Most balancing discussion assumes the mass centre sits conveniently between the bearings and near the correction plane. Real parts often do not oblige. Where the mass centre is offset — overhung beyond a bearing, or far from the only plane you can correct in — the relationship between unbalance and bearing load changes, and a correction that zeroes the reading can leave the assembly worse off.
Drawn from Balancing with an Offset Mass Center by John Haidler.
The Two Conditions
Static unbalance exists when the mass centre does not lie on the spin axis. Dynamic unbalance exists when the principal inertia axes are not aligned with the spin axis. The cases below show how correcting the first can create the second.
Notation
| Symbol | Meaning | Units |
|---|---|---|
| a, b, c, d, r | Dimensions | inches or mm |
| w | Correction weight | ounces or grams |
| ω | Rotational speed | radians per second |
| Ustatic | Static unbalance | oz·in or g·mm |
| Udynamic | Dynamic unbalance | oz·in² or g·mm² |
| K | Unit conversion constant | 1.619 × 10⁻⁴ lb·s²/(in·oz) imperial; 1.000 × 10⁻⁶ (kg·m)/(g·mm) metric |
The unbalance force in every case is F = Ustatic × ω² × K.
Case 1 — Mass Centre Between the Bearings
The straightforward case. The correction plane passes through the mass centre, which lies between the two bearings.
Rleft = (b ÷ d) × Ustatic × ω² × K
Rright = (a ÷ d) × Ustatic × ω² × K
The two bearing reactions are in phase with each other, and each takes a share of the unbalance force set by its distance from the mass centre. With the mass centre midway between the bearings the load splits evenly. Correcting in this plane leaves no residual dynamic unbalance.
Case 2 — Mass Centre Outside the Bearings
An overhung rotor. The same force equations apply, but the geometry produces a different and less intuitive result: the two bearing reactions are 180° out of phase with each other.
More importantly, where the overhang is large relative to the bearing span, the bearing reactions can exceed the unbalance force itself. A modest static unbalance on a long overhang generates bearing loads larger than the force the unbalance is nominally producing, because the bearings are working as a lever pair.
This is the case that catches people out on overhung impellers, blowers and single-end-supported rotors.
Case 3 — Correction Plane Away from the Mass Centre
The common real-world constraint: the part has static unbalance, but the only accessible correction plane is some distance c from the mass centre.
Correcting the static unbalance in that plane zeroes the static reading and creates a residual couple:
Udynamic = w × r × c
Rleft = Rright = (c ÷ d) × Ustatic × ω² × K
| Condition | Meaning | Result |
|---|---|---|
| c = 0 | Correction plane contains the mass centre | No residual unbalance. The ideal case. |
| 0 < c < d | Correction plane offset, but by less than the bearing span | Residual reaction forces are smaller than the original uncorrected static unbalance produced. A net improvement. |
| c > d | Correction plane offset by more than the bearing span | Residual reaction forces exceed those from the uncorrected static unbalance. The correction has made the assembly worse. |
That last row is the finding that matters. Single-plane correction is not automatically an improvement. Where the only available correction plane is far enough from the mass centre, correcting static unbalance in it produces a couple worse than the problem it solved.
What to Do About It
Put the correction plane as close to the mass centre as the part allows — this is a design decision as much as a process one, and it is worth raising while the part is still on the drawing board.
Where that is not possible and c approaches or exceeds the bearing span, two-plane correction becomes necessary rather than optional. The analysis above assumes rigid shafts; flexible rotors need a more involved treatment.
For the general theory behind couples and bearing reactions, see balancing theory. For where to specify the resulting tolerance, see specifying unbalance and tolerance planes.
Frequently Asked Questions
Can single-plane balancing make a rotor worse?
Yes. If the correction plane is further from the mass centre than the bearing span, correcting static unbalance in that plane produces a residual couple whose bearing reactions exceed those of the original unbalance.
Why are bearing reactions out of phase on an overhung rotor?
Because with the mass centre outside the bearing span the two bearings act as a lever pair rather than sharing a load between them. One reacts upward as the other reacts downward.
Can bearing forces be larger than the unbalance force?
On an overhung rotor, yes. Where the overhang is large relative to the bearing span, the lever action amplifies the load at the bearings beyond the centrifugal force the unbalance generates.
Where should the correction plane go?
As near the mass centre as the part geometry permits. Where it cannot be near, plan for two-plane correction.
Source
Adapted from Balancing with an Offset Mass Center by John Haidler, Balance Technology Inc.
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