By Wodek Gawronski
The publication provides and integrates the equipment 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 approach engineers.
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Extra resources for Advanced structural dynamics and active control of structures
Hence, the second derivative of the solution is q Z 2I e jZ t . 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 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. The second part follows from the first part by noting that, for flexible structures with distinct natural frequencies and low damping, Gmj (Zi ) Gmi (Zi ) 2 2 for i z j.
Transfer Function of a Mode. 26), Gmi (Z ) (cmqi jZ cmvi )bmi Zi2 Z 2 2 j] iZiZ . 1. Transfer Function in Modal Coordinates. 30) i i and the structural transfer function at the ith resonant frequency is approximately equal to the ith modal transfer function at this frequency (b) G (Zi ) # Gmi (Zi ) ( jcmqi Zi cmvi )bmi 2] iZi2 , i 1, ! , n. Proof. 28). Structural Poles. 26). The equation s 2 2] iZi s Zi2 0 is the characteristic equation of the ith mode. For small damping the poles are complex conjugate, and in the following form: s1 ] iZi jZi 1 ] i2 , s2 ] iZi jZi 1 ] i2 .
Advanced structural dynamics and active control of structures by Wodek Gawronski