Bearing and shaft misalignments in large turbomachinery:
a few basic questions

Warning

This page holds a series of considerations about bearing misalignments in large turbosets. It intends to foster some debate with people dealing with misalignment issues. Watch its evolution as more information comes in.

Notions of Turbomachinery Alignment.


  Whereas flexible couplings somewhat allow subsequent bearing motions after cold alignment, rigid couplings are mere extensions of shafts. Hot misalignment causing bearing relative motion results in alternate stresses developing at the surface of shaft ends 
 
 
 
 
 
 
 
 
 
 

In the following discussion, one assumes that shaft radial alignment is always perfect. With rigid couplings, this is mandatory to avoid vibrations due to the crankshaft syndrome, as shown left  and introduces unbalances. This can be observed with pairs of eddy probes targeting coupling rims. 
Coupling eccentricies should not be confused with misalignment in the discussion. It can spotted with  traditional cold alignment techniques where clocks rotate with the shaft and this, as long as the coupling faces are not fastened. Afterwards one must use fixed  eddy probes to monitor the relative eccentricities of the rims, banking on the existence of single revolution center line.
1 
Principles of cold alignment

When a single horizontal shaft rests on its two bearings, it bends. As a result its shaft ends are no longer horizontal and the coupling faces are no longer vertical. If one neglects their own weight including the coupling halves belonging to them, they are straight. This assumption is almost true in turbomachinery where the shaft weight is concentrated between bearings. 
One assumes that each bearing supports half of the w shaft weight if shafts and bearings are centrally symmetric. 
Putting two such shafts side-by-side results in coupling faces not being parallel. Should they be bolted together, this would introduce a spurious bending moment that would (un) load bearings so that their reaction would no longer be w/2. 
Cold angular alignment thus consists of bringing coupling faces to be parallel by raising  or lowering some bearings by the appropriate amount, called the bearing cold alignment correction. Note that shaft ends are aligned with aa’ with a certain slope w.r.t. the horizontal. Bearing reactions remain unchanged from the case of single shafts side-by-side. Proceeding in this way along a whole shaft line, one ends up with its characteristic cautionary shape. 
In general the bearings of a well-aligned machine each support the shaft they belong to. In short they behave with some selfishness, not caring for sustaining neighboring shafts on the other side of the nearby coupling. Furthermore, shaft ends are not bent if one neglects their own weights. Neither coupling nor shaft ends transmit no bending moments.
2 
Hot misalignment 
Bearings pedestals may or may not move after cold alignment. The thickness of oil films in sleeve bearings will vary anyway. As a result, shaft ends will normally undergo some bending. The issue of hot misalignment can best be treated via the basic theory of material strength applied to beams. Surprisingly textbooks dealing over shaft alignment never mention this evidence. Yet this tremendously helps design devices to measure angular shaft misalignment directly. 
Beam material strength and hot misalignment. 
The left-hand side figure represents the shaft ends on its two nearest bearings A and B, first when aligned in cold condition (upper part) and then misaligned in cold condition (lower part). L is the distance between bearings 
When aligned, shaft ends are oriented along aa’. Their center line is straight. Bearing B was raised by ya(L) to bring coupling faces to be parallel, where is the axial distance between the bearings A and B nearest to the couplings on either side. The positions of the centers of the shaft cross-sections in a Cartesian system (x, y(x)) of coordinate whose origin coincides with the center of the cross-section containing bearing A. Another system used to described the motion of bearings is obtained by shifting the previous system by -r vertically, where 2r is the diameter of the shaft ends. Without impairing the generality of the reasoning, one assumes that cross-sections do not vary over the span of shaft ends. 
In cold aligned condition, y(x) = ya(x) = (slope of aa’) . x, expressing that shaft ends are straight. Reactions of bearings A and B are both w/2. Distance ya(L) in the bearing system coordinates is the amount by which bearing B was raised to obtain a good cold alignment. 
In hot misalignment, let bearing B rise Dy(L) while bearing remains in the same position. The center line of shaft ends bends. Bearing A unloads by Dw which is transferred to bearing B. A shear force transits through the shaft ends which undergo a linearly varying bending moment. The center line of shaft ends bends. 
The center line is characterized by  
y(x) = ya(x) + Dy(x) 
A straight answer to two important questions over alignment can be supplied via the basic theory of material strength applied to beams, since shaft ends are special examples of them: 
· what is the amount of the load exchange Dw 
· what is the amount Dy(L) by which one should misalign shaft in cold condition so that they would be aligned in hot conditions. 
3

Extensometry and hot misalignment: Shaft alternate stresses.
Shaft ends are a special brand of beams: they can rotate in their supports. If one pastes a strain gauge S on the shaft surface, it measures the axial strain e(x) when up and  -e(x) when down in position S’. In other words, one need not worry about balancing a strain gauge bridge for a position of the shaft known to generate no strain. All one needs to measure to start measuring the vertical misalignment is the difference between strains when the gauge is up and down. The same holds true for horizontal misalignment. One needs to measure strains when the gauge is in the left and right position. 
To this difference of strains correspond differences of stresses via scaling it with the Hooke’s modulus (E) of the shaft material. This difference of stresses is none other than the shaft alternate stress due to misalignment. 

