Vibration and Whirling

ω_n = √(k/m), and a disturbed system oscillates at that frequency while returning to rest. It is the same second-order behaviour as an RLC circuit, with mass for inductance and compliance for capacitance.

Damping sets the decay envelope while barely affecting the frequency. It is measured from the waveform as the logarithmic decrement, and most engineering structures have ζ below 0.05 — which is why they ring for so long.

Forcing at the natural frequency drives the response to a peak whose height is set entirely by damping — at ζ = 0.02 the amplitude is 25 times the static deflection from the same force.

Above √2 × ω_n the transmitted force is less than the applied force, so isolation means making mounts soft enough to put the natural frequency well below the forcing frequency. Below that ratio a mount amplifies, which is why a badly chosen one makes vibration worse.

Any eccentricity produces a centrifugal force that deflects the shaft, which increases the eccentricity, which increases the force. That positive feedback becomes unbounded at exactly the shaft's lateral natural frequency — the critical speed.

Above it the shaft self-centres, rotating about its centre of mass rather than its geometric centre, and the deflection falls. That is why large turbines are accelerated quickly through their criticals and operated above them.

Two inertias oscillating against each other through a shaft's torsional stiffness, with a node between them. Nothing moves laterally and nothing is audible, yet the shaft accumulates fatigue at the node.

That is why crankshafts carry torsional dampers and why ships have engine speed ranges they are forbidden to run in continuously.

Rotating machinery excites at multiples of running speed, and a Campbell diagram plots those orders against the machine's natural frequencies. The crossings are where trouble lives.

There are only three remedies — move the natural frequency, remove the excitation, or add damping — and stiffening blindly can move a frequency straight onto a different order.