4.3 Instability Self-Excited-Vibration Threshold
4.3.1 Symmetric 3-Mass Rotor + 2 Anisotropic Bearings (Same) and Disk
In the previous three examples, for unbalance response, the bearing prop- erties were contrived to be independent of speed just to keep the input shorter. In actual applications involving journal bearings, the dynamic properties of the bearings are usually quite speed dependent and should thus be input as such even for unbalance response computations. In the examples here for instability threshold speed prediction, speed-dependent bearing properties are not optional since they are required to demonstrate the computations. RDA uses bearing dynamic property inputs at a user- selected number of appropriate speeds (maximum of 10) to interpolate
(a) Orbit referenced to static equilibrium position
x/C y/C
+1 -1
+1
-1
x y
Bearing clearance circle C = Radial clearance (b) Orbit referenced to
bearing center
Nonlinear limit cycle
FIGURE 4.11 Transient orbital vibration build-up in an unstable condition: (a) initial linear transient build-up and (b) growth to nonlinear limit cycle.
for intermediate speeds using a cubic-spline curve fit, both for unbalance response as well as instability threshold speed computations.
From the RDA MAIN MENU, option 4 initiates an instability threshold speed computation, and the DATA MENU shown earlier in this chap- ter appears. Using bearing property inputs at five or more speeds is not unusual and the considerable amount of corresponding input certainly suggests that the user use a full-screen editor outside the RDA environ- ment, at least for the speed-dependent bearing properties. That following full-screen input process is applicable for both unbalance response and instability thresholds. Inputs are in free format.
Input Title: 50 spaces for any alpha-numeric string of characters (1 line) No. of: Stations, Disks, Bearings, Pedestals, Extra Weights (integer)
(1 line)
Units Code: “1” for inches & pounds
Shaft Elements: OD, ID, Length, Inertia, Weight (1 line for each element)
Disks: Station No. (integer), OD, ID, Length, Weight, IP, IT(1 line for each disc)
Bearings: Station No. (integer), Weight (1 line for each bearing) Pedestals: Station No. (integer), Weight (1 line for each pedestal) Pedestals: Kxx, Kyy, Cxx, Cyy(1 line for each pedestal)
Added Rotor Weights: Station No. (integer), Weight (1 line for each weight)
Shaft Material: Modulus of elasticity, Poisson’s ratio (1 line) No. of Speeds for Bearing Dynamic Properties: (integer) (1 line) Bearing Dynamic Properties:
RPM (1 line)
Kxx, Kxy, Cxx, Cxy, Kyx, Kyy, Cyx, Cyy
(1 line for each bearing)
⎤
⎦Sequence for each RPM
Unbalances: Station No., Amplitude, Phase Angle (1 line for each station).
This last input group of lines (unbalances) is ignored by RDA when exe- cuting threshold speed runs, but may be retained in the input file. It can therefore also be excluded when executing threshold-speed runs. The input file for this sample can be viewed in file sample04.inp, but is not printed here in the interest of space.
Entering option 1 in the DATA MENU produces the INPUT OPTIONS menu, from which option 1 (file input) prompts the user for the input file name, which is sample04.inp for this example. Input file must reside in RDA99 directory. Upon entering the input file, the user is returned to the DATA MENU, where the user can select any of the six options, including option 5 that executes the previously designated Main Menu option 4 for stability analyses. Three stability analysis options are displayed as follows:
S T A B I L I T Y A N A L Y S I S *********************************************
The Options Are:
1. Do not iterate to find threshold speed.
Energy check will not be performed.
Store the eigenvalues for plotting.
2. Find the threshold speed of instability.
Perform energy check at threshold speed.
Store the eigenvalues for plotting.
3. Find the threshold speed of instability.
Perform energy check at threshold speed.
Do not store the eigenvalues for plotting.
The new user should explore all three of these options. Option 1 pro- vides the complex eigenvalues for a speed range and increment prompted from the user. Plotting the real eigenvalue parts as functions of speed is one way of determining the instability threshold speed, that is, by find- ing the lowest speed at which one of the eigenvalue real parts changes from negative (positively damped) to positive (negatively damped). At this negative-to-positive crossover speed, the two eigenvalues for the threshold (zero-damped) mode are imaginary conjugates and thus provide the natu- ral frequency of the unstable mode. Plotting the first few lowest frequency modes’ eigenvalues versus speed can provide information (see Campbell diagram in Section 4.2.6) to corroborate which modes are shown to be sensitive to rotor unbalance. However, option 3 is more expedient since it automatically “halves in” on the positive-to-negative crossover thresh- old speed to within the user supplied speed convergence tolerance. In this demonstration example, option 3 is selected. Although not often experi- enced by the author, option 3 may skip over an instability threshold speed due to the fact that the bearing coefficients are provided at distinct speeds between which curve fitting of bearing coefficients is used. Small errors stemming from this curve fitting can significantly corrupt the instability threshold search algorithm. Rather than reflecting algorithm shortcomings, this potential difficulty is a result of the extreme sensitivity of the balance of positive and negative energy right at an instability threshold. If this dif- ficulty occurs, use option 1 and plot the real part versus speed for the lower frequency modes and thereby graphically capture the zero crossover speed.
