Two-Plane Balancing: The Fundamentals

This is the second article in a series on the topic of rotor balancing. The first article in this series (Single-Plane Balancing: The Fundamentals) established what unbalance is and how it is quantified, using rotors whose unbalance can be treated as concentrated in a single transverse plane. Many rotors cannot be treated that way. Longer rotors — such as those in multi-stage pumps, compressors, and turbines — can carry significant unbalance at multiple axial stations, and correcting it requires two-plane balancing. This article covers the fundamentals:

• the taxonomy of unbalance: static, couple, dynamic, and the quasi-static special case,
• why couple unbalance is invisible to a single-plane measurement,
• why two correction planes are sufficient for any rigid rotor, and
• the origin of the terms static balancing and dynamic balancing.

Two concepts are central to this topic — the axis of rotation and the axis of inertia — and it is important to distinguish them precisely. The axis of rotation is the line about which the rotor actually turns, established by its journals and the bearings that support them. The central principal axis of inertia is the line about which the rotor's mass is evenly distributed, and therefore the axis about which the rotor would spin freely if unconstrained. In a perfectly balanced rotor, the two axes coincide. Unbalance causes them to diverge, and the shape of the divergence depends on the type of unbalance. The various types of unbalance may be visualized by evaluating the position of the principal axis of inertia in relation to the axis of rotation (Figure 1).

Static Unbalance

Regardless of how unbalance is distributed along a rotor's length, it sums to a single resultant vector — the static unbalance. The static unbalance acts at the same axial position as the rotor's center of mass. As described in the article on single-plane balancing, it is quantified as a mass at a distance from the axis of rotation:

Us = W · e

where:

• Us = static unbalance (g·mm or oz·in)
• W = rotor mass (g or oz)
• e = mass-center eccentricity (mm or in)

A rotor with pure static unbalance behaves as if a single concentrated mass were attached at one angular position at the axial station of the center of mass. The principal axis of inertia is displaced parallel to the axis of rotation. At speed, the rotating force loads both bearings in phase: both bearing reactions point in the same direction at the same instant (Figure 2). Static unbalance also produces a resting tendency. Supported on low-friction rollers, the rotor rolls until the heavy side settles at the bottom.

Couple Unbalance

Couple unbalance is two equal unbalances separated axially and oriented 180° apart (Figure 3). Summed over the rotor, the two mass eccentricities cancel. The center of mass remains on the axis of rotation, the net rotating force is zero, and a single-plane measurement reads zero — yet the unbalances still affect the way the rotor operates. The principal axis of inertia is angled relative to the axis of rotation, with the intersection of the two axes occurring at the center of mass. At speed, the couple forms a rotating moment, rocking the rotor end over end and loading the bearings out of phase. When one bearing reaction points up, the other points down.

The magnitude of a couple unbalance is the product of the unbalance in each plane and the separation between the planes:

C = Uc · d

where:

• C = couple unbalance (g·mm² or oz·in²)
• Uc = unbalance in each of the two planes (g·mm or oz·in)
• d = axial separation between the two unbalance planes (mm or in)

The rocking moment the couple produces at speed follows the same speed-squared law as the rotating force of unbalance:

M = C · ω²

where:

• M = rotating moment (N·m)
• C = couple unbalance (kg·m²)
• ω = angular speed (rad/s)

Note that C enters this formula in consistent SI units; 1 g·mm² = 10⁻⁹ kg·m².

No single-plane correction can remove a couple. A mass added in one plane changes the net force on the rotor, which a pure couple does not have. Removing a couple requires equal and opposite corrections in two planes.

Dynamic Unbalance

When a static component and a couple component are superimposed, the condition is known as dynamic unbalance. This is the general case, in which the two per-plane unbalances are neither in phase nor 180° apart. In reality, all rotors are dynamically unbalanced to some extent; pure static and pure couple unbalance are idealizations. The shape and severity of the dynamic unbalance are determined by the relative magnitudes and phases of the static and couple components. The principal axis of inertia is displaced transversely from the axis of rotation and angled such that the two axes do not intersect at all.

Quasi-Static Unbalance

Quasi-static unbalance is the term applied in the special case that the static and couple components lie in a single axial plane (Figure 4). The name reflects the condition's resemblance to static unbalance: because the components are coplanar, they may be summed to obtain a single equivalent unbalance vector — one heavy spot, like the static case. The difference that earns the prefix quasi is that the heavy spot is located at an axial station away from the center of mass. Thus the principal axis of inertia crosses the axis of rotation instead of running parallel to it.

