A computer programme to determine the bending and pitting

Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol.5, No.3, 2014
50
A Computer Programme to Determine the Bending and Pitting
Stresses of Gears and the Effect of Varying the AGMA Stress
Equation Parameters on the Stress Values
Stephen Kwasi Adzimah*
, Issifu Ibrahim
Department of Mechanical Engineering University of Mines and Technology, Tarkwa Ghana P. O Box 237,
Tarkwa – Ghana *
E-mail the corresponding author: skadzimah@yahoo.com
Abstract
Gears are one of the most critical components in mechanical power transmission systems. The bending and
surface strength of the gear tooth are considered to be the main contributors for the failure of the gears in a gear
set. Thus, analysis of these stresses has become imperative in the area of research on gears to reduce errors and
omissions in order to prevent the failures, and therefore, optimize the design of gears. The objective of this
project is to write a computer programme developed from Microsoft Excel and Matlab softwares, the application
of which is aimed at reducing the errors and omissions when calculating the bending and pitting stresses using
the AGMA methodology effectively, efficiently and quickly in the design and analysis of spur and helical gears.
The procedures employed include extraction of all the figures from the various graphs, obtaining an equation of
the graphs extracted using the curve fitting tool function in Matlab and using the graph equations together with
other equations of AGMA. The computer programme has been tested successfully and it has been established
that its application is capable of determining the bending and pitting stresses of any spur and helical gears and
their corresponding safety factors using the AGMA methodology.
Keywords: Gears; Transmission system; Gear set; Bending and surface strength.
1. Introduction
The crucial requirement of effective power transmission in various machines, automobiles, elevators, generators,
ships, aircrafts, etc, has created an increasing demand for more accurate analysis of the characteristics of gear
systems. This rapid development of these heavy machines requires advanced application of gear technology
(Alemu, 2007).
From the design viewpoint, failure by bending of the teeth as well as pitting failure of tooth surfaces are the most
important criteria when designing a gears et because each gear tooth may experience billions of load cycles
(Stoker etal., 2010).
The manual method of analyzing and designing of spur and helical gears such as using a pen and a paper come
with a lot of inefficiencies apart from it being time consuming when the AGMA methodology is employed. This
increase in errors and spending so much time on calculating these stresses has therefore made it essential to have
a computer software that will be used to calculate the stresses effectively, efficiently and quickly.
2. American Gears Manufacturers Association (AGMA) Standpoint and Contribution to Gear
Manufacturing
2.1 Information about AGMA
AGMA is a voluntary association of companies, consultants and academicians with a direct interest in the design,
manufacture, and application of gears and flexible couplings (Anon., 2012). AGMA was founded in 1916 by
nine companies in response to the market demand for standardized gear products; it remains a member- and
market—driven organization to this day. AGMA provides a wide variety of services to the gear industry and its
customers and conducts numerous programs which support these services.
Some of these services and programs are
Standards: AGMA develops all U.S. gear related standards through an open process under the authorization of
the American National Standards Institute (ANSI) (Anon., 2012).ISO Participation: AGMA is Secretariat to
TC60, the committee responsible for developing all international gear standards. TC60 is an ISO (International
Organization of Standardization) committee.
Market Reports and Statistics: AGMA’s Operating Ratio Report, Wage & Benefit Survey, and Monthly Market
Trend Reports help you stay competitive by giving you up-to-date information on the gear industry (Anon.,
2012).

