How a Small Imbalance Becomes a Large Force

Why this matters

The same residual unbalance is a non-issue on one machine and the reason the bearings fail every eighteen months on another, and the only thing that changed is shaft speed. Techs get this backwards constantly, because a slow machine that shakes visibly feels worse than a fast one that hums. What your hand reads is displacement. What kills the bearing is force, and force does not scale with speed, it scales with the square of speed. This article gives you one rule and runs it against two machines carrying identical unbalance, where it produces opposite calls.

Before you take a reading on a running machine

Readings on a running rotor get taken from outside the guard, full stop. Belt, sheave, coupling and shaft guards stay on while the machine turns; 29 CFR 1910.219 covers mechanical power-transmission apparatus specifically and 29 CFR 1910.212 carries the general machine-guarding duty, and neither has an exception for "just taking a quick reading". Use a magnetic-base sensor or a probe placed on a stationary bearing housing that you can reach without opening anything, with no gloves and no loose sleeves near the shaft line. If the reading requires the guard off, the machine gets shut down, the disconnect locked and tagged, and the rotor confirmed stopped rather than coasting, under 29 CFR 1910.147 and its verification-of-isolation step. A coasting wheel that still has minutes of spin in it is stored energy, not a stopped machine.

The force is the constant, the shaking is the response

Unbalance produces a force that rotates with the shaft. That force exists and has one value at a given speed regardless of what you bolt the machine to. How much the machine visibly moves is the structure's response to that force, and it depends on stiffness, mass and how close a natural frequency sits to running speed.

That split explains the two field observations that seem to contradict each other. A rigid, heavy machine can carry a large unbalance force with barely perceptible movement, and the bearings still eat it. A light machine on soft mounts near a resonance can shake alarmingly on a trivial force. If you judge unbalance by how much the housing moves, you will chase the second machine and ignore the first. The article on why a machine is worse at one speed than at a higher one covers the response side; this one is about the force.

The law, and the geometry it was derived under

For a mass sitting off-center on a rotor, the rotating force is mass times radius times angular velocity squared. Angular velocity is proportional to rpm, so the force goes with the square of shaft speed. In shop units, for a rigid rotor with its unbalance treated as a single point mass in one plane, spinning at a steady speed:

force in pounds = 1.78 x unbalance in ounce-inches x (rpm / 1000) squared

Three conditions live inside that expression and travel with it. It assumes a rigid rotor, meaning one running well below its own first bending critical speed, which covers nearly every fan, pump, blower and motor rotor in field service. It assumes the unbalance is static, a single heavy spot in one plane; a couple unbalance produces a rocking moment rather than a net force and this formula does not describe it. And it assumes steady speed, so it says nothing about what happens during a start or a coast-down.

Work one value to fix the scale. One ounce of extra mass sitting 6 inches out from the centerline is 6 ounce-inches. At 1,750 rpm: 1.78 x 6 x 1.75 x 1.75 = 32.6 lb of rotating force. One ounce. About the weight of four quarters, generating over thirty pounds of pull that reverses direction thirty times a second.

Halve the speed to 875 rpm and it becomes 1.78 x 6 x 0.875 x 0.875 = 8.2 lb. Double it to 3,500 rpm and it becomes 1.78 x 6 x 3.5 x 3.5 = 130.5 lb. Same ounce, same radius, a sixteen-fold spread across a four-to-one speed range.

The gate

Here is the rule the two cases below get run against, and it is deliberately a comparison rather than an absolute number, because a pound of unbalance force means nothing until you know what else that bearing is already carrying.

Unit of analysis: per bearing. Take the rotating unbalance force at running speed, divide it between the bearings, and compare each share against the steady radial load that bearing already carries (its share of rotor weight, plus belt pull or process side load).

• Under about 10 percent of the steady load: the effect on bearing life is inside the noise of everything else, and unbalance is not your fault. Look elsewhere.
• Over about 50 percent of the steady load: unbalance dominates the bearing's duty and correction pays for itself.
• Between the two: the duty cycle decides. A machine that runs continuously earns the correction; one that runs a few hours a week rarely does.

