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Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load
Author(s) -
Shubov Marianna A.,
Belinskiy Boris P.,
Martin Clyde F.
Publication year - 2001
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.266
Subject(s) - eigenfunction , mathematics , controllability , boundary (topology) , constant (computer programming) , constant coefficients , mathematical analysis , boundary value problem , homogeneous , class (philosophy) , ideal (ethics) , eigenvalues and eigenvectors , physics , combinatorics , quantum mechanics , artificial intelligence , computer science , programming language , philosophy , epistemology
We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.