BEAM STRENGTH OF SPUR GEAR TOOTH
The accurate stress analysis of a gear tooth for a particular application is a complex
problem because of the following reasons:
1. There is continuous change in the point of application of load on the tooth
profile.
2. The magnitude and direction of applied load also change.
3. In addition to static load, the dynamic load due to inaccuracy of the tooth
profile, error in machining and mounting, tooth deflection, acceleration and
stress concentration also act on the tooth which are all difficult to model
mathematically.
Wilfred Lewis, in a paper titled, “The investigation of the strength of gear tooth”
published in 1892, derived an equation for determining the approximate stress in a
gear tooth by treating it as a cantilever beam of uniform strength. The beam
strength calculation is based upon the following assumptions:
1. The tangential component of the force on gear tooth, Ft, is uniformly
distributed across the face width; however, in actual force distribution it is
found to be non-uniformly distributed. This assumption is valid for small
face widths b, i.e.
2. The effect of the radial component Fr, which produces direction compressive
stress, is neglected.
9.5m  b  12.5m
3. The maximum stress is assumed to occur when the entire load is at the tip of
the tooth. This is however, not true because when more than one pair of
teeth are in contact, the load is shared between all of them. Further, as the
tooth moves through its path, the magnitude of the force and its moment are
changed.
4. The tooth is assumed to be a simple cantilever beam
5. The effect of stress concentration and manufacturing errors are neglected.
Equation (12.6) gives the relation between the tangential force and the bending
stress produced in the gear tooth. When the bending stress reaches the permissible
magnitude of bending strength of the material, d, the corresponding force Ft is
called the beam strength Fbeam. Thus beam strength is the maximum value of the
tangential force than can be resisted by a tooth without bending failure. Therefore,
Eq.(12.6) can be rewritten as
Fbeam = bmdY
In any gear drive, the average tangential force on a gear tooth can be computed form
Eql (12.4). However, in many practical applications the actual force may be greater
than the average force on account of the poor service conditions and the dynamic
load on the tooth due to profile being inaccurate. In order to include the effect of the
above two conditions, two factors, namely the service factor C s and the velocity
factor Cv are introduced and the maximum force between the two mating teeth is
computed by the following relation:
C F
Fmax  s t
Cv
The values of the factors Cs and Cv may be found from Tables 12.3 and 12.4,
respectively.
In order to avoid failure of the gar tooth due to bending, the beam strength, Fbeam,
should be greater than the maximum force on the gear tooth. Thus,
Fbeam  Fmax
Type of load
Steady
Light shock
Medium shock
Heavy shock
Table 12.3 values of service factor (Cs)
Type of service
Intermittent or 3 h/day
8-10h/day
1.0
1.0
1.0
1.25
1.25
1.5
1.50
1.8
Continuous
1.25
1.5
1.8
2.0
Dynamic load on gear
The maximum force acting on a gear tooth during power transmission is called the
dynamic load, Fdyn, which is the sum of the tangential force Ft and incremental force
Fi. The reasons behind the incremental force F i are the following.
1. Inaccuracies of the tooth profile.
2. Error in tooth spacing resulting in excessive backlash
3. elastic deformation of a tooth under load, affecting kinematic perfection
4. inertial of the rotating masses
5. Misalignment between bearings supporting gears.
Buckingham had found that the maximum effect of the above factors occurs when
the contact is transferred from one pair to the next pair of teeth. According to him,
the dynamic load Fdyn is given by
Fdyn = Ft + Fi
The increment force Fi for average conditions may be found from the following
equation
k 3v cb  Ft 
Fi 
K 3v  cb  Ft
where


e
c is the dynamic load factor  

 K1 1/E1  1/E2  
e is the profile error in action between gears (refer Table 12.5 and 12.6)
K1 = constant = 9 for 20o full depth teeth
K3 = 20.67, a constant.
Further it is suggested that for satisfactory design of the gear under dynamic load
conditions, the dynamic load should be equal to endurance strength Fen.
Endurance strength is computed form the modified Lewis equation as
Fen = Fdyn X FoS = bmenY
Where
en is the endurance limit (=1.75 BHN)
BHN is the Core Brinel hardness number
WEAR STRENGTH
Due to rolling and sliding actions of the gear teeth, the following types of surface
destructions (wear) may occur:
Abrasive wear. Scratching of the tooth surface due to the presence of foreign
materials in the lubricant is called abrasive wear.
Corrosive wear. Chemical reactions on the surface of a gear cause corrosive wear.
Pitting. Repeated application of the stress cycle, known as pitting, cause fatigue
failure.
Scoring. Inadequate lubrication between metal-to metal contact cause scoring.
It is observed from the results of various experiments that abrasion, corrosive, the
scoring are caused by improper lubrication, whereas pitting usually occurs because
of repeated application of Hertz contact stress on the portion of a gear tooth which
has relatively little sliding motion compared to rolling motion. Clearly, the spur gear
will have pitting near the pitch lien where motion is almost all of rolling.
In designing a gear against wear, the material should be able to resist the repeated
contact stresses. In order to obtain a formula for actual surface stress that exists
between tow mating gear teeth, the Hertz equation for contact stress between two
cylinders in rolling contact is used (figure 12.8)
In order to apply the Hertz contact stress equation to spur gears, the force F is
replaced by the allowable wear load Fwear, radii r1 and r2 are considered as pitch radii
of pinion and gear and contact stress is replaced by the allowable surface endurance
limit es.
The limiting wear load equation for spur gear pairs is given as
Fwear = d1bQK
Where
Q is the ratio factor (=2Z2/(Z1+Z2)
  es2 sin   1
1 

K is the load stress factor  


1.4  E1 E 2  

For steel gears the surface endurance limit may be calculated by the following
formula
es = (2.75 BHN – 70) N/mm2
Buckingham has reported that the allowable wear load as obtained by Eq. (12.16) is
clearly a normal force that must be greater than the dynamic load for satisfactory
design.