2.5 Nonlinear Effects in Rotor Dynamical Systems
2.5.2 Journal-Bearing Nonlinearity with Large Rotor Unbalance
Fluid-film journal bearings are a prominent component where dynamic nonlinearity can play a controlling role in rotor vibration when the journal- to-bearing orbital vibration amplitude becomes a substantial portion of the bearing clearance circle. When this is the case, the linear model introduced in Equations 2.60 fails to provide realistic rotor dynamic pre- dictions, as previously explained at the beginning of this section. As detailed in Chapter 5, computation of the fluid-film separating force that keeps the journal from contacting the bearing starts by solving the lubri- cant pressure distribution within the separating film. The film’s pressure distribution is computed by solving the PDE known as the Reynolds lubrication equation, the solution for which other types of CPU-intensive numerical computations are required (e.g., finite difference, finite ele- ment). Performing a numerical time marching integration of the motion equations for a rotor supported by fluid-film journal bearings requires that the fluid-film bearing forces to be recomputed at each time step of the time marching computation. Thus, the fluid-film pressure distri- butions at each journal bearing must be recomputed at each time step.
Therefore, depending upon the level of approximation used in solv- ing the Reynolds equation, it can be quite CPU intensive to perform a time marching integration of the motion equations for a rotor supported by fluid-film journal bearings. For an instantaneous journal-to-bearing {x, y,x,˙ y}, the x and y components of fluid-film force upon the journal˙ are computed by integrating the instantaneous x and y projections of the film pressure distribution upon the journal surface, as expressed by the
following equation:
fx−Wx
fy−Wy
= −R
L/2
−L/2 2π
0
p(θ, z, t) cosθ
sinθ
dθdz (2.87)
Referring to Figure 2.10, Wxand Wyare the x and y components, respec- tively, of the static load vectorW acting upon the bearing. Thus, −Wxand
−Wy are the corresponding static reaction load components acting upon the journal. Just as in linear LRV models, it is convenient in nonlinear anal- yses to formulate the equations of motion relative to the static equilibrium state. In so formulating the nonlinear LRV motion equations, the journal static loads(−Wx and −Wy)are moved to the right-hand side of Equa- tion 2.87, leaving{fx, fy}as the instantaneous nonequilibrium dynamic force upon the journal.
The photographs in Figure 2.15 show some of the aftermath from two 1970s catastrophic failures of large steam turbo-generators. Both these fail- ures occurred without warning and totally destroyed the machines. The author is familiar with other similar massive failures. Miraculously, in none of the several such failures to which the author has been familiarized have any serious personal injuries or loss of life occurred, although the poten- tial for such personal mishap is surely quite possible in such events. The two early 1970s failures led to the author’s work in developing computer- ized analyses to research the vibration response when very large rotor mass unbalance is imposed on a multibearing flexible rotor. For in-depth treatment of computational methods and results for nonlinear LRV, the group of ten papers on nonlinear rotor dynamics topics, listed in the Bibliography at the end of this chapter, are suggested. Some of the author’s reported results are presented here.
The rotor illustrated in Figure 2.16 is one of the two identical low-pressure (LP) turbine rotors of a 700 MW 3600 rpm steam turbo-generator unit. It was used to computationally research the nonlinear vibrations resulting from unusually large mass unbalance. Using methods presented in Sec- tions 1.3 and 2.3, the free–free rotor’s undamped natural frequencies and corresponding planar mode shapes were determined from a finite-element model. All static and dynamic forces acting upon the rotor are applied on the free–free model as “external forces” including nonlinear forces, for example, bearing static and dynamic loads, unbalances, gyroscopic moments, weight, and so on. This approach is detailed in Adams (1980) and supplemented in the other associated papers referenced. The compu- tation essentially entails solving the rotor response as a transient motion, numerically integrating forward in time for a sufficiently large number of shaft revolutions until a steady-state or motion envelope is determined.
Steady-state large unbalance results for the rotor in Figure 2.16 are shown in Figure 2.17a for the rotor supported in standard fixed-arc journal bear- ings, and in Figure 2.17b for the rotor supported in pivoted-pad journal
(a)
(b)
(c)
FIGURE 2.15 (See color insert following page 262.) Photos from the two 1970s catas- trophic failures of large 600 MW steam turbine-generator sets. Using nonlinear rotor dynamic response computations, failures could be potentially traced to the large unbalance from loss of one or more large LP turbine blades at running speed, coupled with behavior of fixed-arc journal bearings during large unbalance. (a) LP steam turbine outer casing. (b) Brushless exciter shaft. (c) Generator shaft. (d) LP steam turbine last stage.
(d)
FIGURE 2.15 Continued.
bearings. In both of these cases, it is assumed that one-half of a complete last-stage turbine blade detaches at 3600 rpm. This is equivalent to a 100,000 pound corotational 3600 cpm rotating load imposed at the last-stage blade row where the lost blade piece is postulated to separate from the rotor.