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The Marketing and Statistical Councils enhance your competitiveness by sharing information and by developing
creative solutions to common industry problems.
Gear Expo: This is the only trade show dedicated solely to the gear industry.
The AGMA Training School for Gear Manufacturing uses current technology to offer.
News Digest, AGMA’s quarterly newsletter, offers you timely, useful information you can use immediately
(Anon., 2012).
2.2 Gears under Consideration: Spur and Helical Gears
2.2.1 General Spur Gears
Spur gears are the most commonly used gear type. These gears have teeth parallel to the axis of the wheel as
shown in Fig.2.1 (Khurmi and Gupta, 2008). Spur gears are by far the most commonly available, and are
generally the least expensive. The basic descriptive geometry for a spur gear is as shown below.
Fig. 1 Spur Gear Fig. 2 Helical Gear
(Source: Suman, 2003) (Source: Suman, 2003)
2.2.1.1 Limitations
Spur gears generally cannot be used when a direction change between the two shafts is required.
2.2.1.2Advantages
Spur gears are easy to find, inexpensive, and efficient.
2.2.2 General Helical Gears
Helical gears are similar to the spur gear except that the teeth are at an angle to the shaft, rather than parallel to it
as in a spur gear. The resulting teeth are longer than the teeth on a spur gear of equivalent pitch diameter. The
longer teeth cause helical gears to have the following differences from spur gears of the same size:
Tooth strength is greater because the teeth are longer,
Greater surface contact on the teeth allows a helical gear to carry more load than a spur gear
The longer surface of contact reduces the efficiency of a helical gear relative to a spur gear.
Helical gears may be used to mesh two shafts that are not parallel, although they are still primarily use in parallel
shaft applications. Helical gears run quieter and have a greater strength hand capacity than spurs (Gretchen,
2001). The basic descriptive geometry for a helical gear is essentially the same as that of the spur gear, except
that the helix angle must be added as a parameter (Suman, 2003).
Limitations: Helical gears have the major disadvantage that they are expensive and much more difficult to find.
Helical gears are also slightly less efficient than a spur gear of the same size (see below).
Advantages of helical gears: Helical gears can be used on non-parallel and even perpendicular shafts, and can
carry higher loads than can spur gears (Suman, 2003).
2.3 Modes of Gear Failure
Gear failure can occur in various modes. If care is taken during the design stage itself to prevent each of these
failure a sound gear design can be evolved. The gear failure is explained by means of flow diagram in Fig.3.

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Fig. 3 Different Modes of Failure
(Source: Gopinath and Mayuram, 2010)
2.3.1 Scoring
Scoring is due to combination of two distinct activities: First, lubrication failure in the contact region and second,
establishment of metal to metal contact. Later on, welding and tearing action resulting from metallic contact
removes the metal rapidly and continuously so far as the load, speed and oil temperature remain at the same level
(Gopinath and Mayuram, 2010).
2.3.2 Wear
As per gear engineer’s point of view, the wear is a kind of tooth damage where in layers of metal are removed
more or less uniformly from the surface. It is nothing but progressive removal of metal from the surface.
Consequently, tooth thins down and gets weakened. Three most common causes of gear tooth wear are metal-to-
metal contact due to lack of oil film, ingress of abrasive particles in the oil and chemical wear due to the
composition of oil and its additives (Gopinath and Mayuram, 2010).
2.3.3 Pitting of Gears
Pitting is a surface fatigue failure of the gear tooth. It occurs due to repeated loading of tooth surface and the
contact stress exceeding the surface fatigue strength of the material. Material in the fatigue region gets removed
and a pit is formed. The pit itself will cause stress concentration and soon the pitting spreads to adjacent region
till the whole surface is covered. Subsequently, higher impact load resulting from pitting may cause fracture of
already weakened tooth. However, the failure process takes place over millions of cycles of running. There are
two types of pitting, initial and progressive (Gopinath and Mayuram, 2010).
2.3.4 Initial / Incipient Pitting
In the helical gear shown in Fig.4 pitting started as a local overload due to slight misalignment and progressed
across the tooth in the dedendum portion to mid face. Here, the pitting stopped and the pitted surfaces began to
polish up and burnish over. This phenomenon is common with medium hard gears. On gears of materials that run
in well, pitting may cease after running in, and it has practically no effect on the performance of the drive since
the pits that are formed gradually become smoothed over from the rolling action. The initial pitting is non-
progressive.