Shaft alternate stresses may not be a safety issue but
Remember that  manufacturers specify within which limits such alternate stresses should remain for an admissible hot alignment. It seems that this limit matches the amount of alternate stresses that one cannot avoid at shaft mid spans without adding an extra bearing, which is totally unpractical. Operating somewhat beyond it does not endanger the machine, because the design of shaft is very conservative. Usually  nominal torques correspond to higher stress values, e.g. 75 Mpa.  Furthermore, the design of bearings which must stand the loss of terminal blades and the subsequent huge centrifugal forces without damage. 
4
Before causing a complete failure, other problems may crop up: 
  • Unloading a bearing may cause its instability.
  •  Due to its lack of symmetry, a coupling may transmit a bending moment unevenly over a revolution thus causing parametric vibrations at even harmonics of the rotating speed.
  •  Bearings flexibility and damping factors may vary according to their loads, causing shifts in critical speeds and in vibrations at nominal speeds, not to interpret as a change of shaft unbalances.
  •  Improper shaft centering.
Misalignment and shaft centering. 
  • A transfer of loads between bearings may alter the proper centering of the shaft by modifying its catenary. Let a bearing transfer its load to its neighbor following a severe misalignment. This is an extreme case well illustrating the problem. This amounts to increase the unsupported span of a shaft by the distance between the bearings with a resulting increase of its deflection.
  •  Bladed shafts not well centered may cause steam whirl and thus self-excited vibrations.
  •  Constant or intermittent rubs between the shaft and stationary parts such as packings may cause vibrations known as dry whirl or Mathieu rub. Incidentally Mathieu rub can also be counteracted by filed balancing, because it cannot be caused by a severe rub but rather by a slight one and thus not too a severe lack of centering.

    •  etc.
5
What happens in shaft lines of large turbosets like a typical European 900 MW 1500 rpm nuclear-powered turbine of  CP1 EDF type? 
There a shaft line is composed of various rotors linked with rigid couplings rests on multiple bearings b1 to b10. 
When aligning shaft ends, one loosens couplings. Then one lowers the heaviest rotor on its bearings. It sags. Its shaft ends are no longer horizontal. At mid span, gravity causes a typically 20MPa surface stress. The neighboring rotor is then laid down. Its bearings are displaced to bring its coupling rims to be concentric with those of the former rotor. If one also brings faces to be parallel, one ends up with a shaft line bending like the aa’ catenary. 
In practice, one anticipates the action of vacuum that tends to lower the bearings belonging to the L(ow) P(ressure) rotors LP1, LP2 and LP3. One lifts the catenary from aa’ to cc’ to allow it to descend to hh’ in nominal operation at maximum vacuum, not far away from the ideal catenary aa’ characterized by zero alternate stresses in shaft ends between rotors and no load transfer DW between (selfish) neighboring bearings. In the process, bearing self-alignment should operate properly from cc’ to hh’ (ask manufacturers about it, you are in for a big surprise) . 
With faces bolted together, a subsequent motion of the bearings supporting the shafts is opposed (or favored in case of misalignment anticipation) by the rigid coupling. Bending moments vary in shaft ends. One can track them with shaft plane stresses and strains (b) to (c). Alternate strains e(x) and -e(x) can be measured with a strain gauge S on the surface of shaft ends. They bring along a heavy and costly instrumentation. 
Thus, bearing misalignments get cast as a problem of beam (shafts are rotating beams) material strength. This has been all along in the minds of shaft designers who consider bearing alignment to be good provided alternate stresses do not exceed 20MPa in shaft ends (40 in stress concentration zones like fillets). 
And yet nobody measures these alternate stresses in practice relying on some kind of model of thermal growth to anticipate bearing motions from a cold start to full operation! What about if this model fails???

Benefits of shaft alternate extensometry.

The main benefits of shaft alternate extensometry are:
  1. It directly relates to a physical well defined bound on shaft alternate stresses that depends on the shaft material and thus has an immediate physical meaning and a generic admissible upper bound. Most conventional “norms” for aligning shafts specify maximum misalignment limits for angular misalignment expressed in fraction of degrees of  hundredths of mm between coupling faces that can be measured only when aligning shafts. One suspects that such limits are based on “good practice” and on what could achieved with existing measuring devices such as clocks (reverse dial) or more modern laser-based methods like OPTALIGN. They lack any hard theoretical background. Furthermore, they become useless once the couplings have been fastened.
  2. From the shaft bow it can predict by how it must be displaced at the bearing level to reach a perfect alignment. This does not precisely tell by how much the bearing pedestal should be moved to reach this target because one generally ignores how the oil film reacts.
  3. Monitoring the relative displacement of bearing pedestals is no direct indication of the shaft hot misalignment. Again note this last approach requires some reference position corresponding to a “perfect shaft alignment. This sort of baseline requires a special team on the site at the unit (re)start and one then knows what the cold alignment is. One could use optical methods to this aim or, else, laser-based methods like PERMALIGN.
  4. Shaft alternate extensometry does not need such a reference starting point. For example, if one plans to measure alternate strains with resistive gauges in a Wheatstone bridge configuration, one need not even balance the bridge accurately first and subsequent bridge unbalance drifts do not alter results. Therefore, one can use alternate extensometry any time to check alignment without being burdened by “following up” the machine from the time when it was first cold-aligned.
Drawbacks and limitations of shaft alternate extensometry.
The foremost reason that has kept people from using alternate strain measurements to address the issue of shaft hot misalignment is that it requires to mount strain gauges on the shaft including the bridge power supply, the conditioning of the bridge output and its transmission to a stationary device via telemetry. This leaves a lot of equipment behind rotating with the shaft and sustaining high centrifugal forces, not to mention either the investment left dormant on shaft ends that, after all, may not exhibit any big misalignment or the hazard such spinning equipùent may represent.

Question to all interested viewers of this page.

Does any of you know of a simpler method to measure alternate stresses et the surface of shaft ends? A method easy to implement in the field, of course.
Then contact us, please.