Why Spinning Things Shake — Rotating Unbalance
No spinning part is ever perfectly balanced — a mass m sitting slightly off-center at radius e and turning at ω creates an outward rotating force F₀ = m·e·ω² that shakes the whole machine. The frightening part is the ω-squared: even a tiny imbalance becomes a huge force at high speed — why a washing machine walks during the spin and an unbalanced wheel shimmies.
Look at the slightly off-center mass inside a rotor. Raise the speed ω and the outward rotating force F₀ = m·e·ω² grows — with the square of speed, so twice as fast is four times the force. That is why a fast rotor cannot tolerate even a tiny imbalance.
That rotating force shakes the whole machine (mass M). Raise the speed ratio r = ωωₙ: it barely moves when slow (the force is small), swings hard at the r=1 resonance, and at higher speed the shaking does not vanish but settles to a constant size.
The response curve differs decisively from other vibrations. Drag r: it swells as r² when low, peaks at the r=1 resonance, and at high speed does not fall to zero but flattens at M·X(m·e) → 1. The amplitude settles to a constant X → m·eM (at high speed the machine effectively whirls about its center of mass).
The fix is to remove the source of the force. Add a counterweight on the opposite side to cancel m·e. The larger the counterweight, the smaller both the rotating force and the machine's shaking. The little lead weights clipped to a wheel do exactly this — balancing m·e toward zero.
The same m·e·ω² causes trouble everywhere. Flip through the examples — a washing-machine spin (bunched laundry raises m·e, so it walks while passing resonance), a car wheel (shimmy at speed, fixed by wheel balancing), a turbine or rotor (precision balancing is essential). The remedy is one of two — reduce m·e (balance it), or get past resonance and run fast.