properties of the oil film between the rotating shaft, and stationary bearings of different configurations. This is an acceptable description of the rotor support system if the bearings are rigidly supported. Industrial machines with heavy cases, and light rotating elements fall within this category. Barrel compressors with internal bearings, rigid gear boxes, high pressure pumps, and many older pieces of equipment operate with structural stiffness that are substantially greater than the oil film stiffness.
However, this is not the case for many other machines that have flexible supports and/or foundations. Units such as induced draft or forced draft fans, steam or gas turbines, horizontally split centrifugal compressors, and pumps with external bearings are just a few examples of machines that operate with flexible supports. For these types of machines the remainder of the mechanical system must be included. In a general case, the effective support stiffness for a typical rotor on a flexible support may be defined by equation (4-15) that describes the relationship as a group of springs in series:
(4-15)
where: Keff = Effective Rotor Support Stiffness (Pounds/Inch) Koil = Oil Film (Bearing) Support Stiffness (Pounds/Inch) Khsg = Bearing Housing Support Stiffness (Pounds/Inch) Kbase = Baseplate Support Stiffness (Pounds/Inch)
Kfnd = Foundation Support Stiffness (Pounds/Inch)
This expression will be subjected to substantial modification if the support structure is in a resonant condition, or if the support is highly flexible. However, these are rare occurrences, and the above equation (4-15) is considered to be gen-erally representative of the normal rotor support parameters.
Quantification of the structural support terms in equation (4-15) is a formi-dable technical feat. The calculation of these individual stiffness terms is diffi-cult at best, and in some cases it is virtually impossible. The most reasonable approach for determination of the support coefficients is a direct measurement of the dynamic stiffness of the support structure. This measurement requires the application of a defined force to the structure, and the determination of the resultant movement. In the simplest case, a dial indicator is used to measure the displacement in thousands of an inch, and a calibrated hydraulic jack provides the force. Division of the applied force by the total movement yields a static
stiff-1 Keff --- 1
Koil --- 1
Khsg --- 1
Kbase --- 1
Kfnd --- 1
etc
---+ + + +
=
ness in Pounds/Inch. This is a zero frequency technique that often fails to provide the correct structural stiffness since the characteristics vary with frequency.
It is possible to measure variable frequency structural stiffness by exciting the system with an appropriate device, and measuring the response with a vibra-tion transducer. A frequency response funcvibra-tion (FRF) measurement (a.k.a., transfer function) may be performed between the signal emitted by a force trans-ducer, and the resultant displacement motion signal. This FRF measurement should include the amplitude relationship between force and motion at each fre-quency bin, plus phase and coherence information. The force applied to the struc-ture would typically be measured in Pounds, and the structural response would be measured in Inches. The vibration or motion measurement could be made with a proximity probe mounted on an isolated stand, or with a seismic trans-ducer that is integrated to displacement. In most cases, the field motion mea-surements are obtained with an accelerometer, and this signal is double integrated to obtain casing displacement. The engineering units for the fre-quency response function are Pounds/Inch, and this measurement is commonly referred to as Dynamic Stiffness.
The device used to excite the structure may vary from an electromechanical shaker to an impact hammer. The use of an electromechanical shaker provides a highly controllable excitation source, whereas an impact hammer is easily applied in a variety of situations. The physical installation of any shaker is often hampered by limited access to the assembled machinery bearing housings. In some cases, an electromechanical shaker with a stinger attached may be used to reach specific mechanical elements. In other situations, the selection of an impact hammer provides the necessary size and flexibility to excite a machine bearing housing with an acceptable and definable impact force. A typical arrangement for measuring the horizontal stiffness of a bearing housing with an
Fig. 4–17 Typical Test Arrangement For Bearing Housing Horizontal Impact Test
Impact Hammer Output to DSA Accelerometer Output to DSA Bearing
Bearing Pedestal Shaft
Bearing Cap
High Sensitivity Accelerometer
Bearing Supports — Measurements and Calculations 181
impact hammer is shown in Fig. 4-17.
