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Axiomatic Design 275

Corollary 8: (Effective Reangularity of a Scalar)
The effective Reangularity*, R, for a scalar coupling
matrix element is unity.

A.2 Axiomatic design theorems†
of general design

Theorem 1: (Coupling Due to Insufficient Number of DPs)
When the number of DPs is less than the number of FRs,
either a coupled design results, or the FRs cannot be
satisfied
Theorem 2: (Decoupling of Coupled Design)
When a design is coupled due to the greater number of
FRs than DPs (i.e., m > p), it may be decoupled by the
addition of new DPs so as to make the number of FRs and
DPs equal each other, if a subset of the design matrix con-
taining p
p elements constitutes a triangular matrix.
Theorem 3: (Redundant Design)
When there are more DPs than FRs, the design is either
a redundant design or a coupled design.
Theorem 4: (Ideal Design)
In an ideal design, the number of DPs is equal to the
number of FRs and the FRs are always maintained inde-
pendent of each other.
Theorem 5: (Need for New Design)
When a given set of FRs is changed by the addition of a
new FR, by substitution of one of the FRs with a new one,
or by selection of a completely different set of FRs, the
design solution given by the original DPs cannot satisfy
the new set of FRs. Consequently, a new design solution
must be sought.

*Reangularity (R) is the R in a measure coupling vulnerability and is defined as the
orthogonality between the DPs in terms of the absolute value of the product of the geo-
metric sine’s of all the angles between the different DP pair combinations of the design
matrix. See Chapter 3, El-Haik (2005) for more details.
†A theorem can be defined as a statement, which can be demonstrated to be true by
accepted mathematical operations and arguments. In general, a theorem is an embodi-
ment of some general principle that makes it part of a larger theory. The process of
showing a theorem to be correct is called a proof.