Lewis Equation For Spur Gear

Lewis Equation For Spur Gear

The beam strength of gear teeth is determined from an equation (known as Lewis equation). In 1892, Wilfred Lewis investigated the strength of gear teeth. He derived an equation which is now extensively used by industry in determining the size and proportions of the gear.

In the Lewis equation certain assumptions were made-

1. Gear was considered as a cantilever beam will load applied at the lip and is uniformly distributed over the entire face width.

2. Radial force effects are neglected.

3. Tooth sliding friction, and stress concentration due to abrupt changes in area were also neglected.

4. Tooth shape is considered a parabolic curve.

5. Stress concentration in the tooth fillet is negligible.

Consider each tooth as a cantilever beam loaded by a normal load (WN) as shown in Fig. It is resolved into two components i.e. tangential component (WT) and radial component (WR) acting perpendicular and parallel to the centre line of the tooth respectively. The tangential component (WT) induces a bending stress which tends to break the tooth. The radial component (WR) induces a compressive stress of relatively small magnitude, therefore its effect on the tooth may be neglected. Hence, the bending stress is used as the basis for design calculations. The critical section or the section of maximum bending stress may be obtained by drawing a parabola through A and tangential to the tooth curves at B and C. This parabola, as shown dotted in Fig, outlines a beam of uniform strength, i.e. if the teeth are shaped like a parabola, it will have the same stress at all the sections. But the tooth is larger than the parabola at every section except BC. We therefore, conclude that the section BC is the section of maximum stress or the critical section. The maximum value of the bending stress (or the permissible working stress), at the section BC is given by 

σw = M.y / I ...(i) 

where M = Maximum bending moment at the critical section BC = WT × h

WT = Tangential load acting at the tooth

h = Length of the tooth

y = Half the thickness of the tooth (t) at critical section BC = t/2

I = Moment of inertia about the centre line of the tooth = b.t3/12, 

b = Width of gear face. 

Substituting the values for M, y and I in equation (i), we get 

In this expression, t and h are variables depending upon the size of the tooth (i.e. the circular pitch) and its profile.

Let t = x ×  pc , and h = k × pc ; where x and k are constants.

Substituting (x2)/6k = y another constant, We Have

WT = σw.b.pc.y = σw.b .πm.y 

Because (pc = π m) 

The quantity y is known as Lewis form factor or tooth form factor and WT (which is the tangential load acting at the tooth) is called the beam strength of the tooth.

Drawbacks of Lewis equation are:

1. The tooth load in practice is not static. It is dynamic and is influenced by pitch line velocity.

2. The whole load is carried by a single tooth is not correct. Normally load is shared by teeth since the contact ratio is near to 1.5.

3. The greatest force exerted at the tip of the tooth is not true as the load is shared by teeth. It is exerted much below the tip when single pair contact occurs.

4. The stress concentration effect at the fillet is not considered.

where Y= Modified Lewis form factoThe Lewis beam strength equation is written as (replacing Ft as Fs and σb as design stress)

Where Fs=Static Strength or Lewis Beam strength of the tootBeam strength of gear must be greater than Lewis dynamic load for safe design.The allowable bending strength Fs of a gear is defined as the allowable tangential force at the pitch circle based on the mutually allowable root stress of two meshing gears under load.Drawback of Lewis equations1. The tooth load in practice is not static. It is dynamic and is influenced by pitch line velocity.