At-Speed Balancing: Machines and Procedure

This is the sixth and final article in a series on the topic of rotor balancing. The article on at-speed balancing fundamentals established that modal unbalance in flexible rotors cannot be adequately mitigated by low-speed balancing. Correcting it requires a facility that can run the rotor up to service speed. This article covers the facilities and the procedure:

• at-speed balancing facilities and their instrumentation,
• selecting correction planes for flexible rotors,
• the sequence of an at-speed balancing job, from setup to acceptance,
• the plane-by-plane influence coefficient method for converting at-speed readings into corrections,
• modal balancing, with influence coefficients applied to proportioned plane sets, and
• specifications and acceptance criteria.

At-Speed Balancing Facilities

An at-speed balancing facility runs the rotor through its criticals to the maximum speed it will see in service while measuring the response at multiple points (Figure 1). The rotor is supported in bearings matching its service bearings, on pedestals whose stiffness is chosen to represent the machine’s bearing supports, so that the critical speeds and responses observed in the facility approximate those in service.

Running to service speed in open air is impractical for rotors carrying blades, impellers, and similar features. The power demand to overcome the windage generated by the aerodynamic components would be significant, and the churned air would heat the rotor and its surroundings. The facility therefore runs the rotor in a vacuum chamber, driven by a variable-speed motor through a shaft seal in the chamber wall, while technicians control and monitor it from outside. With the air removed, a drive of modest power brings the rotor to speed, and the temperature stays under control. The chamber also serves as containment, guarding people and equipment against debris from a rotor failure or a loose item.

Proximity probes at multiple axial stations report shaft displacement as a magnitude and phase, and transducers on the pedestals report the force or motion transmitted there. A once-per-revolution phase reference fixes the angular origin against which every phase is measured, as described in the article on low speed balancing machines. Many practitioners refer to the reference probe as a keyphasor.

Probes are commonly fitted only near the bearings, because that is where the installed machine will have them and where a fully assembled rotor best accommodates them, providing available space and a shaft surface of the finish and low runout a probe requires. However, the nodes of the bending mode shapes lie near the bearings, so the deflection at the bearing probes is small compared with the deflection elsewhere along the rotor. The deflection is greatest at the anti-nodes, near midspan for the first mode and near the quarter-span points for the second. Bearing probes detect each critical adequately, but probes at the anti-nodes measure the deflection where it is largest and where it is more likely to cause rubs at close clearances once the rotor is in service. These considerations may warrant requesting the additional probes during at-speed balancing wherever access allows.

Correction Planes for Flexible Rotors

The at-speed fundamentals article explained why a correction plane’s axial position matters for a flexible rotor. A correction’s authority over a mode is proportional to the mode’s deflection at the plane where the correction is applied. A correction at a node has no effect on that mode. Selecting planes therefore starts from the mode shapes. A plane near midspan addresses the first bending mode. Planes near the quarter-span points, where the second mode deflects most, address the second (Figure 2).

The chosen correction planes must also be accessible on the finished rotor and able to accept a correction. Typical locations include the faces of disks and impellers, where small amounts of material may be removed without harm to the component. Rotor components may also provide threaded holes or balance-ring grooves for adding weights. Coupling hubs and thrust collars may at first seem like convenient correction locations, but they carry limited authority over the first bending mode because they sit near its nodes.

The number of correction planes follows from the number of modes. Balancing a rotor through its flexible modes has commonly required two more correction planes than modes: one plane for each mode, and two more for the rigid resultants that the low-speed balance controls. A rotor that runs through its first critical needs at least three planes, and one that runs through two needs at least four.

The At-Speed Balancing Sequence

An at-speed balance begins where the low-speed balance ends. The rotor arrives at the at-speed facility having been balanced to its rigid residual unbalance tolerance on the low-speed machine. It is installed in the facility in bearings and on pedestals representative of service and instrumented, after which the chamber is evacuated.

The first runs are at low speed. A slow-roll run records the runout each probe reports at a speed too low for any unbalance response, and that runout is subtracted from every reading taken afterward. The rigid unbalance is then measured on the at-speed machine at low speed and compared with the result from the low-speed machine, confirming that the two agree.

