Analysis of lower modes of vibration of rectangular plates of linearly varying thickness

The title problem is solved assuming that the thickness varies symmetrically with respect to the x-axis. The edges defined by $\text{x = ± a/2}$ are elastically restrained against rotation while the remaining edges are clamped or simply supported.Approximate expressions for four of the lower natural frequencies of vibration (including the fundamental) are given.A forced vibrations situation is also dealt with.