Eigenvalue Sensitivities of a Linear Structure Carrying Lumped Attachments

A sensitivity‐based approach to solving the inverse eigenvalue problem for linear structures carrying lumped attachments

“…Because the derivation of this approach is based on the assumed‐modes method, our proposed algorithm can be applied to any linear structure. For definiteness, we will assume the linear structure to consist of a simply supported beam of length L , whose normalized eigenfunctions (with respect to the mass per unit length, ρ , of the beam) are given by such that the generalized masses and stiffnesses are where E is the Young's modulus of the beam and I is the area moment of inertia of the cross‐section of the beam. In the subsequent numerical examples, we will also compare the execution times required by our proposed IEP algorithm and the fsolve‐based algorithm using tests executed on an Intel i7‐6700HQ CPU running at 2.60 GHz.…”

“…We can apply the implicit function theorem to determine the natural frequency sensitivities given Equation . This computation is easily done by hand or by referencing a table of partial derivatives as in the work of Cha and Sabater, but these partial derivatives can also be easily computed using any symbolic manipulation software package. In our experience, pre‐calculating these partial derivatives using MATLAB takes between 1 and 2 seconds.…”

“…The implicit function theorem states that, given a function where f ( x , z )=0 and on some region, not only is z defined in that region by the implicit function z = g ( x ) but the partial derivatives also exist in that region and will be given by …”

“…The application of this theorem here is that, although explicit expressions for the eigenvalues of a combined system might not exist, it is still possible to determine the sensitivities (ie, the partial derivatives) of those eigenvalues with respect to various system parameters, as Cha and Sabater demonstrate. Note that we can describe each lumped attachment by s +1 values: the attachment location x i and s additional parameters β 1 ,…, β s (which might represent the stiffness, mass, damping coefficient, etc, of the attachment).…”

“…When these changes are relatively small, we can apply a first‐order Taylor series approximation where λ 0 is the vector of nominal eigenvalues and [ J ] 0 denotes the Jacobian evaluated at those nominal parameters. When the number of attachments p or basis functions N are very large, recalculating the eigenvalues from scratch can require costly new calculations; however, this approximate updating method is very efficient and reasonably accurate for small perturbations Δ . For large perturbations, this method can still lead to accurate results if an incremental perturbation approach is used (in which the large perturbation is divided into multiple small perturbations applied sequentially) .…”