As a point of magnitude reference, this LP turbine rotor weighs approx- imately 85,000 pounds. The Figure 2.17 results show four orbit-like plots as follows:
• Journal-to-bearing orbit normalized by radial clearance
• Total bearing motion (see bearing pedestal model, Section 2.3.9.2)
• Total journal motion
• Total fluid-film force transmitted to bearing
Journal Journal
FIGURE 2.16 LP rotor portion of a 3600 rpm 700 MW steam turbine.
The normalized journal-to-bearing orbit is simply the journal motion minus the bearing motion divided by the bearing radial clearance. For the cylindrical journal bearing of the Figure 2.17a results, this clearance envelope is thus a circle of unity radius. In contrast, for the pivoted four- pad journal bearing of the Figure 2.17b results, the clearance envelope is a square of unity side. A prerequisite to presenting a detailed explanation of these results are the companion steady-state vibration and dynamic force amplitude results presented in Figure 2.18 for unbalance conditions from zero to 100,000 pounds imposed at the same last-stage blade row of the same nonlinear model.
1 (a)
y y
6 1
6
3 5 4
-1 –1 2
-30 30
30
3
4
2 5
1x
-30
2
1
6 2
3
4 5
Total journal motion (mils)
Total bearing force (pounds)
x y
y
3 4
6 1
5 x 50
–50
–50 50
200,000
–200,000 Journal-to-bearing orbit
(Max e/C = 0.993)
Total bearing motion (mils) Bearing
clearance envelope
x 1
FIGURE 2.17 (a) Steady-state periodic response at bearing nearest the unbalance with force magnitude of 100,000 pounds, rotor supported on two identical fixed-arc journal bearings modeled after the actual rotor’s two journal bearings. Timing marks at each one-half revo- lution, that is, 3 rev shown. (b) Steady-state periodic response at bearing nearest unbalance with force magnitude of 100,000 pounds, rotor supported on two identical four-pad pivoted- pad bearings with the gravity load directed between the bottom two pads. Bearings have same film diameter, length, and clearance as the actual fixed-arc bearings. Timings mark each one-half revolution, that is, 3 rev shown.
Bearing clearance envelope (b)
y
y y
x x
y
x x 4
4
4 3
3
-20 4 20
20
-20
30 85,000
-85,000 -30
-30 30
3
3
6
6
6 6
5
5
5 Journal-to-bearing orbit
(Max e/C = 1.17)
Total journal motion (mils)
Total bearing force (pounds) Totlal bearing motion
(mils) Pivot point
2
2
2
1
1
1 1 5 2
FIGURE 2.17 Continued.
An informative transition between 30,000 and 40,000 pounds unbalance is shown in Figure 2.18, from essentially a linear behavior, through a clas- sic nonlinear jump phenomenon, and into a quite nonlinear dynamic motion detailed by the Figure 2.17 results for a 100,000 pound unbalance force. The explanation for the results in Figures 2.17 and 2.18 can be secured to the well-established knowledge of fixed-arc and pivoted-pad journal bearings concerning instability self-excited vibrations. The x−y signals displayed in Figures 2.17a and b contain sequentially numbered timing marks for each one-half rotor revolution time interval at 3600 rpm. The observed steady- state motions therefore require three revolutions to complete one vibration cycle for both cases shown in Figure 2.17. Thus, these steady-state motions both fall into the category of a period-3 motion since they both contain
Journal vibration amplitude (PAJB) Peak dynamic Bearing force ( PAJB)
Peak dynamicbearing force PPJB
Journal vibration amplitude (PPJB)
Legend:
Pivoted-pad journal bearing-PPJB Partial-arc journal bearing-PAJB Vibration
Dynamic force
(2.54mm) (890, 909 N)
Nonlinear jump phenomenon
200,000
100,000
100,000 (445, 454 N) 50,000
Unbalance force magnitude (pounds) 0
0 0
50 100
Peak dynamic bearing force (pounds)
Journal vibration (peak-to-peak, mils)
FIGURE 2.18 Comparison between partial-arc and pivoted-pad journal-bearing vibration control capabilities under large unbalance operating conditions of an LP steam turbine rotor at 3600 rpm; steady-state journal motion and transmitted peak dynamic bearing force over a range of unbalance magnitudes (data points mark computed simulation cases).
a 1/3 subharmonic frequency component along with a once-per-rev (syn- chronous) component. But these two cases are clearly in stark contrast to each other.
