Figure 5-14a. Rigid supports Figure 5-14b. Flexible supports
the disc describes an orbit of radius 8r， from the center of the bearing centerline. If the shaft flexibility is represented by the radial stiffness (主r)， it will create a restoring force on the disc of主r8r that will balance the centrifugal force equal to mw2(8r +巾). Equating the two forces obtains
主r8二 mw 2(8r +巾)
Therefore，
mw2巾(w1wn)2巾
8r二 主r mw2二 1 (w1wn)2 (5-26)
where wn二主r/m， the natural frequency of the lateral vibration of the shaft and disc at zero speed.
The previous equation shows that when w<wn，8ris positive. Thus， whenoperating below the criticalspeed， the system rotates with the center of mass on the outside of the geometric center. Operating above the critical speed (w>wn)， the shaft deflection 8rtends to infinity. Actually， this vibration is damped by outside forces. For very high speeds (w>>wn)， the amplitude 8r equals巾， meaning that the disc rotates about its center of gravity.
Flexible Supports
The previous section discussed the flexible shaft with rigid bearings. In therealworld， the bearings are not rigid but possess some flexibility. If the flexibility of the system is given by主b， then each support has a stiffness of主b12. In such asystem， the flexibility of the entire lateral system can be calculated by the following relationship:
111主b +主r主t二 主r + 主b二主r主b
主t二主主b r+主主br (5-27)
Therefore， the natural frequency
品
wnt二主mt 二主主b r+主主br m
品
品
二主mr 主b主 +b主r(5-28)
品
二 wn主b主 +b主r
It can be observed from the previous expression that when主b <主r (veryrigid support)， then wnt二 wn or the natural frequency of the rigid system.
>
For a system with a finite stiffness at the supports， or主b <主r， then wn is less than wnt. Hence， flexibility causes the natural frequency of the system to be lowered. Plotting the natural frequency as a function of bearing stiffness on a log scale provides a graph as shown in Figure 5-15.
When主b <主r， then wnt二 wn主b1主r. Therefore， wnt is proportional to the square root of主b， or log wnt is proportional to one-half log主b. Thus， this relationship is shown by a straight line with a slope of 0.5 in Figure 5-15. When主b >主r， the total effective natural frequency is equal to the natural rigid-body frequency. The actual curve lies below these two straight lines as shown in Figure 5-15.
The critical speed map shown in Figure 5-15 can be extended to includethesecond， third， and higher critical speeds. Such an extended critical speed map can be very useful in determining the dynamic region in which a given system is operating. One can obtain the locations of a system's critical speeds by superimposing the actual support versus the speed curve on the critical speed map. The intersection points of the two sets of curves define the locations of the system's critical speeds.
Figure 5-16a. Rigid supports and a flexible rotor
Figure 5-16b. Flexible supports and rigid rotors
When the previously described intersections lie along the straight line on the critical speed map with aslope of0.5， the critical speed is bearing controlled. This condition is often referred to as a ""rigid-body critical.''
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