Two Planes Are Sufficient

A single correction plane has already been shown to be insufficient for the general case: a couple produces a rotating moment that no single-plane correction can remove. Yet unbalance can be distributed across many axial stations along a rotor's length, so it is fair to question why only two planes should be sufficient. The foundation of the answer is that unbalances are vectors, with magnitude and phase, and vectors can be resolved and summed.

For a rigid rotor, any single elemental unbalance vector UN can be converted into an equivalent set of two vectors, UNA and UNB, one in each of the chosen planes, apportioned by their relative proximity to UN, with the nearer plane carrying the larger share:

UNA = UN · b / L

UNB = UN · a / L

where:

• UN = the Nth elemental unbalance, at its axial station (g·mm or oz·in)
• UNA = equivalent unbalance in plane A due to UN (g·mm or oz·in)
• UNB = equivalent unbalance in plane B due to UN (g·mm or oz·in)
• a = axial distance from plane A to the elemental unbalance (mm or in)
• b = axial distance from the elemental unbalance to plane B (mm or in)
• L = axial separation between planes A and B, equal to a + b (mm or in)

A rotor's unbalance distribution is simply an arrangement of many such elemental unbalances at various axial stations — U1, U2, U3, … UN (Figure 5). Each converts into its own pair of vectors, one in each plane: U1A through UNA in plane A, and U1B through UNB in plane B. Summing U1A through UNA vectorially yields a single total vector UA in plane A; summing U1B through UNB yields a single total vector UB in plane B.

Correct those two vectors (UA and UB), and a rigid rotor is balanced, no matter how its unbalance is actually distributed physically. This is the rigid-rotor equivalence that underlies the entire practice: two planes are sufficient for any rigid rotor. The planes need not straddle every element; an element outboard of both planes resolves with components of opposite sense, and the arithmetic holds. The choice of the two correction planes is largely free: they need not lie on opposite sides of the rotor's center of mass, and any two distinct transverse planes will serve, though wider separation is preferred in practice because closely spaced planes demand larger corrections.

The Static-and-Couple Form

Once the rotor's distributed unbalance is reduced to two planes, the two per-plane vectors (UA and UB) can be decomposed into a single equivalent static unbalance, Ueq, obtained from their vector sum, and a single equivalent couple, Ceq, obtained from their vector difference acting across the separation L. The per-plane form (UA and UB) is what a balancing machine reports for the purpose of making corrections, while the static-and-couple form (Ueq and Ceq) provides better perspective on how the rotor's mass distribution affects the way it operates.

Ueq = UA + UB

Ceq = (UA − UB) · L / 2

where:

• Ueq = equivalent static unbalance (g·mm or oz·in)
• Ceq = equivalent couple unbalance (g·mm² or oz·in²)
• UA = total unbalance vector in plane A (g·mm or oz·in)
• UB = total unbalance vector in plane B (g·mm or oz·in)
• L = axial separation between planes A and B (mm or in)

The sum and the difference are vector operations, carried out at the vectors' phase angles. With this decomposition, the equivalent static unbalance is referenced to the plane midway between A and B. This assumed location works mathematically even when it is not coincident with the rotor's center of mass.

Static and Dynamic Balancing

The taxonomy explains a pair of terms that persist in equipment standards, spec sheets, and purchase orders. Historically, balancing without rotation was performed on knife edges or rollers. The rotor rolls until its heavy side settles at the bottom, the resting position marks the angular location of the heavy spot, and correction proceeds by trial. The method detects only static unbalance, and the practice is called static balancing. Couple unbalance produces no resting tendency and reveals itself only in rotation, so balancing that measures a spinning rotor came to be called dynamic balancing. Because two-plane correction is what rotating measurement enables, the terms attached loosely to plane count: static balancing for single-plane work, dynamic balancing for two-plane work.

The loose usage is entrenched and, for rigid rotors, mostly harmless, but the taxonomy is the precise language: static, couple, and dynamic name types of unbalance, not methods. A single-plane correction performed on a spinning balance stand is a dynamic measurement making a single-plane correction. This article and others in the series use plane-count language because it states directly what is measured and corrected.

Companion Calculator

The calculator published with this article converts between the two natural descriptions of a rigid rotor's unbalance: the per-plane form, an unbalance vector in each of two planes as a balancing machine reports, and the static-and-couple form, the resultant displacing the center of mass plus the rocking couple. Enter either form and read the other, in US or SI units. The decomposition shows at a glance how a rotor can present a small single-plane reading while carrying a large couple.

Next in the series: Low Speed Balancing Machines — soft-bearing and hard-bearing architectures, force versus motion sensing, arbors, drives, transducers and phase reference, and machine capability.