With the partial-arc bearings, Figure 2.17a, steady-state motion is domi- nated by the 1/3 subharmonic component and the journal motion virtually fills up the entire bearing clearance circle. However, in the second case which employs a tilting-pad bearing model, the 1/3 subharmonic compo- nent is somewhat less than the synchronous component and the journal motion is confined to the lower portion of the bearing clearance enve- lope. As is clear from Figure 2.18, with partial-arc bearings, motion undergoes a nonlinear jump phenomenon as unbalance magnitude is increased. With pivoted-pad bearings, a nonlinear jump phenomenon is not obtained. This contrast is even clearer when the motions are trans- formed into the frequency domain, as provide by FFT in Figure 2.19. The
journal-to-bearing trajectories in these two cases provide the instantaneous lubricant minimum film thickness. For the partial-arc case, a smallest computed transient minimum film thickness of 0.1 mil (0.0001 in.) was obtained, thus surely indicating that hard journal-on-bearing rubbing would occur and as a consequence seriously degrade the bearings’ catastro- phe containment abilities. For the pivoted-pad case, a smallest computed transient minimum film thickness of 2 mils (0.002 in.) was obtained, thus indicating a much higher probability of maintaining bearing (film) integrity throughout such a large vibration event, especially considering that the pivoted pads are also inherently self-aligning.
The comparative results collectively shown by Figures 2.17 through 2.19 show a phenomenon that is probably possible for most such machines that operate only marginally below the threshold speed for the bearing-induced self-excited rotor vibration, commonly called oil whip. That is, with fixed- arc journal bearings and a large mass unbalance above some critical level (between 30,000 and 40,000 pounds for the simulated case here), a very large subharmonic resonance is a strong possibility.
In linear systems, steady-state response to harmonic excitation forces can only contain the frequencies of the sinusoidal driving forces, as can be rigorously shown from the basic mathematics of differential equations.
However, in nonlinear systems, the response to sinusoidal driving forces has many more possibilities, including periodic motion (possibly with sub- harmonics and/or superharmonics), quasiperiodic motion (two or more noninteger-related harmonics), and chaos motion. Here, as the rotor mass unbalance is progressively increased, the journal-bearing forces become
100
50
Frequency (Hz)
Pivoted-pad bearings Fixed-arc bearings (2.54 mm)
00 20 40 60
Simulated condition:
100,000 pound unbalance force at a last-stage blade row of the low- pressure steam turbine illustrated in Figure 2.15, (3600 rpm)
Journal vibration (peak-to-peak, mils)
FIGURE 2.19 Fast Fourier transform of peak-to-peak journal vibration displacement amplitudes.
progressively more nonlinear, thus increasing the opportunity for dynamic behavior which deviates in some way from the limited behavior allowed for linear systems. Such LP turbines typically have a fundamental corota- tional mode which is in the frequency vicinity of one-third the 3600 rpm rotational frequency. This mode typically has adequate damping to rou- tinely be passed through as a critical speed at approximately 1100−1300 rpm.
However, up at 3600 rpm the speed-dependent destabilizing effect(kssxy) of the fixed-arc bearings upon this mode places the rotor–bearing sys- tem only marginally below the instability threshold speed, as dissected in Section 2.4. Therefore, what is indicated by the results in Figure 2.17a is that the progressively increased bearing nonlinearity allows some energy to “flow” into the lightly damped 1/3 subharmonic, whose amplitude then adds to the overall vibration level and thus adds to the degree of bearing nonlinearity, thus increasing further the propensity for energy to flow into the 1/3 subharmonic, and so on. This synergistic mechanism manifests itself as the nonlinear jump in vibration and dynamic force shown in Fig- ure 2.18. In other words, it is consistent with other well-known dynamic features of rotor–bearing systems.
Because of the emergence of strong nonlinearity in such a sequence of events, an exact integer match (e.g., 3:1 in this case) between the forcing frequency and the linearized subharmonic mode is not needed for the above-described scenario to occur.
Pivoted-pad journal bearings have long been recognized as not produc- ing the destabilizing influence of fixed-arc journal bearings. The four-pad bearing modeled in these simulations has a symmetric stiffness coefficient matrix, consistent with its recognized inherent stability. Therefore, the case with pivoted-pad bearings gives results that are consistent with the prior explanation for the high amplitude subharmonic resonance exhibited with fixed-arc bearings. That is, the inherent characteristic of the pivoted-pad-type journal bearing that makes it far more stable than fixed-arc bearings also makes it far less susceptible to potentially catastrophic levels of subharmonic resonance under large unbalance conditions. If the static bearing load vector is subtracted from the total bearing force, the dynamic bearing force transmissibility is approximately 4 for the Figure 2.17a results and 1 for the Figure 2.17b results. Thus, the pivoted-pad bearing’s superiority in this context is again manifest, in a 4:1 reduction in dynamic forces transmitted to the bearing support structure, the last line of defense.
The topic covered in the subsequent Section 2.5.4 would appear to be related to this type of large unbalance-excited subharmonic resonance non- linear jump phenomenon, which occurs at an exact integer fraction of the unbalance forcing frequency. However, the two phenomena are not exactly the same thing, since the journal-bearing hysteresis-loop phenomenon is self-excited and has its own frequency, being initiated only by a large bump or ground motion disturbance.