The integrated product and process design strategy developed in this research requires that we consider the analysis and Design Sensitivity Analysis (DSA) of structural responses such as an eigenvalue λ and eigenvector Φ. The well-known eigen-problem may be written in terms of a design variable vector b as

[K(b)-λ(K) M(b)] Φ(b)=0

which requires a normalization condition written here as G(Φ(b),b)=0. To obtain a unique set of eigenvectors. These equations may be differentiated with respect to a design variable b_i to yield the design sensitivity equations

[K-λM]dΦ/db_i= -[∂K/db_i-dλ/db_i M -λ ∂M/∂b_i]Φ             ∂G/∂Φ·dΦ/db_i= -∂G/∂b_i

Following mathematical manipulation, the eigenvalue design sensitivity is computed from

dλ/db_i=1/(Φ·MΦ)  Φ· [∂K/∂b_i-λ ∂M/∂b_i]Φ

Note that this eigenvalue sensitivity equation is valid for any eigenvector scaling due to the generalized mass appearing in the denominator, unlike other solutions commonly employed. A unique eigenvector design sensitivity solution is obtained from the eigen-problem design sensitivity equations above where the design derivative of G replaces any one of the equations in the system of equations for dΦ/db_i which resolves the singularity issue associated with the system matrix [K – λM]. This approach is a generalization of Nelson’s Method [Nelson, 1976] which is widely used, but limited to mass normalized eigenvector scaling only.

As an example, a rectangular plate is considered where part of its thickness is defined by the design variable b. The magnitude of the eigenvector Φ varies with G, but its relative shape is unchanged. Note, however, that dΦ/db is altered by choice of normalization condition G.

The Effect of Eigenvector Scaling on Eigenvector Design Sensitivity. D.E. Smith and Siddhi, V. In preparation.

• U.S. National Science Foundation