Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.
§ 111. Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.
It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.
The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.
| (From Balancing of Engines, by permission of Edward Arnold.) |
| Fig. 130. |
§ 112.* Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, “The Balancing of Engines with special reference to Marine Work,” Trans. Inst. Nav. Arch. (1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O
(1) ΣF = 0
(2) ΣFa = 0
(a)
The plane through O to which the shaft is perpendicular is called the reference plane, because all the transferred forces act in that plane at the point O. The plane through the radius of the weight containing the axis OX is called the axial plane because it contains the forces forming the couple due to the transference of F to the reference plane. Substituting the values of F in (a) the two conditions become
| (1) (W1r1 + W2r2 + W3r3 + ...) | α2 | = 0 |
| g |