ON THE SOLVABILITY OF THE BOUNDARY-VALUE PROBLEM FOR THE ELASTIC BEAM WITH ATTACHED LOAD
Marié Grobbelaar‐Van Dalsen · Mathematical Models and Methods in Applied Sciences · 1994
When a flexible rectangular homogeneous beam, which is horizontal in equilibrium state, has an end rigidly fixed while a load is attached to the other end, the transverse vibrations of the beam are described by the elastic beam equation together with, amongst others, a dynamic boundary condition at that end of the beam to which the load is attached. In this paper we introduce various types of energy dissipation for this problem, viz. damping of Kelvin-Voigt type as well as structural damping. The resulting boundary-value problems are studied within the framework of the abstract theories of B-evolutions and fractional powers of a closed pair of operators by formulating an abstract evolution problem in the paired space X1×X, with X a Hilbert space and X1 continuously imbedded in X. This approach yields a unique solution in the strong sense which exhibits exponential decay as time tends to infinity for any initial displacement in X1 and any initial velocity in X.