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COMPUTER AIDED ANALYSISOF INVOLUTE GEAR TOOTH

FOR MINIMIZATION OF

BENDING STRESS

UNDER THE GUIDANCE OF

Sri.P. SRINIVASASSOCIATE PROFESSORG.I.T.A.M COLLEGE OF ENGINEERING.VISAKHAPATNAM.

BY

P. RAMA KRISHNA

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CONTENTS ABSTRACT

INTRODUCTION OF GEARS

LITERATURE SURVEY

INTRODUCTION OF FEM

GEOMETRIC MODELLING AND ANALYSIS OF GEAR

TOOTH

RESULTS AND DISCUSSIONS

CONCLUSIONS

REFERENCES

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ABSTRACT

Sufficiency in bending load carrying capacity is a serious

problem, as regards the carburized or surface quality

improved gears with very high surface fatigue strength,

such as plastic and sintered gears.

There are several ways to solve the problem such as

heat treatments, improving tooth fillet surface quality,

and using a larger radius of cutters tip corner.

The load carrying capacity of transmissions can also

be increased by modifying the involute geometry.

However, an additional alteration though rarely used, is

to make the gears asymmetric with different pressure

angles for power side of the tooth. Contd.,

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The aim of asymmetric tooth is to improve theperformance of gears such as increasing the load carryingcapacity consequently decreasing the bending stressesthereby increasing the durability for the drive side of thetooth.

The involute profile of gear teeth is modeled and theanalysis part is carried by using ANSYS software.The effect of asymmetric gears with greater pressure angleon the drive side than coast side pressure angle has beenanalyzed.

In addition to the above modifications, it has also beenshown that bending stresses can also be reduced by alteringthe root fillet radius.

The decreasing of bending stress makes it possible tomanufacture small gears, in other words lighter ones.This advantage gives wider application of gears inaerospace and automotive industries.

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INTRODUCTION OF GEARS The strength of a gear depends on the bending stressof the gear teeth. The bending stress can be calculatedusing the most popular and dependable equation developedby Lewis and called after his name as Lewis Equation. This isused for finding the static strength of the gear tooth.

GEAR TERMINOLOGY

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FORMS OF TEETH

Different forms of teeth are used for the profile of gearteeth. They are

Cycloidal teeth

Involute teeth

Construction of cycloidal teeth of gear.

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Construction of involute teeth

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Advantages of Involute Gears

The centre distance for a pair of involute gears can be

varied within limits without changing the velocity ratio.

In involute gears, the pressure angle, from the start of the

engagement of teeth to the end of the engagement, remains

constant. It is necessary for smooth running and less wear

of gears.

The involute teeth are easy to manufacture than cycloidalteeth.

In involute system, the basic rack has straight teeth and

the same can be cut with simple tools.

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SYSTEMS OF GEAR TEETH

The following four systems of gear teeth are commonly used in

practice.

14

degree Composite system.14 degree Full depth involute

system.

20 degree Full depth involute system.

20 degree Stub involute system.

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INTERFERENCE IN INVOLUTE GEARS

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BEAM STRENGTH OF GEAR

TEETH LEWIS EQUATION

GEAR TOOTH UNDER LOAD

Ww = M. y / I

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LITERATURE SURVEYKadir Cavdar [1] has presented a method for thedetermination of bending stress minimization of involutespur gears

Alexander Kapelevich [2], suggested a method that

enables to increase the load capacity, reduce weight, sizeand vibration level and the findings of his research in thisdirection are as follows.

The asymmetric tooth geometry allows for an increase inload capacity while reducing weight and dimensions forsome types of gears. It becomes possible by increasing of

pressure angle and contact ratio for drive side.Faydor L. Litvin [3] discussed about asymmetric spur geardrives in which, according to them the asymmetry meansthat larger and smaller pressure angles are applied for thedriving and coast sides respectively.

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Alexander et al. [4] discussed bending stress minimizationby changing fillet radius and given the following conclusions.

