Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above equation becomes
| u = cv ( | 1 | + | 1 | ). |
| r1 | r2 |
To apply this to involute teeth, let c1 be the length of the approach, c2 that of the recess, u1, the mean volocity of sliding during the approach, u2 that during the recess; then
| u1 = | c1v | ( | 1 | + | 1 | ); u2 = | c2v | ( | 1 | + | 1 | ) |
| 2 | r1 | r2 | 2 | r1 | r2 |
also, let θ be the obliquity of the action; then the times occupied by the approach and recess are respectively
| c1 | , | c2 | ; |
| v cos θ | v cos θ |
giving, finally, for the length of sliding between each pair of teeth,
| s = s1 + s2 = | c12 + c22 | ( | 1 | + | 1 | ) |
| 2 cos θ | r1 | r2 |
(64)
which, substituted in equation (63), gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure.