Gear Design by AGMA Theory: Online Calculator

In this article, It is assumed that the reader has a basic knowledge of gear parameters and terminology such as module, pitch circle diameter, outer diameter, base circle diameters, face width, pressure angle, helix angle, number of teeth, Spur Gear, Helical Gear, etc.  

The life of gear depends on multiple factors such as lubrication, manufacturing quality, gear material, finish, noise, and vibration in the machine, temperature, quality, and selection of bearings of the gear shaft, overall design of the machine, and many more. Unfortunately, there is no established direct mathematical relationship between all the mentioned factors.  Novice gear designed should focus on learning basic principles, theory, and formulas. Eventually overall picture of gear applications, practice experience, and manufacturing knowledge into focus. 

History of Gear Design Theory

The gear tooth design equation was first proposed by Wilfred Lewis (an American engineer and inventor) in 1892 and is still the base of the modern equation. Although, there were many equations that existed before the work of Mr. Lewis but those equations did not consider the tooth shape and geometry factors.  
one such equation from an English rule published at Newcastle-under-Lyme in 1868 is mentioned below.  

X= 20,000 p.f  
Where X= breaking load of gear tooth in pounds 
p= pitch of teeth in inches 
f= face width of teeth in inches  

With time and experience of designers, there are many modifications in the Lewis equation, the basic equation is the same but some factors have been added over the years.  
 
In his publication, Lewis has mentioned 2 main assumptions which are critical – 

> First, If the load will come of entire face width or only concentrated at a corner. In a gearbox deflection in the shaft, improper fitment of bearing or errors in the manufacturing of other components in the gearbox may affect the force coming on gear tooth. Lewis has assumed that in first-class modern machinery load can be taken as well distributed across the tooth than at the corner  

> Second, that the load on gear tooth may not come exactly at that pitch point and load may come at the tip of gear. The load will come only at pitch point if the gear is manufactured as perfect means all the teeth are the same and equally spaced with perfection in tooth forming. Moreover, proper prototype testing is required to prove the performance of the gear. So, with all these limitations at that time, Lewis assumed the worst case that the load will come at the tip of the gear tooth.  

Finally, Lewis came to the conclusion that the gear tooth is like a cantilever beam where the load is coming at the free end of the beam.  Now the total load on gear can be divided into 2 components – Radial and Normal (Tangential). The radial load will act as a compressive load and try to crush the gear. Whereas, the tangential load will cause bending stress in the tooth. But radial component does not have much effect on the strength of the tooth (hardly 10% effect). Most of the gears failed by breakage of the tooth by excessive bending stress caused by Tangential load.  

Calculation of Tooth Strength

As already said, that there have been many modifications in the basic Lewis equation with time, improvement in technology, experience, advancements in the manufacturing and quality checking processes and many more factors. So, in today’s time, 2 International Standards are common designing of Gear, ISO and AGMA.  

There is no denying the fact that at the time when AGMA and ISO were published for the first time there were many differences in the calculation of tooth strength and some differences still exist. Over the years, AGMA and ISO standard (from a gear technology point of view) have influenced each other. Some changes in AGMA were adopted from ISO and vice versa. But AGMA has published more standards related to gear than any other organization and also contributed a lot in developing the ISO standard.  

Moreover, in the age of globalization, ISO and AGMA must come at a common platform and no doubt there are several efforts going on in this direction by both AGMA and ISO.  
 
So, a design engineer cannot follow all the standards for designing gear. One has to follow either of the standards and to have knowledge of good software.   

Gear Tooth Loading as per Lewis Equation –

After considering the gear tooth as a cantilever beam, Lewis derived the following equation.  

 

The form factor in the Lewis equation is the unit less factor based on the geometry of the gear tooth. This form factor basically takes into account of the effective strength of the tooth at the root fillet.  He published the Values of Y for gears of different tooth and pressure angles.  

AGMA Method for Gear Design

Lewis equation is now no longer used in its original form, but it is still the base of the modern version of the AGMA bending stress equation. AGMA stands for American Gear Manufacturers Association. The principle of the Lewis equation is still valid but there are some more factors included to account for gear tooth failure mechanisms studied by other researchers at a later stage. Now Lewis form factor has been replaced by the new Geometry factor J, which takes care of the stress concentration factor at the root fillet. An interesting fact is, stress concentration theory did not exist at the time of Lewis! 

