On initial conditions for a boundary stabilized hybrid Euler-Bernoulli beam

Abstract

We consider here small flexural vibrations of an Euler-Bernoulli beam with a lumped mass at one end subject to viscous damping force while the other end is free and the system is set to motion with initial displacement y⁰(x) and initial velocity y¹(x). By investigating the evolution of the motion by Laplace transform, it is proved (in dimensionless units of length and time) that ∫₀¹y²_xtdx ≤ ∫₀¹y²_xxdx,t > t₀, where t₀ may be sufficiently large, provided that {y⁰,y¹} satisfy very general restrictions stated in the concluding theorem. This supplies the restrictions for uniform exponential energy decay for stabilization of the beam considered in a recent paper.