Low Speed Balancing Machines

This is the third article in a series on the topic of rotor balancing. The articles on single-plane balancing and two-plane balancing established what unbalance is, how it is quantified, and why two correction planes are sufficient for any rigid rotor. This article describes the machine that performs the measurement — the low speed balancing machine. It covers:

• the two machine architectures, hard-bearing and soft-bearing,
• the hardware that supports and drives the rotor,
• how the machine measures amplitude and phase, and converts them into unbalance vectors, and
• how to assess whether a balancing machine is adequate for a particular rotor case.

A standing assumption comes first. In low speed balancing, the balancing speed is far below the rotor's own resonant speed, so the rotor behaves as a rigid body — this is what the name low speed balancing means. A spinning unbalanced rotor shakes the supports it runs on, and a balancing machine's ability to measure that effect depends on the ratio of the balancing speed to the natural frequency of the support system. Run the rotor well below the supports' natural frequency and the supports barely move; the unbalance is delivered into them as force. Run the rotor well above the supports' natural frequency and the unbalance expresses itself as motion. Every low speed balancing machine is built on one side of this divide or the other, and this is the key to understanding the two architectures (Figure 1).

Hard-Bearing Machines

A hard-bearing balancing machine supports the rotor on stiff pedestals whose natural frequency lies far above the balancing speed. The rotor runs below the pedestal resonance, the pedestals deflect only microscopically, and the rotating unbalance force passes directly into force transducers built into the load path of each pedestal.

This architecture works because below the pedestal resonance, the relationship between the forces at the two pedestals and the unbalance in the two correction planes is fixed by geometry alone. Just as any elemental unbalance on the rotor can be converted into an equivalent set of unbalances in the correction planes (see the article on two-plane balancing), the forces on the pedestals can be translated into a set of unbalances in the correction planes. At the balancing speed, the unbalance in each correction plane produces a rotating force, and each plane's force apportions to the two pedestals according to the relative proximities of the pedestals to the plane:

F1 = ω² · [ UA · (s − zA) + UB · (s − zB) ] / s

F2 = ω² · [ UA · zA + UB · zB ] / s

where:

• F1, F2 = rotating force vectors at pedestals 1 and 2 (N)
• UA, UB = total unbalance vectors in correction planes A and B (kg·m; 1 g·mm = 10⁻⁶ kg·m)
• ω = angular speed (rad/s)
• s = axial span between the pedestals (m)
• zA, zB = axial distances from pedestal 1 to planes A and B (m)

These equations are vector relations, carried at each vector's phase angle. The machine measures F1 and F2 with its pedestal transducers, knows ω from the phase reference sensor, and takes s, zA, and zB from the dimensions the operator enters. What remains is a pair of equations in the two unknowns UA and UB, and the machine's computer solves that pair continuously. A correction plane located outside the pedestal span makes zA or zB negative or greater than s, and the arithmetic holds.

Because geometry alone fixes the relationship, a hard-bearing machine is permanently calibrated. The operator enters the rotor's dimensions — the axial distances from each correction plane to each pedestal and the correction radii — and the machine computes per-plane unbalance directly from the measured forces, along with the recommended correction mass at each plane's correction radius. No trial masses are required to set up a new rotor, which is why hard-bearing machines dominate general shop and production balancing (Figure 2).

Soft-Bearing Machines

A soft-bearing balancing machine takes the opposite side of the divide. The rotor is supported on a compliant suspension — sprung carriages or flexible pedestals — whose natural frequency lies well below the balancing speed (Figure 3). Above the suspension resonance, the suspension deflects meaningfully due to the rotating force, so the rotor-and-carriage system moves, and motion transducers observe the vibration of the suspension.

Well above the suspension resonance, the vibrating system oscillates with a displacement amplitude that approaches the value given by the following equation:

x = U / (WR + WC)

where:

• x = displacement amplitude of the rotor-and-carriage system (mm)
• U = rotor unbalance (kg·mm)
• WR = rotor mass (kg)
• WC = participating mass of the carriage and suspension (kg)

Note that U enters this formula in consistent units; 1 g·mm = 10⁻³ kg·mm.

The oscillating mass and the dynamics of the suspension change with every rotor mounted, so the scale relating motion to unbalance changes with every rotor, and no permanent calibration is possible. The geometry of the rotor-and-pedestal setup is not sufficient to determine the relationship between the measured motion and the unbalance. A soft-bearing machine must therefore be calibrated for each rotor setup, by attaching trial masses of known magnitude at known positions and measuring the response they produce. The calibration is exact for that rotor on that setup. Soft-bearing machines are capable of excellent sensitivity, but the setup cost is real, and it is the reason the architecture has ceded most production work to hard-bearing machines. Soft-bearing machines came first, and the hard-bearing architecture became more prevalent as force transducers and computation matured.

