Download Active and Passive Vibration Control of Structures by Peter Hagedorn, Gottfried Spelsberg-Korspeter PDF

By Peter Hagedorn, Gottfried Spelsberg-Korspeter

Active and Passive Vibration regulate of buildings shape a topic of very real curiosity in lots of varied fields of engineering, for instance within the car and aerospace undefined, in precision engineering (e.g. in huge telescopes), and in addition in civil engineering. The papers during this quantity compile engineers of alternative heritage, and it fill gaps among structural mechanics, vibrations and sleek regulate thought. additionally hyperlinks among different functions in structural keep watch over are shown.

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The equations of motion therefore read m1 0 0 0 q¨1 + −g m2 q¨2 g q˙1 + 0 q˙2 k1 k3 0 k3 + −n k2 n 0 q1 0 = , (152) 0 q2 where n is not to be confused with the number of degrees of freedom. In can be seen that the characteristic equation m1 m2 λ4 + (g 2 + m1 k2 + m2 k1 )λ2 + 2gnλ + (n2 + k1 k2 − k32 ) = 0 (153) has in fact no term in λ3 , so that according to the Hurwitz criterion not all eigenvalues can have negative real parts. Eigenvalues with zero real parts are possible only for ng = 0, namely only if exclusively even powers of λ are present.

The equations of motion therefore read m1 0 0 0 q¨1 + −g m2 q¨2 g q˙1 + 0 q˙2 k1 k3 0 k3 + −n k2 n 0 q1 0 = , (152) 0 q2 where n is not to be confused with the number of degrees of freedom. In can be seen that the characteristic equation m1 m2 λ4 + (g 2 + m1 k2 + m2 k1 )λ2 + 2gnλ + (n2 + k1 k2 − k32 ) = 0 (153) has in fact no term in λ3 , so that according to the Hurwitz criterion not all eigenvalues can have negative real parts. Eigenvalues with zero real parts are possible only for ng = 0, namely only if exclusively even powers of λ are present.

Here of course we have to examine which of the solutions of the projected system also fulfill (134). For ri∗ N ri = 0, λ2i is complex, so that the quadruple (±λ, ±λ∗ ) contains solutions which are point symmetric about the origin in the complex plane. Therefore all the solutions of (137) are also solutions of (134) and (133). For ri∗ N ri = 0, (137) leads to a pair of purely imaginary solutions. Two of the elements of the quadruple (±λ, ±λ∗ ) are therefore identical but each one fulfills both (134) and (133).

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