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Fig. 4 Initial Pitting
2.3.5 Progressive or Destructive Pitting
During initial pitting, if the loads are high and the corrective action of initial pitting is unable to suppress the
pitting progress, then destructive pitting sets in. Pitting spreads all over the tooth length. Pitting leads to higher
pressure on the unpitted surface, squeezing the lubricant into the pits and finally to seizing of surfaces.
Fig.5 Tooth Surface Destroyed by Extensive Pitting Fig. 6 Whole Tooth is Destroyed by Extensive Pitting
(Source: Gopinath and Mayuram, 2010) (Source: Gopinath and Mayuram, 2010)
Tooth faces are subjected to pitting only in rare cases. Fig. 2.5 shows how in destructive pitting, pitting has
spread over the whole tooth and weakened tooth has fractured at the tip leading to total failure as in Fig.6.
2.3.6 Flaking / Spalling
In surface-hardened gears, the variable stresses in the underlying layer may lead to surface fatigue and result in
flaking (spalling) of material from the surface as shown in Fig.2.7
(a) (b)
Fig.7 Flaking / Spalling
2.3.7 Pitting - Subsurface Origin Failure
Fig.8 shows the subsurface origin failure.

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Fig.8 Subsurface Origin Failure Fig.9 Surface Origin Failure
2.3.8 Pitting - Surface Origin Failure
Failure modes in gear namely the surface origin failure is shown in Fig.9.
2.3.9 Progressive Pitting
The progressive pitting is shown in Fig.10.
(a) (b)
Fig. 10 Progressive pitting
2.3.10 Pitting - Frosting
Frosting usually occurs in dedendum portion of the driving gear first and later on the addendum as shown in
Fig.11. The wear pattern doesn’t have normal metal polish but has etched-like finish.
Fig.11 Frosting Fig.12 Plastic Flow - Cold Flow
Under magnification, surface reveals very fine micro-pits of 2.5µm deep. These patterns follow the higher ridges
caused by cutter marks. Frosting results from very thin oil film and some asperity.
2.3.11 Pitting Failure
Surface endurance strength determines the selection of dimensions and material for almost all gearing operating
under conditions of the best possible lubrication.
2.3.12 Plastic Flow – Cold Flow

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Plastic flow of tooth surface results when it is subjected to high contact stress under rolling cum sliding action.
Surface deformation takes place due to yielding of surface or subsurface material. Normally, it occurs in softer
gear materials. But it can occur even in heavily loaded case hardened gears. Cold flow material over the tooth tip
can be seen clearly in the bevel gear shown in the Fig. 12.
2.3.13 Tooth Fracture
Tooth fracture is the most dangerous kind of gear failure and leads to disablement of the drive and frequently to
damage of other components (shafts, bearings, etc.) by pieces of the broken teeth. Tooth breakage may be the
result of high overloads of either impact or static in nature, repeated overloads causing low-cycle fatigue, or
multiple repeated loads leading to high cycle fatigue of the material.
2.3.13.1 Tooth Breakage – Bending Fatigue
Bending fatigue failure occurs over a long period of time. The initiation of crack takes place at the weakest point,
normally at the root of the tooth or at the fillet where high stress concentration exists together with highest
tensile stress from bending or from the surface defects as shown in Fig.13. The crack slowly propagates over 80
to 90% of the life.
Fig. 13 Root Crack Fig. 14 Tooth Breakage
Then crack propagates fast and suddenly results in fracture of the tooth as shown in Fig. 2.14. The fractured
surface will exhibit beach marks in the slow crack propagation region and brittle fracture behavior in sudden
fracture region. Since time taken for the failure is very long, it is known as high cycle fatigue.
2.3.13.2 Tooth Breakage – High Cycle Fatigue
The tooth breakage in case of high cycle fatigue is shown in Fig..15.
Fig. 15High Cycle Fatigue
2.3.13.3 Tooth Breakage – Low Cycle Fatigue (Over Load)
Overload breakage or short (low) cycle fatigue causes stringy fibrous appearance in broken ductile material. In
harder materials this break has a more silky or crystalline appearance as shown in Fig.16.

