This document describes the design analysis of a spur gear transmission using an advanced computer program. It details the typical steps taken in gear analysis, including geometry calculations, force analysis, bending stress calculations, surface durability analysis, and fatigue analysis. Calculations are shown for gear selection, shaft design, and bearing lifetime. The computer program simplifies the standard mechanical calculation procedures and reduces errors. While not solving all reducer design problems, it demonstrates the advantages of using a computer program for strength computations versus tedious manual calculations.

1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/313869586
DESIGN ANALYSIS OF SPUR GEAR WITH THE USAGE OF THE ADVANCED
COMPUTER
Conference Paper · January 2003
CITATION
1
READS
10,294
2 authors:
Some of the authors of this publication are also working on these related projects:
COST Action: CA16216 - Network on the Coordination and Harmonisation of European Occupational Cohorts View project
Study on the potential and utilization of renewable energy sources in the cross-border region (South-East region in the Republic of Macedonia and South-West region in
the Republic of Bulgaria) View project
Tale Geramitcioski
University "St. Kliment Ohridski" - Bitola
77 PUBLICATIONS 86 CITATIONS
SEE PROFILE
Ljupco Trajcevski
University "St. Kliment Ohridski" - Bitola
20 PUBLICATIONS 18 CITATIONS
SEE PROFILE
All content following this page was uploaded by Tale Geramitcioski on 21 February 2017.
The user has requested enhancement of the downloaded file.

2. DESIGN ANALYSIS OF SPUR GEAR WITH THE
USAGE OF THE ADVANCED COMPUTER
Prof. D-r. Geramitcioski T1
, Ass.M-r. Trajcevski Lj.2
University"St. Clementius of Ohrid"- Bitola University"St. Clementius of Ohrid"- Bitola
Faculty of Technical Sciences, Faculty of Technical Sciences
Macedonia Macedonia
E-mail: tale.geramitcioski@uklo.edu.mk E-mail: papec@mt.net.mk
Abstract:
The design and calculation of a spur gear transmission, one of the often-performed mechanical engineering tasks, requires a lot of long
and complicated tedious computations. With today’s modern technology that process that could be easily simplified by the use of the
computer. However, the result for successful work of the computer programmer will be only as good as the accuracy of data of the input
information as well as the procedure of calculation, provided by mechanical engineering science.
In this paper, is developed the algorithm for design of the gear includes typical steps of analysis and stress calculation of the main
components, such as gears, shafts and bearings. Same different calculations, resulting from the special requirements and conditions of the
application’s exploitations can be also included in mechanical modeling procedure, as a choice. computer program writing in a Visual
C++ was also made. The computer program simplifying standard procedure of tedious mechanical computations is also provided. The
advantage of the time shortening for complicate reiterations and decreasing of the routine mistakes due to use of computer program is
evaluated.
KEY WORDS: GEAR, SHAFT, MODEL, CALCULATION
1.Introduction
The design procedure of the every model of a spur gear
transmission includes the following necessary steps:
1. Gear selection based on evaluation parameters and the gear-
tooth strength.
2. Shaft design with corresponding calculations of the shaft’s
strength and rigidity.
3. Calculation lifetime of the beagings.
In figure 1 is show a gear transmission. All calculations are based
on the three main distances. These distances, same for all shafts,
can be derived from the widths of the elements, installed on them,
such as gears hubs, retaining rings and bearings.
Figure1. Shematics of gear transmission
2. Calculation of gear
The selection of gears, as a major energy transmitting components,
requires serious attention. Even, in the most cases, gears are not
produced for a particular gearbox design and bought from
manufacturer, which specializes in the gear design thorough deep
computatioms are involved in the process of the selection. The
major portion of the spur gear train calculations would be
concerned with geometry and nomenclature, gear force analysis,
gear tooth bending stress, and gear durability.
2.1 Calculation of the Pinion and Gear Geometry
The main formulas for dimensions of standard spur gears are given
in Machinery’s handbook. The results of the geometrical
parameter’s calculations for 16 and 48-teeth geaars are presented in
table 1.
Table 1. Calculation of Gear and Pinion Geometry
2.2 Force and Stress Analysis
The procedure of the force and gear analysis for all gear involves
the same steps:
1. Calculation of rotation speed.
2. Calculation of tangentional velocity at the pitch circle
3. Determining torque.
4. Calculation of tangentional force at the pitch circle.
The rotation speed of the first gear is:
RPM1 = (HP*5252*12)/Tm = (10.5*5252*12)/174=3803.2rev/min
HP – horsepower of the motor, HP=10.5 hp
Tm- torque of the motor, Tm=174 lbs-inch
The tangentional velocity at the pitch circle by equation:
Vt1=(π*RPM1*Dp1)/12=(3.14*3803.2*1.6)/12=1593.1 fpm
Dp1-gear pitch diameter, Dp1=1.6
Torque of the first shaft is the same as torque of motor
Tt=14.5 lbs*ft = 174 lb*inch
The tangentional force at the pitch circle can be found from the
transmitted torque by equation:
Ft1=(2*Tt)/Dp1=(2*174)/1.6=217.5lb

