If we want to know the beam strength of our gear teeth, we use Lewis equation. By using this equation, we get satisfactory results. While deriving the expression Lewis took an assumption that we don’t take load transmission on several teeth because it is not safer for assume. So he assumed that all load is taken and given by only one tooth of the gear.
During load transmission, when load is at the end of driven teeth and the contact comes to an end when it reaches the end of driving teeth. But if we are considering large gears, load transmission is not true. So in the result, he said that proper load distribution in the world does not exist.
So if we have unequal gears, the smaller gear called pinion is week as compared to the large one because it has less number of teeth.
For deriving Lewis equation, consider each tooth of the gear as cantilever beam on which normal load WN acts. This load will resolve into two components radial load WR and tangential load WT.
Tangential load acts perpendicular to the axis of the gear and radial load acts parallel to the axis of the gear. Tangential load produces bending stress in the gear and radial load produce compressive stress. In load transmission, compressive stress is minimum, so we neglect it for the sake of convenience.
The only stress which is left is the bending stress and our whole designing calculation is based on this stress. To get maximum bending stress in tooth, we draw a parabola at point A which is tangential to tooth curves at B and C.
From this, we conclude that at this parabola bending stress is uniform. In reality, our tooth is larger than this curve except BC. This is the point where we have maximum stress. So bending stress at section BC is
σW = My/I
where M is the bending moment (WT x h), y is the half distance of tooth thickness (t/2) and I is the moment of inertia whose value is bt3/12.
After simplifying the whole expression, we will get
WT = σW.b.π.m.y
Where in the above expression m is the module of gear and y is equal to t2/6hpc. If we increase or decrease the size of gear, expression will not change.
But if we take the value of y in terms of teeth then
y = 0.124 – 0.684/T for 14.5° composite and full depth involute system
y = 0.154 – 0.912/T for 20° full depth involute system
y = 0.175 – 0.841/T for 20° stub system