Special Issue Article
Advances in Mechanical Engineering
2017, Vol. 9(8) 1–12
An improved numerical method for  The Author(s) 2017DOI: 10.1177/1687814017721856
journals.sagepub.com/home/ade
the mesh stiffness calculation of spur
gears with asymmetric teeth on
dynamic load analysis
Fatih Karpat1, Oguz Dogan1, Celalettin Yuce1 and Stephen Ekwaro-Osire2
Abstract
Gears are one of the most crucial parts of power transmission systems in various industrial applications. Recently, there
emerged a need to design gear drivers due to the rising performance requirements of various power transmission appli-
cations, such as higher load-carrying capacity, higher strength, longer working life, lower cost, and higher velocity. Due
to their excellent properties, gears with asymmetric teeth have been designed to obtain better performance in applica-
tions. As the rotation speed of the gear transmission increases, the dynamic behavior of the gears has become a subject
of growing interest. The most important contributing factor of dynamic behavior is the stiffness of the teeth, which
changes constantly throughout the operation. The calculation of gear stiffness is important for determining the load dis-
tribution between the gear teeth when two sets of teeth are in contact. The primary objective of this article is to
develop a new approach to calculate gear mesh stiffness for asymmetric gears. With this aim in mind, single tooth stiff-
ness was calculated in the first stage of the study using a finite element method. This study presents crucial results to
gear researchers for understanding spur gears with involute asymmetric teeth, and the results will provide researchers
with input data for dynamic analysis.
Keywords
Spur gear, asymmetric teeth, tooth stiffness, mesh stiffness, pressure angle
Date received: 4 March 2017; accepted: 23 June 2017
Academic Editor: Yi Wang
Introduction rotated in only one direction. In the unidirectional rota-
tion, the geometry of the coast side does not have to be
Recently, due to environmental concerns and air pollu- symmetric with the drive side, allowing for the possibil-
tion associated with energy consumption, there has ity of an asymmetric teeth design. Because it is a non-
been a greater demand for higher efficiency machinery standard design, asymmetric teeth provide variability
in industries. The efficiency of gears, an integral part of
power transmission in industrial machinery, aircraft,
1
and automotive, has been considered as a significant Faculty of Engineering, Department of Mechanical Engineering, Uludağ
University, Bursa, Turkey
factor in decreasing energy consumption. Furthermore, 2Department of Mechanical Engineering, Texas Tech University, Lubbock,
the improvement of gear system efficiency also provides TX, USA
some benefits, like a reduction in gear system failures
and frictional heat generation in the gearbox, as well as Corresponding author:
Fatih Karpat, Faculty of Engineering, Department of Mechanical
a decrease in their operational costs.
Engineering, Uludağ University, Ali Durmaz Building Görükle Campus,
In the operation of several applications, traditional 16059 Bursa, Turkey.
gears with an involute profile that is symmetric can be Email: karpat@uludag.edu.tr
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2 Advances in Mechanical Engineering
to designers in different application fields.1 Asymmetric internal gears with asymmetric involute teeth. Sekar
teeth consist of a standard involute profile but with dif- and Muthuveerappan13 investigated the influence of
ferent pressure angles on the drive and back side of the the gear ratio, transverse contact ratio, top land thick-
teeth. Apart from that, all the other parameters are the ness coefficient, and pinion teeth number on the load-
same as with the symmetric standard spur gears. In par- sharing ratio and the non-dimensional stress number in
ticular, symmetric gears have 20–20 pressure angles, asymmetric helical gears. Spitas et al.14 introduced a
while asymmetric gears, for example, may have 20–25 concept of asymmetric half-involute gear teeth, which
and 20–30 pressure angles. The asymmetric gears are they studied using finite element analysis (FEA).
mainly used in gear pumps, wind turbines, helicopter The tooth surfaces are subjected to different stresses
drivetrain, and turboprop engine. If they are correctly under working conditions. The stress distribution of
designed, asymmetric teeth can offer important contri- gear teeth plays an important role in preventing failure
butions to the improvement of designs in several indus- during operation and aids in the advanced prediction of
tries. The asymmetric profile of these gears provides a failure. When the tooth surfaces are subjected to exces-
good degree of flexibility for obtaining the most favor- sive stress conditions, tooth surface failure may occur.
