Page 77 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

64 Harmonically Excited Vibration Chap. 3

Figure 3.4-2.

and substituting into the first of Eq. (3.4-5) gives the amplitude equation

meo)
r = (3.4-7)

{k —mcü^Ÿ + {cfjjŸ
2^1 —
These equations indieate that the eecentricity line e = SG leads the displace­
ment line r = OS by the phase angle (/>, which depends on the amount of damping
and the rotation speed ratio co/co,^. When the rotation speed coincides with the
critical speed ca,, = yjk/m , or the natural frequency of the shaft in lateral
vibration, a condition of resonance is encountered in which the amplitude is
restrained only by the damping. Figure 3.4-3 shows the disk-shaft system under
three different speed conditions. At very high speeds, ta » a),,, the center of mass
G tends to approach the fixed point (9, and the shaft center .S' rotates about it in a
circle of radius e.
It should be noted that the equations for synchronous whirl appear to be the
same as those of Sec. 3.2. This is not surprising, because in both cases the exciting
force is rotating and equal to meco^. However, in Sec. 3.2 the unbalance was in
terms of the small unbalanced mass m, whereas in this section, the unbalance is
defined in terms of the total mass m with eccentricity e. Thus, Fig. 3.2-2 is
applicable to this problem with the ordinate equal to r/e instead of MX/me.

UJ = Lün
Figure 3.4-3. Phase of different rotation speeds.