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Fig.16 Low Cycle Fatigue (Over Load) Fig. 17 Bending Fatigue
2.3.13.4 Tooth Breakage – Bending Fatigue
Fig.17 shows tooth fatigue by bending fatigue.
2.3.13.5 Tooth Breakage
Breakage is often due to load concentration along the tooth length as a result of errors in machining and
assembly or of large elastic deformation of the shafts; tooth wear leading to weakening of the teeth results in
increased dynamic loads. Shifting of sliding gears into mesh takes place without stopping the rotation of the
shafts. Cracks are usually formed at the root of the teeth on the side of the stretched fibers where the highest
tensile stresses occur together with local stresses due to the shape of the teeth. Fracture occurs mainly at a cross
section through the root of the teeth.
In the case of fatigue failure, the fracture is of concave form in the body of the gear; it is of convex form when
the failure is from overload. The teeth of herringbone or wide-face helical gears usually break off along a
slanting cross section. To prevent tooth breakage, the beam strength of the gear teeth is checked by calculations.
Fatigue pitting of the surface layers of the gear teeth is the most serious and widespread kind of tooth damage
that may occur in gears even when they are enclosed, well lubricated and protected against dirt (Gopinath and
Mayuram, 2010).
2.4 Analytical Methods of Gear Design
2.4.1 The Lewis Bending Equation
The design of gear strength is based on two models: bending stress and contact stress models. The former is
related to the stress at the gear base, while the latter is related to the wear at the contact surface. The bending
equation was introduced by Wilfred Lewis in 1892. Since then, this equation has remained the standard for gear
design (Budynas and Nisbett, 2011; Dudley,1984). Lewis calculated stress in the gear base using a cantilevered
beam under an applied bending moment. By using this simple model, an accurate bending stress could be
determined (Budynas and Nisbett, 2011). This calculation assumes that the load is applied at the location of the
pitch radius (Colbourne, 1987). In practice, however, contact between the gear and pinion occurs at various
locations during the rotation.
To derive the basic Lewis equation, refer to Fig.2.18, which shows a cantilever of cross-sectional dimensions F
and t, having a length l and a load Wt, uniformly distributed across the face width F. The section modulus I/c is
Ft 2/6, and therefore the bending stress is
(1.a)

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Fig. 2.18Gear Tooth as Cantilever Beam
(Source: Budynas and Nisbett, 2011)
Referring to Fig. 18, we assume that the maximum stress in a gear tooth occurs at point a. By similar triangles,
we can write
(1.b)
By rearranging Eq. (2.a)
(1.c)
If we now substitute the value of x from Eq. (2.b) in Eq. (2.c) and multiply the numerator and denominator by
the circular pitch p, we find
(1.d)
Letting y = 2x/3p, we have
(2)
This completes the development of the original Lewis equation. The factor y is called the Lewis form factor, and
it may be obtained by a graphical layout of the gear tooth or by digital computation (Budynas and Nisbett, 2011).
2.11.2 AGMA Stress Equations
Two fundamental stress equations are used in the AGMA methodology, one for bending stress and another for
pitting resistance (contact stress). In AGMA terminology, these are called stress numbers, as contrasted with
actual applied stresses, and are designated by a lowercase letter s instead of the Greek lower case σ. (Budynas
andNisbett, 2011).
The fundamental equations are
(SI unit) (3)
where (for SI units),
Wt
is the tangential transmitted load, (N); KOis the overload factor; Kvis the dynamic factor
Ksis the size factor; F (b) is the face width of the narrower member, (mm)
KHis the load-distribution factor; KB is the rim-thickness factor; (mt)is the transverse metric module. YJis the
geometry factor for bending strength (which includes root fillet stress-concentration factor Kf)
The fundamental equation for pitting resistance (contact stress) is
σ= (S.I)
(4)
where Wt, KO, Kv,Ks, Km, F, and b are the same terms as defined for Eq. (2.2).
For SI units, the additional terms are
ZEis an elastic coefficient,(√N/mm2);ZRis the surface condition factor
dw1is the pitch diameter of the pinion, (mm); ZIis the geometry factor for pitting resistance