The rotor is then run up in stages, at lower speeds first, to confirm that its behavior is approximately as expected and that the setup and the facility are functioning as they should. Before any readings are recorded for balancing, a relaxation run takes the rotor to its maximum allowable momentary speed, the trip speed of a turbine rotor. Its purpose is to settle the rotor. Small adjustments in shaft fits and other subtle changes that occur under the highest centrifugal load are made to occur before the baseline is taken, so that the baseline reflects the settled rotor. The run also confirms the rotor’s integrity at that speed, and nondestructive examination after the at-speed balance is complete is usually performed to confirm that the exposure to high speed has not affected it.

A baseline run then takes the rotor from rest to the maximum continuous speed, with the response at every probe recorded continuously, through each critical that lies within that range. The record identifies the critical speeds and shows how strongly each mode responds.

Balancing proceeds by one of two methods, the plane-by-plane influence coefficient method or modal balancing, both of which are described in detail in the sections that follow. The corrections are computed, applied, and checked by a further run against the acceptance limits at every specified speed, with trim corrections as needed. A final low-speed check confirms that the rigid balance remains within tolerance. The low-speed check merely serves as a record of the rotor’s final rigid residual unbalance and does not include any correction steps.

The Plane-by-Plane Influence Coefficient Method

The influence coefficient method converts readings into corrections by first measuring how the rotor responds to a known unbalance. The process is similar to the calibration of a soft-bearing machine, as described in the article on the low speed balancing procedure: run a baseline, attach a trial mass of known unbalance, run again, and relate the change in response to the trial unbalance. The same operation is performed on rotors during at-speed balancing, and the ratio of the change in response to the trial unbalance is called the influence coefficient.

After the baseline run is complete, a single trial mass is attached in one correction plane at a known radius and angular position, and the rotor is run up again, with readings taken at the same speeds. The trial unbalance Ut is the product of the trial mass mt and its radius rt:

Ut = mt · rt

where:

• Ut = trial unbalance (g·mm or oz·in)
• mt = trial mass (g or oz)
• rt = radius at which the trial mass is attached (mm or in)

The difference between the baseline reading X0 and the reading with the trial mass attached, XT, is the change the trial unbalance produced, ΔX:

ΔX = XT − X0

where:

• ΔX = change in response produced by the trial unbalance (magnitude and phase, in the reading’s units)
• XT = reading with the trial mass attached (magnitude and phase)
• X0 = baseline reading (magnitude and phase)

The subtraction is a vector subtraction. Each reading is resolved into components, the components are subtracted, and the result is expressed again as a magnitude and phase. Every phase is measured in the same convention, positive against rotation from the phase reference. The influence coefficient H is the change in response per unit of trial unbalance:

H = ΔX / Ut

Expressed as a magnitude and a phase, with θt the angular position of the trial mass, the division becomes:

|H| = |ΔX| / Ut
phase(H) = phase(ΔX) − θt

where:

• H = influence coefficient for the plane and probe in question (reading units per g·mm or oz·in, with a phase in degrees)
• |ΔX|, phase(ΔX) = magnitude and phase of the change in response
• Ut = trial unbalance (g·mm or oz·in)
• θt = angular position of the trial mass (degrees)

The influence coefficient’s magnitude and phase are properties of the rotor on that setup at that speed, and both change from speed to speed. The units may be metric or US throughout, so long as they are consistent. The method accepts any once-per-revolution vector measurement, shaft displacement or pedestal velocity alike, provided the same measurement serves for the baseline, the trial run, and every trim.

Every plane-speed-probe combination has its own H. Where probes are fitted in pairs at 90 degrees, each probe is treated as its own sensor with its own influence coefficient. The magnitude of H reflects how much a unit of unbalance in that plane changes the reading at that speed. Its phase is the lag between the heavy spot and the high spot, plus the fixed angular offsets of the probe’s mounting position relative to the phase reference and any electronic phase shift. None of these angles needs to be known separately. The vector arithmetic to determine the correction requires only the baseline reading and the influence coefficient.

