Computer Aided stress analysis Of A Helical Gear pair and Spur Gear pair Using AGMA standards and comparison with CATIA Models.

GUIDE:Lect.Satyak Padhy

Abstract:
This paper introduces a new methodology to test a designed gear material before manufacturing. Since every design needs to be tested for its feasibility with regard to manufacturing material, operating conditions, loading etc. The program Discussed here accepts all parameters relevant to a gear, gear material, operating conditions. Finally the program produces the factor of safety and the critical stresses at portions exposed to deforming stress The problem is also applied to the finite element method so as to compare the stresses introduced in the gear tooth with that of generated values. Various failure conditions are taken care of by introducing parameters to tackle conditions like dynamic loading, irregularity in size and uneven load distribution. Finally conclusions are drawn on the basis of developed program and FEM analysis.

Based on AGMA standards a program is carried out to:
1 Obtain the critical bending and pitting stresses.
2. Acquire enough practice in the safe gear design and in the gear parameters calculation.
3. Obtain application specific information on safe gear material.
4. Compare the developed model with finite element analysis software (CATIA).

Keywords: spur gears, helical gears, CATIA

I. Introduction
Gears are the most used machine element in Modern mechanisms. Transmission of exact velocity ratio and high power transmission are one of the biggest asset of using a gear transmission .Gears are classified on the basis of position of shafts, peripheral velocity and type of gearing. The case presented here deals with the parallel, coplanar & medium velocity gears with external gearing arrangement i.e. spur gear and helical gear. A spur gear has teeth parallel to the axis of the wheel. Another name given to the spur gearing is the helical gearing, in which the teeth are inclined to the axis of the wheel.



Fig 1: screen shot of developed software


Helical Gears and spur Gears suffer from serious problem of Backlash it is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circle. Undercutting occurs under certain conditions when a small number of teeth are used in cutting a gear. These problems are taken care of while design parameter calculation of a gear .A very important part of design is selection of appropriate materials so as to meet the load conditions and the factor of safety expected. Perceiving the need many design factors have been introduced to compensate the errors. In the first phase the material strength of gear materials, Count of tooth, pressure angle and other necessary parameters were recorded into the program. This program helps us to analyze a situation specific problem. The program also allows us to calculate maximum power that could be transmitted with a given set of materials. The Program presents a new methodology to test gear materials before use and it could be applied to other situations with appropriate changes in source code.

All internal formulation is derived from the AGMA standards. The standards are subject to continual improvement, revision, or withdrawal as dictated by increased experience. Any person who refers to AGMA technical publications should be sure that he has the latest information available from the Association on the subject matter. Credit line should read: "Extracted from ANSI/AGMA #2001-C95 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, with the permission of the publisher, American Gear Manufacturers Association, Alexandria,
Virginia."

Fig: failure of helical gear due to choice of improper material

II. PROBLEM STATEMENT:

A 17-tooth 20º pressure angle spur pinion rotates at 1800 rev/min and transmits 3.0 KW to a 52- tooth disk drive. The module is 2.5, the face width 38.0 mm, and quality standard is no 6.the gears are straddle mounted with bearings immediately adjacent. The pinion is a grade 1 steel with a hardness of 240 brinnel tooth surface and through-hardened core. the gear is steel, through hardened also,grade1 material, with a brinnel hardness of 200,tooth surface and core. poissions ratio is 0.30,lewis form factor for pinion =0.30,
Lewis form factor for ear=0.40 and young’s modulus is 2(105) MPa.The loading is smooth due to motor and load. Assume a pinions life of (108 cycles) and reliability
of 0.90 and use stress cycle factor for bending strength yn=1.3558N-0.0178;stress cycle factor for pitting resistance zn=1.4488N-0.023. The tooth profile is uncrowned. This is a commercial enclosed gear unit.

A helical Gear drives with a right hand helix angle of 30º with same all other parameters and materials remaining same as that of spur gear.
The design of these two gear systems was accomplished using the program developed.
Final stress analysis was compared with a CATIA model for accuracy and conclusions were drawn.

Solution algorithm:

 Parameters list:

B Net width of face of narrowest member
Z Elastic coefficient factor
dp Pitch diameter, pinion
dG Pitch diameter, Gear
E Modulus of Elasticity
HBG Brinnel Hardness, Gear
HBP Brinnel Hardness, Pinion
KH Load Distribution factor
KB Rim thickness factor
KS Size factor
KO Overload Factor
K’V Dynamic factor
mn Module
mt Transverse metric module
nP Pinion Speed
NP Number of Teeth On Pinion
Pn Normal Circular Pitch
PX Axial Pitch
QV Transmission Accuracy Level Number
rG Pitch-Circular Radius, gear
rP Pitch-Circular radius, Pinion
rbP Pinion base circle radius
rbG Gear Base Circle Radius
σHP AGMA Surface endurance strength
σFP AGMA Bending Strength
SF Safety factor-bending
SH Safety Factor –Pitting
Wt Transmitted load
YN Stress cycle factor for bending strength
Yө Temperature Factors
YJ Geometry Factor For Bending
YI Geometry Factor For Bending
YZ Reliability Factor
ZN. Stress Cycle factor
ZE Elastic Condition factor
ZR Surface condition Factor
σ Bending strength
σC Contact stress from hertzian relation ship
σc Contact stress from AGMA relationship
σall Allowable Bending Stress
σc,all Allowable contact stress, AGMA

Pressure angle
t
Transverse Pressure angle

Helix angle at standard pitch diameter

 Design Procedure of a Gear Mesh:
A useful Decision Set for a spur Gear and helical gear include (formulae according to ANSI/AGMA 2110-C95 and 2101-c95):

• Function: load, speed, reliability, life, overload.
• Unquantifiable risk: design factor
• Tooth system: Φ, φ, addendum, dedendum, root fillet radius
• Gear ratio: Quality number
• Module
• Face width
• Pinion material, core hardness , case hardness
• Gear material: core and case hardness

 Calculation of load capacity of spur and helical gears.

