World Journal of Mechanics, 2012, 2, 239-245
doi:10.4236/wjm.2012.25029 Published Online October 2012 (http://www.SciRP.org/journal/wjm)
A Computer Program for Dynamic Load Simulation of
Spur Gears with Asymmetric and Symmetric Teeth
Fatih Karpat1, Stephen Ekwaro-Osire2, Esin Karpat3
1Department of Mechanical Engineering, Uludag University, Bursa, Turkey
2Department of Mechanical Engineering, Texas Tech University, Lubbock, USA
3Department of Electronics Engineering, Uludag University, Bursa, Turkey
Received August 2, 2012; revised September 1, 2012; accepted September 12, 2012
ABSTRACT
An important concern in gear design is to reduce the dynamic load and noise of gear systems. It has been found that the
noise generated from gearing is basically due to gearbox vibration excited by the dynamic load. Since one of the situa-
tions that demand high performance is the high rotational speeds, there is a need to understand the dynamic behavior of
the gears at such speeds. Such knowledge would shed light on detrimental characteristics like dynamic loads and vibra-
tions. An efficient way in performing studies on the dynamic behavior of gears is using computer aided analysis on nu-
merical models. In this paper, a developed computer program is introduced to analyze dynamic behavior of spur gears
with asymmetric teeth that have a potential use for higher performance in wind turbine gearboxes. This program can be
used to compare conventional spur gears with symmetric teeth and spur gears with asymmetric teeth. By using this pro-
gram, gear designers can design a gear pair and obtain results, e.g. dynamic Load, transmitted torque, static transmis-
sion error, and frequency spectra of static transmission error etc., just by pressing a command button.
Keywords: Gears; Computer Program; Asymmetric Teeth
1. Introduction
Gear dynamics has been a subject of intense interest to
the gearing area during the last few decades dynamic
loads and vibration are a major concern for gear drives at
high speeds. Simulation of meshing of gear drives per-
formed by application of tooth contact analysis and ex-
perimental tests of gear drives have confirmed that trans-
mission errors (TE) are the prime cause of noise and vi-
brations of the gear drives [1]. The definition of trans-
mission error is made as “The difference between the
actual position of the output gear and the position it
would occupy if the gear drive were perfectly conjugate”.
This may be expressed as angular displacement or as
linear displacement at the pitch point [2]. The causes of
transmission error are elastic deflections under load,
geometrical errors and geometrical modifications.
Transmission errors causing dynamic loads in gears
affect not only the gear vibrations and noise but also
tooth fatigue, and surface failure. Therefore, the most
important objective in gear design is the minimization of
dynamic loads and transmission errors.
There have been many studies on gear design in lit-
erature. The prediction of gear transmission errors and
gear dynamic loads, gear noise and vibration for gear
drives are always main concerns in gear design. Com-
prehensive reviews on the development of a variety of
simulation models for both static and dynamic analysis
of different types of gears are presented in [3-5]. Tea-
ruchi and Hidetaro [5] used the tooth deflection, equiva-
lent composite error, and equivalent mass of gear, in the
calculation of the dynamic loads on gear teeth. The nu-
merical results obtained were shown to be in good
agreement with the experimental results. A similar vi-
bratory model was presented in [6]. A comparison of the
theoretical and the experimental results, obtained for
dynamic characteristics of the heavily loaded spur gears,
was made. A numerical approach for the equations of
motion that contain the excitation terms due to errors and
periodic variation of the mesh stiffness was developed
and presented. This method was adapted and employed
by several researchers [7-14] to calculate the dynamic
contact load or the torsional response, depending on dif-
ferent gear parameters, i.e. tooth errors, addendum modi-
fication, mesh stiffness, lubrication, damping factor, gear
contact factor, and friction coefficient. In gear design, the
dynamic factor is generally used to quantify the dynamic
effects. In this context, the dynamic factor is defined as
the ratio of the maximum dynamic load to the maximum
static load on the gear tooth. Dynamic loads of gears with
low contact ratio (contact ratio is between 1 and 2) are
affected by several parameters, namely, time-varying