able design in several applications. As a result, gears This can cause deformations of the contact points of
with asymmetric teeth can be designed to provide dif- the tooth surfaces, leading to damage and a reduction
ferent pressure angles on the coast side and drive side. in gear tooth stiffness, which can be used to assess the
In this way, key properties, such as low weight, reduc- severity of tooth damage. Tooth stiffness is a key para-
tion in vibration and acoustic emissions, and high load- meter of gear dynamics in determining factors such as
carrying capacity, are obtained.2 dynamic tooth loads, load-carrying capacity of gears,
Many researchers have investigated the stress and and vibration characteristics of geared system. The cal-
deformation analysis of gears with asymmetric teeth in culation of single tooth stiffness requires determining
the literature.3–13 Kapelevich3 derived equations the elastic deflection of the tooth along the direction of
required for asymmetric gear design and developed a tooth load, which primarily contributes to bending,
method for this purpose. The author confirmed that, shear, and rim deformations of the gear tooth.15 Tooth
when a high-pressure angle on the drive side is chosen, stiffness is needed for a variety of reasons. For gears in
the bending and contact stress and vibration levels are mesh, there are different numbers of teeth in contact,
substantially reduced. Different computer programs depending on the contact ratio, during motion. With
were developed by Karpat et al.4 and Di Francesco and more than two sets of teeth in contact, tooth stiffness
Marini5 for optimizing the asymmetric teeth design. must be known to determine the load on an individual
These programs can be used to optimize the degree of tooth.
asymmetry automatically in order to maximize the per- Numerous works in the literature have been con-
formance of the teeth. A method was developed by ducted on methods of calculating gear tooth stiff-
Alipiev6 for the geometric design of gears with asym- ness.15–26 Tooth failures can be estimated through the
metric teeth. calculation of tooth stiffness reduction. Yesilyurt
Litvin et al.7 used numerical examples to investigate et al.15 analyzed the single tooth stiffness of spur gears
static transmission errors for modified asymmetric teeth according to deformations of the rim, bending, and
and found that asymmetric teeth reduced contact stress shear. Chen and Shao18 derived equations for use in
and bending stress. A theoretical method was proposed the study of the effects of tooth errors on transmission
by Cavdar et al.8 to examine the bending stress of errors and mesh stiffness. Li19 investigated the effect of
asymmetric gears. The authors stated that asymmetric the addendum on contact strength, bending strength,
teeth perform better for bending stress minimization and the basic performance parameters of two pairs of
than both symmetric teeth with common pressure spur gears with different addendums and high contact
angles and symmetric teeth with high-pressure angles. ratios. The author stated that mesh stiffness is reduced
Yang9 developed a method for designing helical gears if the addendum becomes longer and the number of
with asymmetric teeth based on a rack cutter and, in contact teeth is not changed. Chaari et al.20 proposed
this study, the three-dimensional (3D) stress analysis an analytical formulation of the time-varying gear
results of the helical gears with asymmetric teeth and mesh stiffness and modeled the effect of tooth cracks
spur gears with asymmetric teeth are compared com- on stiffness. They stated that tooth cracks decrease gear
prehensively. Pedersen10 showed that bending stress mesh stiffness when the affected tooth was in the mesh.
