# The Influence Coefficient Method: How the Instrument Calculates the Correction Weight

> The instrument doesn't see unbalance. It sees vibration at the bearing housings, and standing between unbalance and vibration is the entire machine: rotor, bearings, housings, frame, foundation. The influence coefficient method measures that transfer directly, with a single trial weight. Below: the physics in plain words, vector arithmetic with real numbers, the 2×2 matrix for two planes, and the limits beyond which the method stops working.

**In short:** The influence coefficient method determines exactly how your machine responds to unbalance, instead of pulling its properties from a reference table. You take a run with no weight and a run with a trial weight of known mass at a known radius, and the software subtracts the 1x vectors (vibration at the rotor's rotation frequency) and divides the difference by the trial unbalance. Out comes the influence coefficient: how many mm/s, and at what angle, one gram produces in this plane at this housing. From it, the instrument solves the inverse problem and outputs the correction's mass and angle. For two planes there are four coefficients, which is why two separate trial runs are needed.

Source: https://axiline.pt/en/articles/the-influence-coefficient-method/  
Publisher: AXILINE · Vila Nova de Gaia, Portugal · +351 931 831 229 · axilinegeral@gmail.com

## What Happens Between the Trial Run and the Number on the Screen

Vibration at the rotation frequency is the system's response to the centrifugal force from unbalance. The force grows in proportion to the unbalanced mass and the radius it sits at. But the size of the response also depends on the structure's stiffness, its mass, and its damping. You don't know the housings' stiffness or the frame's compliance in advance, and you won't find them in a reference table.

What you can do instead is query the system directly. Put a weight of known mass at a known location on the rotor, and the unbalance changes by a known amount. The difference in vibration before and after is the machine's response. Divide the response by the input, and you get the influence coefficient. No calculation model of the rotor gets built in the process: the system is measured as it actually is, housings, foundation, and all.

- Run 0 gives the baseline vibration at each housing: 1x amplitude and phase.
- The run with the trial weight gives new vibration for a known change in unbalance.
- The difference between the two vectors is divided by the trial unbalance. That's the influence coefficient.
- The instrument calculates the correction as an inverse problem: what unbalance, through this coefficient, produces a vector equal in length to the original one and opposite in direction.

## Amplitude and Phase Are One Vector, Not Two Separate Numbers

Everything that follows rests on one idea: the running-speed component of vibration is a vector. Amplitude in mm/s sets its length; phase relative to the laser tachometer's marker sets its direction. Neither amplitude nor phase is any good for calculating the correction on its own.

This leads to a consequence that's confusing if you treat amplitudes as ordinary numbers. Vibration dropped from 6.0 to 4.0 mm/s, yet the change came out to 7.2 mm/s. There's no error here: the vectors rotated relative to each other, and the difference vector ended up longer than either one.

Phase by itself doesn't show where the heavy spot sits. Its job is different: making the vector subtraction correct.

> If the phase jumps by more than 5–10° from run to run, you don't have a vector — you have a cloud of points. Subtracting readings like that is pointless: stabilize the speed and check the sensor mounting first.

## What the Influence Coefficient Is, and What It's Measured In

The influence coefficient answers one specific question: what will one gram, added at this correction plane and this radius, do to the vibration. The answer has two parts, because the coefficient is a vector too.

### Magnitude

How many mm/s one gram adds at a given radius. For example, 0.72 mm/s per gram at a 100 mm radius. This is the sensitivity of a specific housing to a specific correction plane.

### Direction

The angle between the direction of the added unbalance and the direction of the response. It's almost never zero: stiffness and damping rotate the response by tens of degrees, and near resonance, by nearly 180°.

### Scope

The coefficient belongs not to the rotor but to the entire "rotor — bearing housings — frame — foundation" system, and specifically at a given speed, at given measurement points, and at a given weight-mounting radius.

## A Numerical Example for One Plane

Let's work through one plane, because everything's visible on paper there. Numbers are rounded so you can check the arithmetic yourself. This is an illustrative example, not a real measurement report.