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2.12 Matlab Programming Language
Matlab is a high-level language and interactive environment for numerical computation, visualization, and
programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications.
The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution
faster than with spreadsheets or traditional programming languages, such as C/C++ or Java (Anon., 2013). Fig
2.19 shows the interface of the Matlab programming language.
Fig.19 Interface of the Matlab Programming Language, IDE
(Source: Anon., 2013)
3. Application Architecture, its Development and Installation
3.1 AGMA Stress Calculator Tool (ASCT)
The AGMA Stress Calculator Tool (ASCT) is a software application that determines the bending and pitting
stresses of spur and helical gears and their corresponding factors of safety. This was designed using Matlab
R2012a, Microsoft excel and Microsoft word.
3.2 Application Architecture
The application is structured to operate with a little user interactions by taking off most of the manual processes
and automating them behind the scene, thereby reducing errors caused by human interactions.
The flow chart below shows the system architecture of the application. It shows a quick way for understanding
the development process for the system.

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A look at the system architecture above shows that the only input required by the application is the data given
which are loaded via the application interface. A click on “Solve variables” on the application interface displays
all the results.
3.2.1 Data Input for ASCT
The minimum input data required are: power (kW), pinion speed (rev/min), Brinell hardness material selection,
quality of gears, pinion life, reliability, diameter gears (mm) and teeth, module (mm), face width (mm) and type
of mesh. Fig. 21 shows the part of the software for loading data
3.2.2 Output of ASCT
The ASCT displays two main results. Firstly, it displays all the stress parameters which comprise the transmitted
load, reliability factor, dynamic factor, geometry factor, pitch-line velocity, etc. as shown on the left hand side in
Fig.21 below. Then finally, the pinion and the gear parameters which comprise of the bending and pitting stresses
and their corresponding factors of safety are also displayed.
3.3 Development of the Application Interface
Microsoft word was used to design the interface f the AGMA stress calculator tool (ASCT) as shown in Fig.21. It
was then developed with Matlab programming language using the GUIDE tools. The basic controls used for the
interface includes, Edit text, List boxes and button controls. The User Interface controls were bound to series of
script and functions that solve for all the parameters that will be used to determine the bending and pitting
stresses and their corresponding factors of safety.
Fig. 21 Part of the Software for Loading Data
Fig.20 Flow Chart of the Application Architecture

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Fig.22 The Software Interface
3.4 Installation of the Application Interface
As introduced earlier, the application (ASCT) was engineered in Matlab using the Matlab functions. After the
development of the application, it was built and packaged into windows executable file (i.e. .exe) which can be
run on any windows OS. The application however cannot run on non-windows OS like Mac OSX, Linux, etc.
The steps for using the application are outlined as follows;
The setup file must be run by the user for the installation of the software to be executed.
Upon installing it, an executable file and an icon is placed on the desktop which when opened, launches
the executable file from the Application folder of the target machine.
The application is called forth by the user by running the application icon file found on the desktop after
installing the application. On running it, the application launches the interface with the input data for
loading as shown in Fig 22.
4. Calculation Analyses
4.1 Solution of a Sample Question to be Compared with that of the Software
4.1.1 Sample Question
A 20o
spur pinion with 70 mm diameter and a module of 2.5 mm transmits 15 kW to a 210 mm diameter
gear. The pinion speed is 1200 rev/min, number of teeth of pinion and gear are 28and 84 respectively
and the gears are grade 1, 21 mm face width, through-hardened steel at 200 Brinell, uncrowned,
manufactured to a No. 10 quality standard, and considered to be of open gearing quality installation.
Find the AGMA bending and contact stresses and the corresponding factors of safety for a pinion life of
109
cycles and a reliability of 0.98.
All Solved Variables Solved Stress and Safety Values
Input Values
from Question

