and (2) the self-excited instabilities that are independent of outside stimuli and independent of the frequency. Table 5-3 is the characterization of the two categories of vibration stimuli.
Forced (resonant) vibration. In forced vibration the usual driving fre-quency in rotating machinery is the shaft speed or multiples of this speed.
This speed becomes critical when the frequency of excitation is equal toone of the natural frequencies of the system. In forced vibration， the system is a function of the frequencies. These frequencies can also be multiples of rotor speed excited by frequencies other than the speed frequency such as bladepassing frequencies， gearmesh frequencies， and other componentfrequencies. Figure 5-20 shows that for forcedvibration， the critical fre-quency remains constant at any shaft speed. The critical speeds occur at one-half， one， and two times the rotor speed. The effect of damping in forcedvibration reduces theamplitude， but it does not affect the frequency at which this phenomenon occurs.
Typical forced vibration stimuli are as follows:
1.
UnbGlGnJ巾. This stimulus is caused by material imperfections， toler-ances， etc. The mass center of gravity is different from the geometriccase， leading to a centrifugal force acting on the system.
2.
A盧ωmm巾triJ.l巾xibilitω . The sag in a rotor shaft will cause a periodic excitation force twice every revolution.
3.
ShG.tmi盧Glignm巾nt . This stimulus occurs when the rotor center line and the bearing support line are not true. Misalignment may also be
.able 5-. .haracteristics of Forced and Self-.xcited Vibration
Forced or Self-.xcited or Resonant Vibration Instability Vibration
Frequency/rpm .F二 .rpm or . or Constant and relatively relationship rational fraction independent of rotating speed. Amplitude/rpm Peak in narrow bands Blossoming at onset and continue relationship of rpm to increase with increasing rpm.
Influence of damping Additional damping Additional damping may defer to Reduce amplitude a higher rpm. Will not No change in rpm at materially affect amplitude.
which it occurs System geometry Lack of axial sym. Independent of symmetry. External forces Small deflection to an axisymmetric system. Amplitude will self-propogate. Vibration frequency At or near shaft Same. critical or natural frequency Avoidance 1. Critical freq. Above 1. Operating rpm below onset. running speed
2. Axisymmetric 2. Eliminates instability.
3. Damping Introduce damping.
caused by an external piece such as the driver to a centrifugal com-pressor. Flexible couplings and better alignment techniques are used to reduce the large reaction forces.
.eriodic loading. This type of loading is caused by external forces thatare applied to the rotor bygears， couplings， and fluid pressure， which is transmitted through the blade loading.
Self-.xcited Instabilities
The self-excited instabilities are characterized by mechanisms， which whirl at their own critical frequency independent of external stimuli. These typesof self-excited vibrations can be destructive， since they induce alternating stress that leads to fatigue failures in rotating equipment. The whirlingmotion， which characterizes this type of instability， generates a tangentialforce normal to the radial deflection of theshaft， and a magnitude propor-tional tothat deflection. The type of instabilities， which fall under thiscategory， are usually called whirling or whipping. At the rotational speedwhere such a force isstarted， it will overcome the external stabilizing damp-ing force and induce a whirling motion of ever-increasing amplitude. Figure 5-21 shows the onset speed. The onset speed does not coincide with any particular rotation frequency.Also， damping results from a shift of thisfrequency， not in the lowering of the amplitude as in forced vibration.Important examples of such instabilities include hystereticwhirl， dry-frictionwhip， oil whip， aerodynamicwhirl， and whirl due to fluid trapped in therotor. In a self-excitedsystem， friction or fluid energy dissipations generate the destabilizing force.
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