can be reduced significantly using asymmetric gear Rincon et al.21 presented a model for assessing the con-
teeth. Kumar et al.11 examined the influence of asym- tact forces between gear pairs and analyzed the effect
metric teeth pressure angles on gear drive quality. of transmitted torque, friction, and modified center dis-
Yang12 presented basic aspects of the geometry of inter- tance on mesh stiffness. Pedersen and Jorgensen22 pre-
nal gears with asymmetric involute teeth and developed sented a method for estimating the stiffness of
a simulation of meshing and the contact of misaligned individual gear teeth as a function of the contact point
Karpat et al. 3
position. The authors stated that an increase in rim deformation, and axial compression. The single tooth
thickness reduced stiffness, while an increase in contact stiffness is required for calculating the mesh stiffness of
length increased stiffness. Chang et al.23 proposed a tooth pairs. The tooth stiffness can be calculated using
model for determining the mesh stiffness of cylindrical the following formulas
gears. The authors stated that a decrease in rim thick-
ness and web thickness results in smaller mesh stiffness. Fkpı = ð1Þ
The literature shows that many authors have dis- xpı
cussed the calculation of tooth stiffness for gears with F
symmetric teeth theoretically by using several equa- kgı = ð2Þ
x
tions. However, no methods or equations could be gı
observed in the literature for the calculation of tooth F
k = ð3Þ
stiffness for asymmetric gears. Therefore, this work pıı xpıı
aims to fill this gap in the literature. Even though other
F
authors in the literature24 have developed a simple kgıı = ð4Þ
equation in order to compute the tooth stiffness of xgıı
asymmetric spur gears, they have examined only one where xpı, xpıı, xgı, and xgıı are the single tooth deflection
pressure angle. In addition to gear design, dynamic in the direction of the applied load and F is the applied
loads that occur during mesh periods are determined load. In the literature, a number of techniques and the-
using calculated tooth stiffness values. There are some oretical equations were developed to identify the single
parameters, such as profile errors, rotational speed, tooth deformation of gears. These methods generally
tooth number, and tooth stiffness, which affect the depend on numerical approaches and theory of elasti-
dynamic load of the gear. Until now, the effects of all city. For the calculation of single tooth stiffness, the
parameters on the dynamic behavior have only been well-known researchers, Kuang et al.,16,17 developed a
considered by researchers for involute symmetric teeth, new approach for calculating the single tooth stiffness
leaving a gap in the literature regarding asymmetric using analytical, numerical, and finite element methods
teeth. Since asymmetric teeth are not standard, the (FEMs). The authors derived equations (5)–(9) to cal-
results of this study pertaining to mesh stiffness and culate the single symmetric tooth stiffness of gears
tooth stiffness will serve as input for designers.
The potential energy method is another powerful
 ðr  RiÞ
way for the calculation of gear mesh stiffness. A num- KiðrÞ= ðAO +AXiÞ+ ðA2 +A3XiÞ ð ð5Þ1+XiÞ  m
ber of studies are present in literature. By using poten-
2 3
tial energy method, the gear mesh stiffness can be A0 = 3:867+ 1:612Zi  0:02916Zi + 0:0001553Zi ð6Þ
calculated analytically.25–27 Moreover, the effect of
A1 = 17:060+ 0:7289Zi  0:01728Z2 + 0:0000999Z3
crack and gear failures on the gear mesh stiffness can i i
be calculated using this method.26 ð7Þ
The primary aim of this article is to develop novel A 2 32 = 2:637 1:222Zi + 0:02217Zi  0:0001179Zi ð8Þ
equations which facilitate the calculation of tooth stiff-
2 3
ness. Hence, a parametric study was conducted to deter- A3 = 6:330 1:033Zi + 0:02068Zi  0:0001130Zi
mine the mesh stiffness of spur gears with asymmetric ð9Þ
teeth. In this context, a two-dimensional (2D) tooth
model with asymmetric teeth was created for FEA. This study started with the creation of a 2D FEA of
Novel equations were developed from the FEA results an asymmetric gear tooth to calculate the deflection
to calculate the approximate tooth stiffness with pres- values. Because more than 1200 analyses were planned
sure angles on the drive side and coast side, and tooth to calculate the single tooth stiffness for a variety of
number. Furthermore, by using the equations devel- tooth numbers and pressure angles, 2D models were
oped in this study, the mesh stiffness of the involute preferred. Certainly, 3D models may be more reliable
spur gear pair with asymmetric teeth was calculated for than the 2D models; however, the FEA running time of
different cases. 2D models is less than 15 times that of 3D models for
each analysis. Furthermore, in the FEM approach, if
the model has a constant cross-sectional area, the 2D
Calculation of mesh stiffness models result as accurate as the 3D models. The
amount of error between the 2D and the 3D models is
Single tooth stiffness
nearly 3%. To eliminate these differences in the study,
The single tooth stiffness is defined as the value of the correction factors for each case were substituted into
transmitted load divided by the total deflection of the the derived formulas. A computer program1 was pre-
tooth, which consists of bending deformation, shear pared using MATLAB. This program enables the
4 Advances in Mechanical Engineering
researcher to change basic gear parameters, such as
module, tooth number, pressure angle, face width,
modulus of elasticity, and Poisson’s ratio. During the
running process of MATLAB, a batch file was gener-
ated for ANSYS, and all FEM procedures from 2D
modeling to post-processing were done automatically
using the batch file. Finally, a text file including the
deflection values of the nodes was produced; the nodal
deflections could also be gained from the software
interface. This process was repeated for each gear pair.
