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.