3. From a simple observation of the particular schematics of the
described train is clear that rotational speed and torque of the
second and thirid gears, installed on the same shaft, are equal, as
well as the tangentional forces and the velocities at the pitch circle
of the matching gears. Therefore, rotational speed of the second
and third gears is:
RPM2=RPM3=RPM1*R=1267.7 rpm
R-gear ration of the first stage of reducer, R=0.33
Torque of these gears can be calculated by formula
T2=(T1*Dp2)/Dp1=(174*4.8)/1.6=522 lb-inch
Dp2-pitch diameter of the gear closest to support B; Dp2=4.8 inch
The tangentional velocities at the pitch circle of the second gear is
Vr1=Vr2=1593.1 fpm
The tangentional force at the pitch circle of the second gear is:
Ft1=Ft2=217.5 lb
The tangentional velocities at the pitch circle for the second and
third gears are equal and calculated by equation:
Vr3=( π*RPM3*Dp3)/12= π*1267.7*1.6)/12=531.0fpm
Dp3-gear pitch diameter,Dp3=1.6
Dp4-gear pitch diameter,Dp4=4.8
All results of calculation are tabulated in following table 2
Table 2. Force Analysis for Gear
2.3. Gear Tooth Bending Stress Calculation
The main question of the gear calculation is how much power or
torque a given pair of gears will transmit without tooth failure.
When set of spur gear is installed and lubrcated properly, they
normally may be subject to three primary modes of failure:
1. Tooth Scoring, that is a scuffing or welding type of tooth failure,
caused by high sliding speed combined with contact stress. Scoring
is not a faticue failure but rather a failure of the lubricant, caused
by increases in lubricant viscocity with pressure.Well-proportioned
commercial gears with a pitch line velocity 7000 fpm will normally
not score, if they have a reasonable-good surface finish and are
properly lubricated.
2. Tooth breakage that is usually tensile fatigue failure at the
weakest section of the gear tooth. The weakest point is normally
the tensile side of the gear tooth fillet, and it may be anticipated in
the gear design by determining the stress at this weakest section of
the gear tooth. Bending strength is measured in terms of the
bending (tensile) stress in a cantilever plate and is directly
proportional to this same load.
3. Pitting, caused by the small crakes first developed on and under
the surfaces of gear teeth as a result of metal failure. Failure usually
occurs at a point just below a pith surface of driving pinion and
may be anticipated in the gear design by determining of the gear set
contact compressive stress. The definition of the acceptable pitting
varies widely with gear application. The main goal of the pitting
resistance calculation is to determine a load rating at which
destructive pitting of the teeth does not occur during their design
life.
The problem of the gear tooth bending fatique requires an
evaluation of the fluctuation stresses in the tooth fillet and the
fatigue strength of the material in the same hihhly localized
location. The basic theory employed in the bending stress analysis
assumes the gear tooth to be rigidly fixed at its base, considered as
a cantilever beam with resultant force applied to the tip.
Calculation of the gear is made for following simplifying
assumptions:
1. The full load is applied on the tip of a single tooth.
2. The radial component Fr is negligible.
3. The load is distributed uniformly across the full-face width.
4. Forces that are due to tooth sliding friction are negligible.
5. Stress concentration in the tooth filler is negligible
6. The gear teeth have a machined surface.
7. The gear-teeth fillet area temperature is less than 160o
F.
8. The gears rotate in one direction.
9. The gear material is homogeneous, isotropic and completely
elastic.
10. Thermal and residual stresses are negligible.
The gear tooth bending fatigue analysis is the same for all gear and
may be illustrated on sample of the first gear. The gear tooth
bending stresses analysis might be done by equation presented by
Wilfred Lewis.
Sl=(Ft1*P)/(B*Yt)
Where:
Ft-tangential force, applied on the gear tooth;
P-dialetral pitch, same for all gears of reducer;
B-face width, same for all gears of reducer;
Y-form factor.
For the first gear Lewis stress is:
Sl1=(Ft1*P)/(B*Yt)=(217.5*10)/(1*0.254)=8562.9 psi
The orginal Lewis beam strength theory assumed a static loading of
the tooth. This appropriate for low speed operations only. To
account the speed and dynamic effects, the allowable Lewis beam
force is reduced by a speed factor.
In the Barth speed factor, calculated by formula:
Kd=A/(A+Vr)
For ordinary industrial gears the constant A is 600.
For the first shaft, the Barth speed factor is:
Kd=600/(600+1593.1)=0.273;
Regarding the dynamic effects, the bending stress in the tooth root
of the gear 1 become:
Sb1=(Ft1*P)/(Kd1*B*Yt)=(217.5*100)/(0.273*1*0.254=31298 psi
Although the Barth speed factor has been recommended for many
years, its use has been largely superseded by Buckingem equation:
where:
C-deformation constant, that depends on the error in action. For
class 3 gears with diametral pitch P>=6, error in action is 0.0005.
For steel gears with tooth form 14.5, C=800.
For found dynamic force the gear bending stress is:
S1=(Ft1*P)/(B*Yt)=(0.944*10)/(1*0.254)=37167 psi
Gear tooth bending strength S1=37167 psi is less than the
endurance limit of the given steel Sn=84750 psi. Consequently,
gear N1 satisfies the bending load.
Results of the bending stress calculations for all four gears are
shown in table 3:
Table 3. Bending Stress Calculation for Gears by Different
Methods
Sl-general bending stress, calculated by Lewis equation,
Sb-bending stress, calculated by Lewis equation, eith account of
the Barth speed factor,
S-bending stress, calculated by Lewis ecuation, with account of
dynamic force.
2.4 Calculation of the Gear Tooth Surface Durability
To evaluate the satisfaction of chosen gear-to-gear fatigue
durability conditions, the gear tooth surface fatigue strength has to
be compared with gear tooth surface fatigue stress.
The calculation of the gear tooth surface durability done with the
consideration of the following assumptions:

4. 1. The surface fatigue endurance limit can be calculated from the
surface hardness.
2. The surface fatigue stress is a maximum at the pitch point.
3. The manufacture quality of the pinion and gear corresponds to
precision shaved and ground.
4. Output of gears experience moderate torsional shock.
5. The characteristics of support include less rigid mounting, less
accurate gears, and contact across the full face.
6. The tooth profiles of the gears are standard involutes. Cylinders
can approximate the contact surface at the pitch point.
7. The gears are mounted to mesh at pitch circles.
8. The effect of surface failure from abrasive wear and scoring are
eliminated by enclosure and lubrication-only pitting needs
consideration.
9. The stresses caused by sliding friction can be neglected.
10. The contact pressure distribution is unaffected by the lubricant.
11. Thermal stresses and residual streses can be neglected. The
surface endurance limit and life factor data available are
sufficiently accurate. The velocity factor Kv, the overload factor
Ko, the mounting factor Km obtained from available data are
reasonable accurate.
2.5 Calculation of Gear Tooth Surface Endurance Limits
The strength ratings are based on plate theory that is modified to
consider:
1. The compressive stress on thooth roots caused by the redial
component of tooth loading.
2. Nonuniform moment distribution resulting from inclined angle
of the load lines of the teeth.
3. Stress concentration at the tooth fillets.
4. The load, sharing between adjacent teeth in contact.
The gear tooth fatigue stress might be evaluated by formula:
Sh=Sfe*Cli*Cr=290*1*1=290 ksi
Where
Sfe-surface fatigue strength. For steel gears,
Sfe=0.4*(Bhn)-10ksi=0.4*750-10=290ksi
Cli-life factor. For steel gears with surface fatigue life 10^7 cycles,
Cli=1;
Cr-reliability factor. For realibility=99%, Cr=1.
2.6 Gear-Tooth Fatigue analysis
The gear tooth fatigue might be estimated by the formula:
For the pinion installed on shaft 1, the gear tooth fatigue is:
where:
I-geometry factor. For gear and pinion diameters ratio R=3, I is
calculated by formula:
Kv1-velocity factor; for the manufacture quality of the pinion and
gear corresponding to precision shaved and ground, and velocity of
the pitch Vr1=1593.07 fpm, Kv1 might be calculate by formula:
Ko-overload factor. For moderate shock of driven machinery and
light shock of source of power Ko=1.5.
Km-mounting factor. For face width<=2 and less rigid mtg.,contact
across the full face Km=1.6.
Cp-elastic coefficient. For gears and pinion made from steel
Cp=2300 sqrt(psi)
The procedure of the surface fatigue calculation for all gears is the
same. For convenience, all results of the calculation surface fatigue
for four gears are tabulated:
Table 4. Surface Fatigue Calculation of the Gears
2.7 Listing of a Calculation of the Gear Tooth Strength and
fatigue

6. 3. Conclusion
The computer program is writing in a Visual C++. The first step of
the procedure is the force and stress analysis. This analysis
involves determination of the main external loadings applied to
components, such as forces, bending moment and calculation the
stresses. The found values of the external loading are further used
to determine consequent strength and fatigue analysis for gear
calculated components. Ideally, the main function of the computer
program for the gear design is of the components currently used by
the particular manufactureer and satisfying the criterion of strength
requirements. The program usually includes the main force, stress
and strength equations, dealing with material properties and the
different sets of relationships between stress concentration factors
and geometry of common stress risers. This program is written for
strength calculation of gear, was not solve all problems related to
the reducer design, but to demonstrate the simplified model of the
strength computation procedure for the gear.
However, despite having some programming imperfections, the
program is a good sample of the important advantages of the usage
of computer technology for the design process, such as the time
needed for tedious reiterations, for selecting required coefficients
and for performing various complicated calculations.
4. Literature
[1] Shigley, J. E., "Standard Handbook for Machine Design”,
McGraw-Hill, New York, 1986
[2] Baumeister T., and Avallone A., Marks “Standard Handbook
for Mechanical Engineers”, McGraw-Hill, New York, 1978
[3] Juvinal R.C., and Marsehek K.M., “Fundamentals of Machine
Component Design”, 3 ed. John Wiley, New York, 2000.
View publication stats
View publication stats