The flowchart of the study is shown in Figure 1.
In this study, a 2D FEM of the asymmetric gear
tooth profile was created for the FEA (Figure 2). First,
a larger pressure angle for the drive side than the coast
side was selected for the asymmetric tooth design, and
then tooth stiffness and mesh stiffness were investi-
gated, respectively. In the FEA, the loads were applied
to six different locations of the tooth (Figure 3). For
each contact point, the applied loads were chosen as
250N to calculate the tooth deflection. In order to
define the Hertzian part of the deflection at the point
of loading, the size of the mesh structure near the point
of loading was used as suggested by Coy and Chao.28
The estimated curves for tooth stiffness were drawn
Figure 1. Flowchart of the study. with regard to the radius of the gears using the deflec-
tion values at the nodes according to the text file pro-
duced by the computer program.
Gear mesh stiffness
Gear mesh stiffness depends on the number of teeth in
contact. As can be seen in Figure 4, the meshing pro-
cess starts at point A, which is the addendum circle of
the driven gear and terminates at point E, which is the
addendum circle of the driver gear. During the mesh-
ing process of low contact ratio spur gears, in some
locations (between BD), the single tooth pair is in con-
tact; however, in some locations (between AB and
DE), the double teeth pairs are in contact (Figure 4).
Thus, the gear mesh stiffness constantly changes
Figure 2. 2D FEM model of asymmetric teeth. between single and double teeth pair zones.
Figure 3. Load application points on asymmetric tooth profile: (a) ac = 20, ad = 30; (b) ac = 18, ad = 26; and (c) ac = 30,
ad = 20.
Karpat et al. 5
Figure 4. Contact line of asymmetric gears.
Figure 5. Changing of mesh stiffness in the gear meshing
period.
The equivalent stiffness of meshing tooth pairs can h   i0:5
rDp= rbp2 +(ðrbp+ rbdÞ tana rad2  0:5rbd2 +pm cosa))2
be written as follows
First tooth pairs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð13Þ
jAEj= rap2  rbp2 + rad2  rbd2  ad sina ð14Þ
kpıkgı
K = ð10Þ jADj= jAEj  pmn cosa ð15ÞI
kpı + kgı
Second tooth pairs j
jAEj
ACj= ð16Þ
2
k k jABj= jAEj  pmn cosa ð17Þpıı gıı
KII = ð11Þ
kpıı + kgıı The contact ratio can be calculated as
Here, the tooth pairs are considered as springs con- jAEj
nected in series: ea = ð18Þ
pmn cosa
 KI¼6 0 and KII=0 in single contact zone In general, the contact ratio is between 1.1 and 1.98
(between |BD|) for gears with low contact ratio. The contact ratio
 KI 6¼ 0 and KII 6¼ 0 in double contact zone should not be more than 1.98 for low contact ratio spur
(between |AB| and |DE|) gears. The contact ratio of the gears directly defines
the dynamic performance of the gear system. When the
In the single tooth contact zone, the gear mesh stiff- contact ratio increases, the dynamic performance of the
ness is equal to KI because only one gear pair is in con- gears also increases, but the vibration levels and noise
tact. In the double tooth contact zone, the gear mesh decrease. The contact ratio also defines the load-sharing
stiffness can be calculated as (KI + KII). Thus, the ratio and the number of teeth in contact. For instance,
double tooth contact zone mesh stiffness values are with the increment of the contact ratio, the number of
higher than the single tooth contact zone. In Figure 4, teeth in contact changes; in low contact ratio gears, one
it is illustrated with cord BD that as the pressure angle gear pair is always in contact, but in high contact ratio
increased on the drive side of the asymmetric gear, the gears, two gear pairs are always in contact. A diagram
length of the single tooth contact zone also increased. of a typical contact point, three mesh processes, and
Thus, the gear pair contact ratio decreased. To define varying mesh stiffness is given in Figure 5 for low con-
the single and double tooth contact positions, the radii tact ration gears. As can be seen in the first position,
of the highest and lowest points of single tooth contact two tooth pairs are in mesh in the region |AB|. When
must be known. The radii of these points were defined the gear reaches point B, the tooth pairs separate from
by Colbourne.29 each other, meaning that only one tooth pair is in con-
The radius of the lowest point of single tooth contact tact in the region |BD|. Thus, the gear mesh stiffness
is decreases sharply. While the gear system is rotating, this
     process continues periodically as in Figure 5.0:5 2 0:5
rBp= rbp2 + rap2  rbp2  pmn cosa) ð12Þ Results and discussion
The radius of the highest point of single tooth con- In this study, there were four different cases for each
tact is contact point. The deformation values and the tooth
6 Advances in Mechanical Engineering
Table 1. Gear pairs’ properties of the cases.