The required correction is the unbalance whose response cancels the baseline reading. Since H relates unbalance to response, the required correction unbalance Ucor is the baseline reading divided by the influence coefficient and reversed in direction:

Ucor = −X0 / H

As a magnitude and a phase:

|Ucor| = |X0| / |H|
phase(Ucor) = phase(X0) + 180° − phase(H)

where:

• Ucor = correction unbalance (g·mm or oz·in, with a phase in degrees; the phase is the angle at which mass is added, and a removal lies at the opposite angle)
• X0 = baseline reading (magnitude and phase)
• H = influence coefficient (magnitude and phase)

The trial mass is removed before the correction is applied. The correction is applied as a mass addition m at the correction radius r, placed at the angular location given by the phase of Ucor:

m = Ucor / r

where:

• m = correction mass to add (g or oz)
• Ucor = correction unbalance magnitude (g·mm or oz·in)
• r = correction radius (mm or in)

Consider a rotor with a correction radius of 250 mm, a baseline probe reading of 40 µm pk-pk at 120°, and an 8 g trial mass attached at 0° on the same 250 mm radius (a trial unbalance of 2,000 g·mm) that changes the reading to 62 µm pk-pk at 150°. The vector that represents the change in response is 33.9 µm pk-pk at 186°, and the influence coefficient is 0.0170 µm pk-pk per g·mm at 186°. The correction is 2,360 g·mm at 114°, a 9.4 g mass added at 114° on the 250 mm radius. The predicted effect of that correction is H · Ucor, equal and opposite to X0 (Figure 3).

Because readings scatter, correction masses are placed imperfectly, and the response is not perfectly linear, the first correction may not cancel the baseline exactly. The reading from the run after the correction becomes the new baseline. It is not necessary to repeat the trial mass procedure. The same influence coefficient may be applied again, because a correction mass is small relative to the rotor and has negligible effect on how the system responds to unbalance. In the worked example, a subsequent probe reading of 6 µm pk-pk at 40° calls for a further correction of 354 g·mm at 34°, a 1.4 g mass added at 34° on the same radius. The sequence repeats until the response meets the acceptance limit at that speed.

A correction may equally be made by removing the same unbalance at the opposite angle, which is how disk and impeller faces are corrected by grinding. When the amount added or removed is not accurately known, as with grinding, welding, or spraying, it may be calculated retroactively from the pre- and post-correction runs’ readings and the influence coefficient. The unbalance actually applied, Uapp, is the change in reading divided by the influence coefficient:

Uapp = ΔX / H

where:

• Uapp = unbalance actually applied by the change (magnitude and phase; a removal lies at the opposite angle)
• ΔX = change in reading from the run before the change to the run after it (magnitude and phase)
• H = influence coefficient (magnitude and phase)

Such corrections therefore proceed in passes, each measured by the run that follows it.

With more than one correction plane, each plane’s influence on each probe at each speed is measured with its own trial run, one plane at a time, and the corrections are solved together. For one probe, one speed, and two planes, the predicted reading X after correction is the baseline X0 plus the effect of each correction through that probe’s influence coefficients HA and HB at that speed:

X = X0 + HA · UcorA + HB · UcorB

where:

• X = predicted reading at the probe after correction (magnitude and phase)
• X0 = baseline reading at the probe (magnitude and phase)
• HA, HB = the probe’s influence coefficients for planes A and B at that speed (magnitude and phase)
• UcorA, UcorB = correction unbalances in planes A and B (magnitude and phase)

Setting the predicted reading to zero at each probe gives one equation per probe, and two probes suffice to solve for two corrections. In practice the facility typically has more probes than planes, and readings are usually taken at more than one speed, so the equations outnumber the unknowns.

A single set of corrections across the various planes can be solved from readings at multiple speeds, and this practice is known as multi-speed balancing. A correction found for one speed alone may not hold at another, because the coefficients change with speed as the modes move in and out of participation. By including readings at every speed of interest in the formulation of a single solution, the corrections that result hold across the range.

The corrections are then chosen by least squares. Each residual reading is first expressed as a fraction of its acceptance limit. The corrections chosen are the combination that makes the sum of the squares of those fractions, across all probes and speeds, as small as possible. No single reading is driven to zero, but all of them are minimized to the extent possible to meet the acceptance criteria.

