# Multi-bearing rotor balancing: a long shaft on three or more bearings

> As long as a rotor has two bearing housings, balancing follows a well-worn scheme: three runs, two planes, two measurement channels. Add an intermediate bearing, and the task changes in kind. You can no longer solve it piece by piece: a weight in any plane shows up at every bearing at once, so correcting one section spoils the next one over. Below, we break down why that happens, how the size of the problem grows, and what's honestly achievable at the site of operation.

**In short:** A multi-bearing rotor is a distinct problem, not an extended two-plane one. For a rigid rotor on two bearings, you're solving a system of two equations with two unknowns, and the influence coefficient matrix is 2×2. Three bearings give a 3×3 system with nine coefficients and a minimum of four runs; four bearings give sixteen coefficients and five runs. You have to measure at every bearing and solve the system as a whole: if you work through the planes one at a time, the vibration will keep running from bearing to bearing, and the process won't converge.

Source: https://axiline.pt/en/articles/multi-bearing-rotor-balancing/  
Publisher: AXILINE · Vila Nova de Gaia, Portugal · +351 931 831 229 · axilinegeral@gmail.com

## A two-bearing rotor: two channels, two planes, three runs

Let's start with what works without qualification. For a rigid rotor on two bearing housings, any distribution of unbalanced mass reduces to two vectors, one per correction plane. There are exactly two unknowns, so you need exactly two equations. Two measurement channels on two bearings give you those.

From there, the influence-coefficient method takes over: the instrument measures how a known trial weight changes the vibration at each bearing, and calculates a correction from those responses. A run with no weight gives the starting 1x vectors — the running-speed component, the part of the vibration at the shaft's rotational frequency — at both bearings. A run with a trial weight in the first plane gives two coefficients: α11 as bearing 1's response and α21 as bearing 2's response. A run with a weight in the second plane gives α12 and α22. Four coefficients, a 2×2 matrix, three runs. The software solves the system and outputs a mass and an angle for each plane.

Notice that α12 and α21 aren't zero. A weight in one plane shows up at both bearings, because a rigid rotor behaves like a beam on two points. That's exactly why one mass can't remove vibration from both bearings at once when the unbalance is a couple unbalance — when the heavy points sit at opposite ends of the rotor and face in opposite directions. We worked through this calculation's vector arithmetic in detail in a separate article on the influence-coefficient method.

> Two channels on the instrument aren't a convenience — they're exactly the number of equations needed for two unknowns. The moment the unknowns become three, two equations fall short in principle, and no amount of careful work makes up for it.

## A third bearing: nine coefficients instead of four

Now add an intermediate bearing housing. A multi-section cardan shaft, a long transmission line, a drum with a steady bearing. The shaft gains a third section and a third accessible correction plane, and you gain a third equation.

The system looks like this: the vibration at each of the three bearings is made up of contributions from all three planes. Nine influence coefficients instead of four, a 3×3 matrix. Every trial run fills in one column of this matrix: you fit a weight in one plane and take the response at all three bearings at once. A minimum of four runs — one starting run and three trial runs.

Four bearings give a 4×4 matrix, sixteen coefficients, and a minimum of five runs. The number of runs grows linearly, the number of coefficients grows quadratically, and the labor on site grows even faster than that. Every run means a stop, repositioning the trial weight, a run-up, reaching the operating regime, a measurement, and a coastdown.

| Bearings and planes | Unknown vectors | Influence coefficients | Minimum runs | Channels at once |
| --- | --- | --- | --- | --- |
| 2 | 2 | 4 (2×2 matrix) | 3 | 2 |
| 3 | 3 | 9 (3×3 matrix) | 4 | 3 |
| 4 | 4 | 16 (4×4 matrix) | 5 | 4 |

> The minimum runs in the table don't include a verification run or follow-up runs. In practice, plan for six to eight runs on a three-plane job.

## Why you can't fix sections one at a time

The temptation is understandable: take a two-channel instrument, balance the front section using bearings 1 and 2, then the rear section using bearings 2 and 3. Logical on paper. It doesn't converge on site.