Case 1 Case 2 Case 3 Case 4
Teeth number (pinion and gear) 20–60 20–60 20–60 20–60
Module (mn), mm 10 10 10 10
Addendum 13mn 13mn 13mn 13mn
Dedendum 1.253mn 1.253mn 1.253mn 1.253mn
Cutter radius 0.33mn 0.33mn 0.33mn 0.33mn
Pressure angle on coast side (ac) 20 25 18 20–30
Pressure angle on drive side (ad) 20–32 25–30 20–26 20
Contact ratio 1.78–1.28 1.55–1.21 1.91–1.38 1.78–1.31
Materials Steel Steel Steel Steel
Elasticity modulus, GPa 215 215 215 215
stiffness values were computed. In all cases, the number For case 1 (ac=20 and ad=20 ... 32)
of teeth selected was between 20 and 60 for pinion and
gear, and the module was 10mm. In addition, different ½r  Ri
a= ð20Þ
modules were used to test for accuracy, and the results mn
of the different modules demonstrated a high degree of
Ki = 566:869a
3 + 223:518a2 + 15751:52a+ 95:11
similarity. The pressure angle on the drive side (ad) and
the pressure angle on the coast side (ac) were variables. ð21Þ
In the first case, ad and ac were 20–32 and 20, respec- K = 0:05213Z30  8:104Z2 + 446:95Z+ 20633:22
tively. In the second case, ad varied from 25 to 30,
and ac was held constant at 25. In the third case, a ð22Þd
and ac were 20–26 and 18, respectively. In the fourth ½f ðadÞ= 0:0002146257a2d + 0:0270155172ad
case, ad was held at 20 and ac varied from 20 to 30. ð23Þ+ 0:5456900534
The gear data for each case are given in Table 1. When 	 
 
the pressure angle increases, the tip width of the gear lðtÞC = 2:6866  106Z3  4:0747104Z2
decreases. Thus, the reduced tip width would be unsafe ð24Þ
+ 0:02155Z+ 0:7120Þb
in practice as the tip is liable to be damaged easily, espe-
cially if the teeth are hardened. For case 2 (ac=25 and ad=25 ... 30)
Tooth stiffness calculation results
Ki = 395:185562160789a
3 + 344:26a2
The primary purpose of this study was to develop a ð25Þ
+ 18900:06a 23:79
new method including some equations for calculating
the tooth stiffness of spur gears with asymmetric teeth. K = 0:176Z30  33:04Z2 + 1826:92Z+ 4382:59 ð26Þ
A parametric study was conducted for this research
3 2
with parameters of teeth number, and pressure angle ½f ðadÞ= 0:00027777ad  0:022738ad ð27Þ
on the coast side (ac) and drive side (ad). Four cases of + 0:632698ad  4:9476
asymmetric gear tooth stiffness were investigated for 	 
  6 3
different contact points. The FEA results of the tooth lðtÞC = 4:983164910 Z  0:0004217893Z2 ð28Þ
stiffness of case 2 are given in Figure 6. By using the + 0:006320586Z+ 1:06264069Þb
FEA results for each case, four different equations were
developed. The new equation (19) was derived to esti- For case 3 (ac=18 and ad=20 ... 26)
mate the tooth stiffness values of the spur gears
	 

K =(K +K )½f ða Þ lðtÞ ðN=mmÞ ð19Þ Ki = 3181:99203a3 + 1406:636455a2ts 0 i d C ð29Þ
+ 13953:7297464a 271:70
where f (ad) and l(t)C are the factors of pressure angle
and stiffness, respectively. K0 and Ki are the empirical K = 0:056949494Z30  8:486753246Z2
values. By using equation (19), the tooth stiffness values ð30Þ+ 449:489Z+ 20041
were obtained for standard addendum (1 3 mn), deden-
dum (1.25 3 mn), and standard cutter tooth radius ½f ðadÞ= 0:000100894a3d + 0:00686a2d
(0.3 3 m ). ð31Þn  0:1371ad + 1:804
Karpat et al. 7
Figure 6. FEA results of the tooth stiffness for contact points of case 2.
