How do bar-end weights change the resonant frequency and reduce vibration?

Resonant frequency is the frequency at which an object "rings" if it is moved. When you hit a tube with a hammer, and listen at the end of the tube, you are hearing the audible portion of its resonant frequency spectrum. The "fundamental" is the lowest note. In handlebars, this is the frequency made by the bars "flapping." By flapping I mean that the bar clamps stay put, while the tips move most. This fundamental frequency is typically the strongest frequency, thus the first one to attack. Weights on the tips of the bars reduce vibration by moving the resonant frequency lower, away from the frequencies generated by the engine. This happens for the same reason that a heavier guitar string sounds lower, under the same tension, than a lighter one. Because the effectiveness of a weight in reducing vibration this way reduces to zero as the weight approaches the bar clamp, I was careful to keep as much weight outboard of the bar ends as possible. All the weight is concentrated where it is most effective- right near the bar-ends. All the weight is within 2.5" of the bar ends. There's no dead weight as with solutions that go all the way through the bars. By moving the resonance lower, it typically moves further from the frequency of the engine, which reduces the build-up of vibration in the bars dramatically. The other way to look at it is in terms of inertia. By solidly mounting the weights to the ends of the bars, the bar-ends gain a great deal of inertia, which means that they are much more reluctant to change speed. Since the flapping motion involves the bar ends accelerating this way and that in rapid succession, any weight on the ends reduces this motion. When you are trying to contain vibration by changing the resonant frequency, solid-mounting is the way to go. Any flexibility to the mounting muddies the effect of the extra weight, hurting the cause.

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