physics//mechanics//vibration//resonance

Resonance is the large amplification of a vibration that occurs when a system is driven at a frequency close to one of its natural frequencies, and it is behind a good share of the mechanical failures of rotating machinery: a balanced motor that still shakes its frame, a pipe that cracks at one pump speed and not another, a drone whose attitude estimate goes noisy at one throttle setting. Each cycle of the excitation arrives in step with the motion and adds to it, and only damping limits how far the amplitude grows.

Resonance is the large amplification of a vibration that occurs when a system is driven at a frequency close to one of its natural frequencies, and it is behind a good share of the mechanical failures of rotating machinery: a balanced motor that still shakes its frame, a pipe that cracks at one pump speed and not another, a drone whose attitude estimate goes noisy at one throttle setting. Each cycle of the excitation arrives in step with the motion and adds to it, and only damping limits how far the amplitude grows.

A two-pole induction motor on 50 Hz turns at about 2,950 rpm, roughly 49 Hz (a little below 50 because of slip), and any residual imbalance pushes the frame once per turn. If the pump bracket has a mode at 47 Hz, that push lands near it, the vibration is amplified many times over and the bearings pay for it. For light damping the amplification at the peak is

Q≈12ζ,Q\approx\frac{1}{2\zeta},Q≈2ζ1​,

the quality factor: with the 2 % damping typical of a welded steel frame, a push at the natural frequency moves the structure about 25 times more than the same force applied slowly. The pump sits 2 Hz off the peak and its push is still amplified about ten times; only a margin of 20 % or so between excitation and mode brings the factor down near two, which is why the remedies are about distance and damping.

Resonance is cured by moving the frequency, damping the mode or keeping the excitation away.

Stiffen or lighten the structure to push its natural frequency out of the operating range, add damping or isolation mounts to lower the peak, or never run there: variable-speed drives can be set to skip the critical speeds of the machine they drive.

On a drone the excitation sweeps. Propeller and motor vibration move with throttle, so a frame mode that is harmless in hover can be hit during a climb; flight software removes it with notch filters that follow each motor's measured speed, which take out a narrow band with much less delay than a heavy low-pass filter (notch filter).

Resonance is a forced, bounded response, proportional to the excitation; instability grows by itself with no excitation at all. A resonant machine calms down when the drive is removed, an unstable loop does not (equilibrium and stability).

Finding the frequencies before building is modal analysis; spotting a resonance in a running machine, as a peak that follows speed in a spectrum, is part of vibration analysis. Sweeping the speed with a drive tells it from plain imbalance: the vibration of an imbalance grows smoothly with speed, while a resonance peaks around one speed and falls off on both sides. The same peak appears in electrical circuits and in control loops with little damping, where it amplifies disturbances near the loop's crossover.