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Showing results for tags 'Coupling of various stresses'.
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A flat plate has no bumps including archings. It is interesting to see what happens in a curved plate, bumps and all. Try to flatten a violin plate and you will see stretching of some wood and compression in other places. If there is excitation at M5, the outer wood is being stretched back and forth and the central region is compressed. You can see that the flatter the arching, the less this happens. The wave equation solutions on a bounded surface are what give the normal modes. There is obviously a "coupling" of in-plane stretching/compressing to the perpendicular vibrating of the surface bumps. The curved shell equation is of higher order than that for the flat plate, and this introduces extra terms. These terms provide for coupling in "cross terms" which involve the mixing of in-plane stretching and the flattening of bumps. (and vice versa for comompression/ raising.) In other words, the in-plane stresses give restoring forces to bumps. It is not surprising that string tension and post tension will affect the sound of a violin. This point has not been mentioned. It is sound math, and you can visualize it easily. The guitar is very much different, by the way. PS: Vibration depends on two things.. a mass is moving, with inertia. Countering that is a "restoring force" which opposes the motion and brings the situation back to zero distortion. This happens when the plate passes through sine(frequency) goes through 0, 180, 360 degrees etc. This is for a single excitation of a normal mode. Vibrations of other modes can be superimposed, but they do so in an additive way. This last point is VERY important. It is called the linear superposition of waves. Radio waves act the same way, otherwise your radio could not separate different signals.