Understanding the Four-Run Method in Rotor Balancing

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The four-run method is a systematic procedure for two-plane balancing that uses four distinct measurement runs to establish a complete set of influence coefficients for both correction planes. It begins by measuring the rotor’s as-found condition, then tests each correction plane independently with a trial weight, and finishes with a fourth run in which both planes carry trial weights at the same time. That fourth run is what distinguishes the method from its faster cousin, the three-run method — it is a deliberate cross-check rather than a strict mathematical necessity.

This thorough approach fully characterises the dynamic response of the rotor-bearing system, allowing accurate calculation of the correction weights that minimise vibration at both bearing locations simultaneously.

1. The Four-Run Procedure

The method consists of precisely four sequential test runs, each with a specific purpose. Throughout, vibration is recorded as a vector — both amplitude and phase — at each of the two bearings.

Run 1 — Initial (baseline) run

The machine runs at its balancing speed in its as-found state. Vibration is recorded at both bearing locations (Bearing 1 and Bearing 2), capturing the baseline signature produced by the original unbalance.

• Record: vibration at Bearing 1 = A₁ ∠θ₁
• Record: vibration at Bearing 2 = A₂ ∠θ₂

Run 2 — Trial weight in Plane 1

The machine is stopped and a known trial weight (T₁) is fitted at a marked angular position in Correction Plane 1. The machine is restarted and vibration is measured again at both bearings. The vector change reveals how a weight in Plane 1 influences both measurement points.

• Trial weight T₁ added to Plane 1 at angle α₁
• Record: new vibration at Bearing 1 and Bearing 2
• Calculate: effect of T₁ on Bearing 1 (primary effect)
• Calculate: effect of T₁ on Bearing 2 (cross-coupling effect)

Run 3 — Trial weight in Plane 2

Trial weight T₁ is removed and a different trial weight (T₂) is fitted in Correction Plane 2. A further run reveals how a weight in Plane 2 influences both bearings.

• Trial weight T₁ removed from Plane 1
• Trial weight T₂ added to Plane 2 at angle α₂
• Record: new vibration at Bearing 1 and Bearing 2
• Calculate: effect of T₂ on Bearing 1 (cross-coupling effect)
• Calculate: effect of T₂ on Bearing 2 (primary effect)

Run 4 — Trial weights in both planes

Both trial weights are now installed together (T₁ in Plane 1 and T₂ in Plane 2) for a fourth run. This supplies extra data that verifies the system’s linearity and can sharpen the calculation when cross-coupling is strong.

• Both T₁ and T₂ installed simultaneously
• Record: combined vibration response at both bearings
• Verify: the vector sum of the individual effects (Runs 2 and 3) matches the combined measurement — confirming linear behaviour

2. Mathematical Foundation

The four-run method populates four influence coefficients that form a 2×2 matrix describing the complete behaviour of the system. The same coefficients underpin every form of multi-plane work, so understanding them here pays off across all dynamic balancing.

The influence-coefficient matrix

• α₁₁: effect of a unit weight in Plane 1 on vibration at Bearing 1 (direct effect)
• α₁₂: effect of a unit weight in Plane 2 on vibration at Bearing 1 (cross-coupling)
• α₂₁: effect of a unit weight in Plane 1 on vibration at Bearing 2 (cross-coupling)
• α₂₂: effect of a unit weight in Plane 2 on vibration at Bearing 2 (direct effect)

Solving for the correction weights

With all four coefficients known, the software solves a pair of simultaneous vector equations for the correction weights (W₁ for Plane 1, W₂ for Plane 2) that cancel vibration at both bearings:

• α₁₁ · W₁ + α₁₂ · W₂ = −V₁ (to cancel vibration at Bearing 1)
• α₂₁ · W₁ + α₂₂ · W₂ = −V₂ (to cancel vibration at Bearing 2)

Here V₁ and V₂ are the initial vibration vectors at the two bearings. The solution combines vector mathematics with inversion of the 2×2 coefficient matrix. Because Runs 1–3 already supply all four coefficients, the system is mathematically determined after three runs; the fourth run is therefore redundant data that buys confidence rather than a missing equation.

3. Advantages of the Four-Run Method

The extra run brings several concrete benefits.