1. **Baseline vibration** — At the bearing housing the instrument shows 6.0 mm/s at a 30° phase. We record: V0 equals 6.0 mm/s at 30°.
2. **Trial weight of 10 g** — You mount 10 g at a 100 mm radius by the reflective marker, and take this position as zero. New reading: V1 equals 4.0 mm/s at 300°. Amplitude changed by a third, phase by 90°. That means the trial weight is valid: the change is clearly above the 20–30% threshold, below which the calculation can't be trusted.
3. **The change vector** — We calculate the difference ΔV = V1 − V0. In components, V0 = (5.20; 3.00), V1 = (2.00; −3.46), and the difference equals (−3.20; −6.46). That's about 7.2 mm/s at 244°: the pure contribution of the trial weight, with everything the machine already had before it stripped out.
4. **The influence coefficient** — We divide the change by the trial mass: 7.2 over 10 gives 0.72 mm/s per gram, direction 244°. Read it like this: one gram at a 100 mm radius produces 0.72 mm/s at this housing, and the response doesn't land where the weight was mounted — it lands 244° further around in the reference frame.
5. **The correction weight** — What's needed is an unbalance that, through this coefficient, produces 6.0 mm/s in the direction of 30° plus 180°, that is, 210°. Mass: 6.0 divided by 0.72 is about 8.3 g. Angle: 210° minus 244° equals minus 34°, that is, 326° in the adopted reference direction from the trial weight's position.

> The software calculates exactly the same thing in complex numbers, without any rounding. It also decides what to do with the trial weight: remove it, or leave it in place and subtract it vectorially. Subtracting a trial weight left in place by simple mass subtraction is wrong — it's a classic manual-calculation mistake.

## Two Planes: The 2×2 Matrix and Cross-Coupling

For a rotor with two correction planes there isn't one coefficient — there are four. The centrifugal force from a weight in the first plane doesn't push on just one housing — it acts on the whole rotor as a rigid body, and a response shows up at both bearings at once.

Picture a between-bearings rotor as a beam on two points. A weight closer to housing 1 will load it more heavily, but it will load housing 2 too, through the shaft, following lever action. On an overhung rotor the coupling is even stronger: a weight on the impeller creates a moment, and at the far housing it sometimes produces more vibration than at the near one. Add in the frame and foundation, through which the housings "talk" to each other.

Hence a system of equations. Vibration at housing 1 equals α11 times the unbalance in plane 1 plus α12 times the unbalance in plane 2. Vibration at housing 2 equals α21 times the unbalance in plane 1 plus α22 times the unbalance in plane 2. All quantities are vectors, and α12 and α21 are exactly the cross-coupling.

| Coefficient | What it links | Where it comes from |
| --- | --- | --- |
| α11 | weight in plane 1 → vibration at housing 1 | Run 1, channel 1. Direct influence |
| α21 | weight in plane 1 → vibration at housing 2 | Same run 1, channel 2. Cross-coupling |
| α12 | weight in plane 2 → vibration at housing 1 | Run 2, channel 1. Cross-coupling |
| α22 | weight in plane 2 → vibration at housing 2 | Same run 2, channel 2. Direct influence |

> Cross-coupling explains a familiar headache: you mount a weight in one plane, vibration at the near housing drops, and at the far one it rises. A single mass won't zero out both vectors if the unbalance is a couple — meaning the unbalanced masses sit at opposite ends of the rotor and twist it, rather than simply shifting it.

## Why You Need Two Separate Trial Runs

There are four unknowns, and every run gives you two measurements, one per channel. One trial run is physically not enough: you can't tell from it how much of the vibration change at housing 1 came from plane 1 and how much from plane 2. You need two independent inputs.

Hence the sequence in the software: run 0 with no weight, run 1 with a trial weight in the first plane, run 2 with the same weight in the second plane. You must remove the weight from the first plane before the second trial run, or the inputs will mix and the matrix columns will end up dependent.

The first trial run fills the matrix's first column; the second run fills the second. From there the software solves the 2×2 system and outputs two masses and two angles. Simultaneous two-channel acquisition isn't a luxury here: both channels have to be measured in the same run, referenced to the same marker.

