Short CommunicationEfficient methods for determining modal parameters of dynamic structures with large modifications
Introduction
Structural dynamic modification (SDM) techniques are very useful for rapidly analysing and applying the effect of structural changes on the dynamic response of structures. That is, given changes in structural parameters, such as stiffness or mass, SDM techniques can efficiently determine the corresponding changes in modal parameters, such as natural frequencies and mode shapes, without solving the generalised eigenvalue problem for the modified dynamic system. Therefore, SDM techniques, which often incorporate finite-element analysis techniques, can greatly reduce the computational effort and increase the efficiency of reanalysis during a structural optimisation process, especially in the cases where complex mechanical and structural systems are considered.
There are various SDM techniques available for the dynamic reanalyses of a structural system, such as sensitivity analyses based on eigenvalue and eigenvector derivatives [1], [2] and Rayleigh quotient iteration [3]. However, the efficiency of sensitivity analysis methods is limited because these methods are complicated and may be only suitable for small modifications of structural parameters [4]. It may be difficult to determine some high-frequency modes using Rayleigh quotient iteration because the predictions of those modes exceeded the limit bound of the Rayleigh quotient [2].
It should be pointed out that the first-order sensitivity analysis techniques and the truncated Taylor's expansion approximation approaches often used for estimating the modified modal parameters may perform properly when the changes in structural parameters from the initial model to the modified model are small. However, for the cases with relatively large modifications of structural parameters, the first order or the truncated Taylor's expansion approximations may be inaccurate. To avoid these shortcomings, a perturbation theory was developed [5], which can provide an exact relationship between the modifications of structural parameters and the associated modal parameters and can be applied to model updating and inverse structural damage identification [6], [7]. Here, based on the developed perturbation theory, an efficient iterative computational procedure is proposed to provide exact predictions of the natural frequencies and the corresponding mode shapes for the modified dynamic system modelled with a large number of degrees of freedom (DOF). A high order approximation approach is also presented without iterative procedures required, which can give excellent estimates of the modified modal parameters. The results of a numerical example show that the iterative procedure presented here converges quickly for evaluating the modified modal parameters and finally gives the exact solution even when structural modifications are large. The proposed computational techniques successfully avoid adopting Taylor series expansion procedure and then the derivatives of modal parameters are not needed. Only limited information on the analytical or experimental modal data of the original structure is required in calculations. Therefore, the proposed methods are well suited for complex structures with a large number of DOFs, especially for the cases where the knowledge of the modified structural parameters is not available and the modified modal data cannot be obtained by solving the generalised eigenvalue problem.
Section snippets
Basic equations for SDM analyses
Assume that modal parameters λi and φi are the ith eigenvalue and the corresponding mass normalised eigenvector of the original dynamic system with structural parameters of the global stiffness matrix K and the global mass matrix M, where i ranges from 1 to N and N represents the total number of DOFs for the system. Then, the ith eigenvalue and the corresponding eigenvector for the modified system can be given by where Δλi and Δφi represent the modifications of ith
Numerical example
A rectangular thin plate model shown in Fig. 1 is utilised to demonstrate the effectiveness of the proposed techniques for calculating the modal data for the modified dynamic system by introducing various levels of structural modifications and comparing different computational approaches. The rectangular aluminium plate is 1000 mm long, 600 mm wide and 10 mm thick, with material properties of Young's modulus E=6.89×1010 N/m2, Poisson's ratio and density ρ=2796 kg/m3. The thin plate is
Conclusions
An improved iterative procedure is proposed for efficiently determining the eigenvalues and the corresponding eigenvectors for a dynamic system with large modifications of structural parameters and a large number of DOFs present. A high order approximation approach is also presented without iterative procedures involved. From the results for the given thin plate bending model problem, it has been shown that the convergence of the proposed iterative procedure can be achieved rapidly, leading to
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