American Institute of Aeronautics and Astronautics 1 An Iterative Method for Large Modification of Vibration and Buckling Characteristics of Plates Simultaneously S. Shahab * , R. Mirzaeifar † GWW School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA H. Bahai ‡ School of Engineering and Design, Brunel University, Uxbridge, UB8 3PH, UK This article investigates a new iterative method based on the first and second order derivatives of eigenvalues which allows large modification of natural frequencies and buckling loads to be carried out simultaneously. A method is proposed for calculating the derivatives of eigenvalues and eigenvectors both in free vibration and buckling conditions. Using these eigen-derivatives, the first- and second-order Taylor expansions are presented for formulating the inverse approximate method or finding the necessary changes in design parameters for achieving simultaneous predefined shifts in natural frequencies and buckling loads. An iterative process is introduced for performing the modification in the case of considering large shifts in natural frequencies and buckling loads. By considering numerical case studies, it is shown that the proposed method can perform the predefined modification with an acceptable accuracy even for large perturbations in the objective functions. Keywords: Natural frequency, Eigenvector, Buckling load, Inverse eigenvalue problem. I. Introduction HE inverse engineering design or behavior modification of a system may be defined as finding the necessary changes in physical or geometrical properties of a structure in order to achieve predefined shifts in structural responses such as natural frequencies or buckling loads. The traditional method for performing such modification is the error and trial method wherein the structural responses are iteratively calculated for different values of design variables till the answer converges to the desired value. This procedure faces more conflicts when simultaneous modification of two parameters is needed. For structural responses that are formulated in eigenvalue form (such as free vibration and linear buckling) the time consuming iterative methods may be replaced by an inverse eigenvalue problem. Aryana et al. 1 and Mirzaeifar et al. 2 presented an inverse approximate eigenvalue method based on eigen- derivatives for modification of free vibration behavior of FGM and laminated composite plates. Simultaneous modification of natural frequencies and buckling loads is a very common and a challenging problem in practice. As an example, this problem is a common problem in aerospace industry in the design of space craft structures where simultaneous consideration of both the vibratory and buckling requirements is of utmost importance. Recently, Mirzaeifar et al. 3 proposed an approximate method for simultaneous modification of natural frequencies and buckling loads of thin rectangular isotropic plates. In the proposed method, the structural modification is considered as an inverse eigenvalue problem and the calculated eigenderivatives are used for transforming this inverse problem to solving a system of algebraic equations. In all the previously reported works for inverse modification which are based on using the eigen-derivatives, the main restriction is the amount of predefined shift which should be small. This work presents an inverse iterative formulation for simultaneous large modification of natural frequencies and buckling loads. The inverse eigenvalue problem is transformed to solving a system of algebraic equations with considering the desired changes in natural frequencies and buckling loads as known terms and the design parameters as unknowns. A repeated procedure is implemented in solving this set of algebraic equations in the case of considering large shifts in the objective functions. In each iteration the values of eigen-derivatives are updated and * PhD Candidate, GWW School of Mechanical Engineering, shima.shahab@gatech.edu. † PhD Candidate, GWW School of Mechanical Engineering, reza.mirzaeifar@gatech.edu. ‡ Professor, School of Engineering and Design, Hamid.Bahai@brunel.ac.uk. T 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th 4 - 7 April 2011, Denver, Colorado AIAA 2011-1927 Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.