- One plane: 2 runs, baseline and one trial.
- Two planes: 3 runs, baseline and two trial.
- Plus a minimum of one verification run either way.
- The weight-mounting radius and the zero reference point are the same across every run.

## Where Linearity Breaks Down, and How You'll See It

The whole method rests on one assumption: the response is proportional to the input. Double the trial unbalance, and 1x doubles too, with the direction unchanged. For a rigid rotor on a sound machine within the working speed range, this holds up well. But not always.

The good news: a breakdown in linearity always gives itself away the same way. The verification run doesn't match the calculation. The software promised a residual of 0.3 mm/s, and you got 2.5 mm/s, or the vibration went off in a completely different direction. That's not an instrument glitch, and it's not an arithmetic error. It's the machine telling you the linear model doesn't fit it right now.

| Cause | What you see | What to do |
| --- | --- | --- |
| Structural or frame resonance | Amplitude and phase drift by more than 10–15% within a single run; vibration jumps with a small change in speed | Move the speed out of the resonance zone, or detune the system's stiffness and mass. You can't balance in resonance |
| Loose mounting, play, soft foot (a casing foot that doesn't sit flush against the frame until torqued) | The coefficient changes from run to run, a comb of 1x, 2x, 3x and subharmonics in the spectrum | Torque the fasteners, eliminate soft foot and play, repeat run 0 |
| Rotor rubbing against stationary parts | Poor repeatability, lots of harmonics, a raised noise floor | Eliminate the rub. While it's present, the coefficients are invalid |
| Nonlinear supports: rubber vibration isolators, the oil wedge in plain bearings | The response depends on the force level: a small trial weight gives one coefficient, a large one another | Check linearity with two different trial weights, keep weights within a similar mass range |
| Thermal changes over the course of balancing | Baseline vibration drifts between runs, a repeated run 0 doesn't reproduce | Hold a stable thermal condition, repeat run 0 until the result settles |
| Aerodynamic or electromagnetic force at 1x | Vibration is normal at the balancing speed, but rises at other speeds | Recognize the method's limit: these forces are proportional to the first power of the rotation frequency, while the centrifugal force is proportional to its square |
| Elastic (flexible) rotor | Coefficients depend noticeably on speed, the result doesn't carry over to a different speed | The rigid-rotor model doesn't apply — you need an approach for rotors in the flexible state |

> Check linearity before it surprises you: run run 0 twice. If the amplitude repeats within ±5% and the phase within ±5°, the coefficients can be trusted. If it doesn't repeat, mechanics first, calculation later. For rotors in the flexible state the guidelines are different — check the applicable part and edition of ISO 21940.

Sources: [ISO 21940-12:2016](https://www.iso.org/standard/50429.html) · [ISO 13373-3:2015](https://www.iso.org/standard/40840.html)

## Saved Coefficients: When You Can Reuse Them

The influence coefficient is a measured property of the machine. If the machine hasn't changed, the property hasn't gone anywhere. The instrument saves coefficients to an archive, and next time the correction is calculated from one run instead of three. Trim balancing rests on the same principle: the coefficients are already known from the current session, so the trim weight is calculated from a single verification run.

The conditions for reuse are strict, and all of them have to hold at once. First, the list of what has to match; then, the cases where old coefficients are already invalid.

- Bearings were replaced or the assembly was overhauled
- The machine was removed and reinstalled, or the foundation was redone
- The impeller, blower wheel, or rotor was replaced, or metal was welded on or cut away
- Sensors were repositioned or the measurement direction was changed
- You're running at a different speed or under a different load condition
- The weight-mounting radius was changed
- There was an incident, an impact, or a significant repair

- [x] The same machine. This exact unit, not "the same model"
- [x] The same speed and the same condition: load, flow, temperature
- [x] The same measurement points, the same sensor direction, the same mounting method
- [x] The same weight-mounting radius and the same zero reference point
- [x] The trial weight's location matches the phase sensor's reflective marker
- [x] The trial weight's mass during the original balancing was entered in grams, not as a percentage
- [x] The mechanical condition is unchanged: bearings, mounting, foundation, rotor geometry

> Not sure? Don't reuse them. Three runs cost less than a weight mounted from a stale coefficient.

