Dimensional Analysis of Synovial Joints

Abstract

Prior to undertaking experiments or developing a mathematical model of a complex phenomenon it is necessary to do some preliminary work in an attempt to gain some understanding of the physics involved and to identify the relevant parameters. Although usually thought of as a tool for physical modelling, dimensional analysis really provides a basic framework for thinking about a problem and this is useful whether the end result is intended to be a physical or a mathematical model. Students of complex systems would thus be well advised to begin with a partial analysis with the intention of obtaining some fundamental understanding of the phenomenon under consideration. Identification of the variables involved leads to the development of functional equations and a set, or sets, of dimensionless parameters. These parameters provide the modeller with a guide to the parameters which might appear, separately or in combination, in any complete solution. In many cases, a complex system may be usefully broken down into a number of subsystems each of which may be treated separately, but with the outputs from some subsystems (the dependent parameters) being the inputs (the independent parameters) to other subsystems or to the main system.

The example given here refers to a study of joints of the human body where dimensional analysis was used as a preliminary tool in a larger investigation designed to lead eventually to a finite element model. The paper attempts to show how such an analysis can be helpful: (1) in thinking about the problem; (2) in identifying the dimensionless parameters which provide a potential framework for a more complete solution; and (3) in breaking down a very complex problem into a number of related subsystems which can be more easily solved.