Mechanics of Beams and Diaphragms

Abstract

In this chapter we discuss and in part derive the equations that describe the mechanical deformation of construction elements found in mechanical sensors, as far as they are needed for the basic design of mechanical sensors. Derivations are given in cases that illuminate the basic ideas and methods of thinking about mechanics. In particular, we shall give derivations of the energy of a bent beam, including an axial load and stretching, and we shall show how partial differential equations can be found from energy expressions. These differential equations will give us the static and dynamic behaviour of membranes and beams. In many cases a closed analytical solution of the differential equation cannot be given, therefore we need ways to find reliable approximations. Therefore we also discuss how to obtain simple approximations for deflection amplitudes by energy minimisation. Finally we shall consider mechanical stability.