Bending Strength Geometry Factor J –  

AGMA 908-B89 has the method for calculation of this factor, this method is a complicated algorithm. J factor account of various aspects related to tooth geometry such as stress concentration at tooth root fillet. The below table mentions the J factor for full depth 20 Degrees pressure angle assuming the load is coming at tooth tip. Although 20 degrees is the most common pressure angle used worldwide for other pressure angles such as 25 and 14.5 please refer AGMA standard. J value will also change if the correction factor is applied on the gear tooth.

Dynamic Factor K_v– 

Dynamic factor K_v accounts for the internally generated vibration loads (additional loads) from tooth to tooth impact due to non-conjugate meshing of gear teeth.  

This factor depends on the Gear Quality Index (Q_v) and the Pitch line velocity. Let’s understand about Gear Quality Index. It is the AGMA quality number and represents the quality of gear in terms of the geometric accuracy of the teeth. In the ANSI/AGMA 2000 A88 Gear Classification and Inspection Handbook, quality numbers from lowest quality Q3 to Highest quality Q16 represent the accuracy of the tooth geometry; the higher the quality number the smaller the tolerance. The four main parameters accounted for the quality index are Tooth lead or tooth alignment; involute profile variation; pitch or spacing variation; and radial runout. 

The tooth lead or tooth alignment criterion applies to spur and helical-type gearing and measures the variation between the specified lead (or helix angle) and the lead of the produced gear.  

Involute profile variation is the difference between the specified profile and the measured profile of the tooth.  

Pitch variation or spacing variation is the difference between the specified tooth location and the actual tooth location around the circumference of the gear.  

Radial runout refers to the disparity in radial position of teeth on a gear – the variation in tooth distances from the center of rotation.  

Geometry inspections are usually made with modern equipment that measures and records all of the critical variations, and can automatically determine the AGMA gear quality level. Most measuring machines have a stylus that follows the tooth form as the part is rotated. The details are charted at high magnification for intensive visual evaluation. 

Gears with high-quality index mean high accuracy in the tooth, which means, driven gear’s angular velocity will be steady smooth and hence no acceleration or jerk loads on the system of gears and associated components. This leads to a quite (very low noise) gear drive without any additional shock and impact load on the gear teeth. On the other hand, If the quality index is low then gear teeth have an additional dynamic load (shock & impact) along with bending load due to torque transmission. Not only this, shock & impact load gets propagated to shaft and bearings – and ultimately into the housing, which then vibrates at all gear mesh frequencies, exciting the structure and the surrounding air to create noise. 

If the actual dynamic loading to transmission error such as shaft misalignment, bearing misalignment, stiffness of the structure supporting the bearing can be taken into account by increasing the tangential load W_t then K_v can be set to 1.  

Load Distribution Factor K_m–  

Any Axial misalignment or axial deviation in the tooth form will cause the tangential load Wt to be unevenly distributed to the face width. This problem is enhanced with bigger face-width gears.  So as a thumb rule face width to module ratio should be 8 to 16. (Face width/ Module = 8 to 16)

Application Factor K_a– 

This factor considers any kind of fluctuating load coming on the tooth. For example- if a gearbox is driving to a stone crusher then shock loads will come on the gear tooth due to crushing loads. For smooth applications where there is no fluctuating load, this factor can be taken as 1.

Size Factor K_s–

 The size factor considers the fact that test samples used to establish the fatigue strength data by experiment to testing may be smaller in size than the actual part. Ks allows the modification of tooth stress to account for such a situation. AGMA has not specified any value or formula for this. So, choosing this factor depends on the designer’s experience.  Generally, this factor is taken care by limiting the maximum allowed stress limit so Ks can be taken as 1. For a conservative approach, Ks 1.25 to 1.5 can be taken.   

Rim Thinness factor K_b –

This factor accounts for the situation of very large gears that are made on rims and spokes rather than on solid disk and have think rim compared to tooth depth. AGMA defines backup ratio m_B as follows. Backup ration less than 0.5 is not recommended, gears on a solid disk will have Kb=1 

Idler Factor T_i –

In a gear train, the idler is subjected to more cycles of stress per unit time than the other gears. To account for this Ki is set to 1.45 for idler and 1 for non-idler.