## Production Rotors, Rigs, and How AXILINE Can Help

Count the runs. Initial two-plane balancing is three runs plus at least one verification run, with two stoppages in between to move the trial weight. With saved coefficients, that comes down to one run plus a verification run. On a rotor that takes ten minutes just to spin up, the difference shows immediately.

On production rotors the effect compounds. When you're balancing identical blower wheels on an arbor on a balancing rig, the influence coefficient describes the rig, not the individual part. Calibrate once, and every part after that runs through the short procedure. For an arbor with eccentricity there's an index-balancing mode: an extra run with the rotor flipped 180° subtracts the arbor's own contribution.

The Balanset-1A handles all of this math. The kit includes two accelerometers for the bearing housings, a laser phase sensor with a reflective marker, a two-channel USB module with preamplifiers, integrators, and an ADC, and Windows software. It measures speed, and vibration velocity amplitude and phase — separately for overall vibration and for 1x — shows the FFT spectrum, solves the problem for one and two planes, and tells you itself whether the trial weight is valid. The software also stores coefficients and results in an archive for reports, splits weight across fixed positions, calculates drilling, and calculates tolerance against balance quality grades G per the applicable part and edition of ISO 21940.

What's left to the person is what the instrument can't verify: rigid rotor behavior at running speed, the absence of resonance and looseness, secure weight mounting, and the choice of measurement points and tolerances. If there's no time to work through vectors on a live site, AXILINE's engineers come and balance it themselves, with the very instruments they design and manufacture. Want to do it yourself instead — get the instrument and our consulting support.

Sources: [Balanset-1A operation manual](https://vibromera.eu/balanset-1a-operation-manual/) · [Balanset-1A manufacturer specification](https://vibromera.eu/product/balanset-1/) · [ISO 21940-11:2016](https://www.iso.org/standard/54074.html)

## Frequently asked questions

**How does the influence coefficient differ from sensor sensitivity?**

These are different things with similar-sounding names. Sensor sensitivity, in mV per g, describes the accelerometer and gets entered into the software once, during setup. The influence coefficient describes the machine: how many mm/s a gram produces at a given radius in a given plane. The first is about the measurement chain; the second is about the mechanics.

**Can the correction be calculated without a trial run at all?**

Not from a single baseline reading. The instrument knows the 1x amplitude and phase, but it doesn't know how many times over this machine amplifies a gram, or at what angle it rotates the response. There's one exception: coefficients for this machine are already saved from a previous balancing job, and nothing has changed since.

**Why doesn't the correction angle come out to exactly 180° from the measured phase?**

Because the 1x phase doesn't show where the heavy spot is. Standing between the direction of the force and the direction of the response is the structure's own phase shift, and it depends on stiffness, damping, and how far the running speed is from resonance. That shift is exactly what's baked into the influence coefficient's direction.

**The trial weight turned out too small. What happens to the calculation?**

The difference vector comes out short, and the relative error in its direction grows. The coefficient drifts off in angle, and the correction weight lands in the wrong place. Guideline: the amplitude should change by at least 20–30%, or the phase by 20–30°. If the change is smaller, increase the weight, enter the actual mass, and repeat the run.

**The verification run came back higher than the software promised. Is the instrument lying?**

Almost never. A mismatch usually means either broken linearity or an error in the physical setup: the angle was read off in the wrong direction, the weight is at a different radius, the trial weight was left on by mistake, the machine is running near resonance, the mounting is loose. Check the weight installation and the repeatability of run 0 first.

**Are saved coefficients good for a different machine of the same model?**

No. Two units identical on paper sit on different foundations, with different mounting torque and different bearing-housing wear. The influence coefficient belongs to the entire system, not to the equipment type. Take your own coefficients for every individual unit.