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4.1.2 Long - hand Solution
Data summarized from the question; Pressure angle, Фt=20o
; Module, m=2.5mm (0.0025m)’ Grade 1 gears,
through-hardened steel; Face width = b = 21mm (0.021m); Brinell hardness, HB=200; Quality number,
Qv=10; Open gearing installation; Pinion life=109
cycles; Pinion speed, np=1200rpm; Power, H=15kW;
Pinion diameter, dp=70mm (0.07m);
Gear diameter, dG=210mm (0.21m); Number of teeth of pinion, NP=28 teeth;
Number of teeth of gear, NG = 84 teeth
General stress equations
J
BH
t
SVO
t
Y
KK
bm
KKKWStressesBending
1
).( =σ (5)
Iw
RH
SVO
t
E
Zbd
ZK
KKKWZstressccontact
1
().(
1
=σ (6)
V
H
WLoaddTransmitte t
=,.
(7)
60
pp dn
V
π
=
(8)
where, V is the pitch-line velocity
60
)07.0)(1200(π
=V
; V=4.3982m/s; 3982.4
1015 3
x
W t
=
;Wt
=3410.4861N; Overload factors, KO
Assuming uniform power source and uniform driven machine, KO=1; ;Dynamic factor, Kv
B
A
VA
Kv )
200
(
+
=
(9)
A=50+56(1-B) and B=0.25(12-Qv (10)
Substituting the quality number Qv into the B equation and solving yields. A=50+56(1-0.3969) and B=0.25(12-
10 ; A=83.7736 and B=0.3969
Then we have
3969.0
)
7736.83
)3982.4200(7736.83
(
x
Kv
+
=
=1.1278
Size factor, KS; AGMA suggests KS=1 (Budynas and Nisbett, 2011), therefore KS= 1
Load distribution factor,KH;; KH =1+Cmc (CpfCpm+CmaCe) (11)
Cmc= 1, (uncrowned teeth); Cpf = b≤ 25mm= (12)
Cpf =0.005; Cpm=1 (straddle mounted pinion)Cma =A+BF+CF2
; (13)
A=0.247, B=0.0167, C=-0.765(10-4
); F= = =0.8268inches
Cma= 0.247 + 0.0167(0.8268) - 0.765(10-4
) (0.82682
); Cma=0.2608; Ce=1
KH = 1+1(0.005×1+0.2608×1), KH =1.2658; Rim thickness
Since the backup ratio is not known we assume KB=1; Bending-strength geometry factor, YJ
(YJ)p = 0.38 (from Fig.14-6in Budynas and Nisbett, 2011)
(YJ)G= 0.44 (from Fig.14-6in Budynas and Nisbett, 2011)
Elastic coefficient; ZE=191 (from Table 14-8 in Budynas and Nisbett, 2011)
Surface condition factor
Standard surface conditions for gear teeth have not yet been established. When a detrimental surface finish effect
is known to exist, AGMA specifies a value of ZR greater than unity (Budynas and Nisbett, 2011)..Therefore,
ZR=1; Surface strength geometry factor, Zi

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Speed ratio ,mG= = =3;
12
sincos
+
=
G
G
n
tt
I
m
m
x
m
Z
φφ
(For external gears) (14)
=
13
3
)1(2
20sin20cos
+
x
; ZI=0.1205
Bending strength, St (Grade 1 through –hardened steel); St=0.533HB+88.3 (15)
=0.533(200) +88.3; St=194.9 MPa; Stress cycle factors, YN and ZN
(YN)=1.3558N-0.0178
(for both pinion and gear) (16)
(YN) p=1.3558(109
)-0.0178
(YN) p=0.8999; (YN)G =1.3558( ; YN)G=0.91767
(ZN)=1.4488N-0.023
(for both pinion and gear) (17)
(ZN) p=1.4488(109
)-0.023
=0.85311; (ZN)G=1.4488( =0.87494; Hardness-ratio factor, Zw
ZW=1, ( )<1.2(from Budynas and Nisbett, 2011)
Contact-fatigue stress (grade 1, through-hardened steel); Sc=2.22(HB) +200 (18)
Sc=2.22(200) +200; Sc=644 MPa
Reliability factor; Yz=0.658-0.0759ln (1-R) (19)
Yz=0.658-0.0759ln (1-0.98); Yz=0.9549
Temperature factor, Yθ
For oil or gear-blank temperatures up to 250°F (120°C), use KT = Yθ = 1.0. For higher temperatures, the factor
should be greater than unity. Heat exchangers may be used to ensure that operating temperatures are
considerably below this value, as is desirable for the lubricant.(Budynas and Nisbett, 2011)..Therefore Yθ=1
Bending stresses for pinion, from equation (5)
(σ)p= ; (σ)p = 244.0454 MPa
Bending stresses for Gear
(σ)G= ; (σ)G = 210.766 MPa
Corresponding factor of safety for the pinion
SF= = = 0.7526 (20)
Corresponding factor of safety for the gear
9262.0
9549.0766.210
9177.09.194
=
×
×
=
××
×
=
Z
Nt
F
YY
YS
S
θσ
Pitting stresses for the pinion
(σc)= = 1001.4 MPa
Pitting stresses for the gear
(σC)G= = 1001.4 MPa
Corresponding factor of safety for the pinion
Zc
WNC
H
YY
ZZS
S
θσ
=
=
Corresponding factor of safety for the gear
Zc
WNC
H
YY
ZZS
S
θσ
=
= = 0.621607 (21)

