This document summarizes a study on the simulation and analysis of transmission error in helical non-circular gear models. The study used ANSYS software to analyze stresses and transmission errors in helical elliptical gear models with different numbers of teeth. Von-Mises stresses were calculated using ANSYS and Lewis equations, showing good agreement between the two methods. Transmission error was analyzed for gear models with 18 and 19 teeth. Total deformation, stresses, and principal stresses were obtained from the ANSYS simulations. The results indicate that finite element analysis can accurately simulate contact and bending stresses in helical elliptical gears.

1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
128
SIMULATION AND ANALYSIS OF TRANSMISSION
ERROR IN HELICAL NON CIRCULAR GEAR MODEL
Dr.SABAH KHAN
Department of Mechanical Engineering, Faculty of Engineering and Technology,
Jamia Millia Islamia, New Delhi-25
ABSTRACT
Gears are one of the most critical components in mechanical power transmission systems.
Transmission error is considered to be one of the main contributors to noise and vibration in a gear
set. Transmission error measurement has become popular as an area of research. To estimate
transmission error in a gear system, the characteristics of involute gears were analyzed using
ANSYS. The contact stresses were examined using 2-D FEM models. The bending stresses in the
tooth root were examined using a 3-D FEM model.
1. INTRODUCTION
Non Circular gears are used by manufacturers of automatic machines in all applications
where variable speed rotary motion is required from a constant speed input. Typical applications
include machines in the printing and textile industry where a controlled in- feed is required with a
rapid return. Elliptical gears are commonly used in packaging and conveyor applications etc (13).
Elliptical gears can be manufactured in a large range of materials, including aluminum, steel, bronze
and stainless steel and some company design and manufacture non-standard gears for special
applications. Generation of planar and helical elliptical gears by application of rack-cutter, hob, and
shaper (1-2).
The most important criteria for proper maintenance and operation of a gear pair is the level of
their noise and bearing contact. A main source of gear noise is transmission errors. Traditional
methods of gear synthesis that provide conjugate gear tooth surfaces with zero transmission errors
and an instantaneous line of contact are not acceptable for the gearing industry due to errors of
manufacturing and assembly. Taking into account such errors, the bearing contact is shifted to the
edge of the tooth, and transmission errors of an unfavorable shape occur. The new trend of gear
synthesis is to localize bearing contact and absorb the transmission errors.
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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A pinion and gear with perfect involute profiles running with no torque should theoretically
run with zero transmission error. However, when these same gears transmit torque, the teeth deflect
causing the gear to “lag” behind the pinion. The difference in angular position between where the
gear is in relation to the pinion as opposed to where it should theoretically be is what is termed
transmission error. This error in position occurs during each mesh cycle, or when each new tooth
pair comes in and goes out of contact. This constant variation in the rotational position of the gears is
normally converted to linear units (microinches or microns) so that gears of varying sizes can be
compared. Even though the error is relatively small, it is these slight variations which can cause
noise and vibration. Under normal operating conditions, the main source of vibration excitation is
from the periodic changes in tooth stiffness due to non-uniform load distributions from single to
double contact zone in each meshing cycle of the mating teeth. This indicates that the variation in
mesh stiffness can produce considerable vibration and dynamic loading of gear teeth in mesh.
Predicting the static transmission error (TE) is a necessary condition for the reduction of the
noise radiated from the gearbox. To obtain TE the contact problem was seldom included because the
nonlinear problem made the model too complicated. This paper deals with estimation of static
transmission error including the contact problem and the mesh stiffness variations of spur gears. For
this purpose, an FEA numerical modelling system has been developed. For spur gears a two
dimensional model can be used instead of a three dimensional model to reduce the total number of
elements and the total number of nodes in order to save computer memory. This is based on a two
dimensional finite element analysis of tooth deflections (4-5). Two models were adopted to obtain a
more accurate static transmission error, for a set of successive positions of the driving gear and
driven gear. Two different models of gear pairs have been generated to analyze the effects of gear
body deformation and the interactions between adjacent loaded teeth.
It is generally accepted that the noise generated by a pair of gears is mainly related to the gear
transmission error. The main source of apparent excitation in gearboxes is created by the meshing
process. Researches usually assume that transmission error and the variation in the gear mesh
stiffness are responsible for the noise radiated by the gearbox (9-10).
The term transmission error is used to describe the difference between the theoretical and
actual relative angular rotations between a pinion and a gear. Its characteristics depend on the
instantaneous positions of meshing tooth pairs. Under load at low speeds (static transmission error)
these situations result from the tooth deflections and manufacturing errors. In service, the
transmission error (7) is mainly caused by:
• Elastic deformation: local contact deformation from each meshing tooth pair and the
deflections of the teeth because of bending and shearing due to transmitted load.
• Tooth geometry errors: including profile, spacing and run out errors from the
manufacturing process;
• Imperfect mounting: geometric errors in alignment, which may be introduced by static and
dynamic elastic deflections in the supporting bearings and shafts.
2. CALCULATION OF STRESSES USING LEWIS EQUATION
The stresses are calculated by Lewis formula using standards of AGMA. The various
coefficients, such as the dynamic factor, are set at 1.1( 11-12). Here analysis of gears with different
numbers of teeth is carried out. First, the number of teeth is 16. The meshing helical gear has pitch
radii of 60.5mm and a pressure angle of 20 degree. The face width =10 mm, transmitted load is 1020
N.

