Myth: The End is Near … I Have Rotordynamic Cross-Coupled Forces

I (Klaus) often joke that turbomachinery aerodynamicists and rotordynamicists hate each other and are natural enemies. The aerodynamicist wants a tiny thin shaft with gigantic, long blades while the rotordynamicist doesn’t even care about the existence of blades and is only happy when the shaft is fat enough. While this may be a slight exaggeration, there is some truth to this, especially when discussing certain topics of rotordynamics such a cross-coupled forces, i.e., when mentioning cross-coupled forces to average turbomachinery designers, their facial expression become terrified, their demeanor transitions to dread, and any meeting becomes a doomsday atmosphere. So, what are these cross-coupled forces that we are so scared of, and how scared should we really be? Some basic discussion may be beneficial.

The functional explanation of cross-coupling is that a force acting on an object in one coordinate direction causes a reacting force of the object in a different coordinate direction. This sounds weird but is quite normal in most complex mechanical systems. For example, if you ride a bike and press with your foot on the pedal in the downward direction, the resulting force transmission leads to a stabilizing forward force and motion via the gear, chain, and wheel. At the same time, since the foot pedal force is applied off the vertical center of the bike, the force creates a destabilizing sideward force that can result in the bike falling over if there is no stabilizing forward motion. For example, if you push down on the pedal but the bike is not moving forward, you usually lose balance. This type of cross-coupling and associated instability happens in all mechanical systems where forces are not just transmitted linearly—which are basically all practical everyday dynamic systems.

There are many causes for cross-coupled forces in rotating machinery, but the most common one is that of cross-coupling in fluid film bearings. In a fluid film bearing specifically, a force in any radial direction will create a cross-coupled force if the shaft is not perfectly centered and/or the pressure profile of the hydrostatic support system is not symmetrical to the bi-normal force vector. Again, sounds a bit complicated but basically all journal bearings can produce cross-coupled forces, and the magnitude of these forces usually increases with the radius of the rotor orbit inside the bearing. Other items in a turbomachine that can produce cross-coupling effects are gear boxes, seals, balance pistons, impeller aerodynamics, etc.—basically everything that is connected or creates a force on the shaft.

From a classical dynamic analysis perspective, we treat a rotor system just like any other mechanical system: We must solve Newton’s first law in two or three dimensions based on mass, stiffness, damping, and applied forces (ma+cv+kx=f). Cross-coupled forces within this analysis then appear as cross-coupled stiffness and damping. This is also the best and easiest way to assess the impact of cross-coupled forces on the stability of the rotor, and this is where the evil magic of the rotordynamicist comes in (Did I mention I am an aerodynamicist?). Unfortunately, I have to work with rotordynamcists on a daily basis, so I will refrain from further adjectives, but rest assured that the field of rotordynamics is basically witchcraft, sorcery, and skullduggery.

In general, you don’t need a rotordynamicist to get a subcritical turbomachine to work, and you don’t need a very good rotordynamicist to get a machine to work that operates between 120% and 200% of the first critical speed. However, as we push above two times the first critical speed, we start encroaching upon the second critical speed and start getting into stability problems that become increasingly difficult. Serious rotordyanmic capabilities and skill are required at this level of witchcraft.

So, when dealing with a rotordynamicist, here are a couple of tips on how to respond to the assertion that there are cross-coupled forces and that you therefore need a thicker shaft:

• All turbomachinery rotor systems have some cross-coupled forces, but that does not automatically mean that the system is unstable. As with any other dynamic system, the damping, c, is usually beneficial and enhances stability while low shaft stiffness, k, can lead to instability. Large cross-coupled stiffness (kxy/kyx) can drive instabilities while cross-coupled damping (cxy/cyx) is usually benign with a large uncertainty in magnitude. The potentially destabilizing cross-coupled stiffness kxy/kyx results from the fluid behavior in bearings, gas (or liquid) seals, and/or aerodynamic flow paths.
• As noted, not all cross-coupling is destabilizing. When the stiffness terms kxy and kyx have different signs, they drive instability. If that cross-coupled stiffness is occurring at or near a resonance mode shape, then that particular mode is not strongly influenced by the cross-coupling. Or if the cross-coupled stiffness is located near an anti-node of a mode shape, then the effect on stability can be pronounced.
• But there are cases where kxy and kyx are of the same sign, and the system is usually stable even if the stiffness is of different magnitudes. One potential source of kxy and kyx being the same sign can come from coordinate transformations. For example, a tilting pad bearing may have no apparent cross-coupled stiffness. But when translating the dynamic coefficients to a different coordinate system, cross-coupled stiffness of both kxy and kyx becomes evident. This can be misleading, but it is not a rare phenomenon, especially with geared systems.

Ultimately, cross-coupled forces and stiffness are poor parameters to judge rotor system stability. They are simply one of the many inputs required for the analysis. As difficult as this is to admit for an aerodynamicist, a full (evil magic) rotordynamic stability analysis should be performed, and the predicted outputs, such as the log-dec or rotor orbits, are much better indicators whether a turbomachine will be stable over its desired operating range.

(Karl Wygant and Brian Pettinato—both rotordynamicists and masters of the dark arts—contributed to this article.)

Klaus Brun is the Director of R&D at Elliott Group. He is also the past Chair of the Board of Directors of the ASME International Gas Turbine Institute and the IGTI Oil & Gas applications committee.

Rainer Kurz is the Manager of Gas Compressor Engineering at Solar Turbines Incorporated in San Diego, CA.He has been an ASME Fellow since 2003 and the past chair of the IGTI Oil and Gas Applications Committee.

Any views or opinions presented in this article are solely those of the authors and do not necessarily represent those of Solar Turbines Incorporated, Elliott Group, or any of their affiliates.