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63
4.1.3 ASCT Solution
The software was used to solve the same example as above; these are the results captured in Fig.22
Fig.23 The Software Displaying the Results
Table 1 Results of the Solution of the Sample Question Compared with that of the Software
Solved Parameters MANUALLY ASCT
Transmitted Load, Wt 3410.4861 3410.4631
Pitch Line Velocity,V 4.3982 4.3982
Dynamic Factor,Kv 1.1278 1.1278
Load Distribution Factor,KH 1.2658 1.2658
Pinion Bending-Strength geometry factor,Yj 0.38 0.3818
Gear Bending-Strength geometry factor,Yj 0.44 0.43632
Bending strength, St (Grade 1) 194.9 194.9
Pinion Stress cycle factors, YN 0.9000 0.8999
Gear Stress cycle factors, YN 0.91767 0.91767
Pinion Stress cycle factors, ZN 0.85311 0.85312
Bending stresses for pinion 244.0454 242.887
Bending stresses for Gear 210.766 212.538
Bending factor of safety for the pinion 0.7526 0.756198
Bending factor of safety for the gear 0.9262 0.881243
Pitting stresses for the gear 1001.4 1001.24
Pitting factor of safety for the pinion 0.57455 0.574628
Pitting factor of safety for the gear 0.621607 0.589332
5 Conclusions and Recommendations
5.1 Conclusion
In conclusion, the following has been achieved:
An effective software for determining the bending and pitting stresses for spur and helical gears has
been developed.
Accurate results will be achieved when determining bending and pitting stresses of spur and helical
gears.
A lot of time will be saved in selecting parameters for a gear design.

