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

By Peter Hagedorn, Gottfried Spelsberg-Korspeter

Active and Passive Vibration keep an eye on of constructions shape a topic of very real curiosity in lots of various fields of engineering, for instance within the automobile and aerospace undefined, in precision engineering (e.g. in huge telescopes), and likewise in civil engineering. The papers during this quantity assemble engineers of other heritage, and it fill gaps among structural mechanics, vibrations and smooth keep watch over conception. additionally hyperlinks among different purposes in structural keep an eye on are shown.

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Substituting this in (201), along with B1 = B3 = 0 and (211), the eigenfunctions of a simply-supported uniform Rayleigh beam can be written as Wn (x) = B sin nπx , l n = 1, 2, . . , ∞, (213) 52 P. Hagedorn where B is an arbitrary constant. These eigenfunctions are clearly orthogonal, and can be normalized to make them orthonormal. In the case of a simply-supported uniform Euler-Bernoulli beam, we have the same expression for βn given by (211), as one can easily check. Therefore, the circular natural frequencies of an Euler-Bernoulli beam are obtained by substituting the expression of β from (211) in (204), and solving for ωn .

We now project the eigenvalue problem (λ2i M + λi G + K)ri = 0 (113) 26 P. Hagedorn on ri∗ and get λ2i ri∗ M ri + λi ri∗ Gri + ri∗ Kri = 0. (114) The positive symmetric mass mass matrix and the semidefinite stiffness matrix imply ri∗ M ri > 0 and ri∗ Kri ≥ 0. Moreover, ri∗ Gri is purely imaginary, since G is skew-symmetric. Therefore 1 ri∗ Gri ± 2 ri∗ M ri ⎛ 1 jri∗ Gri =j⎝ ± 2 ri∗ M ri λi = − 1 ri∗ Gri 2 ri∗ M ri 2 1 jri∗ Gri 2 ri∗ M ri − ri∗ Kri ri∗ M ri ⎞ ri∗ Kri ⎠ + ∗ ri M ri 2 (115) is purely imaginary, since jri∗ Gri is real and the radicand is non-negative.

103) s=1 3 Named after the Scottish engineer and physicist Thomas K. Caughey, *1927 in Rutherglen, Scottland,†2004 in Pasadena, CA, USA. Mechanical Systems: Equations of Motion and Stability 23 It is clear that (103) satisfies the commutativity condition, since each term in the sum satisfies (94). Therefore n ˜ = RTDR = D αs RT M M −1 K s−1 R (104) s=1 is a diagonal matrix. If the eigenvectors are normalized via riT ri = 1, i = 1, . . , n, then RTR = E and (104) implies n αs RTM R RT M −1 R RT K ˜ = D s−1 s=1 n αs RTM R RT M −1 R RT KR = R s−1 s=1 n ˜ M ˜ ˜ −1 K αs M = s−1 (105) s=1 and finally n ˜ −1 D ˜ = M ˜ ˜ −1 K αs M s−1 (106) , s=1 where the diagonal elements can also be written in the form of a linear system of algebraic equations ⎤ ⎡ 0 ⎡˜ ˜1 d1 /m ω1 ⎥ ⎢ ω20 ⎢ d˜2 /m ˜ 2 ⎥ ⎢ ⎢ ⎢ .

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