One‐dimensional approximations of the eigenvalue problem of curved rods

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AbstractIn this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ϵ of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ϵ tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.