Page 14 - Mechanics of Microelectromechanical Systems

Chapter 1

STIFFNESS BASICS

1. INTRODUCTION

Stiffness is a fundamental qualifier of elastically-deformable mechanical
microcomponents and micromechanisms whose static, modal or dynamic
response need to be evaluated. This chapter gives a brief introduction to the
stiffness of microeletromechanical structural components by outlining the
corresponding linear, small-deformation theory, as well as by studying
several concrete examples. The fundamental notions of elastic deformation,
strain, stress and strain energy, which are all related to stiffness, are briefly
outlined. Energy methods are further presented, specifically the Castigliano’s
theorems, which are utilized herein to derive stiffness or compliance
equations.
A six degree-of-freedom lumped-parameter stiffness model is proposed
for the constant cross-section fixed-free straight members that are sensitive to
bending, axial and torsion loading. A similar model is developed for curved
members, both thick and thin, by explicitly deriving the compliance
equations. Composite beams, either sandwiched or in serial/parallel
configurations, are also presented in terms of their stiffnesses. Later, the
stiffness of thin plates and membranes is approached and equations are
formulated for circular and rectangular members. Problems that are proposed
to be solved conclude this chapter.

2. STIFFNESS DEFINITION

MEMS mainly move by elastic deformation of their flexible components.
One way of characterizing the static response of elastic members is by
defining their relevant stiffnesses. The simple example of a linear spring is
shown in Fig. 1.1, where a force is applied by slowly increasing its
magnitude from zero to a final value over a period of time such that the
force is in static equilibrium with the spring force at any moment in time.
The force necessary to extend the spring by the quantity is calculated
as: