By Peter Hagedorn, Gottfried Spelsberg-Korspeter
Active and Passive Vibration keep an eye on 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 in addition in civil engineering. The papers during this quantity collect engineers of alternative history, and it fill gaps among structural mechanics, vibrations and glossy keep watch over conception. additionally hyperlinks among the various functions in structural regulate are shown.
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Extra info for Active and Passive Vibration Control of Structures
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 . This yields ωnEB = n2 π 2 l2 EI , ρA n = 1, 2, . . , ∞. , sr = l2 A/I lower modes given by the Rayleigh beam model tend to be the same as those of those obtained from the Euler-Bernoulli beam model (since 1 + n2 π 2 I/l2 A ≈ 1).
28 P. Hagedorn n3 m, r g b3 q2 Ω b2 q1 b1 n1 n2 Figure 2: Rolling Penny The Rolling Penny As an example for gyroscopic stabilization consider the rolling penny represented in Figure 2. The linearized quations of motion are mr2 3 4 0 0 0 q¨1 + mr2 5 3 q ¨ − 2 4 2Ω 3 2Ω 0 0 q˙1 + 0 q˙2 0 q1 0 = . (123) −mgr q2 0 Obviously the system has a positive deﬁnite mass matrix, a negative semidefinite stiﬀness matrix and is gyroscopic. We ﬁrst treat the case Ω = 0. The two equations of motion are then uncoupled and the system has the eigenvalues λ1 = 0, λ2 = 0, λ3 = 4 g , 5 r λ4 = − 4 g 5 r (124) and therefore an unstable trivial solution.
From elementary theory of elasticity (see, for example, Timoshenko and Goodier (1970)), it is known that when the beam is deﬂected, certain hypothetical longitudinal lines or ﬁbers are elongated, while others are compressed. There, however, exist ﬁbers which are neither elongated nor compressed, but are merely deﬂected. Such a ﬁber is called the neutral ﬁber, and is shown in Fig. 5. It Mechanical Systems: Equations of Motion and Stability 39 is assumed that cross-sections orthogonal to the neutral ﬁber before deformation are are also orthogonal to the neutral ﬁber in the deformed beam.
Active and Passive Vibration Control of Structures by Peter Hagedorn, Gottfried Spelsberg-Korspeter