Mathematical Exposition of the Design Axioms

Abstract

Axiomatic design (AD) offers designers two fundamental principles to follow for a successful design: (1) identify and define the design objectives, i.e., functional requirements (FRs), in such a way that they are inherently independent; and (2) conceive solutions for the FRs that comply with two design axioms: the independent axiom and the information axiom. In the previous chapter, the rationale and origin for the axiomatic nature of the design axioms were provided. In this chapter, the two axioms are given a deeper mathematical understanding, thereby strengthening their value. Starting with the formal definition of functional independence, the criterion for functional independence of FRs in a design is derived as the Jacobian determinant |J| ≠ 0. Since | J | ≠ 0 implies independence of FRs and existence of design solutions, the |J| criterion corroborates the declaration of independence axiom that a good design must “maintain the independence of the functional requirements.” The |J| criterion further reveals that AD criterion for functional independence—design with single input–single output—is only a sufficient condition. For rigor and completeness, the |J| criterion is shown to be necessary and sufficient. In implementing information axiom, AD assessment of uncertainty in design should cover a larger extent than it currently does. AD has not and should begin to recognize and identify the sources of variability and the countermeasures to them. The chapter ends with a summary of implementation steps in AD expressed in mathematical terms.

Appendices

Appendix A1: FR Decomposition of Door-to-Body System

Appendix A2: DP Decomposition of Door-to-Body System