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Design of rotating equipment for high-speed operation requires careful analysis. The discussion in the preceding section presents elementary analy-sis of such problems. Once a design is identified as having a problem， it is an altogether different matter to change this design to cure the problem. The following paragraphs discuss some observations and guidelines based on the analysis presented in the previous sections.
Natural frequency. This parameter for a single degree of freedom is
J
given by wn二王1m. Increasing the mass reduces wn， and increasing the spring constant王 increases it. From a
J study of the damped system， the damped natural frequency w生二 wn 1〈2 is lower than wn.
Unbalances. All rotating machinery is assumed to have an unbalance. Unbalance produces excitation at the rotational speed. The natural fre-quency of the system wn is also known as the critical shaft speed. From thestudy of the forced-dampedsystem， the following conclusions can be drawn:

Figure 5-1.a. Amplitude factor as a function of the fre.uency ratio . for various amounts of viscous damping
J
(1) the amplitude ratio reaches its maximum values at wm二 wn 12〈2， and (2) the damped natural frequency w生 does not enter the analysis of the forced-damped system. The more important parameter is wn， the natural frequency of the undamped system. In the absence of damping the amplitude ratio becomes infinite at w二 wn.For this reason， the critical speed of a rotating machine should be kept away from its operating speed. Small machinery involves small values of mass m and has large values of the spring constant王 (bearing stiffness). This design permits a class ofmachines， which are small in size and of low speed in operation， to operate
in a range below their critical speeds. This range is known as subcriticaloperation， and it is highly desirable if it can be attained economically.

Figure 5-1.b. Phase angle as a function of the fre.uency ratio for various amounts of viscous damping
The design of large rotating machinery-centrifugal compressors， gas andsteam turbines， and large electrical generators-poses a different problem.The mass of the rotor is usually large， and there is a practical upper limit to the shaft size that canbe used.Also， these machines operate at high speeds.
This situation is resolved by designing a system with a very low critical speed in which the machine is operated above the critical speed. This is known as supercritical operation. The main problem is that during start-upand shut-down， the machine must pass through its critical speed. To avoiddangerously large amplitudes during thesepasses， adequate damping must be located in the bearings and foundations.
The natural structural frequencies of most large systems are also in the low-frequency range， and care must be exercised to avoid resonant couplings between the structure and the foundation. The excitation in rotating machinery comes from rotating unbalanced masses. These unbalances result from four factors:
1.  An uneven distribution of mass about the geometric axis of the system. This distribution causes the center of mass to be different from the center of rotation.

2. A
deflection of the shaft due to the weight of therotor， causing further distance between the center of mass and the center of rotation. Add-itional discrepancies can occur if the shaft has a bend or a bow in it.

3.  Static eccentricities are amplified due to rotation of the shaft about its geometric center.

4. If
supported by journal bearings， the shaft may describe an orbit so that the axis of rotation itself rotates about the geometric center of the bearings.

These unbalance forces increase as a function of w2， making the design and operation of high-speed machinery a complex and exacting task. Balan-cing is the only method available to tame these excitation forces.
Application to Rotating Machines
　
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