This page goes a level deeper than the fundamentals of unbalance. It covers the mechanics underneath: how inertia defines the axis a part wants to spin about, how centrifugal force scales, why the metric and imperial formulations differ, and the four distinct types of unbalance including the one most often missed.
Drawn from The Basics of Balancing 202 by Gary K. Grim, John W. Haidler, Joel M. Book PhD and Bruce J. Mitchell Jr. The original paper is available as a PDF.
Why Balance at All
Balancing is the process of aligning a part’s principal inertia axis with its geometric axis of rotation, by adding or removing material. Everything else follows from that single statement.
Inertia and the Axis a Part Wants to Spin About
Mass moment of inertia describes how a body’s mass is distributed about an axis:
I = m × r² for a particle, and I = ∫ r² dm for a rigid body
where I is mass moment of inertia, m is mass and r is distance from the axis. Every rigid body has a set of principal axes about which it will rotate without producing a couple. The one lying closest to the intended rotation axis is the central principal axis, and the mass centre sits on it.
Unbalance is simply the condition where the central principal axis and the axis of rotation do not coincide. The quality of the mounting datums matters enormously here: the axis of rotation is defined by bearing journals or mounting surfaces, so error in those surfaces becomes unbalance regardless of how well the part itself is made.
Centrifugal Force
F = m × r × ω² or equivalently F = U × ω²
where ω is angular velocity in radians per second. To convert from rev/min, multiply by 0.1047 (which is 2π/60).
The critical point, and the one most often misunderstood on a shop floor: the unbalance quantity itself does not change with speed. A rotor with 1 oz·in of unbalance has 1 oz·in at rest and at 10,000 rev/min. What changes, dramatically, is the force that unbalance produces — and it rises with the square of speed.
Moments and Couples
ΣM = F × d
A couple is two equal and opposite parallel forces separated by a distance, producing rotation without net translation. How a body responds depends on how it is restrained: unrestrained, restrained at one point, or restrained at two. For a rotor in two bearings, the reaction at each support follows from the geometry:
R = F × (d ÷ s)
This is why the position of a correction plane relative to the bearings matters as much as the magnitude of the correction. See balancing with an offset mass centre for what happens when that geometry is unfavourable.
Weight, Mass and Which Formula to Use
The two unit systems are not interchangeable and mixing them is a reliable source of error.
| System | Formula | Units |
|---|---|---|
| Metric | F = m × r × ω² | F in newtons, m in kilograms, r in metres, ω in rad/s |
| Imperial | F = (w ÷ g) × r × ω² | F in pounds, w in pounds weight, g = 386 in/s², r in inches, ω in rad/s |
In the imperial formulation the weight term must be divided by gravitational acceleration to yield mass. For reference, 1 slug = 1 lb·s²/ft = 0.0833 lb·s²/in.
A conversion worth committing to memory: 1 oz·in = 720 g·mm.
For other pairings, our unit conversion calculator handles the arithmetic.
The Four Types of Unbalance
| Type | Condition | Expression | Units | Correction |
|---|---|---|---|---|
| Static (force) | Mass centre displaced from the rotation axis; central principal axis remains parallel to it | U = w × r or U = w × e | oz·in, g·mm | One weight in the mass centre plane |
| Couple | Central principal axis intersects the rotation axis at the mass centre but is not parallel to it | U = w × r × d | oz·in², g·mm² | Two equal weights 180° apart in two planes |
| Dynamic | Central principal axis neither parallel to nor intersecting the rotation axis. The general case. | Vector sum of force and couple | Two-plane values | Two unrelated weights in two planes |
| Quasi-static | Special case where the static and couple vectors lie in the same plane; the axis intersects the rotation axis away from the mass centre | — | — | A single, well-positioned correction |
Quasi-static unbalance is the one worth knowing about, because it looks like it needs two-plane correction and in fact does not. Where the static and couple components happen to share a plane, a single correction placed correctly resolves both. This is common in assemblies where a well-balanced component is mounted with some eccentricity onto a well-balanced shaft.
Two Ways to Express a Correction
Right-left correction gives four numbers: an amount and an angle for each of the two planes. It is the direct form and the one most machines display by default. Because each of those readings is a magnitude paired with an angle rather than a single number, the tolerance around it is an area in a plane rather than an interval on a line — see unbalance tolerance in vector space for what that means for how a specification should be written.
Force-couple correction separates the result into a force component, halved and applied at the same angle in both planes, and a couple component applied 180° apart. It is the more useful form when you want to know whether a part’s problem is a heavy spot or a tilt, and it is how axle and driveline specifications are often written.
The two are mathematically equivalent. Removing weight at one angle is identical to adding the same weight 180° opposite.
Amplitude, Frequency and Damping
Machine behaviour is best understood as a ratio of amplitude to frequency. Plotting amplitude ratio (m·x/U) against frequency ratio (ω/ωn) for different damping ratios produces three distinct regions.
| Region | Behaviour | Consequence |
|---|---|---|
| Below about half resonance | Displacements are small; displacement and force are in phase; the part is firmly constrained | Hard suspension machines work here, measuring force |
| At resonance | Centrifugal and spring forces are 90° out of phase; with light damping the amplitude becomes very large | Historically exploited for sensitivity, now avoided |
| Above about twice resonance | Phase difference approaches 180°; the part rotates about its own mass centre, and Xp = U ÷ W | Soft suspension machines work here, measuring displacement |
Machines deliberately built to run near resonance for the mechanical gain — sometimes called quasi-hard or quasi-soft — are non-linear and sensitive to small changes in speed control. Modern electronics make the trade worth avoiding.
Frequently Asked Questions
Does unbalance increase with speed?
No. The unbalance quantity is a property of the part and does not change. The force it produces increases with the square of speed, which is why a part acceptable at low speed can be unacceptable at high speed.
What is quasi-static unbalance?
A special case of dynamic unbalance where the static and couple vectors happen to lie in the same plane. It can be corrected with a single well-positioned weight rather than a full two-plane correction.
What is the difference between right-left and force-couple correction?
They are two ways of expressing the same result. Right-left gives an amount and angle for each correction plane. Force-couple separates the answer into a force component applied at the same angle in both planes and a couple component applied 180° apart.
How many gram-millimetres are in an ounce-inch?
720 g·mm to 1 oz·in.
Source
Adapted from The Basics of Balancing 202 by Gary K. Grim, John W. Haidler, Joel M. Book PhD and Bruce J. Mitchell Jr., Balance Technology Inc. Download the original paper (PDF).
Browse all BTI technical resources or talk to an application engineer.
