Asymptotics of eigenfrequencies and eigenmodes of non-homogeneous inextensible filament with an end load

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...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2001-10, Vol.24 (15), p.1139-1167
Main Authors: Shubov, Marianna A., Belinskiy, Boris P., Martin, Clyde F.
Format: Article
Language:English
Subjects:
Approximations and expansions
Exact sciences and technology
Fourier analysis
ideal filament
integral equation
Mathematical analysis
Mathematics
non-selfadjoint operator
Ordinary differential equations
Partial differential equations
Sciences and techniques of general use
successive approximations
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Summary: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.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.266