By Peter Hagedorn, Gottfried Spelsberg-Korspeter

Active and Passive Vibration keep watch over of constructions shape a subject matter of very genuine curiosity in lots of diversified 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 history, and it fill gaps among structural mechanics, vibrations and smooth regulate thought. additionally hyperlinks among the several functions in structural keep an eye on are shown.

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110) The determinant of a matrix is equal to the one of its transposed, so that for all values of λ det(λ2 M + λG + K) = det(λ2 M − λG + K). (111) This means one has det(λ2 M + λG + K) = det((−λ)2 M + (−λ)G + K), (112) which is only possible if exclusively even powers of λ occur, so that, if λ is an eigenvalue, so is −λ. This is diﬀerent from the damped systems, already examined, in which we had the symmetric matrix D instead of G, and in which the eigenvalues (λ, λ∗ ) occurred in complex conjugate pairs; now we have quadruples (±λ, ±λ∗ ) of eigenvalues!

For higer modes, since cosh z is an exponentially divergent function, the characteristic equation can be approximated by cos βl = 0. The solution of (231) can be represented in the form βn = ω n ⇒ ωn = ρA = EI 2n + 1 π + en 2 2n + 1 π + en 2 2 1 l2 EI , ρA 1 l (232) n = 1, 2, . . , ∞, (233) Mechanical Systems: Equations of Motion and Stability 3 57 cosh z 2 −1/ cosh z 1 Π cos z 2Π 3Π 4Π z 1 Figure 14: Graphical representation of the solutions of the characteristic equation of a free-free beam where en are small correction terms.

For the ﬂutter of an aircraft wing or the self-excited vibrations leading to brake squeal. 6 M -G-K-N -Systems If in addition to the circulatory forces also gyroscopic forces are present in a system, we have the case of a M -G-K-N -system. These systems are 9 It can easily be checked in the example that for p = p CK the stiﬀness matrix has a zero eigenvalue and is positive semi-deﬁnite, for p > pCK it is indeﬁnite. p=0 √ ω = 1/ 4 2 Im λ P. Hagedorn Im λ 36 pCF ω p=0 0 4,9 −ω p=0 4,9 0 p=0 pCK p=0 pCF p=0 −ω 0 ω Re λ 0 Re λ Figure 4: Eigenvalues of the double pendulum with follower force (left, solution of (143)) and with conservative load (right, ﬁrst solution of (148)) rather common in engineering applications and therefore merit our attention.