By Peter Hagedorn, Gottfried Spelsberg-Korspeter
Active and Passive Vibration keep watch over of constructions shape a subject matter of very real curiosity in lots of assorted fields of engineering, for instance within the car and aerospace undefined, in precision engineering (e.g. in huge telescopes), and likewise in civil engineering. The papers during this quantity compile engineers of other heritage, and it fill gaps among structural mechanics, vibrations and glossy keep an eye on idea. additionally hyperlinks among the various purposes in structural keep watch over are shown.
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Extra info for Active and Passive Vibration Control of Structures
E. to purely imaginary eigenvalues, can be chosen real, while real eigenvectors are not possible for ri∗ N ri = 0 in general. In any case the eigenvectors associated to the eigenvalue pair ±λ can be chosen to be identical, if we continue to assume that all eigenvalues are simple. The eigenmotions can be written in real form (25b). A M -K-N -System with Two Degrees of Freedom In order to illustrate the basic ideas, we consider the double pendulum with follower force as shown in Figure 3 on the right.
Therefore, β˜ has four solutions give as β˜ = ±β1 , ±iβ2 , where 1 ω 2 ρI + β1 = √ 2EI and 1 − ω 2 ρI + β2 = √ 2EI 1/2 ω 4 ρ2 I 2 + 4ω 2 EIρA (198) , 1/2 ω 4 ρ2 I 2 + 4ω 2 EIρA . (199) 50 P. Hagedorn Thus, the general (complex) solution of (183) is obtained as W (x) = A1 eβ1 x + A2 e−β1 x + A3 eiβ2 x + A4 e−iβ2 x , (200) where Ai , i = 1, . . , 4 are (complex) constants. Alternatively, the solution may also be expressed in the real form as W (x) = B1 cosh β1 x + B2 sinh β1 x + B3 cos β2 x + B4 sin β2 x, (201) where Bi , i = 1, .
N for the description of the damping of a system with n degrees of freedom with given mass and stiﬀness matrix. The question arises, how D can be expressed as a function of n parameters, so that on one side the commutativity condition (94) is fulﬁlled, and on the other hand n arbitrary modal damping rations ϑ˜i can be represented. It can be shown that both requirements are fulﬁlled if D is represented as a Caughey sum3 n αs M M −1 K D= s−1 . (103) s=1 3 Named after the Scottish engineer and physicist Thomas K.
Active and Passive Vibration Control of Structures by Peter Hagedorn, Gottfried Spelsberg-Korspeter