Critical speed and flexible rotors: why a correction made at one speed doesn't work at another

You balanced a rotor at 2000 rpm and got 1.5 mm/s. You raised the speed to the operating value and saw vibration higher than before the work. The calculation wasn't lying, and the weights sit exactly where the instrument worked out. That's how a rotor behaves once it stops being rigid at operating speed, and you need a different set of rules to deal with it.

Updated 27 August 2026 · by AXILINE · Vila Nova de Gaia

Critical speed: resonance of the shaft itself

A rotor's critical speed is the rotational speed at which the rotation frequency coincides with one of the shaft's natural bending-vibration frequencies. As the shaft approaches it, it stops being a straight line. It bows outward, the deflection rotates along with the rotor, and centrifugal force grows not just from the residual unbalance anymore but from the deflection itself. Vibration at 1x — the running-speed component, the part of the vibration whose frequency equals the rotation frequency — spikes, the 1x phase swings by roughly 180°, and once past the peak, the machine settles down again.

The first critical speed isn't the only one. A rotor with mass distributed along its length has a first, second, and third critical speed, each with its own deflection shape. The first shape, for a shaft between two bearings, looks like a simple arc with its maximum at the middle of the span. The second looks like an S-shaped curve with a node in the middle and two bulges at the ends. Beyond that, the shapes get more complex.

You're unlikely to calculate the exact figure yourself, but keep the dependencies in mind. The first critical speed drops as the span between bearings grows and rises with the shaft diameter, so a long, thin shaft runs into trouble sooner than a short, thick one. Mounted masses (impeller, pulley, half-coupling, flywheel) pull the frequency down. Bearing and support stiffness feeds into the result on an equal footing with the shaft's own stiffness, so a 'rotor's critical speed' actually belongs to the whole shaft-plus-bearings-plus-supports system.

That leads to an unpleasant property: critical speed isn't a constant. On a machine with sleeve bearings, the oil film's stiffness changes with speed, temperature, and load, so the first critical speed will land at a different rpm on a cold start than on a warmed-up machine. On rolling-element bearings the spread is smaller, but fit, preload, and wear shift the number too.

In conversation, 'critical speed' sometimes gets used for the maximum speed a machine can withstand structurally. That's a different thing. Here we're talking only about bending-vibration resonance.

How critical speed differs from support resonance

On the surface, the two phenomena look nearly indistinguishable. Both structural resonance and critical speed produce a narrow peak at 1x and a fast phase swing as speed changes. What tells them apart is what's actually deforming.

In structural resonance, the stationary part vibrates: the bearing housing, a pedestal, the frame, a platform, pipe bracing. The shaft stays essentially straight and rides along with the support. At critical speed, the shaft itself bends, and its shape changes along its length. You fix the first with stiffness, mass, or a change of operating speed. The second can't be fixed that way, because you can't change the shaft's stiffness on site.

A bump test answers honestly for the structure and poorly for the shaft. An impact on the stopped machine excites the natural frequencies of the supports and the frame. A rotor at rest sits in its bearings with no oil film and no gyroscopic effects, so a 'shaft frequency' found this way drifts away from the real critical speed. Critical speeds are found on run-up and coastdown.

Rigid and flexible: it's about the operating regime, not the metal

The words 'rigid' and 'flexible' don't describe the steel or the construction, they describe the operating regime. The same rotor counts as rigid at low speed and flexible above its first critical speed.

Formally, a rotor is in the rigid state if its deflection at operating speed is small enough that the unbalance distribution along its length can be treated as unchanging. In that case, the entire unbalance reduces to two vectors in two chosen planes, and a correction in those planes works at any speed. This assumption is exactly what the G-grade tolerance calculation, and the whole one- or two-plane scheme, rest on.

The example below is composite, meant to illustrate the point, not a description of an actual job. A long rotor was checked on a low-speed balancing machine at 400 rpm, and the residual unbalance came within grade. The machine was assembled, brought up to its operating 9000 rpm, and the vibration turned out to be twice the norm. There's no contradiction: at 400 rpm the rotor was rigid and honestly passed acceptance, while at 9000 it's operating past its first critical speed and distributing unbalance along its length in a completely different way.

