Page 75 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

62 Harmonically Excited Vibration Chap. 3

Figure 3.4-1. Whirling of shaft.

friction in bearings, and so on. The whirling of the shaft can take place in the same
or opposite direction as that of the rotation of the shaft and the whirling speed
may or may not be equal to the rotation speed.
We will consider here a single disk of mass m symmetrically located on a
shaft supported by two bearings, as shown in Fig. 3.4-1. The center of mass G of
the disk is at a distance e (eccentricity) from the geometric center S of the disk.
The center line of the bearings intersects the plane of the disk at G, and the shaft
center is deflected by r = OS.
We will always assume the shaft (i.e., the line e = SG) to be rotating at a
constant speed co, and in the general case, the line r = OS to be whirling at speed
6 that is not equal to co. For the equation of motion, we can develop the
acceleration of the mass center as follows:

Be + a G/S (3.4-1)
where is the acceleration of S and ^c/s ^^e acceleration of G with respect to
S. The latter term is directed from G to S, because co is constant. Resolving a<^ in
the radial and tangential directions, we have

a<^ = [(^ ” ~ (ioi' - 0)]i + -h 2r0) - eco^ sin (wf - 0)jj
(3.4-2)
Aside from the restoring force of the shaft, we will assume a viscous damping force
to be acting at S. The equations of motion resolved in the radial and tangential