Bending stress balance allows equalizing the tooth strengthand durability for the pinion and the gear. Optimization of

the fillet profile allows reducing the maximum bending stressin the gear tooth root area by 10 to 30 %. It work equallywell for both symmetric and asymmetric gear tooth profiles.

Roderick E. Kleiss, Dr. Alexander L. Kapelevich [5]discussed about an alternative method of analysis anddesign of spur and helical involute gears

Direct gear design allows analysis of a wide range of parameters for all possible gear combinations in order tofind the most suitable solution for a particular application.

This optimum gear solution can exceed the limits of traditional rack generating methods of gear design.

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Thomas M. Mc Namara, Alexander L. Kapelevich [6]discussed about an alternative method of analysis anddesign of involute gears which separates gear geometrydefinition from tool selection to achieve the best possible

performance for a particular product and application. Thismethod has successfully been applied for a number of automotive applications.

Direct gear design results in a 15-30% reduction in stresslevel when compared to traditionally designed gears. Thisreduction can be translated into, increased load capacity

(15-30%), Size and weight reduction (10-20%), longer life,Cost reduction, increased reliability, noise and vibrationreduction, increased Gear efficiency and Maintenance costreduction.

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GENERAL PROCEDURE FOR

SOLVING A PROBLEM BY FEM

Model Building

Discretization of the domain

Size of the element

Number of elements

Elements assumed

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CRITERIA OF FINITE ELEMENTS

SELECTION

The number and type of elements to be used isdependent on physical problem itself. The following areimportant considerations in selection of type of element.

The number and degrees of freedom required.

The expected accuracy.

Ease of developing the governing equation.

The degree to which the physical structure can be modeledwithout approximation.

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TYPES OF ELEMENTS

Truss & Beam Elements

Plane Stress & Plane Strain Element

Three Dimensional Elements

Analysis Types

Static Analysis

Dynamic Analysis

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GENERAL PROCEDURE OF THE FINITEELEMENT METHOD

Descretize the given continuum.

Select the solution approximation

Develop element matrices and equations

Assemble the element equations

Solve for the unknowns at the nodes

Interpreting the results

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GEOMETRIC MODELING ANDANALYSIS OF GEAR TOOTH

Alternative I

TO MINIMIZE THE BENDING STRESS IN THE GEARTOOTH, THE ASYMMETRIC GEAR TOOTH IS

CONSIDERED AS FIRST ALTERNATIVE DESIGN.

INPUT PARAMETERS

Gear Ratio = 10: 1

Distance between centers = 660 mm,No. of teeth on gear = 150,

Module (m) = 8 mm,

Pitch circle diameter = 1200 mm,

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GEOMETRIC MODELING ANDANALYSIS OF GEAR TOOTH

Theoretical calculations

The following formulae are used to calculate bending stress. Themaximum value of the bending stress (or the permissible workingstress), at the section BC is given by

Bending = M y / I

= ( WT * h ) t/2

b t*t*t / 12

= ( WT * h ) 6b t*t

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Theoretical calculations for 20-20.Force applied W = 50 N

Now Resolution of force in

Wt = W* Cos 20

= 50 * Cos 20 = 46.98 N

Wr = W * Sin 20

= 50 * Sin 20= 17.10 N

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Bending stress calculationBending stress equation

Bending = (Wt * h * 6) / ( b * t *t )

Here

h = 10 mm

b = 40 mm

t = 23.09 mm

Bending = ( 46.98 * 9.81* 10 * 6) / (40 * 23 * 23)

= 1.29 kgf / mm2

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Involute profile of gear teeth when

pressure angles are 20&20deg.

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Analysis of Involute profile of gear teeth

when pressure angles are 20&20deg.

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3D View:

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pressure angles are 20 &25deg.

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Analysis of Involute profile of gear teeth

when pressure angles are 20&25deg.

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pressure angles are 20&30deg.

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Analysis of Involute profile of gear teethwhen pressure angles are 20&30deg.

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Involute profile of gear teeth whenpressure angles are 20&35deg.

