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Title: Asymptotic distribution of the eigenvalues of the bending‐torsion vibration model with fully nondissipative boundary feedback

Asymptotic and spectral results on the initial boundary‐value problem for the coupled bending‐torsion vibration model (which is important in such areas of engineering sciences as bridge and tall building designs, aerospace and oil pipes modeling, etc.) are presented. The model is given by a system of two hyperbolic partial differential equations equipped with a three‐parameter family of non‐self‐adjoint (linear feedback type) boundary conditions modeling the actions of self‐straining actuators. The system is rewritten in the form of the first‐order evolution equation in a Hilbert space of a four‐component Cauchy data. It is shown that the dynamics generator is a matrix differential operator with compact resolvent, whose discrete spectrum splits asymptotically into two disjoint subsets called the α‐branch and the β‐branch, respectively. Precise spectral asymptotics for the eigenvalues from each branch as the number of an eigenvalue tends to ∞ have been derived. It is also shown that the leading asymptotical term of the α‐branch eigenvalue depends only on thetorsioncontrol parameter, while of the β‐branch eigenvalue depends on twobendingcontrol parameters.

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Author(s) / Creator(s):
Publisher / Repository:
Date Published:
Journal Name:
Studies in Applied Mathematics
Page Range / eLocation ID:
p. 996-1025
Medium: X
Sponsoring Org:
National Science Foundation
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