Wt is the tangential force that acts in the transverse plane and tangent to the pitch circle of the gear and that causes the torque to be transmitted from the driver the driven gear. Therefore, this force is often called the transmitted force.
If the torque being transmitted (T) and the size of the gear (D) are known, we can compute Wt as follows:


For the unit-specific situation where power is expressed in KW and the rotational speed is in r•p•m, the torque


If the power being transmitted(P) and the rotational speed (n) are known, torque (T) can be computed as follows




 Stress calculation of spur gear
Terminology of gear teeth.:A, addendum circle; B, pitch circle; C, clearance circle; D, dedendum circle; E, bottom land; F, top land; G, flank; H, face; a = addendum distance; b = dedendum distance; c = clearance distance; = circular pitch; t = tooth thickness; u = undercut
distance.

Every Gear Design needs to be checked for following stresses in order to attain a satisfactory degree of safety in the design.(1)Bending and (2)contact stresses are calculated by product of various empirical terms with the LEWIS EQUATION. These empirical terms are obtained from corresponding graphs and Tables in the Reference book (REF1).

The Fundamental stress Equation used in AGMA methodology, one for bending stress:

σ = (Wt. KO. K’V. KS KH KB )/(b* mt YJ)

-eqn(14-15,REF )

The Fundamental stress Equation used in AGMA methodology, for Pitting resistance (Contact Stress) is:

σ = ZE √( (Wt. KO. K’V. KS KH ZR )/(b* dP ZI) )
-eqn(14-16,REF)

The equation for allowable bending stress is:

σall= (σFP.YN )/( SF .YZ. Yө)
-eqn(14-16,REF)

The equation for allowable contact stresses:
σc,all=(σHP.ZN. Zw )/( SH .YZ. Yө)
-eqn(14-16,REF)


 Stress calculation of helical gears:

For designing a helical gear, first of all
Some conversions from normal to transverse plane are required.


WN is the true normal force that acts perpendicular to the face of the tooth in the plane normal to the surface of the tooth. wN






We seldom need to use the value of WN because its three orthogonal components.
Normal pressure angle:fn;
Transverse pressure angle: ft;
Helix angle: y
For helical gears, the helix angle and one of the other two are specified. The third angle can be computed from:

The value of the tangential load is the most fundamental of the three orthogonal components of the true normal force. The calculation of the bending stress number and the pitting resistance of the gear teeth depends on Wt.

After conversion of normal component to transverse components.Various parameters are calculated in a way similar to that of a spur gear. Finally stress calculation is done using the same set of formulas as mentioned for the spur gears and the factor of safety was calculated.
III. VB.net Code for Spur Gear Stress Analysis:
(Setup file requires .net 2.0 driver, with updated installer package by windows.)


Note: The Variables post fixed with G indicate that they are gear parameter and those with P indicates pinion. Other variables are for the pair.


Public Class Form1
Dim teethP, teethG, pdiaP, pdiaG, bhnP, bhnG, modulepair, modulepair1, facewidth, pinionlife, speedratio, velo, rpmP, loadtangential, powertrans As Double
Dim prangle As Double
Dim lewisP = 0.303, LewisG = 0.412
Dim bendingstrgthP, bendingstrgthG, surfendurancestrgthP, surfendurancestrgthg,

....................<<>>>>>>>>.................

IV. Comparison of results with CATIA model:

In a continuum problem of any dimension the field variable possesses infinitely many values because it is a function of each generic point in the body or solution region. Consequently, the problem is one with an infinite number of unknowns. The finite element discretization procedures reduce the problem to one of a finite number of unknowns by dividing the solution region into elements and by expressing the unknown field variable in terms of assumed approximating functions within each element.
A similar approach was applied to the subject problem with appropriate boundary conditions. Finally gears were generated using following constraints after sufficient analysis.
For both gears types:
Transmitted Power 3 KW
Speed 30rps
Face width 38 mm
Pinion Teeth 17
Gear Teeth 52
BHN for pinion material 240
BHN for gear material 200
Young’s Modulus 2(105)
Quality standard 0.6


Spur Pitch circle diameter ,gear 130 mm
Spur Pitch circle diameter ,pinion 42.5 mm
Helical Pitch circle diameter ,pinion 49.075 mm
Helical Pitch circle diameter ,gear 150.111 mm
Tangential force, spur gear 750 N
Tangential force, helical gear 649N
(Obtained as per appropriate calculations according to REF)







V. Conclusion
The stress values for spur gear when compared in case of catia model and that of program come in close agreement. Though the choice of boundary condition makes the catia model vulnerable to errors.still the obtained stress values from the von-misces analysis validate the computer generated model. thus similar analysis could be carried out in case of helical gears also.
This method of analysis opens an entirely new methodology for validation of materials for manufacturing an engineering component.Such analysis could be done for any component of machine assembly giving us a pre-calculated factor of safety.
References
[1]Prasad N.S,Shunmugam M S,Krishnamurthy S, MECHANICAL ENGINEERING DESIGN ,Indian adoption,TATA Mc GRAWHILL

.[2] Calculation of load capacity of spur and helical gears. Calculation of surface durability.
[3 shigley J.E and Mischke C.R ,Mechanical Machine Design , e-book
[4] Auto-learning system for the calculation of spur and helical gears using ISO 6336