F. KARPAT ET AL.
240
mesh stiffness, tooth profile error, contact ratio, friction,
and sliding. The static transmission errors change in a
periodic manner due to the variation of gear mesh stiff-
ness during contact. This is the source of vibratory exci-
tation in gear dynamics. The static transmission error has
basic periodicities related to the shaft rotational frequen-
cies and the gear mesh frequency. The mesh frequency
and its first harmonics are the predominant contributors
to the generation of noise. Many researchers investigated
the effects of different parameters (e.g. design load and
tooth profile modification) in decreasing the static trans-
mission errors [11-13]. In addition, the Fast Fourier
Transform (FFT) can be used to perform the frequency
analysis of static transmission error.
Since the variation of gear-pair meshing tooth stiffness
causes static transmission errors, one of the most impor-
tant method to minimize transmission errors is to change
the gear tooth stiffness. In involute gears, high contact
gear ratio gears and non standard gears are used to vary
the gear tooth stiffness. Recently, involute spur gears
with asymmetric teeth provide flexibility to designers for
different application areas due to non-standard design. If
they are correctly designed, they can make important
contributions to the improvement of designs in aerospace
industry, automobile industry, and wind turbine industry.
This often relates to improving the performance, increa-
sing the load capacity, reduction of acoustic emission,
and reduction of vibration.
A number of studies on the design and stress analysis
of asymmetric gears are available in literature. A large
number of studies have been performed over the last two
decades to assess whether asymmetric gears are an alter-
native to conventional gears in applications requiring
high performance. In these studies, some standards (e.g.
ISO 6336 and DIN 3990), analytical methods (e.g. the
Direct Gear Design method and the tooth contact analy-
sis), and numerical methods (Finite elements method-
FEM) have been used to compare the performance of
conventional and asymmetric gears under same condi-
tions [15-18].
In order to utilize asymmetric gear designs more ef-
fecttively, it is imperative to perform analyses of these
gears under dynamic loading. In some studies [19-22],
preliminary results related on the response of asymmetric
gears under dynamic loading are presesented. The effect
of some design parameters, such as pressure angle or
tooth height on dynamic loads, was shown. Although
asymmetric tooth is emerging as a major concern in gear
researches, in literature, by now there was not any virtual
tool for design of spur gears with asymmetric teeth.
Therefore, in this study, a MATLAB-based virtual tool
called DYNAMIC to analyze dynamic behavior of spur
gears with asymmetric teeth depending on various tooth
parameters. The objective of this paper is to introduce a
developed MATLAB-based virtual tool called DY-
NAMIC and to demonstrate its capabilities.
2. Dynamic Analysis of Gears
2.1. Dynamic Model
During one mesh period the tooth contact load in one
gear pair does not stay constant. This load varies de-
pending on the transition from double tooth contact to
single tooth contact.
To determine the variation of dynamic load as a func-
tion of the contact position (or time), it is necessary to
derive the equations of motion for gear tooth pair in a
mesh. Considering the free body diagrams of the gear
and pinion shown in Figure 1, the equations of motion
can be formulated as:


g
gbgIIIgII IgIIII IIbgD
J
rFFF FrF   
&& (1)


p
pbpDb pIIIp IIIp IIIIII
J
rFr F FFF   
&&
 (2)
where Jp and Jg represent the polar mass moments of in-
ertia of the pinion and gear, respectively. The dynamic
contact loads are FI and FII, while 
I and

II are the in-
stantaneous coefficients of friction at the contact points.

p and

g represent the angular displacements of pinion
and gear. The radii of the base circles of the engaged
gear pair are rbp and rbg, while the radii of curvature at
the mating points are 
pI,II and

gI,II.
In above equations, if the speed of the pinion tooth is
greater than the speed of the gear tooth, the sign of the
friction force is positive, otherwise it is negative. The
static tooth load is defined as:
p
g
Dbp bg
TT
Frr  (3)
If the angular coordinate is converted into the coordi-
nate along the line of action, the displacements of the
Figure 1. Engaging teeth pairs along the line of action.

F. KARPAT ET AL.
241
undeformed tooth profiles, along the line action, can be
written as: rpg
x
yy


&&&
(7)
g
gg
yr

 (4) rpg
x
yy


&&&&&& (8)
Including viscous damping, the equations of motion
reduce to:
p
pp
yr

 (5)
The relative displacement, velocity, and acceleration
can then be cast as:
22
2
rrrs
x
xx   
&&& x (9)
rpg
x
yy (6) The loaded static transmission errors can be obtained:





 
g
pDI IpIggIpIIIIpIIggIIp
s
IpIggIp IIpIIggIIp
MMFKSMSM KSMSM
xKSMSMK SMSM
 

The effective gear masses are:
2
p
pbp
M
Jr (11)
2
g
gbg
M
Jr (12)
The equivalent stiffness of meshing tooth pairs can be
written as:
p
IgI
I
p
IgI
kk
Kkk

 (13)
p
II gII
II
p
II gII
kk
Kkk

 (14)
The friction experienced by the pinion and the gear
can be expressed as:
1
I
pI
pI bp
Sr

 
 (15)
1
I
gI
gI bd
Sr

 
 (16)
1
I
IpII
pII bp
Sr

 
 (17)
1
I
IgII
gII bd
Sr

 
 (18)
The signs in the above expressions are positive (+) for
the approach and negative (−) for the recess. The dy-
namic contact loads, which include tooth profile error,
can then be written as:

I
Ir I
FKx

 (19)

I
IIIrII
FKx

 (20)
where 
I and

II are the tooth profile errors. In this paper,
the effects of profile errors on the dynamic response of
gears are not considered. Thus, the tooth profile errors
are assumed to be zero. The developed computer pro-
gram has a capability of using any approach for the de-
termination of errors.
The reduced equation of motion in Equation (1) and (2)
is solved numerically using a method previously detailed
in Reference [19-20]. This method employs a linearized
iterative procedure that involves dividing the mesh pe-
riod into many equal intervals. In this study, a MATLAB
program is developed. The flowchart of this computa-
tional procedure used for calculating the dynamic re-
sponses of spur gears, is shown in Figure 2. The time
interval, between initial contact point and the highest
point of single contact is considered as a mesh period. In
the numerical solution, each mesh period is divided into
200 points for good accuracy. Within a small interval,
various parameters of equations of motion are taken as
constants and an analytical solution obtained. The calcu-
lated values of the relative displacement and the relative
velocity after one mesh period are compared with the
initial values xr and vr. Unless the differences between
procedure is repeated by taking the previous calculated
Figure 2. The structure of developed computer program.

F. KARPAT ET AL.
242
them are smaller than a preset tolerance, the iteration xr
and vr at the end point of single pair teeth contact as new
initial conditions. Then the dynamic loads are calculated
by using the calculated relative displacement values.
After the gear dynamic load has been calculated, the
dynamic load factor can be determined by dividing the
maximum dynamic load along contact line to the static
load. The dynamic factor indicates the instantaneous in-
crease of gear tooth load over the static load. It is one of
the most important parameters used for understanding the
dynamic responses of gear drives.
In literature, different methods and empirical equations
are used to calculate the tooth deflections of spur gears.
These methods are often based on the classical theory of
elasticity and numerical approaches. However, all of
them are derived for symmetric tooth. Therefore, in this
study, a 2-D finite element model is developed to calcu-
late the deflections of both the asymmetric and the sym-
metric gear teeth (Figure 3). A computer program, which
saves time and provides a means to carry out a paramet-
ric study with the gear parameters, was developed using
MATLAB. This program generates batch files for input
into ANSYS. When this file is executed in ANSYS, the
general procedure of FEA (i.e. 2D modeling, meshing,
loading, solution, and post processing) is automatically
performed. At the end, an output file, that contains nodal
deflection for loaded nodes, is created. This process is
Figure 3. 2-D finite element model.
α L2
α L1
α L3
α L4 F
F
F
F
F
α L4
Figure 4. Load application.
repeated for each gear. It should be noted that in this
analysis the loads are applied at five locations on the gear
files, the approximate curves for the single tooth stiffness,
along the contact line, are obtained with respect to the
radius of the gears.
To facilitate the calculation of the Hertzian component
of the deflection at the point loading, the size of the grid
near the point of loading is chosen as recommended by
[19,21] using the following equation:
e0.2 1.2
h
c
be