Complete system characterisation

Testing each plane alone and then both together fully captures both direct effects and cross-coupling. That matters when planes sit close together or when bearing stiffness differs markedly between ends.

Built-in verification

Run 4 is a linearity check. If the combined effect of both trial weights does not match the vector sum of their individual effects, the system is behaving non-linearly — a symptom of looseness, bearing play, or foundation problems that should be cured before balancing continues.

Improved accuracy

When cross-coupling is significant — one plane strongly affecting the far bearing — the redundant data yields a more robust result than a bare three-run solution.

Redundant data and error tolerance

Four measurements against effectively four unknowns provide redundancy, letting the software detect and partly average out measurement scatter.

Confidence in results

The systematic sequence and the built-in check give the technician justified confidence that the calculated corrections will work the first time.

4. When to Use the Four-Run Method

The four-run method is particularly appropriate when:

• Cross-coupling is significant: closely spaced planes or asymmetric stiffness make one plane influence both bearings strongly.
• Precision is demanding: tight balancing tolerances — fine G-grades under ISO 21940-11 (the modern successor to ISO 1940-1) — must be met.
• System behaviour is unknown: a machine is being balanced for the first time and its response is not yet understood.
• The equipment is critical: high-value critical machinery where one extra run is cheap insurance.
• Permanent calibration is being established: when storing permanent calibration coefficients for repeated future use, the method’s thoroughness ensures the saved data is accurate.

5. Comparison with the Three-Run Method

The four-run method is best understood against the simpler three-run method, which omits the combined run.

Three-run sequence

• Run 1: initial condition
• Run 2: trial weight in Plane 1
• Run 3: trial weight in Plane 2
• Corrections calculated directly from the three runs

What the fourth run adds

• Linearity verification: Run 4 confirms the system behaves linearly.
• Better cross-coupling characterisation: richer data when cross-coupling is strong.
• Error detection: anomalies stand out more readily.

What the three-run method gives up — and keeps

• Time savings: one fewer run cuts balancing time by roughly 20%.
• Sufficient accuracy: for many machines, three runs are entirely adequate.
• Simplicity: less data to handle and fewer weight changes.

In practice the three-run method is the workhorse for routine balancing, while the four-run method is reserved for high-precision jobs or problem machines. Both rest on the same physics; for either approach a portable two-channel analyser such as the Balanset-1A records the amplitude and phase at each bearing, computes the influence coefficients automatically, and — for the four-run sequence — flags any failed linearity check before you commit to a correction. Sizing the trial weights themselves is made simpler by a trial weight calculator.

6. Practical Execution Tips

For a clean four-run result, pay attention to three areas.

Trial-weight selection

• Choose trial weights that produce a 25–50% change in vibration from baseline.
• Use similar magnitudes in both planes for consistent measurement quality.
• Make sure every weight is securely attached for all runs.

Measurement consistency

• Hold identical operating conditions — speed, temperature, load — across all four runs.
• Allow thermal stabilisation between runs where needed.
• Keep the same sensor locations and mounting for every measurement.
• Take several readings per run and average them to suppress noise.

Data-quality checks

• Confirm each trial weight produces a clearly measurable change (at least 10–15% of the initial level).
• Check that Run 4 roughly matches the vector sum of the Run 2 and Run 3 effects (within about 10–20%).
• If the linearity check fails, investigate mechanical issues before proceeding.

7. Troubleshooting

Two failure modes account for most difficulties with the method.

Run 4 does not match the expected response

Possible causes:

• Non-linear behaviour — looseness, soft foot, or bearing play.
• Trial weights too large, driving the system into a non-linear regime.
• Measurement errors or inconsistent operating conditions.

Solutions:

• Find and correct the mechanical problem.
• Use smaller trial weights.
• Verify the measurement chain’s calibration.
• Hold operating conditions constant across all runs.

Poor final balance results

Possible causes:

• Calculated corrections installed at the wrong angles.
• Errors in weight magnitude.
• System characteristics shifting between the trial runs and correction installation.

Solutions:

• Carefully verify the correction-weight installation.
• Ensure mechanical stability throughout the procedure.
• Consider repeating the job with fresh trial-run data, and finish with a trim balance if a small residual remains.

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