The unstable mode theoretically has exactly zero net damping at the instability threshold, so its eigenvector at the threshold speed (and only at the threshold speed) is not complex. Thus, a real mode shape can be extracted from the threshold-speed eigenvector. The “energy check”
referenced in the STABILITY ANALYSIS menu uses the eigenvector com- ponents for the mode at the determined stability threshold speed to construct that mode’s normalized x and y harmonic signals at the bearings to perform an energy-per-cycle computation at each bearing, as provided by Equation 2.81. This computation provides a potential side check for solution convergence of the threshold speed, because exactly at a thresh- old of instability the sum of all energy-per-cycle “in” should exactly cancel all energy-per-cycle “out.” However, the second example in this section demonstrates that in some cases inherent computational tolerances in eigenvector extraction can make the energy-per-cycle residual convergent to a relatively small but nonzero limit. STABILITY ANALYSIS option 3 prompts for the following (inputs shown):
Input lower speed (rpm) 0 (RDA starts at the lowest bearing data speed)
Input upper speed (rpm) 4000
Desired accuracy (rpm) 1
The user is next prompted with an option to change the speed tolerance.
With present PCs being so much faster than the early PCs for which RDA was originally coded, the user should answer the prompt with “N” for
“No.” The user is next prompted to select from the following three choices.
The bearing coefficients will be fitted by a cubic spline.
Three types of end conditions could be used:
1. Linear 2. Parabolic 3. Cubic
Option 1 (Linear) is used in this demonstration example.
The user is next prompted to select from the following three choices pertaining to shaft mass model formulation, just as in unbalance response cases.
Shaft mass model options:
1. Lumped mass 2. Distributed mass 3. Consistent mass
The consistent mass option is usually the preferred choice, and is chosen here. For any given rotor, curious users may compare model resolution accuracy or convergence of these three options by varying the number of shaft elements.
The last user prompt is to give a name to the output file that will be generated (here sample04.out is provided). The complete output file for this example is provided with the CD-ROM that comes with this book.
An abbreviated portion of that output file is given here as follows:
Stability Analysis Results
Threshold speed 2775.6 rpm±1.00 rpm Whirl frequency 1692.5 cpm
Whirl ratio 0.6098
Energy Per Cycle at the Onset of Instability Damping Part, Stiffness Part,
Bearing No. Rotor Location Csij Kssij Net Energy
1 1 −349.9 350.0 0.148
2 3 −348.5 348.6 0.143
Energy/cycle of the bearings total 0.291
As can be observed from the quite small bearing energy-per-cycle residual, the user provided 1-rpm convergence criteria for the instabil- ity threshold speed provides an eigenvector indicative of a zero-damped mode. The energy-per-cycle output tabulations reflect that the model (including bearing coefficient inputs) is symmetric about the rotor mid- plane. In the next example, where the bearings are somewhat different, it is seen that the total energy per cycle residual does not approach “small- ness” to the same degree as this example, even though the threshold speed iteration has essentially converged to the solution. One can conclude that the energy-per-cycle criteria for convergence are much more stringent than the speed tolerance. The “whirl ratio” (whirl frequency/threshold speed) is always less than “one” for this type of instability, that is, the associated self-excited vibration is always subsynchronous.
The normalized threshold (zero-damped) mode used for the energy- per-cycle computations is essentially planar, which can be deduced from the following RDA output for this example. The mode for this exam- ple is indicative of the typical nearly circular orbit shapes at instability thresholds.
Normalized Self-Excited Vibration Mode
Coordinate Amplitude Phase (rad) Phase Angle (◦)
x1 1 0.1016590 0.9276607E−03 0.0
y1 2 0.8729088E−01 −1.822578 −104.4
θx1 3 0.1160974 1.317930 75.5
θy2 4 0.1352273 −0.9837417E−04 0.0
x2 5 1.000000 0.0000000 0.0
y2 6 0.8586232 −1.823513 −104.5
θx2 7 0.2455765E−03 1.304664 74.8
θy2 8 0.2896766E−03 0.9317187E−02 0.0
x3 9 0.1014572 0.8898759E−03 0.0
y3 10 0.8711492E−01 −1.822611 −104.4
θx3 11 0.1163277 −1.823615 −104.5
θy3 12 0.1355006 3.141485 180.0
Namely, the orbits are “fat ellipses” or “almost circular,” and there is an insight to be gleaned from this. Referring to Equation 2.79 for the energy- per-cycle input from the skew-symmetric part of the bearing stiffness matrix, the integrated expression is the orbit area. Thus, the destabilizing energy is proportional to the normalized orbit area, which is a maximum for a purely circular orbit. A major European builder of large steam turbo- generator units used this idea “in reverse” by making the journal bearings much stiffer in the vertical direction than in the horizontal direction, to create “very flat” modal orbit ellipses (i.e., small normalized orbit areas),
with the objective of increasing the instability threshold power for steam- whirl-induced self-excited vibration. This design feature unfortunately made these machines difficult to balance well, and was thus subsequently
“reversed” in the power plants as per customers’ request (Adams and Makay, 1981).