Modal Balancing

Modal balancing is the influence coefficient method applied to proportioned plane sets rather than to individual planes. A plane set is a distribution of unbalance across two or more correction planes, in proportions fixed in advance to match one mode shape. The set for the first bending mode has a midspan unbalance. The set for the second has an opposed pair near the quarter-span points (Figure 2). Each also includes end-plane unbalances that cancel the net force and moment the set would otherwise add, so that the rigid balance is undisturbed. The proportions are proportions of unbalance, not of mass. Where planes have different correction radii, the masses differ so that the unbalances hold the intended proportions, since unbalance is mass times radius. Where the end planes lie at different axial distances from the midspan plane, the end unbalances of the first-mode set differ so that the moment cancels as well as the force. In the second-mode set, the opposed quarter-span pair adds no net force but a couple, so the end unbalances form an opposing couple of equal size, and their magnitude follows from the ratio of the quarter-span spacing to the end-plane spacing. The Plane Set Calculator determines the proportions for a given set of planes and a target mode.

The set is trialed as a whole, deliberately. Where the plane-by-plane method determines one influence coefficient for each plane-speed-probe combination, the plane-set method determines one for each set-speed-probe combination: the set’s effect on each probe’s reading at each speed of interest. The math is otherwise the same. A job proceeds in one framework or the other, but the mode shapes inform both, guiding the choice of planes in the plane-by-plane method and defining the sets in the modal method. A set is a generalized plane, with one unknown, its scale as a magnitude and angle, and one influence coefficient per probe and speed, exactly like a plane. Corrections are applied in sets of the same unbalance proportions, scaled by the same arithmetic used for an individual plane.

Because each set is built to move one mode and to leave the others and the rigid balance undisturbed, the modes are worked one at a time from the lowest upward. The first mode’s set is trialed and corrected, then the second mode’s.

Specifications and Acceptance

For a new rotor, at-speed balancing should be specified whenever the rotordynamic model indicates that a bending critical speed is likely to lie within or near the range of speeds the rotor will experience, from rest to its trip speed.

Beyond new rotors, at-speed balancing may be called for whenever a flexible rotor’s mass distribution may have changed or needs to be confirmed: after reblading, a replaced disk, impeller, or other major component, a shaft repair, or any disassembly and reassembly; after a long period in service, to confirm that the balance remains acceptable; or after a history of vibration at speed that a low-speed balance could not cure.

Acceptance is judged differently from the low-speed balance. A low-speed balance is accepted against a residual unbalance. An at-speed balance is accepted against the vibration the rotor shows at specified speeds, most often the maximum continuous speed and each critical within the operating range, measured at the probes as a peak-to-peak shaft displacement or at the pedestals as a velocity.

Acceptable response at the bearing probes does not necessarily ensure acceptable response at every point along the rotor. If probes are not positioned where maximum deflection is expected, the bearing-probe readings should be interpreted through the rotor’s rotordynamic analysis to confirm that deflection at the close-clearance locations will be acceptable. For each speed of interest, the mode shape that dominates there gives the ratio of the response at the point of maximum deflection to the response at the bearing probes, so the maximum deflection can be estimated from the readings at the bearings. When probes are placed at the points of maximum deflection, these interpretation steps are unnecessary.

Companion Calculator

The At-Speed Balancing Calculator takes every plane, plane set, probe, and speed of a job, sizes the trial masses, and solves them together. For modal balancing, the Plane Set Calculator may be used as a supplement to determine appropriate unbalance proportions. In setup, the user defines the correction planes, the probes with their acceptance limits, and the speeds. Each run then records the masses added or removed in any plane and the reading at every probe and speed. From all the runs entered, the calculator fits the influence coefficients, reports every reading as a fraction of its acceptance limit, and recommends the next correction in each plane by least squares, with the predicted residual after that correction. A polar plot shows the run history at any probe and speed, and the job can be saved, reloaded, and printed as a report.

Next in the series: Balancing Knowledge Check — an open-book check across the whole series, auto-scored, with a printable certificate for premium subscribers.