Consider a square plate of side a that is simply supported along all four edges. (i) Obtain expressions for the first three natural frequencies of the plate. (ii) Obtain expressions for the natural...

Consider a square plate of side a that is simply supported along all four edges. (i) Obtain expressions for the first three natural frequencies of the plate. (ii) Obtain expressions for the natural frequencies of a circular plate of radius R and simply supported along the periphery from the known roots of the associated characteristic equation. (iii) Determine the radius of an equivalent circular plate of the same thickness and material with exactly the same first natural frequency as the square plate. Like the square plate, the circular plate is also simply supported along its periphery. (iv) Repeat the calculations in (i), (ii) and (iii) for the case of square and circular plates clamped along the edges. You should employ any valid and approximate method to obtain the natural frequencies of the square plate.