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Advanced Structural Dynamics & Active Control of Structures by Wodek Gawronski

By Wodek Gawronski

The booklet provides and integrates the equipment of structural dynamics, indentification and keep an eye on right into a universal framework. It goals to create a typical language among structural and regulate procedure engineers.

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For this system, the nodal model consists of the mass matrix M diag(m1 , m2 , m3 ) I 3 , and the stiffness matrix: M, Special Models 43 K ª k2 « k « 2 «¬ 0  k2 k2  k3 k3 ª 3 3 0 º « 3 6 3» . 8165» . 4083»¼ The modes are shown in Fig. 1. Note that the first mode does not have flexible deformations (springs are neither expanded nor compressed). This is the rigid-body mode. Note also that the corresponding natural frequency is zero. 1. Modes of a simple system (see (a)): The first mode is the rigid-body mode (without spring deformation).

A mode without flexible deformations is called a rigid-body mode. Corresponding zero frequency implies that the zero frequency harmonic excitation (which is a constant force or torque) causes rigid-body movement of the structure. Structural analysts sometimes ignore this mode, as there is no deformation involved. However, it is of crucial importance for a control engineer, since this mode is the one that allows a controller to move a structure and track a command. 42 Chapter 3 We obtain the rigid-body modes by solving the same eigenvalue problem as presented for the standard models.

9 with a solid line) and on other states (through the gains Amij shown in Fig. 9 with a dashed line). In modal coordinates the cross-coupling gains Amij are zero, thus each state is independent and depends only on itself. 9. Block diagram of the state-space representation of a structure with two modes: The modal cross coupling (marked with a dashed line) is nonexistent. We consider three modal representations in this book. 53) x Modal model 3: Ami ª 0 « 2 «¬ Zi 1 º », 2] iZi »¼ Bmi ª 0 º «b » , ¬ mi ¼ cmvi º¼ .

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