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A system for studying the effects of varying one gear parameter on other parameters.
5.2 Recommendation
The application should be further developed to include the determination of the bending and pitting
stresses of bevel and worm gears since this project only focused on spur and helical gears.
It is strongly recommended for gear designers, lecturers and the like to use this software whenever
dealing with the bending and pitting stresses of spur , helical, bevel and worm gears, which are the basic
gears in operation and also for more complex gears like hypoid, spiral, novikoff and others.
6 References
Alemu, N, (2007),” Analysis of Stresses in Helical Gears by Finite Element Method”,
Published MSc Thesis,School of Graduate Studies of Addis Ababa University, 64pp.
Anon. (2012), Technical Publications Catalog, American Gear Manufacturers
Association, 1001 N. Fairfax Street, 5th Floor Alexandria, VA 22314—1560, 75pp
Anon. (2013), “Matlab”,Matlab Central,www.mathworks.com/products/matlab/.Accessed: March 14, 2013.
Budynas, R.G. and Nisbett, J.K. (2011), Shigley's Mechanical Engineering Design, 9th
edition, McGraw-Hill Company Publishers Inc. 1221 Avenue of the Americas, New York, NY 10020, 1088
pp.
Colbourne, J.R. (1987), The Geometry of Involute Gears, Springer- Verlag, pp.5-8
Dudley, D.W. (1984), Handbook of Practical Gear Design, McGraw-Hill Company
Publishers Inc., pp. 10-20
Gopinath, K.and Mayuram, M.M. (2010),"Machine Design II", Published Lecture Notes,
Indian Institute of Technology, Madras, 20 pp.
Gretchen, R. (2001),"Gear Manual", Published by Martin Sprocket and Gear, Inc., 62pp
Khurmi, R, S. and Gupta, J.K. (2008), A Text Book of Engineering Mechanics, S.Chand
and Company Publishers, Ram Nagar, New Delhi-110055, 1230pp.
Stoker, K, Chaudhuri, A and Kim, N.H,“Safety of Spur Gear Design under Non-Ideal
Conditions with Uncertainty”,Proceedings of the ASME International Design Engineering Technical
Conferences and Computers and Information in Engineering Conference, IDETC/CIE, August 15-18, 2010,
Montreal, Quebec, Canada pp. 1-5
Suman, M.M (2003), “Gear Classification”
http://www.mae.ncsu.edu/eischen/courses/mae415/docs.pdf,13pp.
APPENDIX
Code Used for the Application of the Software Interface
The code used for this project which was written in Matlab R2012a is as shown below. The code is to
calculate the bending and pitting stresses of spur and helical gears and their corresponding safety factors.
% --- Executes on button press to solve variables.
Function solve variables_Call back(h Object, event data, handles)
%0h Object handle to solve_variables (see GCBO)
% event data reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA
global ZE A B C YJpYJgZEgZEp Ko YNpYNgZNpZNg mg ZI phi gear_type V Wt ...
Kv Cmc Cma Cpm Cpf Ce F KH KB mt Zr Kr dw1 Ks pBSgBSpPSgPS grade gStpSt ...
YtgScpSc Zw pBSFgBSFpSHgSH
% k = (get(handles.listbox1,'String'));
K = cell(0,1);
k2=cell(0,1);
selected_row=get(handles.gear_condition,'Value');
allstrings=get(handles.gear_condition,'String');
in put value = all strings (selected_row);

Vol.5, No.3, 2014
65
% Read the CMA CONDITION TABLES
[A B C]=cma_condition(inputvalue{1:end});
% Read the Elastic coefficient (ZE)
mrow=get(handles.pinion_material,'Value');
mcol=get(handles.gear_material,'Value');
ZEg= Elastic_coefficient_ZE(mrow,mcol);
ZEg=ZEg * 1000;
ZEp=Elastic_coefficient_ZE(mrow,7);
% Read the Overload Factor (Ko)
mrow=get(handles.power_source,'Value');
mcol=get(handles.driven_machine1,'Value');
Ko = overlaod_factor_ko(mrow,mcol);
% Solve for the Geometry Factor of Pinion
pteeth=str2double(get(handles.pinion_teeth,'String'));
gteeth=str2double(get(handles.gear_teeth1,'String'));
YJp= yj(pteeth, gteeth );
YJg= yj(gteeth, pteeth );
% Read the mg value
pdiameter=str2double(get(handles.pinion_diameter,'String'));
gdiameter=str2double(get(handles.gear_diameter,'String'));
mg=mg_value(gteeth,pteeth,gdiameter,pdiameter);
% Solve for Stress Cycle Factors (bending) of Gear n Pinion
P Load Cycle=str2double(get(handles.pinion_life,'String'));
hb=str2double (get (handles.hb_p,'String'));
YNp=Stress_Yn (hb, pLoad Cycle);
YNg=Stress_Yn (hb, pLoad Cycle/mg );
%Solve for Pitting Stress Cycle Factor of Gear and Pinion
ZNp = Stress_ZN(pLoadCycle );
ZNg = Stress_ZN(pLoadCycle/mg );
% Solving for Surface Strength Geometry factor (ZI)
gear_type = get (handles.gear_type,'Value');
phi=str2double(get (handles.pressure_angle,'String'));
ZI=ZI value (gear type, mg, phi);

A computer programme to determine the bending and pitting

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