Resonance — stiffening moves the peak, it does not lower it

The structure is being shaken at its own natural frequency. Drag upward to add damping; drag down to take it away. The arrow keys do the same. End is as damped as this goes; Home is almost none. The pale curve is the same structure made stiffer — note where it moves to, and note how tall it still is. Sound is off until you turn it on.

What you are looking at

Shake a structure slowly and it simply follows the shaking. Shake it fast and it barely moves at all. Shake it at its own natural frequency and it magnifies the motion — each push arrives exactly when it does the most good, and the amplitude builds until something takes energy out as fast as the shaking puts it in. That something is damping, and it is the only thing in the problem that does it.

So the height of the peak is set by the damping and by nothing else. Stiffness is not in that expression. Make the structure stiffer and the whole curve slides to the right: the resonance now happens at a higher frequency, and the peak there is exactly as tall as it was.

And away from the peak, damping does almost nothing. At three times the natural frequency the response is the same whether the structure is barely damped or heavily damped. Damping is not a general reduction in vibration; it is a local one, at one frequency.

The two things worth taking away

Stiffening relocates the problem. On a fixed-speed machine, moving the resonance out of the way is a genuine fix. On anything with variable speed, or with harmonics of the running speed to land on, it is a rearrangement — and the peak that is found later is just as tall as the one that was moved.

A very small amount of damping is worth a great deal. Because the peak goes as one over the damping, the first fraction of a per cent is worth more than everything after it. Going from almost nothing to a little is the difference between magnifying the shaking a hundredfold and magnifying it tenfold; going from a lot to more barely registers.

What is not verified

Stated plainly, because a piece whose value is trustworthiness should not overclaim. The first list is the part that would change how you read the result. The second is scope — true, worth saying, and not load-bearing.

• No engineer has reviewed this. If you are one and would check it, REVIEW.md says what would actually help and is meant to cost about fifteen minutes. The relation is the standard single-degree-of-freedom response and the built-in checks confirm it exactly — but that check is the author's own, not a vibration engineer's.
• Viscous damping, and a constant amount of it. Real damping comes mostly from joints, friction and material behaviour, is not proportional to velocity, and changes with amplitude. The viscous model is a convenience that fits measurements near resonance better than it deserves to.
• The excitation is held constant with frequency. Out-of-balance forces grow with the square of speed, which tilts the whole picture and is the usual case in rotating machinery.

Scope, for completeness. True of the model, and none of it changes the answer above.

• One mode, one degree of freedom. Real structures have many modes, and the one you stiffened out of the way is often replaced by the next one down rather than by nothing.
• Steady state only. Nothing here shows the build-up, and a machine that passes quickly through a resonance on run-up may never reach the amplitude drawn. That is a real and commonly used mitigation this piece does not represent.

Commissions

The subject changes; the method does not. If you have a failure mode, a mechanism, or a process that people keep getting wrong because the diagram cannot move, that is the shape of problem this suits. You get one file, the checks that hold it honest, and the notes explaining every decision including the ones that were reversed. Commissions start at £4,000 and take three to four weeks from an agreed brief. yinxueshi1017@gmail.com

Not commissioning anything? Still worth an email. If your version of this failed differently — other geometry, other numbers, a mechanism this piece leaves out — tell me in one line. Corrections are the main thing that decides what gets built next.