THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

6.8 Vibrating Thin Plates 125Figure 6.4. Plot showing the average (normalized) displacement response as a functionof frequency. The effect of damping is also shown.phase with the driving force, while that of the membrane’s outer portion remains inphase. As the driving frequency increases and the nodal circle shrinks, the averagedisplacements of the two zones tend to cancel each other out. The cancellationbecomes complete at ka = 5.136, and no displacement occurs across the entiresurface of the membrane.6.8 Vibrating Thin PlatesThe principal difference between the vibration of a membrane and a thin plateis the restoring force in the former is due entirely to the tension acting on themembrane and in the latter there is no tension applied, and the restoring force isattributed entirely to the inherent stiffness of the plate.To keep matters simple, we consider only symmetrical vibrations of a uniformcircular diaphragm. The appropriate equation, essentially equivalent to Equation(6.36) but modified to include the effect of stiffness, is∂ 2 z∂t = κ 2 E2 ρ(1 − μ 2 ) ∇2 (∇ 2 z) (6.38)where ρ is the density of the material, μ is the Poisson’s ratio, E is Young’smodulus, and κ is the surface radius of gyration. For a circular plate of uniformthickness b, the radius of gyration is given byκ =b √12