Single-plane balancing on a low-speed stand is among the most frequently performed activities in a rotating equipment repair shop (Figure 1). Some practitioners refer to it as static balancing. This article covers the fundamentals:

• what unbalance is physically,
• the units and conventions used to quantify it,
• which components qualify for single-plane treatment,
• how API and ISO tolerance criteria are constructed and how they relate to one another.

Unbalance as a Rotating Force

Unbalance is mass eccentricity; that is, when the center of mass is not coincident with the axis of rotation. A rotor whose center of mass lies off its axis of rotation by distance e produces a rotating force:

F = W · e · ω²

where:

• F = rotating unbalance force (N)
• W = rotor mass (kg)
• e = mass-center eccentricity (m)
• ω = angular speed (rad/s)

Note that W enters this formula as a mass in consistent SI units (kg). The practical formulas below absorb the necessary conversions.

The angular speed follows directly from the rotational speed:

ω = 2π × N / 60

where:

• ω = angular speed (rad/s)
• N = rotational speed (RPM)

The product of mass and eccentricity is the unbalance — the quantity measured and corrected on the balance stand:

U = W · e

where:

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

A convenience of this definition is that the same unbalance can be represented by any mass-and-radius pair whose product equals U. Rather than picturing the entire rotor mass acting at the small eccentricity e, the unbalance can be treated as a single concentrated mass at a radius of our choosing:

U = m · r

where:

• U = unbalance (g·mm or oz·in)
• m = equivalent concentrated mass (g or oz)
• r = radius at which the mass is located (mm or in)

This mass-at-radius form is how unbalance is handled in practice. Reducing the unbalance by modifying the rotor's mass distribution is called correction. Because corrections are applied at an accessible radius on the component, balance tolerances are likewise converted to a mass at that radius.

The force is therefore a function of the unbalance U and the rotational speed ω (in consistent units):

F = U · ω²

Force increases with the square of speed, so a residual unbalance that is inconsequential at balancing speed may be significant at operating speed.

On the balance stand, the component is run at the stand's balancing speed — typically several hundred RPM, well below service speed. This is valid for a rigid component because unbalance is a property of mass distribution and does not vary with speed; only its force consequence does.

In practical SI units, the rotating force generated by an unbalance is:

F ≈ 0.011 × U × (N / 1,000)²

where:

• F = rotating unbalance force (N)
• U = unbalance (g·mm)
• N = rotational speed (RPM)

In US units, the rotating force generated by an unbalance is:

F ≈ 1.77 × U × (N / 1,000)²

where:

• F = rotating unbalance force (lbf)
• U = unbalance (oz·in)
• N = rotational speed (RPM)

The force follows the rotation of the rotor, occurring at the same frequency as the rotation itself. Frequency and rotational speed are related as:

f = N / 60

where:

• f = frequency of rotation (Hz, cycles per second)
• N = rotational speed (RPM)

This is the synchronous frequency, the frequency at which unbalance-driven vibration appears in measured vibration data — the "1×" component in vibration analysis parlance.

At 3,560 RPM, each ounce-inch of residual unbalance produces approximately 22 lbf of rotating force applied to the bearings at 1× running frequency (approximately 59 Hz). This is the mechanism by which unbalance manifests as synchronous vibration.

Two terms should be distinguished precisely:

• The heavy spot is the angular location of the mass eccentricity — the quantity to be corrected.
• The high spot is the angular location of peak rotor displacement, as would be measured by a vibration proximity probe.

Like all physical structures, rotors have resonant frequencies at which they naturally tend to vibrate when excited by a force. Below the first resonance, the angular locations of the heavy spot and high spot nearly coincide; through and above resonance they separate by up to 180° of phase lag. On a low-speed stand operating well below any structural or rotor resonance, the distinction has little practical effect, but it is fundamental to interpreting vibration data at operating speed and becomes central when performing at-speed balancing (Figure 2).

Units and Conventions

The units of unbalance follow from its definition as a mass-radius product. US practice uses ounce-inches (oz·in) and, increasingly, gram-inches (g·in), since correction weights and shop scales read in grams even where drawings are otherwise in US customary units. International standards and most balancing machine displays use gram-millimeters (g·mm). The governing conversions:

• 1 oz·in = 720 g·mm
• 1 g·in = 25.4 g·mm
• 1 oz·in = 28.35 g·in

Angular convention requires the same rigor as magnitude. Phase angles are stated in degrees, while angular speed ω is expressed in radians per second; the two measures are related by 2π radians = 360°, so one radian ≈ 57.3°.

Every balance job requires a defined zero-angle reference on the component (a keyway, a stamped mark, a designated hole) and a defined direction of angular measurement relative to rotation. These conventions must be recorded on the balance documentation and held constant throughout the job.

Applicability of Single-Plane Correction

Unbalance in a physical rotor is distributed along its length, but measuring and correcting it at every axial station would be impractical. Instead, for many rotors the distributed unbalance can be represented as a single equivalent mass-radius product in one transverse plane, and correction in that plane restores acceptable balance.

