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and if these ends be not in one straight line with that axis, then S, L − R, and L + R, are the three sides of a triangle, having the angle opposite S at that axis; so that, if θ be the supplement of the arc between the dead-points,
S2 = 2 (L2 + R2) − 2 (L2 − R2) cos θ,
| cos θ = | 2L2 + 2R2 − S2 | . |
| 2 (L2 − R2) |
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| Fig. 111. |
§ 66. Coupling of Intersecting Axes—Hooke’s Universal Joint.—Intersecting axes are coupled by a contrivance of Hooke’s, known as the “universal joint,” which belongs to the class of linkwork (see fig. 111). Let O be the point of intersection of the axes OC1, OC2, and θ their angle of inclination to each other. The pair of shafts C1, C2 terminate in a pair of forks F1, F2 in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at O. This cross is the link; the connected points are the centres of the bearings F1, F2. At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity ratio of the points F1, F2 (which, as the forks are equal, is also the angular velocity ratio of the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity ratio is that of equality, for each successive quarter-turn is made by both shafts in the same time; but its actual value fluctuates between the limits:—
| α2 | = | 1 | when F1 is the plane of OC1C2 |
| α1 | cos θ |
| and | α2 | = cos θ when F2 is in that plane. |
| α1 |
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