 

 for 0.9 3
e
c
 (21)
where c and e are the length and width of the element,
respectively. And bh is the Hertzian contact width:



2
2.15 pdp d
h
F
bE
    
 (22)
tooth (Figure 4). 8-noded parabolic isoparametric ele-
ments are used for meshing of the 2D model. By using
the nodal deflection values that are read from the output
where F is applied load per unit length and E is Young’s
modulus of gear material.
2.2. Virtual Tool: DYNAMIC
Physics-based modeling and simulation is important in
all engineering problems. The current mature stage of
computer software and hardware makes it possible for
complex mechanical problems, such as gear design, to be
solved numerically. In-house prepared codes to handle
individual research projects, graduate, and/or PhD stud-
ies; commercial packages for engineers in industry are
widely used to solve almost every engineering problem.
Tailored with graphical user interfaces (GUIs) and easy-
to-use design steps, anyone-even a beginner can design a
gear pair and obtain results, e.g. Dynamic Load, Trans-
mitted Torque, Static Transmission Error as a function of
time, and Static Transmission Error Harmonics etc., just
by pressing a command button. Lecturers have been in-
creasingly using these packages to increase their teaching
performance and student understanding.
Based on and triggered by these thoughts, a virtual
tool DYNAMIC is prepared that can be used for educa-
tional and research purposes. The DYNAMIC is a gen-
eral purpose tool for gear analysis (Figure 5).
There are six blocks and a figure block on the front
panel of the tool. Three blocks on the right side of the
front panel, belong to the parameters which will be de-
fined by the users. Pinion and Gear blocks are reserved
for the tooth parameters and Mechanism block is for the
parameters related to the mechanical variables. The two
blocks above the figure are Simulation and Figure Selec-
tion panels. Once the user inputs the needed parameters,
he/she clicks the CALCULATE pushbutton to obtain the

F. KARPAT ET AL.
243
3. Conclusion solution for the specified parameters. In the Figure Se-
lection block, from the pop-up menu, user can select
which solution to be plotted: Dynamic Load, Transmitted
Torque, Static Transmission Error or Static Transmission
Error Harmonics. Then the required figure can be plotted
with the PLOT button. Once the solutions are calculated,
it is not needed to run the program again and again for
each figure option. CLEAR is to clean the figure axes
before each plot.
In this paper, a MATLAB-based virtual tool, DYNAMIC,
is introduced to analyze dynamic behavior of spur gears
with asymmetric tooth design. The DYNAMIC can be
used to compare conventional spur gears with symmetric
teeth and spur gears with asymmetric teeth. The results
for dynamic load, dynamic factor, transmitted torque,
static transmission error and static transmission error
harmonics can be obtained for various tooth parameters
to show the powerful aspects of asymmetric teeth. Influ-
ence of various parameters (e.g. the pressure angle on
drive side or coast side, addendum, and teeth number) on
the static transmission errors, the amplitudes of harmon-
ics of the static transmission errors and the dynamic
loads can be investigated. By using this program, gear
designers can design a gear pair and obtain results, e.g.
dynamic load, transmitted torque, static transmission
error, and frequency spectra of static transmission error
etc., just by pressing a command button.
The variation of dynamic load, static transmission er-
rors, static transmission error harmonics and, transmitted
torque with respect to time can be seen in Figures 6(a)-
(c). The solutions for different variables can be plotted in
one figure, for comparison. Figure 6(a) shows a sample
result for the variation of the static transmission error
during a mesh period. In Figure 6(b), the frequency
spectra of the static transmission errors obtained by using
the fast Fourier transform (FFT) are depicted. DY-
NAMIC program can also provide transmitted torque
results respect to time for sample gear pairs.
Figure 5. The front panel of the DYNAMIC tool.

F. KARPAT ET AL.
244
(a)
(b)
(c)
Figure 6. The variation of static transmission errors (a),
static transmission error harmonics (b) and, transmitted
torque (c) with respect to time.
4. Acknowledgements
The authors gratefully acknowledge the support of Ulu-
dag University under grant BAP-YDP(M) 2010/10 and
Texas Tech University under grant DE-FG36-06-GO
86092.
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