Manual impact hammers come in various sizes for different testing applica-tions. For instance, small hammers weigh between 1 and 2 Grams, and exhibit a frequency limit of nominally 900,000 CPM (15 KHz). These miniature impact hammers are used for static testing of items such as turbine blades. At the other end of the scale a 12 pound sledge hammer, or an instrumented battering ram may be used for low frequency tests on large structures such as foundations or buildings. For bearing housing measurements, a typical impact hammer weighs between 0.3 and 3 Pounds, and it is capable of producing a concentrated 5,000 pound force upon the test element. The dynamic force produced by the hammer is generally measured with an integral piezoelectric force transducer. Frequency response characteristics typically vary from 300 CPM (5 Hz), to a usable maxi-mum frequency of about 60,000 CPM (1,000 Hz). Thus, the dynamic characteris-tics of a small to medium sized force hammer adequately cover the operating speed range of most machines.
During structural impact tests, the casing response is usually measured with an accelerometer attached to the bearing cap. The accelerometer should have a frequency range that is compatible with the force transducer. In addition, the accelerometer signal must be double integrated to convert acceleration to casing displacement. This double integration may be performed in an external analog device, or by application of wave form math in the DSA. In either case, the final FRF measurement between the applied force (Pounds) and the result-ant displacement response (Inches) yields an equivalent support stiffness for the bearing housing (Pounds/Inch). In all cases, the validity of the frequency response data is checked with the coherence function, and the relative phase between signals should be examined.
For measurements of structural natural frequencies, the test setup is iden-tical to the dynamic stiffness measurements, but the transducers are reversed.
In this type of test, a FRF is performed between the measured acceleration divided by the input force. Double integration of the acceleration signal is nei-ther required, nor desirable — and the FRF output units are typically G’s/Pound.
This type of data is commonly referred to as Inertance, and it should be per-formed whenever a structural resonance is suspected.
Case History 8: Measured Steam Turbine Bearing Housing Stiffness For demonstration purposes, a typical data set is presented in Fig. 4-18.
This information was obtained with a field instrumentation setup identical to Fig. 4-17. This test was performed on the exhaust bearing housing of an 8,000 HP steam turbine that normally operates at 8,520 RPM. The data was acquired with a 3 pound impact hammer, and a high sensitivity accelerometer mounted in a horizontal plane. Both the impact hammer and the accelerometer were directly connected to an HP-35665A Dynamic Signal Analyzer (DSA). The power source within the DSA was used to drive both piezoelectric transducers, and the result-ant data was stored on a floppy disk. The data was later examined on an HP-35670A, and the data displayed in Fig. 4-18 committed to hard copy format with
an HP-5L LaserJet printer. The FRF yields the dynamic stiffness plot at the bot-tom of Fig. 4-18. Since this data covers a wide amplitude range, a log scale was used for the stiffness. It is noted that a reasonably flat region exists between 4,800 and 13,000 CPM. At the normal operating frequency of 8,520 RPM the FRF reveals a dynamic stiffness value of 1,210,000 Pounds/Inch. This is judged to be a realistic value for the heavy cast steel bearing housing. The FRF also shows a substantial drop in stiffness at frequencies of 660 and 16,560 CPM.
If the center phase plot in Fig. 4-18 is examined, the large phase shift at 16,560 CPM might be interpreted as a structural resonance. However, when the coherence plot at the top of Fig. 4-18 is considered, it is evident that coherence between force and motion signals has dropped to below 0.2 at 16,560 CPM. This indicates that the FRF data is not valid, and the significance of the change at 16,560 CPM should be removed from further consideration.
At the turbine speed of 8,520 RPM the computed coherence was 0.97. Gen-erally, coherence values greater than 0.9 are indicative of acceptable FRF data.
