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

ISBN-10: 1447120264

ISBN-13: 9781447120261

ISBN-10: 1447120280

ISBN-13: 9781447120285

**Dynamic Stiffness and Substructures** types a fancy dynamic procedure and provides an answer to the complex dynamical challenge linked to the results of wind and earthquakes on constructions. because the process matrices are necessarily frequency dependant, these are solely thought of during this book. The relation among the frequency matrices by way of the Leung's theorem is most vital within the improvement of effective algorithms for the traditional modes. This new strategy was once constructed through the writer over the last 15 years. It deals training 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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**Example text**

Csc 2 1/! - cot I/! csc I/! - I/! csc I/! cot I/! cscl/! - I/! cscl/! ] I/! csc2 I/! - cot I/! 6a) and [k] = EAI/! = [ cotl/! I -cscl/! 1. 6b) Since the shape functions are mathematically exact solutions of the governing equation for the free vibration of a bar, the element matrices can be used to predict an infinite number of modes accurately with a minimum number of elements. As an example, the frequency equation of the previously mentioned fixed-free bar formed with only one element is cot I/!

Flutter of frames is studied by Leung [28] using a generally non-conservative dynamic substructure method. A survey of the methods used up to 1981 can be found in Craig [29] and Meirovitch and Hale [30]. 1. 1) where [D(w)] = [K] - w 2 [M] [D] is the dynamic stiffness matrix [K] and [M] are the stiffness and mass matrices, respectively These matrices may be functions offrequency depending on the method of analysis. {F}e iwt and {X}e iwt are the force excitation and displacement response vectors respectively.

D mm ] by an amount equal to the number of negative entries in [dss ]. Therefore. the Sturm number of Eq. 3) equals the Sturm number of Eq. 4) when {qm} = {O}. and this is defined as the number of partial frequencies. The Wittrick-Williams algorithm uses the fact that the Sturm number of a continuum is equal to the Sturm number of its dynamic stiffness matrix plus the number of partial frequencies. The algorithm ensures that no natural frequencies are missed during a frequency search. 11. Derivatives of the Dynamic Stiffness It is known that the mass matrix is equal to the negative of the first derivative of the dynamic stiffness matrix with respect to the square of the vibration frequency [8].

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