By Wodek Gawronski
The booklet provides and integrates the tools of structural dynamics, indentification and regulate right into a universal framework. It goals to create a standard language among structural and keep an eye on method engineers.
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Extra resources for Advanced structural dynamics and active control of structures
Sample text
328 « ». 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.
In this chapter we use both second-order and state-space models, and show their interrelations. 14 Chapter 2 Besides the choice of form of equations, we represent the analytical model in different coordinates. The choice of coordinates in which the system model is represented is rather arbitrary. However, two coordinate systems, nodal and modal, are commonly used. Nodal coordinates are defined through displacements and velocities of selected structural locations, called nodes; and modal coordinates are defined through the displacements and velocities of structural (or natural) modes.
37) y Cmq : 1 x1 Cmv x2 . 1), with the state triple as follows: : º ª 0 A « », ¬ : 2 =: ¼ B ª 0 º «B » , ¬ m¼ C ªCmq : 1 Cmv º . 4) and the following transformation: R ª I 0º « = I » . 36) we obtain a new state variable 32 Chapter 2 x :qm ½ ® ¾. 40) The corresponding state representation is : º ª =: A « », 2 «¬ : = : =: »¼ ª 0 º «B » , ¬ m¼ B C ªCmq : 1 Cmv = ¬ Cmv º . , such that = 2 # 0 ) it simplifies to : º ª =: A « », : =: ¬ ¼ ª 0 º «B » , ¬ m¼ B C ªC : 1 Cmv = ¬ mq Cmv º .