On the problem of semiinfinite beam oscillation with internal damping

TL;DR

This paper analyzes the spectral properties and solution existence for a class of damped beam oscillation equations modeled by a second-order operator pencil in a Hilbert space, relevant to semiinfinite beam dynamics.

Contribution

It provides a spectral analysis and proves existence and uniqueness of solutions for a generalized class of damped oscillation equations including semiinfinite beam models.

Findings

Spectral properties of the operator pencil are characterized.

Existence and uniqueness of solutions are established.

The model applies to semiinfinite beam oscillations with internal damping.

Abstract

We study the Cauchy problem for the equation of the form $$ \ddot{u}(t) + (\aa A + B)\dot{u}(t) + (A+G)u(t) = 0,\tag* $$ where $A$, $B$, and $G$ are \o s in a Hilbert space $\Cal H$ with $A$ selfadjoint, $\sigma(A)=[0,\infty)$, $B\ge0$ bounded, and $G$ symmetric and $A$-subordinate in a certain sense. Spectral properties of the correspondent operator pencil $L(\lambda) := \lambda^2I + \lambda (\alpha A + B) + A + G$ are studied, and existence and uniqueness of generalized and classical solutions of the Cauchy problem are proved. Equations of the type (*) include, e.g., an abstract model for the problem of semiinfinite beam oscillations with internal damping.