Minimum-mass design of a plate-like structure for specified fundamental frequency
The purpose of this paper is to present a new approach to the problem of the minimum-mass design of two-dimension al continuous structures which are required to satisfy a constraint of a dynamic or aeroelastic nature, expressed in the form of one or more partial differential equations. The minimization of a functional subject to constraints of this form belongs to a wider class of problems, encountered in the theory of optimal control of systems with distributed parameters. Classical methods of optimal control theory are here extended to two dimensions in order to derive the set of necessary conditions for an extremum. They are then applied to the theoretical case of the minimum-mass design of a simply-supported shear plate for a given fundamental frequency of vibration under the structural mass assumption. An equation for the optimal displacement, which is the expression of a general optimality criterion in structural design due to Prager and rendering the necessary conditions also sufficient, is derived and solved uniquely in closed form for any shape of the plate. Results are presented for a square and rectangular shape, and for a circular plate. The case of an inequality constraint applied to the thickness (minimum thickness) is also examined.