Rotor Balancing

The unbalance force

A perfectly balanced rotor has its center of mass exactly on the spin axis, and its principal axis of inertia coincident with that spin axis. Real rotors never do: machining tolerances, casting porosity, keyways, uneven coatings and accumulated dirt all leave a small mass off-center. When that mass spins, it demands a centripetal force to hold it on its circular path, and by Newton's third law it pulls the bearings outward with an equal rotating force:

F = m · r · ω²

where:
  m = unbalanced (heavy-spot) mass      (kg)
  r = radius of that mass from axis     (m)
  ω = angular speed = 2π·N/60           (rad/s, N in rpm)
  F = rotating centrifugal force        (N)

The product U = m·r is the "unbalance",
quoted in gram-millimeters (g·mm) or ounce-inches.

The force vector rotates with the shaft, so a fixed bearing feels a sinusoidal force once per revolution. This is why mass unbalance always shows up in a vibration spectrum as a peak at exactly 1× running speed — the single most useful diagnostic fact in rotating machinery. The crucial term is ω²: balancing is forgiving at low speed and unforgiving at high speed.

Why speed is everything

Because force scales as ω², the same physical imbalance behaves completely differently across the speed range. Consider a 1-gram heavy spot 50 mm out from the axis:

m = 0.001 kg,  r = 0.05 m,  U = m·r = 50 g·mm

At 3,000 rpm:  ω = 2π·3000/60 = 314 rad/s
  F = 0.001 × 0.05 × 314²  = 4.9 N    (about half a kilogram-force)

At 30,000 rpm: ω = 3,142 rad/s
  F = 0.001 × 0.05 × 3142² = 494 N    (about 50 kgf)

A tenfold speed increase makes the shaking force a hundred times larger from an identical 50 g·mm unbalance. This is exactly why a turbocharger core (up to ~250,000 rpm), a dental drill (~400,000 rpm) or a centrifuge demands balancing to fractions of a gram-millimeter, while a slow ceiling fan tolerates clip-on weights you eyeball into place.

Three kinds of imbalance

Not all imbalance is the same. The mass distribution determines whether you can fix it in one plane or must work in two.

	Static (force) unbalance	Couple unbalance	Dynamic unbalance
Center of mass	Off the axis	On the axis	Off the axis
Principal inertia axis	Parallel to spin axis, offset	Tilted, crosses spin axis at center	Tilted and offset (skewed)
Visible at rest?	Yes — rolls to heavy side on knife edges	No	Partly
Correction planes	1	2 (equal, opposite)	2 (independent)
Effect at bearings	Both bearings shake in phase	Bearings shake 180° out of phase (rocking)	Both, mixed phase
Typical on	Thin discs, grinding wheels, pulleys	Long rotors with end-to-end mass error	Almost all real long rotors
Measurement	Static (knife edges / single transducer)	Spinning machine, two planes	Spinning machine, two planes

A useful rule of thumb: a disc-shaped rotor whose axial length is less than about one-fifth of its diameter can usually be single-plane (static) balanced, because there is no meaningful lever arm for a couple. Anything longer — a motor armature, a crankshaft, a multistage compressor — almost always needs two-plane dynamic balancing.

Worked example: a single correction weight

A grinding wheel runs at 1,800 rpm. A balancing run finds an unbalance of U = 1,200 g·mm. The only place to add correction mass is a balancing ring at radius r_c = 120 mm. What correction mass, and where?

Place the correction weight 180° opposite the heavy spot
so its centrifugal force directly opposes the unbalance force.

For cancellation:   m_c · r_c = U
  m_c = U / r_c
      = 1,200 g·mm / 120 mm
      = 10 g

So: a 10-gram weight at 120 mm radius, exactly opposite the
heavy spot, cancels a 1,200 g·mm unbalance — at ANY speed,
because both forces scale identically with ω².

The elegance is that the angle and the speed both drop out. Because the unbalance force and the correction force share the same ω², you balance once and it holds across the whole speed range — provided the rotor stays rigid (see below). The remaining art is measuring U accurately, especially its angle.

The influence-coefficient method

On a real machine you cannot see the heavy spot — you can only measure vibration at the bearings. The standard solution treats the rotor as a linear system and uses a trial weight:

• Run 1 — original: measure the vibration vector V₀ (amplitude and phase, e.g. 6.0 mm/s at 40°).
• Add a trial weight W_t of known mass at a known angle.
• Run 2 — with trial: measure the new vibration vector V₁.
• Influence coefficient: α = (V₁ − V₀) / W_t. This complex number captures both how much and at what angle a unit weight in that plane moves the vibration.
• Correction weight: W_c = −V₀ / α. The negative sign means "place it to drive the vibration to zero." The result gives both the mass and the angle.

