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The optimal design of structures with frequency constraints is extremely useful in

manipulating the dynamic characteristics in different ways. For example, in most low-

frequency vibration problems, the response of the structure to dynamic excitation is pri-

marily a function of its fundamental frequency and mode shape. In such cases, the ability

to manipulate the selected frequency can significantly improve the performance of the

structure. Similarly, the aero-elastic characteristics of an aircraft wing which primarily

depends on its torsional and bending properties can be in best way studied by the

torsional and bending modes.

Many different techniques are available, enabling successful application to the

dynamic reanalysis of mechanical structures. One of the most popular is the sensitivity

analysis which has been developed and applied by several experts in the general

eigenvalue problem /29-34/, and, more specifically, in the applications of structural

dynamic modification analysis, Ref. /35-38/.

The areas where sensitivity analysis has been successfully applied include

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system identification,

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development of insensitive control systems,

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use in gradient-based mathematical programming methods,

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approximation of system response to a change in a system parameter, and

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assessment of design changes on system performance.

In this area, both first- and higher-order eigenvalue and eigenvector sensitivities have

been investigated in order to improve the prediction regarding the response of a modified

structure based on known of its spatial and modal original properties, or initial, unmodi-

fied, state. The sensitivity analysis of a mechanical structure is based on a Taylor’s

expansion of eigenvalues and eigenvectors of the unmodified structure. Traditionally, a

truncated Taylor or matrix power series evaluated at a nominal design point is used to

approximate the eigen parameters of modified structures /21,22/. Previous studies indi-

cated that the computation of the higher-order terms of these series is difficult and time

consuming, so the effectiveness of this method is limited to small modifications. Even the

use of higher-order terms in the local approximation series can’t guarantee convergence

for moderate to large perturbations in the structural parameters. The implication of this

observation in the context of structural optimization is that limits posed at severe level

have to be imposed additional troubles to ensure requested convergence in design. Very

few studies in the literature have addressed the structural dynamic reanalysis problem for

moderate to large modifications in the structural parameters. Direct and iterative

approaches present two classes of them, currently mostly applied. The objective of most

direct approaches is to increase the range of validity of local approximation techniques.

Inamura /1/ proposed an approximate procedure in which the eigen pair perturbation

equations are interpreted as differential equations in terms of the perturbation parameters.

A procedure using the eigen sensitivity equations was developed by Pritchard and

Adelman /24/ based on a similar approach.

The sensitivity method is a best representative of the updating approach which allows

selection of updating parameters, but does not require full experimental evidence. For

that, this method seems to be suitable for updating of large models. Also, it is worth no-

ting that updating models based on control methods, such as eigen structure assignment

method proposed by Minas and Inman /39,40/ are quite promising since they can be

defined so that full experimental mode shape matrix is not requiring. The generally

accepted perturbation procedure is presented in a diagram form in Fig. 3.