Table 2. Tooth stiffness FEA results for case 2 point-1 (N/mm).
Z Pressure angle on drive side (ad) (degree)
25 26 27 28 29 30
20 11148.75 11260.23 11483.21 11594.70 11706.18 11929.16
25 13703.17 13840.21 14114.27 14251.30 14388.33 14662.40
30 15224.66 15376.91 15681.40 15833.65 15985.90 16290.39
35 15920.45 16079.65 16398.06 16557.26 16716.47 17034.88
40 16535.18 16700.54 17031.24 17196.59 17361.94 17692.65
45 15899.28 16058.27 16376.26 16535.25 16694.24 17012.23
50 15005.72 15155.77 15455.89 15605.94 15756.00 16056.12
55 14745.87 14893.32 15188.24 15335.70 15483.16 15778.08
60 14068.32 14209.01 14490.37 14631.06 14771.74 15053.11
	 

lðtÞC = b ð32Þ Four cases were investigated in this study, and the
tooth stiffness results of case 2 are given in Tables 2–7.
For case 4 (ac=20 ... 30 and ad=20) The results show that both the pressure angle on the
drive side and the number of teeth affect the single
tooth stiffness, depending on the contact points. The
Ki = 1620:00320a3  4970:27199a2 ð Þ number of teeth increased from 20 to 60, and the pres-33
+ 16210:5089a+ 182:96 sure angle increased from 25 to 30. This process was
repeated for different modules. Consequently, in this
K0 = 0:0618Z
3  9:454Z2 + 514:28025493Z+ 22739:4 study, 1200 cases were analyzed and 7200 displacement
ð34Þ values were gained from the FEA.
Figure 7 shows the effect of the drive-side pressure
½f ða 6 3 2dÞ= 4:9647321410 Z + 0:0078002Z + 0:841 angle on the single tooth stiffness, as calculated accord-
ð35Þ ing to the tooth stiffness results. A correlation was	 
 found between the drive-side pressure angle and the
lðtÞC = b ð36Þ single tooth stiffness of the gear. The average increase
8 Advances in Mechanical Engineering
Table 3. Tooth stiffness FEA results for case 2 point-2 (N/mm).
Z Pressure angle on drive side (ad) (degree)
25 26 27 28 29 30
20 18174.22 18355.96 18719.45 18901.19 19082.93 19446.42
25 20345.44 20548.90 20955.81 21159.26 21362.71 21769.62
30 21611.46 21827.57 22259.80 22475.92 22692.03 23124.26
35 22115.64 22336.79 22779.11 23000.26 23221.42 23663.73
40 22794.24 23022.19 23478.07 23706.01 23933.96 24389.84
45 21966.74 22186.40 22625.74 22845.41 23065.07 23504.41
50 20945.44 21154.89 21573.80 21783.25 21992.71 22411.62
55 20941.06 21150.47 21569.29 21778.70 21988.11 22406.93
60 20455.12 20659.67 21068.77 21273.32 21477.88 21886.98
Table 4. Tooth stiffness FEA results for case 2 point-3 (N/mm).
Z Pressure angle on drive side (ad)(degree)
25 26 27 28 29 30
20 25090.86 25341.77 25843.58 26094.49 26345.40 26847.22
25 26884.81 27153.65 27691.35 27960.20 28229.05 28766.74
30 27899.31 28178.30 28736.29 29015.28 29294.27 29852.26
35 28214.85 28497.00 29061.30 29343.45 29625.59 30189.89
40 28956.34 29245.90 29825.03 30114.59 30404.15 30983.28
45 27940.19 28219.60 28778.40 29057.80 29337.20 29896.01
50 26793.14 27061.07 27596.93 27864.86 28132.79 28668.66
55 27040.27 27310.67 27851.48 28121.88 28392.29 28933.09
60 26742.97 27010.40 27545.26 27812.69 28080.12 28614.98
Table 5. Tooth stiffness FEA results for case 2 point-4 (N/mm).
