis for pinion and wheel are shown in Fig. 9.
Fig. 9. Tooth root stress for pinion (σF01) and wheel (σF02) for the i th
point of contact
For standard unmodified model stands that when double contact exceeds into
single contact (point B on path of contact) and reverse (point D on path of contact) gear

9. 157
tooth root stress changes rapidly i.e., the wheel speed changes at two shifting points,
and causes the additional dynamic load as visible in Fig. 9.
Instead of the first contact between meshing gears with linear tip relief profile
modification on the pinion tooth tip (point A on path of contact), it occurs lower on
tooth flank (point A’ on path of contact). The same situation appears at point E. Gear
tooth root stress increment between points A’ and B’ (double contact) and decrement
between points D’ and E’ (double contact) are almost linear. There aren’t rapid stress
changes at the shifting points so gears run smoother then standard gear pair without
additional dynamic load.
The analysis also showed that the highest values of the tooth root stresses appear in
point B on path of contact for standard and in point B’ for modified model.
8 CONCLUSION
The standard gear model and also modified one have been developed and analyzed by
using finite element method. Nonlinear analysis has been used because it gives the
most accurate results. Numerical calculation methods, such as finite element method,
provides easier stress calculations on teeth with no limits in gears’ geometrical
specifications and also allows determination of stress distribution on whole path of
contact.
Obtained results show that in case of standard unmodified model when double
contact exceeds into single contact and reverse gear tooth root stress changes rapidly
i.e., the wheel speed changes at two shifting points, and causes the additional dynamic
load, unlike, in case of modified model wheel speed don’t change rapidly so there
aren’t rapid stress changes at the shifting points. Also, instead of the first contact
between meshing gears with linear tip relief profile modification on the wheel tooth tip
it occurs lower on tooth flank. The same situation appears at the end of contact
between meshing gears with linear tip relief profile modification. This phenomenon
results in a way that gear tooth root stress increment and decrement on double contact
zones are almost linear so gear pair with linear tip relief profile modification runs
smoother then standard gear pair.
NOTATION
A,B,C,D auxiliary factors for calculating tooth deflection, -
A,A’,B,B’,D,D’,E,E’ characteristic points on path of contact, -
b facewidth, mm
bH half of the Hertzian contact width between the meshing teeth, mm
c* bottom clearance factor, -
da tip diameter, mm
dk diameter at the beginning of correction, mm
df root diameter, mm
Ca profile relief at tooth tip, mm
E modulus of elasticity, N/mm2
Fbt transverse load in plane of action (base tangent plane), N
Fr radial force, N
Ft tangential force, N
*
0ah tool addendum factor, -

10. 158
mn normal module, mm
rb base radius, mm
rP distance between point of application of the force and centre of gear, mm
x addendum modification coefficient, -
Φ auxiliary angle, rad
YF tooth form factor, -
yP bending arm, mm
YS stress correction factor, -
Yβ helix angle factor, -
z number of teeth, -
αb auxiliary angle, °
αFY angle of action of nominal transverse load, °
αn normal pressure angle, °
Δs removed material, mm
δ deflection, mm
δb bending deflection, mm
δH Hertzian contact deformation, mm
εα transverse contact ratio, -
ν Poisson’s ratio, -
ρ roll distance, mm
*
a0ρ tip radius of the tool factor, -
σF0 nominal tooth root stress, N/mm2
ωb auxiliary angle, °
Indexes 1 pinion
2 wheel
i i th
point of contact
References:
[1] Obsieger, J. (1989). Some considerations to the choice of profile correction of involute
gears, STROJARSTVO 31(1989)1, pp. 17-23, ISBN 0562-1887
[2] Terauchi, Y. & Nagamura, J. (1981). On tooth deflection calculation and profile
modification of spur gear teeth, Intern. Symp. Gearing and Power Transmission, Proc.
Vol. II, pp. C-27 (159-164), Tokyo, 1981
[3] Franulović, M. (2003.) Influence of base pitch deviation on stresses in involute gearing,
Masters Thesis, University or Rijeka, Faculty of Engineering, Rijeka, 2003
[4] ISO 6336 (1996.), Calculation of load capacity of spur and helical gears, International
standard, 1996
Part 1: Basic principles, introduction and general influence factors
Part 2: Calculation of surface durability (pitting)
Part 3: Calculation of tooth bending strength
[5] ANSYS Structual analysis Guide // Canonsburg: ANSYS Inc. 2004
[6] Zienkewich, O.C. (1997). The Finite Element Method, Mc Graw-Hill, London, 1977
[7] Basan, R.; Franulović, M. & Križan, B. (2008.). Numerical model and procedure for
determination of stresses in spur gears teeth flanks, Proceedings of XII International
conference on mechanical engineering, Starek, L. & Hučko, B. (Ed.), Bratislava, 2008
[8] Franulović, M.; Križan, B. & Basan, R. (2006.) Calculation methods of load carrying
capacity of spur gears, Advanced Engineering Design AED 2006, Musilek, L. (Ed.), Prag,
2006
Received: 2009-07-15