Fundamental Schemes of Discretized Formulation of Continuously Distributed Models

Abstract

For continuums such as elastically vibrating substances, e. g., beams and plates, mathematical models are discussed with respect to the major problems involving the discretized formulations : (1) To decrease the order of spatial dimension. (Cantor's theorem of the infinite set theory and the continuous cardinality are referred to help imagining the idea.) The example of decreasing is given to show how the principle applies by means of the comparison of two cases which are formulations of modal controls of bending vibrations. (2) Finite freedom forms for infinite eigenmodes of distributed parameter systems. An approximating method using spectral decompositions and resolvent representations of the linear operator is shown. The necessities of the conditions for the state observability and the state controllability in the designs of finite freedom controls provided with the eigenmodes for continuously distributed objects are pointed out.