Rotating Unbalance Force

If a rotor's mass centre sits even slightly off its axis of rotation, the offset mass is being accelerated in a circle, and the force required is $F = m_u e\omega^2$, rotating with the shaft. This is the dominant excitation on almost every rotating machine there is: fans, pumps, motors, grinders, centrifuges. When a vibration spectrum shows a large peak at exactly one times running speed, unbalance is the first suspect and usually the right one.

The $\omega^2$ is the whole character of the thing. Double the speed and the force quadruples. A rotor that feels perfectly civil at half speed can destroy its bearings at full speed, and that single fact is why balancing tolerances tighten as speed rises instead of staying fixed, why a machine trimmed on a slow-speed balancing stand can still fail in service, and why the run-up is where unbalance problems announce themselves.

Only the PRODUCT $m_u e$ can be measured, never either factor by itself. A balancing machine reports an unbalance in gram-millimetres, and 10 g at a 10 mm radius is completely indistinguishable from 1 g at 100 mm. That is also why correction works the way it does: a balancer does not find the offending mass, it adds or removes a compensating one anywhere convenient on the rotor that produces an equal and opposite $m_u e$. Field balancing on a fan wheel is a few trial weights, a phase reading and this arithmetic.

The balance quality grades in ISO 21940-11 — the successor to ISO 1940-1 — are written as $e\omega$ in millimetres per second, which looks odd until you see why: dividing the permissible unbalance by the rotor mass and multiplying by the service speed gives a single figure that captures "how tight does this class of machine need to be", independent of size. G 6.3 covers general machinery, G 2.5 turbines and pumps, G 1 and finer precision spindles. Two practical notes to close on. This force ROTATES: it sweeps every radial direction once per revolution, so a mount that is stiff vertically and soft horizontally responds quite differently to the same force twice per turn, and the resulting orbit is an ellipse rather than a circle. And single-plane balancing only fixes a rotor whose unbalance acts through one plane — a long rotor generally needs two-plane balancing, because two equal offsets at opposite ends and opposite sides give zero net force and a very energetic rocking couple.