A comparison of techniques for identifying the configuration state of statically indeterminate rotor bearing systems

Turbomachinery rotors are frequently supported on several hydrodynamic bearings and so are statically indeterminate. In such cases, the relative locations of the bearing centres (viz. the system configuration state) affect the bearing reaction forces and hence their stiffness and damping properties, thereby significantly influencing the vibration behaviour of the rotor bearing system. Since this configuration state may differ from its value at time of installation, due to thermal effects and/or foundation settlement, it would be useful to identify its value under operating conditions. This paper illustrates how this can be done in principle, regardless of the unbalance, by measuring the locations of the rotor journals relative to their respective bearing housings at any speed at which the system has reached steady state operating conditions, provided one has good models of the rotor and the foundation. Two identification procedures are compared. Both methods rely, to varying degrees, on using the Reynolds equation for hydrodynamic lubrication to obtain the bearing reaction forces. The first procedure uses the Reynolds equation to evaluate both the magnitudes and directions of the forces (the ‘magnitude and direction’ or MAD method), whereas the second procedure uses the Reynolds equation to evaluate only the directions of the forces (the ‘direction only’ or DO method). Numerical experiments on a flexibly supported statically indeterminate four bearing flexible rotor prove that both the MAD and DO identification procedures are sound in principle, being able to identify the locations of the two inboard bearings relative to the two outboard bearings to within 0.1 μm assuming seven-digit accuracy in journal orbit eccentricity measurements. On the other hand, three-digit measurement accuracy, felt to be the best accuracy practically achievable, restricts identification of the bearing locations to within 10 μm, with somewhat better identification being achieved with the MAD procedure. Such identification accuracy presupposes that the Reynolds equation correctly predicts the bearing reaction forces and could be in error owing to the temperature dependence of the bearing clearance, the assumption of a mean lubricant viscosity and the uncertainty of the cavitation boundaries. It is shown that error in lubricant viscosity may introduce significant errors into the identification achievable with the MAD procedure, but has no effect on that achievable with the DO procedure; and error in clearance introduces more error into the identification achievable with the MAD procedure than the DO procedure. Identification errors due to assumed cavitation conditions still need to be addressed.