Advanced structural dynamics and active control of by Wodek Gawronski

By Wodek Gawronski

The booklet offers 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 keep an eye on approach engineers.

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8. 42). 3280» . 3 Modal Models Although the above representations were derived using modal displacements, qm , they are not considered modal state representations. 48) where Bmi and Cmi are 2 u s and r u 2 blocks, respectively. 49) and each component consists of two states xi ­ xi1 ½ ® ¾. 50) 36 Chapter 2 The ith component, or mode, has the state-space representation ( Ami , Bmi , Cmi ) independently obtained from the state equations xi Ami xi  Bmi u , yi Cmi xi , y ¦ yi . 51) n i 1 This decomposition is justified by the block-diagonal form of the matrix Am , and is illustrated in Fig.

4) into the above equation. 2 Second-Order Structural Models In this and the following sections we will discuss the structural models. One of them is the second-order structural model. It is represented by the second-order linear differential equations, and is commonly used in the analysis of structural dynamics. Similarly to the state-space models the second-order models also depend on the choice of coordinates. Typically, the second-order models are represented either in the nodal coordinates, and are called nodal models, or in the modal coordinates, and are called modal models.

X The phase of a transfer function of a flexible structure shows 180 degree shifts at natural frequencies, see Fig. 10(d). x Poles of a flexible structure are complex conjugate. Each complex conjugate pair represents a structural mode. The real part of a pole represents damping of the mode. The absolute value of the pole represents the natural frequency of the mode. Consider two different structures, as in Fig. 11(a). The first one has poles denoted with black circles (x), the second one with crosses (u).

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