Preface

The following case study concerns the resonant oscillations of an elastic plate in a fluid. At first sight one might expect that any analysis would get difficult quite quickly, forcing numerical solutions. However, simple beam theory suffices for the motion of the plate since it is thin, and potential theory for the motion of the fluid suffices since little viscous shear is generated. Moreover, amplitudes are very small, linear theory works, and a very simple solution generates the form of the answer that the client was seeking. The numerical results for this solution do not fit data at all well, but re-examination of the device indicates that a change in the boundary condition is required. This modification then provides results in remarkable agreement with data. The latter instructs on the importance of correctly modelling boundary conditions.

What follows is part of the analysis generated in projects completed by Claremont Mathematics Clinic teams for ITT-Barton, a company manufacturing the densitometer. The basic analysis was completed in 1982–3; various extensions were analyzed in 1983–4, see [8], [4]. A formal presentation was prepared, [2]. Only the basic analysis is presented here; extensions are suggested as projects for the interested student in section 3.7.

Introduction

As its name implies the densitometer is a device which measures density. (The reader is challenged to design a procedure which measures density to high accuracy – here the tolerance is 0.1% error – to operate safely in a hostile environment for long periods.)