By Wodek Gawronski
The e-book offers and integrates the tools of structural dynamics, indentification and keep an eye on right into a universal framework. It goals to create a standard language among structural and keep watch over method engineers.
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Extra info for Advanced structural dynamics and active control of structures
6, where the transfer function of the structure was constructed as a sum of transfer functions of individual modes. The impulse response of the structure is shown in Fig. 19). 1210 rad/s. The Standard Models 27 harmonics are shown on the impulse response plot, but are more explicit at the impulse response spectrum plot, Fig. 7, as the spectrum peaks at these frequencies. 1 can be written in time domain as n h(t ) ¦ hi (t ) i 1 where h(t) is the impulse response of a structure and hi (t ) is the impulse response of the ith mode.
7) by )T , obtaining )T M )qm )T D) qm )T K ) qm y Coq )qm Cov )qm . 16) we obtain the above equation in the following form: M m qm Dm qm K m qm )T Bo u , y Coq )qm Cov )qm . Next, we multiply (from the left) the latter equation by M m1 , which gives qm M m1Dm qm M m1K m qm M m1)T Bo u , y Coq )qm Cov )qm . The obtained equations look quite messy, but the introduction of appropriate notations simplifies them, qm 2 =:qm : 2 qm Bm u , y Cmq qm Cmv qm . 19) : is a diagonal matrix of natural frequencies, as defined before.
Such that = 2 # 0 ) it simplifies to : º ª =: A « », : =: ¬ ¼ ª 0 º «B » , ¬ m¼ B C ªC : 1 Cmv = ¬ mq Cmv º . 42) we see that, although the state matrices A and the output matrices C are different, they actually are very close to each other. 43) and it differs from the identity matrix by a small off-diagonal element =. In the third model the state vector consists of modal displacements and velocities, x1 qm , and x2 qm . This is the most straightforward approach and it has direct physical interpretation, therefore it is the most popular model; however, its properties are not so useful as the first and second models, as will be seen later.
Advanced structural dynamics and active control of structures by Wodek Gawronski