By Wodek Gawronski
The ebook offers and integrates the tools of structural dynamics, identity and keep an eye on right into a universal framework. It goals to create a typical language among structural and regulate approach engineers.
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Extra resources for Advanced Structural Dynamics and Active Control of Structures
2. Transfer Function in Modal Coordinates. 62) i i where Gmi (Z ) Cmi ( jZ I Ami )1 Bmi (cmqi jZ cmvi )bmi Zi2 Z 2 2 j] iZiZ i 1,! 63) is the transfer function of the ith mode. The value of the transfer function at the ith resonant frequency is approximately equal to the value of the ith mode transfer function at this frequency: ( jcmqi Zi cmvi )bmi G (Zi ) # Gmi (Zi ) 2] iZi2 . 64) Proof. 45) to the definition of the transfer function we obtain G (Z ) Cm ( jZ I Am ) 1 Bm n ¦ i 1 Cmi ( jZ I Ami ) 1 Bmi n ¦ Gmi (Z ), i 1 which proves the first part.
8) gives ( K Z 2 M )I e jZ t 0. 9) This is a set of homogeneous equations, for which a nontrivial solution exists if the determinant of K Z 2 M is zero, det( K Z 2 M ) 0. 10) The above determinant equation is satisfied for a set of n values of frequency Z. , n d nd . The frequency Zi is called the ith natural frequency. ,In ` that satisfy this equation. The ith vector Ii corresponding to the ith natural frequency is called the ith natural mode, or ith mode shape. The natural modes are not unique, since they can be arbitrarily scaled.
The natural modes are not unique, since they can be arbitrarily scaled. 9), so does DIi , where D is an arbitrary scalar. 11) and the matrix of mode shapes, or modal matrix ) , of dimensions nd u n, which consists of n natural modes of a structure ) >I1 I2 ... In @ ª I11 «I « 12 «" « ¬«I1nd I21 I22 " I2 nd ! In1 º ! In 2 »» , " "» » ! 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 ).
Advanced Structural Dynamics and Active Control of Structures by Wodek Gawronski