• Operating speed below 0.5 times the first critical speed: the rotor behaves reliably rigid. This is where almost all plant fans and induced-draft fans, pumps, crushers, mulchers, screw conveyors, electric motors, pulleys, and drums live.
• From 0.5 to 0.7 times the first critical speed: a gray zone. The rigid-rotor scheme usually still works, but the influence coefficients — the measured relationship between 'trial weight' and 'change in vibration' — already depend noticeably on speed, and the result needs to be checked at a second speed.
• From 0.7 to 1.3: the operating point sits inside the bending resonance. Balancing here is pointless, because the vector doesn't repeat from run to run.
• Above the first critical speed: flexible, compliant behavior. The rotor operates past resonance, the deflection shape depends on speed, and a different method is needed.

These boundaries are an engineering rule of thumb, not figures from a standard. ISO 21940-11 is written for rotors in the rigid state. The behavior of rotors that deform while running, and the procedure for balancing them, is covered by a separate part of the series, ISO 21940-12 (formerly ISO 11342). Check the applicable part and edition for your machine.

Sources: ISO 21940-11:2016 · ISO 21940-12:2016

Why a correction from one speed doesn't work at another

Here's the core of the problem. The deflection shape depends on speed, and a correction weight acts not on 'the rotor in general' but on one specific shape.

Take a shaft between two bearings. In the first shape, the maximum deflection sits at the middle of the span, so a weight in the middle damps that shape well. In the second shape, the middle is a node, where the deflection is zero. That same weight at that same point does almost nothing for the second shape. Conversely, a pair of weights in antiphase at the ends works on the second shape and barely touches the first. As long as you're balancing at one speed, you're working with whichever shape dominates at that speed, and you know nothing about the rest.

The second blow lands on the influence-coefficient method itself. The method relies on linearity: double the trial weight, get double the change in the vibration vector. For a rigid rotor this holds, because it doesn't deform. For a flexible rotor, added mass at the heavy spot increases the deflection, the radius of that mass grows along with the deflection, and the response comes out larger than proportional. The linear model starts lying, and the correction calculation stops converging.

• The influence coefficient stops being a property of the rotor and becomes a property of the rotor at that particular speed. A coefficient saved from a previous visit won't work for a machine like this.
• The trial-weight criterion (a 20–30% change in the vector) can technically be met, and the next run still pulls the result off to one side.
• Two correction planes physically aren't enough. Controlling several shapes needs more planes and more runs.
• An improvement at operating speed can turn into higher vibration during run-up, if you've reinforced the shape the machine passes through on its way to the operating point.

Sources: Balanset-1A operation manual

Signs you're dealing with a flexible rotor

How to find critical speeds on site

1. 01

   Record the transient, not the operating point

   Steady-state operation doesn't show critical speeds, the machine passed through them long ago. The information is carried by run-up and coastdown — the free deceleration after the drive is switched off. Two accelerometers on the bearing housings, mounted rigidly, magnet on a clean pad. A laser phase sensor on the reflective tape, so every recorded point has both speed and phase.

2. 02

   Plot amplitude and phase against speed

   You need two curves: 1x amplitude against speed, and 1x phase against speed. This is a Bode plot, and on it, critical speed shows up as an amplitude peak with the phase passing through the middle of its swing at the same time. A peak with no phase swing usually means a change in excitation, not resonance.

3. 03

   Separate the shaft from the structure

   Compare the peak at both bearing housings and on the frame, the feet, the platform. If the surrounding structure is clearly moving on its own while the bearings stay smoother, you've found structural resonance. If the peak shows up strongly at both bearings and the shaft's relative displacement grows, that's critical speed. A bump test on the stopped machine helps rule the first option in or out.

4. 04

   Check the influence coefficient at two speeds

   This is the most direct test for rigidity. Fit a trial weight and take the response at 60–70% and at 100% of operating speed. If the vectors match in magnitude and angle once corrected for the square of the speed, the rotor is behaving rigidly and balances the usual way. If they diverge, the flexible-rotor scheme takes over from there.

5. 05

   Know your instrument's limits

   The Balanset-1A records overall vibration on two channels in blocks lasting 1, 5, 10, 15, or 20 seconds, plus 1x with phase and speed, spectrum, and time waveform separately. There's a Run Down coastdown mode that plots amplitude and phase against frequency, and the manual flags it as experimental. On a large machine, coastdown takes minutes, so take a series of recordings. The instrument doesn't perform a full modal test with a calibrated hammer, transfer function, and coherence, and it's worth knowing that up front.

Sources: Balanset-1A operation manual · Balanset-1A manufacturer specification

Modal balancing and multi-speed balancing

For a flexible rotor, the task gets reframed. You're not balancing 'the rotor' — you're balancing each natural mode shape that falls inside the operating range or the run-up range.