Hence, the information in the vicinity of the turbine running speed is considered
Fig. 4–18 Frequency Response Function (FRF) Of Steam Turbine Bearing Housing Horizontal Dynamic Stiffness
Bearing Supports — Measurements and Calculations 183
to be excellent data. When coherence drops to levels below 0.9, the FRF data should be cautiously applied. If coherence drops below 0.7, the FRF data should generally be ignored.
The data array shown in Fig. 4-18 is easily acquired, and rapidly processed.
From the previous discussion it is summarized that the dynamic stiffness at tur-bine speed of 8,520 RPM was obtained directly from the FRF plot, and verified by the coherence. The validity of the amplitude and phase change at 16,560 CPM was found to be highly questionable due to the low coherence. However, the drop in FRF amplitudes at low frequencies was not fully explained. For an improved understanding of this behavior, and examination of the component force and dis-placement signals is required. This supplemental data is presented in Fig. 4-19 over the same frequency range used for the FRF data in Fig. 4-18.
The upper diagram in Fig. 4-19 is the force (in Pounds) applied across the frequency domain of 0 to 24,000 CPM. If this same data was viewed in the time domain, a sharp initial pulse would be observed. Within the frequency domain, this pulse provides a reasonably uniform excitation across the selected analysis bandwidth. Hence, it may be properly concluded that the low frequency drop off of the FRF data is not due to any significant variations in the applied force.
However, the measured displacement presented in the bottom diagram of Fig. 4-19 reveals a large increase in the response at 660 CPM. There might be a tendency to consider the 660 peak as a resonance, but this conclusion is not sup-ported by the differential phase data of Fig. 4-18. Furthermore, structural reso-nances have a narrow bandwidth, and the 660 CPM peak show in Fig. 4-18 does not display this characteristic. In all probability, the 660 CPM peak is due to a measurement anomaly. More specifically, an accelerometer was used to make the bearing housing response measurements. The acceleration signal was double integrated to obtain displacement. This conversion is accomplished within the DSA by dividing the acceleration signal by frequency squared.
At the boundary condition of zero frequency, the integrated displacement would have a value of infinity. This does not appear in the data because there is
Fig. 4–19 Force And Dis-placement Data Used to Develop Steam Turbine Frequency Response Function (FRF) Dynamic Stiffness Plot
virtually no measurable acceleration output until the vibration transducer becomes active around 180 CPM (3 Hz). However, it is a fundamental fact that displacement amplitudes at low frequencies may be abnormally amplified due to the double integration process. This is further complicated by any noise in the acceleration signal that might also be erroneously amplified during double inte-gration. Hence, the mechanical significance of the 660 CPM peak is eliminated.
In addition, the displacement drop at 16,560 CPM is not meaningful information due to the previously mentioned low coherence at this frequency.
Since the dynamic stiffness FRF consists of force divided by displacement, the increased displacement at low frequencies produces a reduction in the dynamic stiffness. This is common behavior in all of these measurements, and low frequency data is generally ignored. This is acceptable since the low speed stiffness is considerably less important than the housing stiffness within the operating speed domain.
From a measurement standpoint, it should be mentioned that the casing displacement resulting from an impact hammer test is very small. For example, the peak displacement at 8,520 CPM on Fig. 4-19 is only 3.36 x 10-7 Inches,o-p. This value is equivalent to 0.000672 Mils,p-p. Using equation (2-21), this dis-placement converts to an accelerometer output of 0.000693 G’s,o-p at the machine frequency. Fortunately, a high sensitivity accelerometer was used for this data, and the scale factor of 10,000 mv/G resulted in a signal strength of only 6.93 mil-livolts,o-p. If the DSA is set for a full scale range of 1.0 volt,o-p, the acceleration signal would appear at -43 dB. Although this is a small voltage, it is still within the range of most analyzers.
As an alternate scenario, if the response accelerometer had a scale factor of only 100 mv/G, the electrical signal would be proportionally reduced. In this situ-ation, the analyzer would have to accommodate a low level signal of -83 dB.
