With the aid of Eq. (b) of Problem 6.14 formulate the Rayleigh quotient for axisymmetric circular plates. Now obtain an approximation of the lowest frequency of vibration of a clamped circular plate. The exact result is given as
Problem 6.14:
Derive an exact power series solution to a simply-supported rectangular plate (see Fig. 6.24) loaded uniformly with loading q0. Show that the solution can be given as follows:
Show that for a plate where b/a → 0, the above solution degenerates to that of a beam uniformly loaded by loading p0 and given as:
where I is the moment of inertia per unit width. Also (q0)(1) ¼ p0 and v ¼ 0 in the development. Finally take a/m = ∞
Thus we have derived the deflection of a beam by considering that of a rectangular plate by letting one dimension become very large compared to the other dimension.