Z Pressure angle on drive side (ad) (degree)
25 26 27 28 29 30
20 31994.84 32314.78 32954.68 33274.63 33594.58 34234.47
25 33412.20 33746.33 34414.57 34748.69 35082.81 35751.06
30 34175.65 34517.41 35200.92 35542.68 35884.44 36567.95
35 34302.91 34645.94 35331.99 35675.02 36018.05 36704.11
40 35107.15 35458.23 36160.37 36511.44 36862.51 37564.66
45 33902.72 34241.75 34919.80 35258.83 35597.86 36275.91
50 32630.14 32956.44 33609.04 33935.34 34261.64 34914.25
55 33128.33 33459.61 34122.18 34453.46 34784.74 35447.31
60 33019.31 33349.51 34009.89 34340.09 34670.28 35330.67
in the stiffness was found to be 10% when the pressure Second, time-varying mesh stiffness results were
angle was 25. When the drive-side pressure angle compared. The comparison of FE analysis results and
reached 32, the single tooth stiffness increased by Kuang et al.’s results16,17 is given in Figure 9. The dif-
nearly 20%. ferences between the two methods are acceptable. Thus,
In this study, in order to verify the results obtained it can be said that the FE analysis procedure can be
from FEA, the results were compared with the findings used for the definition of gear mesh stiffness.
in the literature.16,17 The FEA results and Kuang
et al.’s16,17 equations, which are well known in the liter-
Mesh stiffness calculation results
ature, were compared. First, the single tooth stiffness
results were compared. As can be seen in Figure 8, both After tooth stiffness values were obtained from
the results from the literature and FE analysis matched ANSYS, the mesh stiffness was calculated using equa-
well. tions (10) and (11). In the condition of single tooth
Karpat et al. 9
Table 6. Tooth stiffness FEA results for case 2 point-5 (N/mm).
Z Pressure angle on drive side (ad) (degree)
25 26 27 28 29 30
20 38541.98 38927.40 39698.24 40083.66 40469.08 41239.92
25 39602.23 39998.26 40790.30 41186.32 41582.34 42374.39
30 40127.60 40528.88 41331.43 41732.71 42133.98 42936.54
35 40076.30 40477.06 41278.59 41679.35 42080.11 42881.64
40 40940.07 41349.47 42168.27 42577.67 42987.07 43805.87
45 39557.07 39952.64 40743.79 41139.36 41534.93 42326.07
50 38165.45 38547.11 39310.41 39692.07 40073.72 40837.03
55 38901.72 39290.74 40068.77 40457.79 40846.80 41624.84
60 38971.26 39360.98 40140.40 40530.11 40919.83 41699.25
Table 7. Tooth stiffness FEA results for case 2 point-6 (N/mm).
Z Pressure angle on drive side (ad) (degree)
25 26 27 28 29 30
20 43320.52 43753.73 44620.14 45053.34 45486.55 46352.96
25 45459.44 45914.04 46823.23 47277.82 47732.41 48641.60
30 46720.63 47187.84 48122.25 48589.46 49056.66 49991.07
35 47197.41 47669.38 48613.33 49085.31 49557.28 50501.23
40 48723.67 49210.90 50185.38 50672.61 51159.85 52134.32
45 47572.15 48047.87 48999.32 49475.04 49950.76 50902.20
50 46397.25 46861.22 47789.17 48253.14 48717.11 49645.06
55 47829.62 48307.91 49264.51 49742.80 50221.10 51177.69
60 48479.19 48963.98 49933.57 50418.36 50903.15 51872.74
Figure 7. Pressure angle effects on the single tooth stiffness
(module = 3.18 mm, Z = 20, b = 25.4 mm). Figure 8. Comparison of single tooth stiffness by using
different methods (module = 3.18 mm, ad =ac = 20, Z = 20,
b = 25.4 mm).
