Acoustics and Vibration Animations
Daniel A. Russell
Graduate Program in Acoustics, The Pennsylvania State University

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Mode Shapes for a Hanging Chain

Take about 30 paper clips, and connect them end-to-end in a long chain. Hold one end of the chain in your fingers and let the other end dangle. Gently swing (or twirl) the chain and you should find that the chain will "lock" in on a very specific mode shape which occurs at a particular natural (resonance) frequency.


The shapes of vibration which the chain will "lock" onto are defined by Bessel Functions. The figures below show the first three mode shapes for a hanging chain.

Mode 1
Mode 2
Mode 3
animation of the first bending mode of a hanging chain.
animation of the first bending mode of a hanging chain.
animation of the first bending mode of a hanging chain.