">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.
Copyright © 2012 SciRes. WJM
F. KARPAT ET AL.
Copyright © 2012 SciRes. WJM
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
Copyright © 2012 SciRes. WJM
F. KARPAT ET AL.
Copyright © 2012 SciRes. WJM
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.
REFERENCES
[1] F. L. Litvin, D. Vecchiato, K. Yukishima, A. Fuentes, I.
Gonzalez-Perez and K. Hayasaka, “Reduction of Noise of
Loaded and Unloaded Misaligned Gear Drives,” Com-
puter Methods in Applied Mechanics and Engineering,
Vol. 195, No. 41-43, 2006, pp. 5523-5536.
doi:10.1016/j.cma.2005.05.055
[2] M. Åkerblom, “Gear Noise and Vibration—A Literature
Survey,” Report, Maskinkonstruktion, Stckholm, 2001, p.
26. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-9891
[3] H. N. Ozguven and D. R. Houser, “The Mathematical
Models Used in Gear Dynamics—A Review,” Journal of
Sound and Vibration, Vol. 121, No. 3, 1988, pp. 383-411.
doi:10.1016/S0022-460X(88)80365-1
[4] A. Parey and N. Tandon, “Spur Gear Dynamic Models
Including Defects: A Review,” Shock Vibration Digest,
Vol. 35, No. 6, 2003, pp. 465-78.
doi:10.1177/05831024030356002
[5] Y. Tearuchi and M. Hidetaro, “Comparison of Theories
and Experimental Results for Surface Temperature of
Spur Gear Teeth,” Journal of Engineering for Industry,
Transactions of the ASME, Vol. 96, No. 11, 1974, pp.
41-50.
[6] K. Ichimaru and F. Hirano, “Dynamic Behavior of
Heavy-Loaded Spur Gears,” Journal of Engineering for
Industry, Transactions of the ASME, Vol. 96, No. 2, 1974,
pp. 373-381.
[7] K. L. Wang and H. S. Cheng, “Numerical Solution to the
Dynamic Load, Film Thickness, and Surface Tempera-
tures in Spur Gears—Part 2 Results,” Journal of Me-
chanical Design, Vol. 103, No. 4, 1981, pp. 188-194.
doi:10.1115/1.3254860
[8] K. L. Wang, H. S. Cheng, K. L. Wang and H. S. Cheng,
“Numerical Solution to the Dynamic Load, Film Thick-
ness, and Surface Temperatures in Spur Gears—Part 1.
Analysis,” Journal of Mechanical Design, Vol. 103, No.
4, 1981, pp. 177-187. doi:10.1115/1.3254859
[9] S. M. A. Arikan, “Dynamic Load and Contact Stress
Analysis of Spur Gears,” 17th Design Automation Con-
ference Presented at the 1991 ASME Design Technical
Conferences, Miami, 22-25 September 1991, pp. 85-91.
[10] J. H. Kuang and A. D. Lin, “The Effect of Tooth Wear on
the Vibration Spectrum of a Spur Gear Pair,” Journal of
Vibration and Acoustics-Transactions of the Asme, Vol.
123, No. 3, 2001, pp. 311-317. doi:10.1115/1.1379371
[11] J. H. Kuang, A. D. Lin, J. H. Kuang and A. D. Lin,
“Theoretical Aspects of Torque Responses in Spur Gear-
ing Due to Mesh Stiffness Variation,” Mechanical Sys-
tems and Signal Processing, Vol. 17, No. 2, 2003, pp.
255-271. doi:10.1006/mssp.2002.1516
[12] A. Andersson, “An Analytical Study of the Effect of the
Contact Ratio on Spur Gear Dynamic Response,” Journal
of Mechanical Design, Vol. 122, No. 4, 2000, pp. 508-
514. doi:10.1115/1.1320819
Copyright © 2012 SciRes. WJM
F. KARPAT ET AL. 245
[13] R
. Kasuba and J. W. Evans, “An Extended Model for
Determining Dynamic Loads in Spur Gearing,” Journal
of Mechanical Design, Vol. 103, No. 2, 1981, pp. 398-
409. doi:10.1115/1.3254920
[14] A. S. Kumar, T. S. Sankar and M. O. M. Osman, “On
Dynamic Tooth Load and Stability a Spur-Gear System
Using the State-Space Approach,” Journal of Mecha-
nisms, Transmissions, and Automation in Design, Vol.
107, No. 1, 1985, pp. 54-60. doi:10.1115/1.3258695
[15] K. Cavdar, F. Karpat and F. C. Babalik, “Computer
Aided Analysis of Bending Strength of Involute Spur
Gears with Asymmetric Profile,” Journal of Mechanical
Design, Vol. 127, No. 3, 2005, pp. 477-484.
[16] A. Kapelevich, “Geometry and Design of Involute Spur
Gears with Asymmetric Teeth,” Mechanism and Machine
Theory, Vol. 35, No. 1, 2000, pp. 117-130.
doi:10.1016/S0094-114X(99)00002-6
[17] F. Karpat and S. Ekwaro-Osire, “Influence of Tip Relief
Modification on the Wear of Spur Gears with Asymmet-
ric Teeth,” Tribology Transactions, Vol. 51, No. 5, 2008,
pp. 581-588. doi:10.1080/10402000802011703
[18] F. Karpat, S. Ekwaro-Osire and M. P. H. Khandaker,
“Probabilistic Analysis of MEMS Asymmetric Gear
Tooth,” Journal of Mechanical Design, Vol. 130, No. 4,
2008, pp. 042306.1-042306.6.
[19] F. Karpat and S. Ekwaro-Osire, “Influence of Tip Relief
Modification on the Wear of Spur Gears with Asymmet-
ric Teeth,” Tribology & Lubrication Technology, Vol. 66,
No. 6, 2010, pp. 50-60.
[20] F. Karpat, S.Ekwaro-Osire, K.Cavdar and F.C. Babalik,
“Dynamic Analysis of Involute Spur Gears with Asym-
metric Teeth,” International Journal of Mechanical Sci-
ences, Vol. 50, No. 12, 2008, pp. 1598-1610.
doi:10.1016/j.ijmecsci.2008.10.004
[21] J. J. Coy and C. H. C. Chao, “Method of Selecting Grid
Size to Account for Hertz Deformation in Finite Element
Analysis of Spur Gears,” Journal of Mechanical Design,
Vol. 104, No. 4, 1982, pp. 759-766.
doi:10.1115/1.3256429
[22] R. Muthukumar and M. R. Raghavan, “Estimation of
Gear Tooth Deflection by the Finite Element Method,”
Mechanism and Machine Theory, Vol. 22, No. 2, 1987,
pp. 177-181. doi:10.1016/0094-114X(87)90042-5
Copyright © 2012 SciRes. WJM