Rotor Support and Drive

Between the rotor and the measurement sits ordinary but consequential hardware. Rotors with their own journals ride directly in roller carriages or vee blocks at the pedestals. Rotors without journals — impellers, discs, couplings, tooling — are mounted on an arbor (also called a mandrel), a precision shaft that provides journals for the balancing run. The arbor becomes part of the measured system, and any eccentricity between the arbor's mounting surface and its journals is indistinguishable from rotor unbalance. Arbor quality therefore sets a floor on the achievable result.

The drive that spins the rotor also participates in the measurement. Three methods are common. A belt drive wraps a flat belt around the rotor body itself, adding no hardware to the rotor ends; it disturbs the measurement least and is preferred for sensitive work, but it requires a smooth cylindrical surface to run on. An end drive connects a driveshaft, usually through a universal joint, to one end of the rotor. It delivers torque positively and suits heavy rotors, but the coupling hardware adds its own unbalance and constraint to the measured system, and its contribution must be accounted for. Air drives, which spin the rotor with compressed air, exist for small high-speed rotors such as turbocharger cores but are rare in plant work.

Measuring Amplitude and Phase

A two-plane machine carries one transducer per pedestal and processes the two channels together. The plane-separation computation — resolving what each pedestal senses into what belongs to each correction plane — runs continuously, so the operator reads per-plane vectors directly (Figure 4).

The transducers report only vibration — a signal oscillating once per revolution whose size tracks the magnitude of the unbalance. A vibration signal can only be converted into an unbalance vector when it is tied to the rotor's angular position, and that is the job of the phase reference. A photocell aimed at a reflective mark on the rotor, or a probe observing a keyway or notch, produces one pulse per revolution at a known rotor angle. The instrumentation measures the timing of the vibration signal against that pulse, and the result is a phase angle.

Transducer Phase Correction

A phase angle is meaningless until one more question is answered: measured on what quantity? The same vibration can be expressed as displacement, velocity, or acceleration, and the three are not in phase with one another. Velocity leads displacement by 90°, and acceleration leads velocity by a further 90° (Figure 5). A machine whose transducer measures velocity will indicate an angle 90° different from one measuring displacement on the identical rotor with the identical unbalance. Neither reading is wrong; they are the same vibration reported in different quantities.

A force-sensing hard-bearing machine avoids this complication entirely: it measures the transmitted unbalance force, which is fixed to the heavy spot, so a transducer-type correction is unnecessary. A motion-sensing soft-bearing machine, on the other hand, measures the vibration as displacement, velocity, or acceleration, so the indicated phase depends on which parameter its transducer measures. The machine's instrumentation applies a fixed correction for its transducer type so that the indicated angle marks the heavy spot's true angular location, and the operator never sees the offset.

The transducer offset described here is a property of the measured parameter (displacement, velocity, or acceleration), and it has a fixed value in degrees. This is distinct from the phase lag between the heavy spot and the high spot — the rotor's peak displacement, as defined in the article on single-plane balancing. The heavy-spot/high-spot phase lag is a property of the rotor-and-support system's response, and it varies from near 0° well below the system's resonance toward 180° well above it. A balancing machine runs deliberately far from its support resonance, on one side or the other, precisely so that this response lag sits at a known, stable value.

Machine Capability

Two standardized metrics establish a balancing machine's precision. The first is the minimum achievable residual unbalance, written Umar — the smallest residual unbalance the machine can reliably achieve and indicate. It is the machine's noise floor. Below it, further correction is indistinguishable from measurement scatter. Because the floor scales with the size of the rotor, it is specified as a specific unbalance, denoted emar, and the floor for a particular rotor follows from the rotor's mass:

Umar = emar · WR

where:

• Umar = minimum achievable residual unbalance per plane (g·mm or oz·in)
• emar = minimum achievable residual specific unbalance, a machine characteristic (g·mm/kg or oz·in/lb)
• WR = rotor mass (kg or lb)

The second metric is the unbalance reduction ratio, or URR — the fraction of a known initial unbalance that one measure-and-correct cycle removes. It expresses, in a single number, the combined accuracy of the machine's magnitude indication, angle indication, and plane separation.

Both metrics are established by test on a standardized proving rotor, at machine acceptance and periodically thereafter. The practical use of the metrics is a simple comparison: the machine's Umar value must sit comfortably below the rotor's residual unbalance tolerance.

Companion Calculator

The calculator published with this article performs the hard-bearing machine's central computation, in either direction. Enter the balancing speed and the rotor geometry, then either enter (a) the two pedestal force readings, to calculate the unbalance vectors in the two correction planes, or (b) the per-plane unbalance vectors, to calculate the pedestal forces they produce. The calculator is a teaching-grade model of what the machine's computer does every revolution; it models the machine's calibration and does not replace it, since a real machine's calibration also accounts for effects beyond rigid-rotor geometry.

Next in the series: Low Speed Balancing Procedure — the shop balancing procedure from setup through correction, including correction splitting and the residual unbalance verification test that proves a balance without taking the machine's word for it.