The reason lies in the cross-coupling coefficients. A weight in the first plane shows up at the third bearing, and a weight in the third shows up at the first. On a long shaft with an intermediate bearing, these cross-coupling contributions are comparable to the direct ones, not negligibly small. Solving the problem two equations at a time, you're treating the third contribution as zero every time. It isn't zero.

What follows is the picture typical of this approach. You closed out bearings 1 and 2, and vibration at bearing 3 went up. You fit a weight in the third plane, bearing 3 calms down, and bearings 1 and 2 come back close to their starting numbers, sometimes with a different phase. Vibration keeps running from bearing to bearing, the number of runs keeps growing, and there's no overall result. That isn't the operator's mistake — it's a property of solving a system that isn't being solved as a whole.

The cost isn't only in time. The intermediate bearing is often the most heavily loaded and the hardest of all to access for replacement. Dynamic bearing load feeds into its calculated life under ISO 281 to the third power: exceeding the calculated load cuts into the life nonlinearly. Unbalance left on the intermediate bearing works exactly that way.

Sources: [ISO 281:2007](https://www.iso.org/standard/38102.html)

## Where you'll run into this problem

All these machines share one trait: there are intermediate bearings between the outer ones, and each section of the shaft has its own accessible correction plane. The length-to-diameter ratio is high, operating speed sits closer to the first critical frequency, and access to the planes is usually uneven.

### Multi-section cardan and transmission shafts

Three or four bearings, intermediate bearings, two or three correction planes along the tubes. The classic case multi-plane algorithms were developed for in the first place.

### Conveyor and dryer drums

A long shell, a high L/D ratio (length to diameter), sometimes steady bearings in the middle. On top of the geometry, material buildup changes the unbalance while the machine is running.

### Paper machine shafts and web-processing line shafts

Long, thin shafts at high speed, running near the first critical frequency or above it. Here, multi-bearing behavior is mixed in with rotor flexibility.

### Multistage units

Pumps and compressors with a long stack of stages, spacers, and intermediate bearings. There are many correction planes, and access to them is usually only possible on disassembly.

### Shaft lines and transmissions with several couplings

Dust extraction and ventilation lines, drives running through intermediate bearings. Shaft misalignment at the couplings is almost always mixed in with the unbalance.

## What needs to be in order before the first run

On a multi-bearing line, the preparation requirements are stricter than on a two-bearing machine, and the reason is arithmetic: the more equations you're solving, the more ways there are for outside causes to corrupt them.

There are two main sources of interference. The first is shaft misalignment at the intermediate couplings. It produces its own 1x component with an axial part, the coefficient matrix will faithfully measure it, and you'll end up with weights that compensate for an assembly error. Fix the shaft alignment, and the vibration will drift right back. The second source of interference is uneven support stiffness. If the intermediate bearing is more compliant than the outer ones, or its mounting has come loose, the influence coefficients there will jump around from run to run, and the system will solve with a large error.

Take a baseline vibration map before any balancing at all: every bearing, three directions, overall level and 1x recorded separately. From the overall level in mm/s RMS in the 10–1000 Hz band, assess the machine's condition against the applicable part and edition of ISO 20816, and from the 1x share, work out whether balancing is even worth doing. The procedure for diagnostic measurement and data processing is described in the ISO 13373 series.

- [x] Every bearing's mounting to the foundation or frame is tight, and none of them has 'soft foot' — a mounting foot that doesn't sit flush against the frame.
- [x] The intermediate bearings are in good condition: clearances are within spec, the fits aren't worn loose, and there are no impact signatures in the spectrum.
- [x] Shaft alignment at every coupling has been done and documented before balancing, not after.
- [x] Bearing alignment has been checked along the whole line: a skewed base produces a constant load that weights can't remove.
- [x] Splined and telescoping joints on cardan lines have no axial play.
- [x] Every section of the shaft has access to its correction plane with the machine stopped.
- [x] Sensor pads are marked out and prepared at every bearing, with the same measurement direction everywhere.
- [x] There's a single tachometer mark for the whole job: it can't be repositioned between runs, or phases from different runs stop being comparable.

Sources: [ISO 20816-1:2016](https://www.iso.org/standard/63180.html) · [ISO 13373-3:2015](https://www.iso.org/standard/40840.html)

## Rigid or flexible: where the matrix stops helping

All the arithmetic above rests on one assumption: the rotor behaves as a rigid body at operating speed. In that case, the unbalance is described by a finite number of vectors tied to the planes, and it doesn't depend on speed.

