Symmetry in the problem of vibration of a polar-orthotropic non-homogeneous plate on an elastic foundation

The problem of bending vibration of a polar-orthotropic non-homogeneous elastic plate on an elastic foundation is considered, and the invariance of the problem-equation under an inversion transformation with respect to a circle is proved. As a corollary, the optimization problem (the ''best'' position of the point mass or point support, which optimizes the plate fundamental frequency) is considered and certain geometrical inequalities, which reduce the optimization domain, are proved. Computational time economy generated by the inequalities for the optimization problem is illustrated with numerical examples: (a) the ''best'' position of the point mass on segment and ring-sector plates; (b) the ''best'' radius of the ring support for annular plates. Some other corollaries from the problem-equation inversion invariance are given. (C) 1997 Academic Press Limited.