Mathematics of tubular wind chimes[edit]
A wind chime constructed of a circular tube may be modelled as a freely vibrating Euler–Bernoulli beam[10] and the dominant frequency in cycles per second is given by:
where L is the length of the tube, E is the Young's modulus for the tube material, I is the second moment of area of the tube, and μ is the mass per unit length of the tube. Young's modulus E is a constant for a given material. If the inner radius of the tube is ri and the outer radius is ro, then the second moment of area for an axis perpendicular to the axis of the tube is:
The mass per unit length is:
where ρ is the density of the tube material. The frequency is then
where W=ro-ri is the wall thickness and D is the average diameter D=ro+ri. For sufficiently thin-walled tubes the W2 term may be neglected, and for a given material, the main frequency is inversely proportional to L2 and proportional to the diameter D.
For the main mode of vibration, there will be two nodes on the tube, where the tube is motionless during the vibration. These nodes will be located at a distance of 22.416% of the length of the tube from each end of the tube. If the tube is simply supported (not clamped) at one or both of these nodes, the tube will vibrate as if these supports did not exist. A wind chime will give the clearest and loudest tone when it is hung using one of these node points as the attachment point. These attachment points are also the same as used by other similar instrument such as the xylophone and glockenspiel.
*Enola Gay
- Manufacturer: Glenn L. Martin Company, Omaha, Nebraska
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Bong
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