Is all subsynchronous vibration an instability? Is it always at or close to half running speed? Either way, what physical phenomena would drive such below-running-speed vibration?
The short answer is that, no, not all subsynchronous vibration is a rotordynamic instability, or even at half running speed. However, in nearly all cases, stable or unstable, whirling fluid motion is the root cause of the vibration, as we will see.
First, let’s consider what we mean by “instability”. A physical process is unstable if it cannot maintain steady operation, and with the slightest “perturbation” (nudge) will be forced by the physics of the situation to change its condition. This generally involves some sort of “feedback”, in which a deflection causes a force, which causes more deflection, which causes more force, etc.
In a column holding up a great weight, instability is caused by a slight bow in the column leading to a bending moment in the column that leads to more bending, that leads to more moment, etc., until the column collapses. Engineer/builders have known for centuries that this is a function of the column’s bending stiffness, not (surprisingly) dependent on the column’s material strength. So, trying to fix a problem with a building’s column buckling by using a stronger material is a waste of effort, not to mention dangerous, unless the material changes lead to greater stiffness as well.
In the case of a rotor system, the intent is for the shaft to perform its duty while orbiting within a relatively tight “orbit”, maintaining adequate radial clearance between the rotating surfaces and the stationary surfaces of the bearings and annular (e.g., labyrinth) fluid seals. When a rotor passes through a natural frequency on its way to its ultimate running speed, its orbit amplitude (size) increases because of “resonance”. The orbit in such a case becomes amplified by storing up the motion, each revolution’s response to any residual imbalance adding onto the response of previous revolutions. If not for energy-absorbing damping, usually mostly from the bearings, then when the frequency of a strong excitation force becomes the same as the shaft natural frequency, vibration will increase without limit, quickly damaging the machine with a severe rub in one or more components. However, thank goodness, damping is present in all practical rotor systems. Thus, vibration does become elevated, at least at the excitation frequency such as 1x running speed excitation provided by imbalance. However, if the exciting force such as balance is reasonably adequate (for example as required by the ISO 1940 standard, or API 617 standard) the orbit retains sufficient radial clearance, avoiding overloading the bearings or causing a fatal rub.
Is there any situation in which this damping might be lost, or overcome by a competing phenomenon? The answer unfortunately is “yes”, and this is what causes rotordynamic instability. In modern turbomachinery, the phenomenon is nearly always sourced in fluid whirl within the close running clearances of the machine. The most typical culprit components are the fluid film bearings or annular seals, but sometimes the issue can be associated with the action of the periphery of an axial bladed disk or the periphery of shroud sidewalls of an impeller. API standards such as RP 684 do a good job of describing this situation, and how to conservatively predict it. The root explanation is that fluid whirls in the close running clearances, at a speed typically about half running speed (usually 42 to 49% rpm), such that the whirl is a bit closer to the zero speed of the stationary walls rather than the 1x rpm right at the surfaces of the somewhat lesser diameter rotor surfaces. This whirl results in static pressure build-up due to the “dam” of whirling fluid in front of a locally tighter radial clearance when the rotor is not perfectly centered. This in turn leads to a force known as “cross-coupling” acting perpendicular to the rotor deflection in the stator bore, tending to drive the rotor in the direction of the path the orbit is already whirling in.
Unfortunately, this force is equal and opposite to the damping force, which is pointed in the direction discouraging the whirl motion. If this cross-coupling force becomes greater than the damping force, then the damping can be overcome, and in fact the apparent net damping then can be negative. This situation is facilitated when the whirl speed has increased to be equal to (or, afterwards, greater than) a poorly damped rotor natural frequency. At this point, the phase lag between cause-and-effect (i.e., whirl force versus the vibration it produces) results in the cross-coupling continuously acting to close the radial clearance (force reduces gap, which increases force, which further reduces gap, etc.), until the clearance is fully used up, resulting in serious rub and rapid machinery damage. In rare circumstances, the bearing support (e.g., a lightweight machine casing) can move more than the rotor, and be the prime source of the feedback-producing motion.
The scenario above describes classic rotordynamic instability. It is recognized by its typical roughly half running speed vibration being dominant, growing quickly once initiated. Unlike a resonance, increasing the rotor speed to “drive through” the situation does not work. For some examples and discussion of fixes, stay tuned for the next installment!
