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Eigenvalue formulas for the uniform Timoshenko beam: the free-free problem. (English) Zbl 0897.34071

Timoshenko’s equation for coupled transverse vibrations which consider shear and inertia due to rotation of cross-sections of the beam is written as a pair of coupled differential equations \[ (EI\psi_x)_x+ AkG(y_x- \psi)- I\varrho\psi_{tt}= 0, \]
\[ (kAG(y_x- \psi))_x- \varrho Aw_{tt}= P(x,t). \] \(w\) is the transverse displacement, \(\psi\) is the angle of rotation of the cross-sectional area. Free-free end conditions are assumed. The usual separation of variables leads to coupled eigenvalue problems \[ (EI\psi_x)_x+ kAG(w_x- \psi)= p^2 I\varrho\psi, \]
\[ (kAG(y_x- \psi))_x= p^2\varrho Aw. \] A computation of eigenvalues \(p_i\) as functions of the design parameters, and of the sensitivity with respect to changes in these parameters is difficult because their nonlinear dependence on the constants \(E\) (Young’s modulus), \(kG\) (where \(G\) is the shear modulus), \(A\) (cross-sectional area), \(I\) (moment of inertia of cross-sectional area about the neutral axis), \(\varrho\) (material density).
The authors introduce new variables to decouple the second-order equations and rewrite them as two fourth-order equations. They are still coupled through boundary conditions.
Finally, the authors derive asymptotic formulas for eigenvalues. These formulas are essential in any future studies of optimization of density distribution to maximize the lowest eigenvalue, or in similar optimization efforts.
Reviewer: V.Komkov (Roswell)

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
74M20 Impact in solid mechanics
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References:

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