The following notation is used throughout the chapter on Shafts:
A0 = Angular deflection (degrees).
B = Simple bending moment (inch-lbs.).
B0=Equivalent bending moment (inch-lbs.).
c = Distance from neutral axis to outer fiber (inches).
d, d0, d2, d3, d4 - Diameters of shaft (inches).
d1=Internal diameter of shaft (inches).
E= Direct modulus of elasticity (a ratio).
e = Transverse deflection (inches).
G=Transverso modulus of elasticity (a ratio).
H=Horse-power (33,000 ft.-lbs. per minute).
I = Moment of inertia.
K=Distance between bearings (inches).
L =Length along shaft (inches). L1, L2 = Length of bearings (inches.) M = Distance between bearings (feet.) N = Number of revolutions per minute.
P = Driving force of belt (lbs.).
P1=Load applied as stated (lbs.).
R = Radius at which load as stated acts (inches).
T = Simple twisting moment (inch-lbs.).
Te=Equivalent twisting moment (inch-lbs.).
Tn=Tonsion in tight side of belt (lbs.).
T0=Tension in loose side of belt (lbs.).
W = Load applied as stated (lbs.).
The simplest case of shaft loading is shown in Fig. 26. The equal forces W, similarly applied to the disc at the distance R from its center, tend to twist the shaft off, the tendency being equal at all points of the length L between the disc and the post, to which the shaft is rigidly fastened. The fastening to the post, of course, in this ideal case, takes the place of a resisting member of a machine. A state of pure torsion is induced in the shaft; and any element, such as ca, is distorted to the position cb, aob being the angular deflection for the distance L.
The case of Fig. 27 is illustrative of what occurs when a belt pulley is substituted for the simple disc. Here the twisting action is caused by the driving force of the belt, which is Tn - T0 = P, acting at the radius R. Torsion and angular deflection exist in the shaft, as in Fig. 20. In addition, however, another stress of a different kind has been introduced; for not only does the 'shaft tend to be twisted off, but the forces Tn and To acting together, tend to bend the shaft, the bending moment varying with every section of the shaft, being nothing at the point o, and maximum at the point c. This combined action is the most common of any that we find in ordinary machinery, occurring in newly every case with which we have to deal.
In Fig. 27, if the forces Tn and T0 be made equal, there will be no tendency at all to twist off the shaft, but the bending will remain, being maximum at the point c. This condition is illustrative of the case of all ordinary pins and studs in machines. In this sense, a pin or a stud is simply a shaft which is fixed to the frame of the machine, there being no tendency to turning of the pin or stud itself. The same condition would be realized if the disc in Fig. 27 were loose upon the shaft. In that case, the bending moment would be caused by Tn + T0 acting with the leverage L. Of course there would have to be some resistance for Tn-T0 to work against, in order that torsion should not be transmitted through the shaft. This condition might be introduced by having a similar disc lock with the first one by means of lugs on its face, thus receiving and transmitting the torsion.
If the distance L becomes very great, both the angular deflection due to twisting, and the sidewise deflection due to bending, become excessive, and not permissible in good design. This trouble is remedied by placing a bearing at some point closer to the disc, which, as it decreases L, of course, decreases the bending moment and therefore the transverse deflection. The angular deflection can be decreased only by bringing the resistance and load nearer together.
The above implies, of course, that the diameter of the shaft is not changed, it being obvious that increase of diameter means increase of strength and corresponding decrease of both angular and transverse deflection.
If the speed of the shaft be very high, and the distance between bearings, represented by L, be very great, the shaft will take a shape like a bow string when it is vibrated, and smooth action .cannot be maintained.
It is necessary to carry the cases of Figs. 26 and 27 but a single step farther to illustrate the actual working conditions of shafting in machines. Suppose the rigid post to have the shaft passing clear through it, and to act as a bearing, so that the shaft can freely rotate in it, the resistance being exerted somewhere beyond. The twisting moment will be unchanged, also the bending moment; but the effect of the bending moment will be on each particle of the shaft in succession, now putting compression on a given particle, and then tension, then compression again, and so on, a complete cycle being performed for each revolution. This brings out a very important difference between the bending stress in pins and the bending stress in rotating shafts. In the one case the bending stress is non-reversing; in the other, reversing; and a much higher fiber stress is permissible in the former than in the latter.