Unfortunately, many instruments do not have an adequate dynamic range to handle this variation in amplitudes. In all cases, it is recommended that a high sensitivity accelerometer (1,000 or 10,000 mv/G) be employed for this type of measurement. In addition, the data should be processed with a DSA that has a sufficiently large dynamic range (e.g., HP-35670A).
If an electromechanical shaker was substituted for the impact hammer, the applied force would be greater, and the resultant casing motion measured by the accelerometer would also increase. Thus, the measurement problems would diminish. However, the larger signal amplitudes must be compared with the potential difficulty and time required to properly mount an electromechanical shaker in the field. Regardless of the excitation source, the data processing and examination techniques are essentially the same. Overall, the user must be fully aware that this type of measurement is subject to a variety of errors, and all aspects of the FRF must be validated.
In retrospect, the primary objective of this exercise is directed at a mea-surement of bearing housing dynamic stiffness. As discussed, the bottom plot of Fig. 4-18 depicts the variation of this parameter with frequency. In many cases it is desirable to develop an equation that describes this behavior. By using
Bearing Supports — Measurements and Calculations 185
priate curve fitting software within the DSA, a suitable polynomial equation may be defined that relates frequency to stiffness. This is an important consideration during the accurate modeling of rotating equipment as discussed in chapter 5.
However, there are situations when this type software is not available, and another approach must be used to develop the characteristic equation.
Case History 9: Measured Gas Turbine Bearing Housing Stiffness
The data presented in Fig. 4-20 was acquired on the inlet bearing housing of a natural gas fired 40,000 HP gas turbine. This machine operates at 5,300 RPM, and casing stiffness information was required to enhance the accuracy of the analytical rotor model. Data was obtained in the vertical direction (Y-Y) with a vertical accelerometer, and a vertical impact hammer excitation. Information was also acquired in the horizontal plane (X-X) with a horizontal accelerometer, and a horizontal impact. For consistency with previous examples, the horizontal FRF data is shown in Fig. 4-20. This information was derived with a three pound
force hammer directly connected to an HP-3560A portable analyzer. Casing motion was measured with a high sensitivity accelerometer that was subjected to analog double integration prior to DSA processing. A comparison of this FRF with the previous example reveals a somewhat jagged curve in Fig. 4-20. After examining the various supplemental plots, it was concluded that the deviations are attributed to the analog double integration. Fortunately, coherence was above 0.9 at all frequencies above 1,000 CPM. Hence, the FRF data was consid-ered to be acceptable, but a polynomial equation of this FRF was still required.
Since the HP-3560A does not have curve fitting capabilities, the FRF data was exported to a Microsoft® Excel spreadsheet, and dynamic stiffness values were listed at 30 CPM intervals. This data was then subjected to a sixth degree poly-nomial curve between frequencies of 1,000 and 6,000 CPM, and the following characteristic equation was generated:
Fig. 4–20 Horizontal Stiff-ness Measurement Of Gas Turbine Inlet End #1 Bearing Housing
Within the specified frequency range, this polynomial expression may be used to calculate the horizontal structural stiffness of this inlet end #1 gas tur-bine bearing housing as a function of speed. As an example, the horizontal stiff-ness at the machine running speed of 5,300 RPM may be computed in the following manner.
This calculated value of 909,000 Pounds/Inch is consistent with the FRF measurement in Fig. 4-20. This type of curve fitting may also be used to deter-mine the oil film stiffness as a function of rotating speed. The calculated stiffness versus speed curves (e.g., Fig. 4-1) may be converted to a polynomial equation, and the resultant expression used within the rotor response programs. In most cases, a third or fourth degree polynomial is sufficient to describe the stiffness curves, but the use of fifth or sixth degree equations are common.
The oil film stiffness, and various structural stiffness are combined in a reciprocal manner as described in equation (4-15). However, the field FRF tests essentially combine the stationary structural elements into a single housing stiffness. Thus, it is reasonable to simplify the overall or effective rotor support equation into the following common format:
(4-16)