ON THE SOLVABILITY OF THE BOUNDARY-VALUE PROBLEM FOR THE ELASTIC BEAMWITH ATTACHED LOAD

When a flexible rectangular homogeneous beam, which is horizontal in e quilibrium 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 bo undary condition at that end of the beam to which the load is attached . In this paper we introduce various types of energy dissipation for t his problem, viz. damping of Kelvin-Voigt type as well as structural d amping. The resulting boundary-value problems are studied within the f ramework of the abstract theories of B-evolutions and fractional power s of a closed pair of operators by formulating an abstract evolution p roblem in the paired space X1 x X, with X a Hilbert space and X1 conti nuously imbedded in X. This approach yields a unique solution in the s trong sense which exhibits exponential decay as time tends to infinity for any initial displacement in X1 and any initial velocity in X.