ABSTRACT
Newton’s laws provide solutions to dynamic systems with unprecedented precision. However, the limits of application of these laws are often due to lack of precision in the manufacture and assembly of mechanical mechanisms that leads to complex behaviour such as hysteresis, friction and non-linearity. Much of the rest of the book is concerned with methods for optimising precision by eliminating these undesired features of machines and mechanisms.
All dynamic systems for small perturbations can be modelled as linear systems and understood in terms of frequency response functions and characteristic roots. Consequently, while dynamics is a large topic developed over many centuries, this current chapter has focused only on this aspect of the theory.
Ideal objects such as springs, rigid masses and frictionless pendulums are used to illustrate the basic principles of linear dynamics. These model objects can be used as ideal building blocks of a larger and more complex system of interacting bodies, such as may be encountered in a typical precision measurement system. An understanding of the basic concepts of motion in simple systems of one or two bodies illuminate foundational principles that apply to more complex, multiple-component systems. As examples of approximately linear, single-input single-output, dynamic processes, two commonly used contact measurement systems have been investigated: the properties and motion of a heavily damped contact probe from a coordinate measurement system, used widely in metrology; and the amplitude and frequency characteristics of a typical cantilever sensor from a scanning force microscope, which is used to analyse material surface properties on a much smaller scale. Such approximations are often adequate to model the response of the system.
Methods of calculating the motions of multi-body dynamical systems, and matrix calculations, with the use of eigenanalysis to obtain solutions of the system response to a variety of applied forces, are introduced towards the end of this chapter.
110Notwithstanding the narrow limits of this chapter, the use of generalised coordinates and the Lagrange equation forms a very broad foundation of all of classical mechanics and justifies the effort required to appreciate the underlying principles of this approach.