A long shaft breaks this assumption more often than a short one. It operates closer to the first critical frequency, sometimes above it, and bends noticeably. The bending shape changes with speed. Multi-plane balancing performed at one speed reduces vibration at that speed and guarantees nothing at other regimes.

Here's how to recognize it. Run a slow run-up or coastdown and watch 1x at every bearing at once. For a rigid rotor, the ratio of amplitudes and phases between bearings stays roughly constant across the whole operating range. For a flexible one, the amplitude shows pronounced maxima, the phase swings, and the picture across the bearings diverges differently at different speeds.

Flexible rotors are balanced with different methods: by mode shape, at multiple speeds, sometimes with schemes using more planes than the number of shapes being accounted for. The classification of rotors into rigid and flexible, and the requirements for balancing flexible rotors, sit in the ISO 21940 series; check the applicable part and edition separately. The honest limit here is this: on-site balancing with a two-channel instrument doesn't solve this problem, and it's not worth promising otherwise.

Sources: [ISO 21940-11:2016](https://www.iso.org/standard/54074.html) · [ISO 21940-12:2016](https://www.iso.org/standard/50429.html)

## What this looks like step by step

Regime repeatability matters more here than on a two-bearing machine. Keep the speed within a narrow band, let the machine reach its thermal regime, and don't change the load or move the sensors between runs in the same cycle. Every deviation introduces an error into an entire column of coefficients at once, and a system of three equations is more sensitive to errors than one of two: a small measurement error can turn into a large error in the weight's mass.

1. **A complete measurement map** — You mount sensors at every bearing housing, same direction, rigid mounting. You take overall vibration and 1x with phase at each bearing. If you have fewer channels than bearings, work through the points over several runs at the same speed: phase is measured from the tachometer mark, so measurements from different runs are comparable.
2. **Assessing feasibility** — You work out the 1x share of overall vibration at each bearing, check for resonance on coastdown, and look at the repeatability between two consecutive measurements at the same regime. If 1x is small or the readings jump around, it's too early to balance.
3. **Number of planes and access** — You decide how many correction planes you can actually use. A plane you can't access doesn't belong to the problem, and that changes the whole setup, not just a little.
4. **Trial runs, one plane at a time** — Each run carries one trial weight in one plane, and the measurement is taken at every bearing. That's how a column of the matrix gets filled in. The criterion for an acceptable trial run is the same as always: a change in 1x amplitude of at least 20–30%, or in phase of at least 20–30°, at a minimum of one bearing.
5. **Solving the system as a whole** — Only once you have the full set of coefficients do you calculate the correction for all planes at once. You fit the weights in all planes at once too, not one at a time with checks in between.
6. **A verification run at every bearing** — You check the result at every bearing, not just the noisiest one. Any remainder is taken out with a trim correction — a targeted follow-up adjustment using the coefficients already obtained, with no repeat trial runs.

> The arithmetic of time. Three planes means four runs for the coefficients, one verification run, and usually one follow-up run. Six runs through the cycle of 'stop, fit the weight, run up, reach the regime, measure, coast down.' On a line with a long coastdown, that's a full working day for one machine.

## A two-channel instrument and a multi-bearing shaft: what's actually possible

Let's say it plainly: the Balanset-1A solves the problem for one and two planes. A 3×3 or 4×4 matrix needs a different calculation algorithm and more channels measured at once. That's a specialized method, not an extended setting of the two-plane mode. Algorithms like this for three- and four-bearing shafts exist and are implemented in multichannel systems, but a two-plane calculation shouldn't be passed off as a multi-plane one.

Here's what a two-channel instrument does well in this situation. It builds a complete map of the machine: speed, overall vibration, 1x with amplitude and phase, FFT spectrum, and time waveform at each bearing in turn, with phase referenced to the laser mark. This map shows which planes dominate, whether there's resonance, whether the readings are stable, and whether the line is misaligned. The problem often simplifies right here.

The problem genuinely does split apart when the cross-coupling coefficients are small. That happens when the intermediate bearing carries almost no load from the unbalance of neighboring sections, or the shaft sections are decoupled in stiffness. This is checked with one trial weight: fit it in the first plane, look at the response at the far bearing. Small — you can work section by section. Comparable to the direct response — section-by-section won't work, and it's better to find that out on the first run than on the tenth.

From there, it depends on the situation. Sometimes the shaft is removed and balanced on a multi-bearing machine in a shop, where every bearing is measured at once, the machine's own drive sets the regime, and repeatability is incomparably higher. For stands like this, there's a Balanset-1A OEM version without the case, built into the machine's measurement system. We covered the comparison between a site visit and a shop in a separate article, along with the criteria for choosing between them.

Sources: [Balanset-1A operation manual](https://vibromera.eu/balanset-1a-operation-manual/) · [Balanset-1A manufacturer specification](https://vibromera.eu/product/balanset-1/)

## What we do when a long shaft comes our way

We measure first, promise second. There's no other way on a multi-bearing machine: until there's a measurement map covering every bearing, nobody can say whether your problem reduces to two planes.

From there, three honest outcomes are possible. First: the cross-coupling is small, the planes are accessible, the rotor behaves rigidly. We balance on site, in two planes, in a single visit. Second: the problem is three-plane, but it can be broken into stages with a check at every bearing after each one. We do it, agreeing the number of runs and stops in advance. Third: the rotor is flexible, or there are more planes than channels, or the planes aren't accessible. We say so plainly and offer a shop balancing machine or a different method instead of runs that won't get you anywhere.

The people doing the work are the engineers who design and manufacture the Balanset instruments and do the balancing themselves on site visits. If you'd rather handle it yourself, we provide the instrument and consulting support on your measurements. What to prepare before the visit is covered in a separate article on preparing equipment for on-site balancing.

Send us the type of machine, the number of bearings, the operating speed, and the current vibration figures for each bearing with the 1x share stated. From this data, we'll tell you what can honestly be tackled on site and what's better left alone.

## Frequently asked questions

**Can a three-bearing shaft be balanced with a two-channel instrument?**

Not the full three-plane problem: three unknown vectors need three equations, meaning simultaneous measurement at three bearings and a 3×3 matrix. With a two-channel instrument, you build a map across all the bearings over several runs, assess the cross-coupling, and decide whether the problem splits into two planes. If it splits, the job is feasible on site. If it doesn't, you need a multichannel system or a shop balancing machine.

**How many runs are needed for three correction planes?**

A minimum of four: one starting run and one trial run for each plane. Add a verification run and usually one follow-up run to those. Plan for six to eight runs. On a rotor with a long coastdown, count time instead of runs: the cycle of 'stop, fit the weight, run up, reach the regime, measure, coast down' is the real unit of work.

**Is it mandatory to mount sensors on every bearing at the same time?**

Doing it at the same time is more accurate and faster, but not strictly mandatory if the regime is repeatable. 1x phase is measured from the tachometer mark, not from a second sensor, so measurements from different runs at the same speed are comparable. The conditions: the same speed, the same thermal regime, the same load, the same measurement direction, and the same mark on the shaft.

**After correction, vibration at the middle bearing went up. What does that mean?**

Usually it means you solved part of the system instead of the whole thing. A weight chosen from two bearings also changed the load at the third, and its contribution wasn't accounted for in the calculation. Go back to the complete map: take the influence coefficients for every plane at every bearing, and calculate the correction as one solution, not section by section.

**How does multi-bearing balancing differ from flexible-rotor balancing?**

A multi-bearing problem stays a rigid-rotor problem, it just has more equations. A flexible rotor changes its bending shape with speed, so a correction found at one speed is already incomplete at another. There, modal methods and multi-speed schemes apply. The requirements for balancing flexible rotors sit in the ISO 21940 series; check the applicable part and edition separately.

**The shaft is coming out for repair anyway. Should we balance it on site or on a machine?**

If the shaft is coming out anyway, a balancing machine is usually the better deal: every bearing is measured at once, the machine's own drive sets the regime, repeatability is higher, and the number of runs isn't limited by shop logistics. On-site balancing wins when disassembly is expensive or impossible, and the problem reduces to accessible planes and is confirmed by measurement.