Modal balancing works shape by shape. Weights are placed along the length so their distribution matches the shape you're damping and disturbs the others as little as possible. The work proceeds in sequence: it starts with the first shape and measures near the corresponding critical speed, where that shape shows up most strongly.

Multi-speed balancing stays within the logic of influence coefficients, but takes responses at several regimes at once. There end up being more equations than unknowns, and the system is solved by least squares. The result is a compromise: not a perfect zero at one speed, but an acceptable level across the whole range, including passage through the critical speeds.

More than two correction planes are needed. Controlling N shapes calls for on the order of N+2 planes, and each plane means additional trial runs. That's where both the labor involved and the access requirements come from: you usually can't get to the middle of a turbine rotor at its site of operation.

• One trial run per plane doesn't work here: the number of runs grows as the number of planes plus a verification run at each speed.
• Many rotors like this are balanced not in their own bearings but on a spin-balancing test stand, often with a vacuum chamber, to bring the rotor up to operating speed without aerodynamic drag and heating.
• Balancing large rotors directly at their installation site is covered by a separate part of the ISO 21940 series, and the equipment requirements there are noticeably higher than 'showed up with a two-channel instrument and magnetic sensors.'
• Some of these jobs aren't about balancing at all: if the operating point sits inside a bending resonance, the first move is to change the speed, the stiffness of the bearing assemblies, or the rotor's construction.

Sources: ISO 21940-12:2016 · ISO 21940-11:2016

What to do in practice: a short list

• Find where your machine's first critical speed sits: in the manufacturer's documentation, or from a run-up recording. Without that figure, you're working blind.
• Maintain detuning — a frequency margin between the operating regime and the resonances. Keep the operating speed at least 15–20% in frequency away from any critical speed and from the natural frequencies of the bearing housings and the frame.
• Pass through the resonance zone quickly, without lingering in it. Run-up and shutdown are the most heavily loaded moments for a flexible rotor.
• If a variable-frequency drive is fitted, add the resonance band to the skip frequencies, so the automation doesn't leave the machine running inside it.
• Don't balance at the critical speed. The phase swings there within a few percent of speed, the influence coefficient is unstable, and the result won't repeat.
• Balance in a stable zone, and always check the result at a second speed. If it holds there, the rotor is behaving rigidly and you're done.
• Assess the condition by overall vibration velocity in mm/s RMS in the 10–1000 Hz band, per the applicable part and edition of ISO 20816, not by 'it sounds quieter.'
• If the result depends on speed, stop. Don't keep adding weight blindly: what's needed from here is a different method, not one more trial run.

Sources: ISO 20816-1:2016

Where on-site balancing ends, and what we do

Let's say it plainly. On-site balancing in a rotor's own bearings is built for rotors that stay rigid across the whole operating range. That's most plant machinery: fans and induced-draft fans, crushers and mulchers, pumps, electric motors, pulleys, drums, screw conveyors, conveyor shafts. For these, the one- or two-plane scheme and the influence-coefficient method deliver a repeatable result within a working day, with no disassembly and no sending the rotor to a shop.

If the rotor is flexible, we don't pretend it's the same job. We come out and measure: we take vibration at the bearing housings, separate overall from 1x, record run-up and coastdown, look for where the critical speeds and the supports' natural frequencies sit, and check the influence coefficient at two speeds. From this data, you get a report with numbers: where the first critical speed sits, whether the machine is running past it, what on-site balancing will and won't achieve.

From there, three outcomes are possible. First: the rotor is actually rigid, and the high vibration was coming from support resonance, shaft misalignment, or loose fasteners, and the job gets closed out on site. Second: the rotor is rigid, but the operating point sits close to resonance, so the frequencies need separating first, with balancing coming after. Third: the behavior really is flexible, and we say so honestly — specialized conditions and a different method are needed — and we explain what data to hand over to whoever takes it on, so they don't have to start from zero.

The same engineers who design and manufacture the Balanset instruments also use them on site visits, so we work through cases like this from measurements, not from a description over the phone. If you're building your own test stand or building a measurement system into an existing machine tool, there's the Balanset-1A OEM version without the case, and consulting support is available for setting up the measurement.

Related topics are covered separately: signs of support resonance and the bump test, the influence-coefficient method and reusing saved coefficients, and choosing between on-site balancing and shop-machine balancing. See the corresponding articles in this section.

Sources: Balanset-1A manufacturer specification

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