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
130
Then,
If the number of teeth is changed to 19 and the load applied is 1050 N
Then,
The above calculations of the Von-Mises stresses on the teeth were carried out in order to
know if they match the results from ANSYS. The results are shown in table 4.1. This result match in
the present study, effective methods to estimate the root bending stress by ANSYS software are
proposed. The accuracy of the present method for the bending stresses, 3-D models were built in this
chapter. The results with the different numbers of teeth and different load conditions were used in
comparison.
Table 2.1 Von –Mises Stress of 3-D Model
Number of Teeth Stress 3D (ANSYS) Stress (LEWIS EQ.) Difference in Stress
16 21 24.5 3.5
19 28.45 28.33 0.12
It can be observed that the difference between the ANSYS stress value and that calculated by
Lewis equation is much less than 5% and thus it can be analyzed that the analysis So the analysis of
helical elliptical gear using ANSYS software is good enough for stress analysis.
3. ANALYSIS OF TRANSMISSION ERROR OF HELICAL ELLIPTICAL MODEL
Under normal operating conditions, the main source of vibration excitation is from the
periodic changes in tooth stiffness due to non-uniform load distributions from single to double
contact zone in each meshing cycle of the mating teeth. This indicates that the variation in mesh
stiffness can produce considerable vibration and dynamic loading of gear teeth in mesh.
11 6
0 . 1 3 2
1 2 1
d
N
P m m
d
−
= = =
t d a s m
t
j v
F P K K K
b y K
σ
× × × ×
=
× ×
1 0 2 0 . 1 3 2 1 . 1 1 . 1 1 . 1 3
2 4 . 5 4
1 0 0 . 5 1 . 5
M P a
× × × ×
= =
× ×
t d a s m
t
j v
F P K K K
b y K
σ
× × × ×
=
× ×
1 0 5 0 . 1 4 8 1 . 1 1 . 1 1 . 1 3
2 8 . 3 3
1 0 0 . 5 1 . 5
M P a
× × × ×
= =
× ×
119
0.148
121
d
N
P mm
d
−
===

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
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3.1 ANALYSIS OF GEARS HAVING 18 TEETH
Fig 3.1 Helical elliptical gear model under meshing condition having 18 teeth
Fig 3.2 Helical elliptical gear model under meshing condition having 18 teeth (meshed)
Fig 3.3 Total deformation of the gear teeth under meshing condition

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
132
Fig 3.4 Directional deformation of the gear teeth under meshing condition
Fig 3.5 Equivalent stress of the gear teeth under meshing condition
Fig 3.6 Maximum principal stress of the gear tooth under meshing condition

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
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3.2 ANALYSIS FOR 19 TEETH
Fig 3.7 Helical elliptical gear model under meshing condition having 19 teeth
Fig 3.8 Helical elliptical gear model under meshing condition having 19 teeth (meshed)
Fig 3.9 Total deformation of the gear tooth under meshing condition

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
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Fig 3.10 Directional deformation of the gear tooth under meshing condition
Fig 3.11 Equivalent stress of the gear tooth under meshing condition
Fig 3.12 Total deformation of the gear tooth under meshing condition

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 2, February (2015), pp. 128-136© IAEME
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TABLE 3.1: Results calculated from meshing condition of gears having 18 teeth
Object name
Total
deformation
(m)
Directional
deformation
(m)
Equivalent
stress(Pa)
Maximum
principal
stress(Pa)
Minimum value 0 -5.639e-7
7.936 4.90e6
Maximum value 1.368e-6
5.3241e-7
1.0639e7 -2.25e6
TABLE 5.2: Results calculated from meshing condition of gears having 19 teeth
Object name
Total
deformation
(m)
Directional
deformation
(m)
Equivalent
stress(Pa)
Maximum
principal
stress(Pa)
Minimum value 0 6.0409e-7
1.3741e7
1.1807e7
Maximum value 4.9255e-6
-2.001e-6
0 -3.3885e6
5.3 CONCLUSION
The FEA model is used to simulate contact between two bodies accurately by verification of
contact stresses between two helical elliptical gears in contact and comparison is made with the
results of AGMA analysis. The average difference of results between ANSYS and AGMA approach
are very small and equal to 1.71%. It was shown that FEA model could be used to simulate bending
between two bodies accurately. The transmission error for two sets of meshing helical elliptical gears
are analyzed for transmission error.
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