Design sensitivity analysis for frequency analysis

The theory presented below assumes that all the eigenvalues are distinct (i.e., no repeated eigenvalues). If this is not the case, further manipulations are required to obtain the eigenvalue and eigenvector sensitivities corresponding to the repeated eigenvalues, and the following equations for the sensitivities will be incorrect. The repeated eigenvalue case is considered in the next section.

Performing a frequency analysis means solving the following eigenvalue problem (see Eigenvalue extraction):

where $\lambda$ represents the eigenvalues, ${\varphi }_N$ represents the eigenvectors, and $M^{NM}$ is the mass matrix. In addition, the eigenvectors are scaled such that either

or

for each mode. The default is the first scaling scheme. To obtain eigenvalue and eigenvector sensitivities, first differentiate Equation 1 with respect to design parameter $h_p$ to obtain the following equation:

where $\alpha$ represents a particular mode number.

Pre-multiplying by ${\varphi }_{\alpha }^N$, making use of Equation 1 and manipulating the result gives the eigenvalue sensitivities:

Except for the mass and stiffness derivatives, all quantities in this equation are known once the eigenvalue problem has been solved.

This section outlines the formulation used to obtain eigenvalue sensitivities for repeated eigenvalues. Further information can be found in the papers by Mills-Curran (1988) and Shaw (1991). When an eigenvalue $\lambda$ repeats R times, the eigenvectors $\Phi =\lfloor {\varphi }_1\dots {\varphi }_R\rfloor$ associated with $\lambda$ are linearly independent but not unique—any linear combination of these eigenvectors is also an eigenvector. Because of this non-uniqueness, the eigenvectors may not be continuous or differentiable in the design parameter. However, a set of R eigenvectors $Z=\lfloor z_1\dots z_R\rfloor$ that are continuous and differentiable can be obtained by an orthogonal transformation:

where $A^{T}A=I$, and $A$ is to be determined. Replacing the eigenvectors $\varphi$ with the eigenvectors $z$ in Equation 4 and premultiplying by ${\Phi }^T$ yields in matrix notation:

where $\Lambda =\lambda I$, $d\Lambda /d{h}_p$ is a diagonal matrix,

and

Equation 7 is recognized as an $R\times R$ reduced eigenvalue problem whose R eigenvalues $d\Lambda /d{h}_p$ are the eigenvalue sensitivities associated with the repeated eigenvalue $\lambda$ and whose eigenvectors are $A$. N sensitivities are obtained for the single eigenvalue $\lambda$. The physical interpertation of this is that a perturbation in the design parameter may cause the original repeated mode to branch into as many as N distinct modes (imagine a beam with a circular cross-section perturbed into an elliptical section; the repeated modes associated with the original symmetry of the section now split into distinct modes associated with the minor and major axes of the ellipse).

A semi-analytic approach is used to compute the eigenvalue sensitivities. The basic idea of this approach, as outlined in the section on static DSA, is to compute some of the required intermediate derivatives using finite differencing. In the context of DSA for frequency procedures this means that the derivatives of the mass and stiffness matrices are computed using finite differencing.