José A. Vázquez

A Flexible Rotor on Flexible Bearing Supports: Stability and Unbalance Response

An experimental study of the effects of bearing support flexibility on rotor stability and unbalance response is presented. A flexible rotor supported by fluid film bearings on flexible supports was used with fifteen support configurations. The horizontal support stiffness was varied systematically while the vertical stiffness was kept constant. The support characteristics were determined experimentally by measuring the frequency response functions of the support structure at the bearing locations. These frequency response functions were used to calculate polynomial transfer functions that represented the support structure. Stability predictions were compared with measured stability thresholds. The predicted stability thresholds agree with the experimental data within a confidence bound for the logarithmic decrement of ±0.01. For unbalance response, the second critical speed of the rotor varied from 3690 rpm to 5200 rpm, depending on the support configuration. The predicted first critical speeds agree with the experimental data within -1.7 percent. The predicted second critical speeds agree with the experimental data within 3.4 percent. Predictions for the rotor on rigid supports are included for comparison.

Reconciliation of Rotordynamic Models With Experimental Data

A computationally efficient strategy is presented for adjusting analytic rotordynamic models to make them consistent with experimental data. The approach permits use of conventional rotordynamic models derived using finite element methods in conjunction with conventional plant identification models derived from impact or sine sweep testing in a transfer function or influence coefficient format. The underlying assumption is that the predominant uncertainties in engineered models occur at discrete points as effects like shrink fits, seal coefficients or foundation interactions. Further, it is assumed that these unmodeled or poorly modeled effects are essentially linear (at least within the testing and expected operating domains). Matching is accomplished by deriving a dynamic model for these uncertain effects such that the resulting composite model has a transfer function which matches that obtained experimentally. The derived augmentations are computationally compatible with the original rotor model and valid for stability or forced response predictions. Further, computation of this augmentation is accomplished using well developed and widely disseminated tools for modern control. Background theory and a complete recipe for the solution are supported by a number of examples.

Model Identification of a Rotor With Magnetic Bearings

The experimental identification of a long flexible rotor with three magnetic bearing journals is presented. Frequency response functions are measured between the magnetic bearing journals and the sensor locations while the rotor is suspended horizontally with piano wire. These frequency response functions are compared with the responses of a rotor model and a reconciliation process is used to reduce the discrepancies between the model and the measured data. In this identification, the wire and the fit of the magnetic bearing journals are identified as the sources of model error. As a result of the reconciliation process, equivalent dynamic stiffness are calculated for the piano wire and the fit of the magnetic bearing journals. Several significant numeral issues that were encountered during the process are discussed and solutions to some of these problems are presented.

Including the Effects of Flexible Bearing Supports in Rotating Machinery

Unbalance response and stability analyses of a flexible rotor on three lobe journal bearings on flexible supports are presented. The influence of the support structure was included in the analyses using polynomial transfer functions. These transfer functions were extracted from measured dynamic compliance data of the support structure, measured at the bearing locations. Numerical predictions using polynomial transfer functions and single mass supports are compared to the experimental data. Predictions using the transfer function representation of the support structure show a clear improvement over the predictions using single mass supports without over-complicating the problem. The predicted critical speeds are within 2.9% of the measured critical speeds. The predicted stability threshold agrees with the measured stability threshold within 1%. The effects of cross talk between supports and cross coupling between horizontal and vertical directions are investigated. The cross talk between supports was found to have a strong influence in the results while the influence of cross coupling between the vertical and horizontal directions is negligible.

Representing Flexible Supports by Polynomial Transfer Functions

Flexible bearing supports may have a great influence in the calculation of forced response and stability of rotor systems. However, this effect is not always included in rotor analyses since an accurate model of the foundation and pedestals may be difficult and costly to obtain. It is common practice to use either a one degree of freedom model or a full modal analysis to represent the bearing supports. While the one degree of freedom model is easy to set up for computer calculations, it often requires experience to determine values for the stiffness, mass and damping of the model that will accurately represent the support under study. This model, however, fails to capture the dynamics of the system for stability analyses when more than one mode of the support structure is in the range of interest. On the other hand, modal representation provides much more information and can be measured experimentally, but requires measurement of the mode shapes. Even though modal representation can include all the dynamics of the system in the frequency range of interest, it provides much more information than is required for calculation of the rotor response and it is more difficult to use in calculation programs. This paper presents a procedure to include the support characteristics using transfer functions. Transfer functions permit modeling of multi-degree of freedom systems while maintaining the size of a one degree of freedom system (2×2 matrix if rotation at the bearing is not considered). Another advantage of transfer functions is that they can be obtained from existing discrete models, from modal information or can be measured directly. The fixed size of the transfer function matrix permits the method to be easily incorporated into rotor dynamic stability and forced response programs. The method is applied to stability calculations of models of typical industrial machines.© 1998 ASME

Synchronous Response Estimation in Rotating Machinery

Synchronous response estimation attempts to determine the forced response (displacement) of a rotor at critical points which cannot be measured directly. This type of prediction, if accurate and reliable, has broad potential use in the rotating machinery industry. Many machines have close clearance points on their shafts, such as seals, which can easily be damaged by excess vibration. Accurate estimates of the actual level of vibration at these points could usefully assist machine operators in troubleshooting and in protecting the equipment from expensive damage. This type of response information can be used both to generate less conservative alarm limits and, if magnetic bearings are available, to directly guide the bearing controllers in restricting the rotor motion at these critical points. It is assumed that the disturbance forces acting upon the rotor are predominantly synchronous. The response estimate is constructed using the measurable response in conjunction with an estimator gain matrix derived from a model of the transmissibilities of the rotor system. A fundamental performance bound is established based on the single-speed set of measurements by bounding the response to the unmeasurable component of the disturbance force. Acknowledging that some model uncertainty will always exist, a robust performance analysis is developed using structured singular value (μ) analysis techniques. Assuming some reasonable levels of uncertainty for the model parameters (natural frequencies, modal dampings, mode shapes, bearing stiffnesses, and dampings) the results of the estimator construction and analysis establish feasibility of the proposed estimation. Two reference rotor models that are representative of industrially sized machines are used to demonstrate and evaluate the estimation. The unmeasurable response estimation errors consistently lie below 25 μm for the examples examined.

Comparison Between Calculated and Measured Free-Free Modes for a Flexible Rotor

Many industrial machines nowadays are sold based on analysis performed on mathematical models of the rotors, bearings, substructures, and other components. The validity of the analysts therefore depends on the accuracy of the models themselves. When the rotor is available, modal testing may be used to validate the model of the rotor by comparing the calculated and measured free-free natural frequencies and mode shapes. This work presents additional tools for the verification of analytical models against experimental data. These tools use models of the rotor constructed from the measured data and the analytical model. A comparison of the first six calculated and measured free-free natural frequencies and mode shapes for a multi-mass flexible rotor is presented. The natural frequencies compare within 1.8%. The calculated and measured mode shapes were used to construct independent reduced order models of the rotor. These models were used to perform forced response and stability analyses. Forced response functions are presented comparing the forced response characteristics obtained from the two models. This provides a comparison between the measured and calculated forced response functions for the same number of modes. For the stability analysis, identical bearing models were added to both reduced order models. The eigenvalues were calculated using both models for a range of bearing stiffness and damping coefficients and were plotted for comparison.Copyright