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Excessive vibration can be caused by shaking forces (“excitation forces”) that are higher than normal. For example, maybe the rotor imbalance is too high. Such shaking forces could be mechanically sourced (such as the imbalance) or hydraulically based (such as from piping pressure pulsations). They can even be electrically based (such as from uneven air gap in a motor, or from VFD harmonic pulses). In all these cases, high rotor vibration is typically just rotor increased oscillating displacement “x” in response to the shaking force “F” working against the rotor-bearing support stiffness “k”. In equation form, F = k*x, and calculating x for a given F is known as “forced response analysis”.
This article is an excerpt from the paper, "An end-user's guide to centrifugal pump rotordynamics" by William Marscher of Mechanical Solutions, Inc.
However, sometimes all of the shaking forces are actually reasonably low, but still excessive vibration is encountered. This can be an unfortunate circumstance during system commissioning, leading to violation of vibration specifications, particularly in variable speed systems where the chances are greater that an excitation force’s frequency will equal a natural frequency over at least part of the running speed range. This situation is known as resonance.
A key reason for performing rotordynamic analysis is to check for the possibility of resonance. Rotordynamic testing likewise should include consideration of possible resonance. In rotor vibration troubleshooting, it is recommended to first investigate imbalance, then misalignment, and then natural frequency resonance, in that order, as likely causes, unless the specific vibration vs. frequency plot (the “spectrum”) or vibration vs. time pulsations indicate other issues (some of these other issues will be discussed in some detail later).
An important concept is the "natural frequency", the number of cycles per minute that the rotor or structure will vibrate at if it is "rapped", like a tuning fork. Pump rotors and casings have many natural frequencies, some of which may be at or close to the operating speed range, thereby causing “resonance”. The vibrating pattern which results when a natural frequency is close to the running speed or some other strong force’s frequency is known as a "mode shape".
Each natural frequency has a different mode shape associated with it, and where this shape moves the most is generally the most sensitive, worst case place for an exciting force such as imbalance to be applied, but similarly is the best place to try a “fix” such as a gusset or some added mass. In resonance, the vibration energy from previous "hits" of the force come full cycle exactly when the next hit takes place. The vibration in the next cycle will then include movement due to all hits up to that point, and will be higher than it would have been for one hit alone (the principle is the same as a child’s paddle-ball). The vibration motion keeps being amplified in this way until its large motion uses up as much energy as that which is being supplied by each new hit. Unfortunately, the motion at this point is generally quite large, and is often damaging to bearings, seals, and internal running clearances (e.g. wear rings).
It is desirable that the natural frequencies of the rotor and bearing housings are well separated from the frequencies that such “dribbling” type forces will occur at. These forces most often tend to be 1x running speed (typical of imbalance), 2x running speed (typical of misalignment), or at the number of impeller vanes times running speed (so-called “vane pass” vibrations from discharge pressure pulses as the impeller vanes move past a volute or diffuser vane “cut-water”).
In practice, the vibration amplification due to resonance is usually between a factor of two and twenty five higher than it would be if the force causing the vibration was steady instead of oscillating. The level of Q depends on the amount of energy absorption, called "damping", which takes place between the force oscillation high points. In an automobile body, this damping is provided by the shock absorbers. In a pump, it is provided mostly by the bearings and the liquid trapped between the rotor and stator in “annular seals” like the wear rings and balance piston. If the damping is near the point where it just barely halts oscillating motion (this is how automobile shocks are supposed to operate, to provide a smooth ride), the situation is known as “critical damping”. The ratio of the actual to the critical damping is how a rotor system’s resistance to resonant vibration is best judged.
In other terms that may be more familiar, for practical values of the damping ratio, 2 times pi times the damping ratio approximately equals the logarithmic decrement or “log dec” (measures how much the vibration decays from one ring-down bounce to the next). Also, the amplification factor Q equals roughly 1/(2*damping ratio). One way to live with resonance (not recommended for long) is to increase the damping ratio by closing down annular seal clearances, or switching to a bearing that by its nature has more energy absorption (e.g. a journal bearing rather than an antifriction bearing). This may decrease Q to the point where it will not cause rubbing damage or other vibration related deterioration.
For this reason, the API610 Centrifugal Pump Standard does not consider a natural frequency a “critical speed” (i.e. a natural frequency of more than academic interest) if its Q is 3.3 or less. The problem with any approach relying on damping out vibration is that whatever mechanism (such as tighter wear ring clearance) is used to increase damping may not last throughout the expected life of the pump. A counter-intuitive but important concept is the "phase angle", which measures the time lag between the application of a force and the vibrating motion which occurs in response to it.
A phase angle of zero degrees means that the force and the vibration due to it act in the same direction, moving in step with one another. This occurs at very low frequencies, well below the natural frequency. An example of this is a force being slowly applied to a soft spring. Alternately, a phase angle of 180 degrees means that the force and the vibration due to it act in exactly opposite directions, so that they are perfectly out of step with each other. This occurs at very high frequencies, well above the natural frequency. Phase angle is important because it can be used together with peaks in vibration field data to positively identify natural frequencies as opposed to excessive excitation forces. This is necessary in order to determine what steps should be taken to solve a large number of vibration problems. Phase angle is also important in recognizing and solving rotordynamic instability problems, which typically require different solutions than resonance or excessive oscillating force problems.