mechanical vibrations explained

rotating unbalance vibration

An eccentric mass makes F₀=m·e·ω² (grows with speed²); response M·X/(m·e)=r²/√((1−r²)²+(2ζr)²), at high speed X→m·e/M, fixed by balancing

No rotor is perfectly balanced. A mass sitting a little off center, turning at a radius, throws an outward force F₀ = m·e·ω² that shakes the machine. What is frightening is that the force goes as the square of speed, so a small imbalance becomes a huge force at high speed. That is why a washer walks during the spin and an unbalanced wheel trembles. The square of speed is the whole danger.

Look at the mass sitting a little off center inside the rotor. Raise the speed ω and the outward force F₀ = m·e·ω² grows, and it grows as the square of speed. Go 2 times as fast and the force is 4 times as large. A high-speed rotor cannot leave even a tiny imbalance alone. The same offset grows fiercer as the speed rises.

That turning force shakes the whole machine (mass M). Raise the speed ratio r = ωωₙ. When it is slow the force is small and the machine hardly shakes. At r=1, resonance, it rocks hard. Go faster still and the shaking does not vanish. It settles to a steady size. Faster does not mean quiet.

The response curve differs from other vibration in a decisive way. Move r and at low speed it swells as r². At r=1 a resonance peak stands up. At high speed it does not fall to 0. It stays flat at M·X(m·e) → 1. The amplitude becomes steady, X → m·eM. At high speed the machine is turning about its center of mass.

The cure is to remove the source of the force. Add a balance weight on the opposite side and cancel m·e. As the balance weight grows, the turning force and the machine's tremble shrink together. The small lump of lead on a wheel is exactly this. It balances the wheel and brings m·e to 0. Erase the source and the force goes with it.

The same turning force makes trouble everywhere. Its shape is m·e·ω². Step through the examples. A washer spin, when the clothes clump, m·e↑ → walking as it passes resonance. A car wheel, tremble at high speed → wheel balancing. A turbine or a rotor, where precise balancing is required. The prescription is one of two. Reduce m·e, or pass resonance quickly and run at high speed.

In PracticeRotating unbalance is a kind of forced vibration (A4), but the force is not applied from outside. It makes itself. A mass m offset by e from the center, turning at ω, gives F₀ = m·e·ω², a rotating force that grows as the square of speed. The dimensionless response of the machine (mass M) is M·X(m·e) = r²√((1−r²)²+(2ζr)²), with r = ωωₙ. Three regions matter. Low speed (small, ∝r²) → resonance (r=1 peak) → high speed (→1, that is X→m·eM, steady). The last one is the difference from other vibration. The amplitude does not die to 0. It stays at a steady value, because at high speed the machine turns about its center of mass, a self-centering. Two cures. Reduce m·e, with a balance weight, balancing, the lead on a wheel, or pass resonance quickly and run at high speed. Startup and shutdown pass through resonance once. Washers, car wheels, turbines, and rotors all sit on this one formula.

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