For two-plane dynamic balancing the same idea becomes a 2×2 matrix of complex influence coefficients relating two correction planes to two measurement bearings, solved simultaneously. Modern balancing instruments do all of this automatically; the operator only attaches weights and reads off the answer. The method's one assumption — linearity — is excellent below the first critical speed and breaks down as resonance approaches.

Balance quality: how good is good enough?

You never balance to zero — it is neither possible nor worth the cost. ISO 1940-1 sets allowable residual unbalance through a balance quality grade G, defined as the permissible center-of-mass eccentricity (in micrometers) times the angular speed, giving a velocity in mm/s. Lower G means tighter balance.

Grade	e·ω (mm/s)	Typical application	Why this level
G40	40	Car wheels, crankshaft drives	Cheap, tolerant mounts, low speed
G16	16	Drive shafts, agricultural machinery	Moderate speed, robust bearings
G6.3	6.3	Pump impellers, electric motor armatures, fans	Workhorse industrial default
G2.5	2.5	Gas/steam turbines, machine-tool drives, compressors	High speed, long bearing life
G1	1.0	Tape/CD drives, small high-speed armatures	Precision, low audible noise
G0.4	0.4	Gyroscopes, grinding-machine spindles	Sub-micron-class precision

The permissible residual unbalance is U_per = (G × M × 1000) / ω, where M is the rotor mass in kg and ω is in rad/s. The 1/ω term means a high-speed rotor of the same grade must be balanced far more precisely — another consequence of the ω² force law. For a 50 kg rotor at G2.5 running 3,000 rpm (ω = 314 rad/s), U_per ≈ 2.5 × 50 × 1000 / 314 ≈ 398 g·mm, split between the two correction planes.

Rigid vs flexible rotors

Everything above assumes a rigid rotor — one that does not bend appreciably at its operating speed because it runs well below its first bending critical speed. For such rotors, two correction planes are mathematically sufficient, and a balance done at low speed holds at high speed.

A flexible rotor — a long turbine shaft, a large generator, a centrifuge bowl that runs above its first critical speed — actually bows into different mode shapes at different speeds. The mass distribution that balances the first bending mode does not balance the second. These rotors require modal balancing or the N+2 plane method, balancing each mode in turn at speeds near each critical, often on a high-speed balancing bunker. This is one of the most demanding tasks in rotating machinery and is why a large turbine rotor can take days to balance.

Failure modes and trade-offs

• Balancing a non-unbalance fault. The classic mistake: a rotor vibrates, so it gets balanced, but the real cause was misalignment (2× signature), looseness or a bad bearing. The weights chase a phantom and can mask the true fault until it fails catastrophically. Always confirm the vibration is dominantly 1× first.
• Running near a critical speed. Near resonance the amplitude and phase swing wildly with tiny speed changes, the linearity assumption fails, and influence coefficients become unstable. A rotor must be balanced away from its criticals; if it must pass through one on the way up, it is balanced for that transient separately.
• Thermal and assembly unbalance. A rotor balanced cold can go out of balance hot as it bows under uneven temperature (common in steam turbines), or shift when re-assembled because keys, couplings and rotor stacks repeat differently. Field balancing in the as-run condition often beats shop balancing.
• Lost or migrating mass. Fan blades shed dirt or ice, pump impellers erode, compressor blades foul unevenly, and welded balance weights can detach. Any of these re-introduces unbalance in service — balancing is a maintenance item, not a one-time fix.
• Over-tight grade. Specifying G0.4 where G6.3 would do wastes balancing-machine time and money, and a tighter shop balance can be wiped out the instant the rotor is coupled to its own bearings. Match the grade to the duty.
• Correction-plane limits. If the only accessible balance planes are far from where the unbalance lives, large correction weights are needed and cross-effect between planes grows — sometimes the rotor cannot be adequately balanced without adding a third plane.

Where balancing shows up

• Turbomachinery: jet-engine spools, gas and steam turbines, turbochargers — high speed makes the ω² penalty brutal, so these get the tightest grades and high-speed balancing.
• Electric machines: motor and generator armatures are two-plane dynamically balanced after the windings and laminations are stacked.
• Automotive: wheels (clip-on weights opposite the heavy spot), crankshafts, driveshafts, clutch assemblies and brake discs.
• Consumer and medical: washing-machine drums (active counterweight balancing during spin), hard-disk and optical drives, dental and surgical drills, lab centrifuges.
• Process industry: centrifugal pumps and fans, blowers, separators and decanter centrifuges — usually G6.3, field-balanced in place when accessible.