Advanced Structural Dynamics & Active Control of Structures by Wodek Gawronski

By Wodek Gawronski

The ebook provides and integrates the equipment of structural dynamics, indentification and regulate right into a universal framework. It goals to create a standard language among structural and keep watch over procedure engineers.

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Extra resources for Advanced Structural Dynamics & Active Control of Structures

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12) Standard Models 19 Ii ­Ii1 ½ °I ° ° i2 ° ® ¾. 14) Km )T K ). 15) The obtained diagonal matrices are called modal mass matrix ( M m ) and modal stiffness matrix ( K m ). 16) gives the modal damping matrix Dm , which is not always obtained as a diagonal matrix. However, in some cases, it is possible to obtain Dm diagonal. In these cases the damping matrix is called a matrix of proportional damping. The proportionality of damping is commonly assumed for analytical convenience. This approach is justified by the fact that the nature of damping is not known exactly, that its values are rather roughly approximated, and that the off-diagonal terms in most cases—as will be shown later—have negligible impact on the structural dynamics.

2. 1. 5910 » . 7370 »¼ ) The modes are shown in Fig. 2. 1. Pole location of the ith mode of a lightly damped structure: It is a complex pair with the real part proportional to the ith modal damping; the imaginary part approximately equal to the ith natural frequency; and the radius is the exact natural frequency. 0156 0 º «0 ». 3280 » . 2. Modes of a simple system: For each mode the mass displacements are sinusoidal and have the same frequency, and the displacements are shown at their extreme values (see the equation (a)).

26) n y ¦ yi , i 1 where bmi is the ith row of Bm and cmqi , cmvi are the ith columns of Cmq and Cmv , respectively. The coefficient ] i is called a modal damping of the ith mode. In the above equations yi is the system output due to the ith mode dynamics, and the quadruple (Zi , ] i , bmi , cmi ) represents the properties of the ith natural mode. Note that the structural response y is a sum of modal responses yi, which is a key property used to derive structural properties in modal coordinates.

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