contact, the mesh stiffness is directly equal to
Ksingle =Kı. When the gears are in the double contact
zones, the gear pairs are considered as springs con- Gears with asymmetric teeth for which the coast-side
nected in parallel. Thus, the mesh stiffness for the dou- angle was constant at 20 and drive-side angle varied
ble tooth contact zone was calculated as from 20 to 32 were investigated in terms of gear mesh
K =K +K . stiffness in case 1. The drive-side pressure angle wasdouble ı ıı
10 Advances in Mechanical Engineering
Figure 9. Comparison of mesh stiffness by using different
Figure 11. Effect of drive-side pressure angle on gear mesh
methods (module = 3.18 mm, ad =ac = 20, Z = 20, b = 25.4 mm). stiffness for case 2 (ac = 25, ad = 25–28–30).
varied from 25 to 30were investigated in terms of
gear mesh stiffness in case 2. This type of gear is espe-
cially used in the United States. The mesh stiffness var-
iations of the gears were obtained from the enhanced
equation (20). In this case, the results were similar to
case 1. When the drive-side pressure angle increased,
the mesh stiffness of the gears also increased (Figure
11), although not as much as in case 1, because the ref-
erence gear pairs (ac=25, ad=25) in case 2 were
stiffer than the reference gear pairs (ac=20, ad=20)
in case 1. Also, in case 2, it was clearly seen that the
augmentation of the drive-side pressure angle reduced
the gear contact ratio.
In case 3, a different type of gear pair was investi-
gated. Both coast-side and drive-side pressure angles
Figure 10. Effect of drive-side pressure angle on gear mesh (ac, ad) were the smallest gear type in this study. Thus,
stiffness for case 1 (ac = 20, ad = 20–25–32). the lowest stiffness values were gained in case 3.
However, the characteristics of the mesh stiffness were
the same as in the other cases, in which the mesh stiff-
found to be an effective parameter on the mesh stiff- ness of the gear pair increased when the drive-side pres-
ness. When the gear drive-side pressure angle increased, sure angle increased. The amount with which the mesh
the root of the gear also increased; thus, the single gear stiffness increased according to the drive-side pressure
tooth stiffness increased. When the tooth pressure angle angle was greatest in case 3 (Figure 12).
on the drive side was higher than 20, it became stiffer Case 4 is similar to case 1 but the coast-side pressure
than the symmetric gear. When the gear pressure angle angle (ac) increased from 20 to 30 and the drive-side
increased, the contact stiffness increased gradually. In pressure angle (ad) was held constant at 20. In this
Figure 10, three different gear pairs are given. When case, the effect of the coast side on the mesh stiffness
the drive-side pressure angle increased, both the single was investigated. The results were also similar to case 1
and the double zone mesh stiffnesses increased. but with slight differences. In Figure 13, the mesh stiff-
Furthermore, the double contact zones become shorter ness of the gear pairs could be seen. When the coast-
and the single tooth contact zone becomes larger. This side pressure angle increased, the mesh stiffness also
means that the contact ratio was also affected by the increased, although not as much as in case 1. Moreover,
drive-side pressure angle. When the drive-side pressure the contact ratio decreased when the coast-side pressure
angle increased, the contact ratio decreased. angle increased. The increased stiffness and low contact
Gears with asymmetric teeth for which the coast-side ratio lead to an increase in dynamic force.
angle was held constant at 25 and the drive-side angle
Karpat et al. 11
gears indicate that FEA results and empirical equation
results are compatible with each other, with an accepta-
ble margin of error of 5% between the FEA and the
equations results. For the asymmetric teeth, when the
pressure angle increased (drive side or coast side) due
to an increase in the tooth thickness of any radius, the
single tooth stiffness increased slightly. Consequently,
the gear mesh stiffness increased. Also, the pressure
angle affected the contact ratio of the gear pair, which
decreased as the pressure angle increased because the
single tooth contact length increased as the pressure
angle on the drive side increased. The results obtained
from this study may provide important input for
designers. Because the dynamic gear loads are affected
by mesh stiffness, designers may use the mesh and
tooth stiffness results for the dynamic analysis of gears
Figure 12. Effect of drive-side pressure angle on gear mesh with asymmetric teeth.
stiffness for case 3 (ac = 18, ad = 20–24–26).
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial sup-
port for the research, authorship, and/or publication of this
article: This study was supported in part by Uludağ
University (The Scientific Research Project; project no:
OUAP (MH)-2014/25).
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