In wheels that have their teeth cut to form in a gear-cutting machine the thickness of tooth at any point in the depth is equal at any point across the breadth; hence, supposing the wheels to be properly keyed to their shafts so that the pitch line across the breadth of the wheel stands parallel to the axis of the shaft, the contact of tooth upon tooth occurs across the full breadth of the tooth.
As the practical result of these conditions we have three important advantages: first, that the stress being exerted along the full breadth of the tooth instead of on one corner only, the tooth is stronger (with a given breadth and thickness) in proportion to the duty; second, that with a given pitch, the thickness and therefore the margin for safety and allowance for wear are increased, because the tooth may be increased in thickness at the expense of the clearance, which need be merely sufficient to prevent contact on both sides of the spaces so as to prevent the teeth from locking in the spaces; and thirdly, because the teeth will not be subject to sudden impacts or shocks of tooth upon tooth by reason of back lash.
Fig. 180.
Fig. 181.
In determining the strength of cut gear-teeth we may suppose the weight to be disposed along the face at the extreme height of the tooth, in which case the theoretical shape of the tooth to possess equal strength at every point from the addendum circle to the root would be a parabola, as shown by the dotted lines in [Fig. 180], which represents a tooth having radial flanks. In this case it is evident that the ultimate strength of the tooth is that due to the thickness at the root, because it is less than that at the pitch circle, and the strength, as a whole, is not greater than that at the weakest part. But since teeth with radial flanks are produced, as has been shown, with a generating circle equal in diameter to the radius of the pinion, and since with a generating circle bearing that ratio of diameter to diameter of pitch circle the acting part of the flank is limited, it is usual to fill in the corners with fillets or rounded corners, as shown in [Fig. 129]; hence, the weakest part of the tooth will be where the radial line of the flank joins the fillet and, therefore, nearer the pitch circle than is the root. But as only the smallest wheel of the set has radial flanks and the flanks thicken as the diameter of the wheels increase, it is usual to take the thickness of the tooth at the pitch circle as representing the weakest part of the tooth, and, therefore, that from which the strength of the tooth is to be computed. This, however, is not actually the case even in teeth which have considerable spread at the roots, as is shown in [Fig. 181], in which the shape of the tooth to possess equal strength throughout its depth is denoted by the parabolic dotted lines.
Considering a tooth as simply a beam supporting the strain as a weight we may calculate its strength as follows:—