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Percentage change in bending stress

Pressure

Angle ( )in

degrees

Stress ( ) in

Kgf /mm2

% reduction

in b

20 -20 1.09 Reference

20 - 25 0.975 10.5%

20-30 0.874 20.18%

20-35 0.814 25.32%20-40 0.756 31.19%

20-45 0.629 42.29%

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Comparison of theoretical andcalculated stresses

S.N

O.

Pressure

angle ( )

in degrees

Theoretical

Stress(

Bending )Kgf/mm2

Stress shown by

ANSYS

Kgf/mm2

1 20-20 1.290 1.090

2 20-25 1.117 0.975

3 20-30 1.027 0.875

4 20-35 0.904 0.814

5 20-40 0.762 0.756

6 20-45 0.638 0.629

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Alternative II:Minimization of Bending stress in the gear

tooth by modifying the root fillet.

Keeping all parameters of alternative-1 except thefillet radii, an attempt is made to explore thepossibility of obtaining better bending stresses bysuitable curve fitting method. (Best curve fit method)

Bending stress minimization is the result of thedefinition of the fillet profile that provides minimumbending stress concentration and satisfies certainconditions (eg. manufacturability).

They are based on a curve fitting technique when the

trochoid filet profile, typical for the rack or matinggear generative method, is replaced by a parabola,ellipsis, chain line, or other curve reducing thebending stress.

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Involute profile of symmetric gear toothwith designed fillet:

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Analysis of involute profile of symmetricgear tooth with designed fillet.

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Involute profile of symmetric geartooth modified with circular root fillet:

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Analysis of involute profile of symmetricgear tooth modified with circular root fillet:

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Involute profile of symmetric gear toothmodified with Elliptical-1 root fillet:

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Analysis of involute profile of symmetricgear tooth modified with Elliptical-1 rootfillet:

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Involute profile of symmetric gear toothmodified with Elliptical-2 root fillet:

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Analysis of involute profile of symmetricgear tooth modified with Elliptical-2 rootfillet:

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Bending stresses at different root filetshape of 200 involute symmetric teeth.

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Results and Discussions:

Contd.,

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Contd.,

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Conclusions In the suggested alternative -1, in which the

asymmetric teeth are considered with increased pressureangle on the drive side from 200 - 200 to 200 - 450 , itis noted that the bending stresses in the gear tooth isreduced by 42.29 %. As the pressure angle on the drive

side is further increased beyond 450

the top land of thetooth is narrowed and finally resulted in a single linebecause of intersection of the two involute profiles oneither side of the tooth.

In the suggested alternative- 2, it is observed thatthe bending stresses are reduced by 48.51% when the

fillet radius is elliptical. From the two alternatives, it is found that the

alternative- 2 is better option to employ in themanufacture of gears for reduced bending stresses byemploying the modified fillet radius.

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REFERENCES

[1]Kadir Cavdar, Fatih Karpat and Fatih C. Babalik, Computeraided analysis of bending strength of involute spur gears withasymmetric profile.

Journal of mechanical design, May 2005, volume 127/477-484. [2] Alexander L. Kapelevich and Yuriy V. shekhtman, Direct

gear design: Bending stress minimization , Gear technology,sep/oct-2003, pg.No.s 44-48 [3]Alexander L. Kapelevich and Thomas M.Mc Namara, Direct

gear design for automotive applications, 2005 SAEInternational paper 05p-149.

[4]Alexander L. Kapelevich and Roderick E.Kleiss, Directgear design for spur and Helical Involute gears , Geartechnology, sep/oct 2002- pg.No.s29-36

[5] Alexander L. Kapelevich Geometry and design of Involutespur gears with asymmetric teeth .

[6]Faydor L. Litvin, Qiming Lian, Alexander L. KapelevichAsymmetric modified spur gear drives: reduction of noise,localization of contact, simulation of meshing and stressanalysis, Computer methods in applied mechanics andEngineering 188 (2000) pg.No.363-390.

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