The traditional screening rule holds that disc-form components — single-stage pump impellers, fan wheels, coupling hubs, thrust collars — qualify for single-plane correction when the length-to-diameter ratio is below approximately 0.5 (Figure 3). As a rule for individual components on a balancing arbor, this serves well enough. However, it may be misleading as a criterion for assembled rotors, because L/D is only a proxy. The governing question is the axial distribution of the rotor's mass and of its credible sources of unbalance. Where the large masses and potential sources of unbalance are concentrated near a single axial station, single-plane correction may be appropriate regardless of the rotor's overall proportions. Ultimately, the adequacy of single-plane balancing can be verified directly on the balancing machine by observing the residual unbalance indicated in a second plane, rather than evaluated based on geometry alone. Two-plane balancing will be covered in a subsequent article.

Balance Tolerances: API and ISO Criteria

A balance tolerance defines the residual unbalance below which the rotor is considered acceptable for service. Practically speaking, some amount of unbalance will always remain. US practitioners in refining, petrochemical, and pipeline services will most commonly encounter the API criterion. The ISO grade system is broader in scope and analytically more general, and the relationship between the two is worth understanding.

The API criterion

Maximum allowable residual unbalance per correction plane has commonly been specified in API machinery standards as:

U = 4W / N

where:

• U = maximum allowable residual unbalance per correction plane (oz·in)
• W = static journal loading attributable to the plane (lb); for a single-plane component, the component weight
• N = maximum continuous speed (RPM)

Allowable unbalance scales directly with rotor weight and inversely with speed.

Applied to a representative case — a 50 lb single-stage impeller for a 3,560 RPM pump — the criterion yields 4 × 50 / 3,560 = 0.056 oz·in (approximately 40 g·mm). At a practical correction radius of 7 inches, this corresponds to a correction mass resolution of 0.008 oz, or approximately 0.23 g.

The ISO grade system

ISO 21940-11 (successor to ISO 1940-1) constructs the tolerance from the kinematics rather than fixing a mass-radius product directly. The balance quality grade is defined as:

G = e · ω

where:

• G = balance quality grade (mm/s)
• e = mass-center eccentricity (mm)
• ω = angular speed (rad/s)

The grade therefore fixes the velocity at which the rotor's mass center orbits the rotation axis. Every rotor balanced to G6.3 exhibits a mass-center velocity of 6.3 mm/s at its rated speed, whatever that speed is, regardless of rotor size, which is why the grade correlates with vibration severity across dissimilar machinery. Improving the balance quality — lowering the G value — at a given speed therefore means bringing the center of mass closer to the axis of rotation.

Permissible residual unbalance for a selected grade is:

U = 9,549 × G × W / N

where:

• U = permissible residual unbalance (g·mm)
• G = balance quality grade (mm/s)
• W = rotor mass (kg)
• N = maximum continuous speed (RPM)

The standard assigns grades by machinery class on the basis of accumulated industry experience: G6.3 for pump impellers, fans, and general machinery components; G2.5 for turbine and compressor rotors; G1 and G0.4 for precision spindle applications. The system's principal advantages are generality — it provides a defensible tolerance for any rotating component, including those outside the scope of any API standard — and physical transparency, since the grade maps directly to mass-center eccentricity at a given speed.

Dividing through by the rotor mass expresses the tolerance as a specific unbalance — unbalance per unit of rotor mass:

U/W = 9,549 × G / N

where:

• U/W = specific unbalance (g·mm/kg)
• G = balance quality grade (mm/s)
• N = maximum continuous speed (RPM)

Specific unbalance is numerically equal to the permissible mass-center eccentricity in micrometers (µm), and it is the quantity plotted against service speed in the ISO grade selection chart (Figure 4).

The relationship between the criteria

Converting 4W/N to ISO terms demonstrates that it is a constant-grade criterion: API 4W/N is equivalent to approximately G0.67 evaluated at maximum continuous speed, independent of rotor size and speed. This places the API criterion roughly a factor of four below (more stringent than) the ISO G2.5 recommendation for the same machinery class.

For the 50 lb impeller above, G2.5 permits 9,549 × 2.5 × 22.7 / 3,560 = 152 g·mm (0.21 oz·in) against the API figure of 40 g·mm — a ratio of 3.8, which is simply 2.5 / 0.665. Neither criterion is more correct; they encode different judgments. The API figure reflects the economics of unscheduled outages in continuous process service and the need for margin against effects a low-speed balance cannot capture. The ISO grades reflect broad field experience of what machinery classes tolerate. A specification writer should be able to state both, convert between them, and justify the selection.

Companion Calculator

The calculator published with this article implements the computations discussed above: balance tolerance determination under API 4W/N or the standard ISO G-grades, in US or SI units, evaluated at your actual rotor weight, speed, and correction radius; the equivalent specific unbalance and mass-center eccentricity; and the rotating force the residual unbalance produces at service speed.

Next in the series: Two-Plane Balancing: The Fundamentals — the unbalance taxonomy of static, couple, and dynamic unbalance; why couple unbalance is invisible to a single plane; and why two planes are sufficient for any rigid rotor.