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Volume 2,     Number 1,     Winter 1994

 

DAMPED VIBRATIONS OF BEAMS
AND RELATED SPECTRAL PROBLEMS
PETER LANCASTER AND A. SHKALIKOV

Abstract. A class of quadratic eigenvalue problems is considered which includes an abstract model for transverse motions of thin beams in the presence of both internal and external damping forces. The eigenvalue problem concerns operator functions of the form

where A, B, and the identity I act on an appropriate Hilbert space, and A and B are generally unbounded. The parameter α ≥ 0 determines the magnitude of internal damping. Detailed analysis is made of the distribution of eigenvalues in the complex plane, their accumulation points, and their dependence on α. Inclusion domains are also obtained.
   Solvability of the Cauchy problem for the equation

is established as well as regularity of the solutions and expansions in terms of eigenfunctions.

 

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