Dynamic Stiffness and Substructures by Andrew Y. T. Leung MSc, PhD, CEng, FRAeS (auth.)

By Andrew Y. T. Leung MSc, PhD, CEng, FRAeS (auth.)

Dynamic Stiffness and Substructures versions a fancy dynamic approach and provides an answer to the complex dynamical challenge linked to the results of wind and earthquakes on buildings. because the method matrices are unavoidably frequency dependant, these are completely thought of during this ebook. The relation among the frequency matrices via the Leung's theorem is most vital within the improvement of effective algorithms for the usual modes. This new method used to be constructed via the writer during the last 15 years. It bargains working towards engineers and researchers a large selection for structural modelling and research. considerable numerical examples let the reader to appreciate the theory and to use the methods.

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However, it should be solved by subspace iteration [34] if the method is to be modified Dynamic Substructures 58 to generate a few modes at a time in order to avoid the difficulties of repeated natural frequencies. Convergence to the modes of interest may be safeguarded by invoking the Sturm theorem associated with the dynamic stiffness matrix. Let sED] denote the Sturm number of the matrix [D], that is the number of negative elements on the diagonal of the triangularized form of [D] obtained using Gauss elimination without interchanges, then the Sturm sequence properties of [D] ensure that the number of natural modes below w* is equal to s[D(w*)].

Also, when solving an eigenvalue problem using Newton's method, the derivatives of the "dynamic stiffness matrix are required. If A. and {vol are approximations of an eigenvalue problem, better approximations A. + AA. and {vo + &v} are given by the following inverse iteration, [D(A. o°A. )] {v} + neglected higher order terms Since {vo + Av} will be normalized, AA. is irrelevant and a new approximation of A. 1t can therefore be shown that for vibration prob- and for stability problems, References 1.

Solve the eigenvalue problem for the lowest absolute eigenvalue p and mode shape {X",} [D*] {X",} = p[M*] {X",} If [plw o] < e, an acceptable error, then take the required frequency as J(w~ + p), otherwise replace Wo by J(w~ + p) and go to step 2. 7. Multiply 6. {X,} = - [Z] {Xm} The required mode shape is given by {X} = {~:} The condensed eigenvalue problem may be solved by inverse iteration [33]. However, it should be solved by subspace iteration [34] if the method is to be modified Dynamic Substructures 58 to generate a few modes at a time in order to avoid the difficulties of repeated natural frequencies.

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