The reason the gate is expressed as a ratio rather than a force is that bearing rating life falls roughly with the cube of equivalent dynamic load for ball bearings and a 10/3 power for roller bearings, per the ISO 281 rating-life method. That exponent applies to the total equivalent load on the bearing, not to the unbalance component alone, which is exactly why the comparison against the existing load is the load-bearing part of the rule.

Case A: the 875 rpm belt-driven exhaust fan

A shop measures a wheel with about 6 ounce-inches of residual unbalance, belt-driven at 875 rpm on two pillow-block bearings.

Force at speed: 8.2 lb, split roughly between two bearings, so 4.1 lb each.

What each bearing already carries: the fan's own rotor share, say 22 lb per bearing, plus static shaft load from the belt drive, which the drive tension chart puts at about 33 lb per bearing. Steady load per bearing: 55 lb.

4.1 against 55 is 7.5 percent. That is under the 10 percent line, so the call is: leave it. Correcting this rotor would change the bearing's calculated life by a margin smaller than the error in the belt tension itself. If this machine has a bearing problem, the cause is in the belt tension, the alignment, the lubrication or the environment, not in the wheel's balance. Sending the wheel out for balancing here buys the customer a clean report and no change in outcome, and it burns the trust you need for the recommendation that actually matters.

Case B: the 3,500 rpm direct-drive blower

Same 6 ounce-inches of residual, on a two-pole direct-drive machine at 3,500 rpm.

Force at speed: 130.5 lb, split between two bearings, so 65 lb each.

What each bearing already carries: no belt pull on a direct drive, and a lighter rotor, so about 30 lb per bearing of steady load.

65 against 30 is 217 percent, more than four times the 50 percent line. Unbalance is not a contributor here, it is the dominant load on the bearing. Treating the equivalent dynamic load as rising by roughly a factor of 2.5 (an approximation, because a rotating load does not simply add to a steady one and the two sum vectorially over each revolution), the cube relationship puts calculated rating life at somewhere near a fifteenth of what the selection assumed. That is the difference between a bearing that outlives the machine and one that becomes a scheduled replacement.

Same rotor condition. Same number of ounce-inches. Opposite calls, and the only variable that moved was speed.

What this changes about speed adjustments

Once you internalize the square, drive changes stop looking neutral. Turning a fan down 10 percent on a variable-frequency drive, from 1,750 to 1,575 rpm, cuts the unbalance force by about 19 percent, because 1.575 squared over 1.75 squared is 0.81. Speeding a fan up 15 percent by changing a sheave raises it about 32 percent. Neither of those shows up on anyone's paperwork as a mechanical change, and both are real changes to bearing duty.

The trap is assuming the relationship runs smoothly both ways. It does not, because response and force are different curves. Slowing a machine down always reduces the force, but if the lower speed lands on a structural natural frequency, the machine can shake noticeably more while the bearings see less. That is not a contradiction and it is not a fault appearing; it is amplification, covered in the companion articles on resonance and on working out whether you are near a critical speed. Check for it before you promise a customer that turning the fan down will quiet the roof.

How to verify you got this right

Re-derive the force for the machine in front of you before you quote a balance job, and then answer one question out loud: what fraction of this bearing's existing load is that? If you cannot name the existing load even roughly, you are not ready to make the call, and the fastest way to get it is the rotor weight from the nameplate or the parts data plus the drive's own tension figures.

Two tells that you have applied the rule outside its conditions. First, if correcting a genuine unbalance barely changes the vibration reading, the response side is dominating and you were never looking at a force problem. Second, if the shake at one bearing drops and the other rises after a single-plane correction, you had couple unbalance, which the force formula above does not describe at all.

References

• ISO 281 rating-life method for the load-to-life exponents used in the bearing-duty comparison
• 29 CFR 1910.219 and 29 CFR 1910.212 for guarding of power-transmission apparatus while readings are taken on a running machine
• 29 CFR 1910.147 for isolation and verification before a guard is removed or a rotor is handled
• See related: What Balance Actually Means on a Rotating Part; Why a Machine Is Worse at One Speed Than at a Higher One; How to Work Out Whether You Are Near a Critical Speed