Vibrations of extensible beams: Unilateral problem

The present work is dedicated to study a unilateral problem relating to the operator Lu(x, t) = ∂2u ∂t2 − ˆa(t)+ ˆb(t) β(t) α(t) ∂u ∂x 2 dx ∂2u ∂x2 +q ∂4u ∂x4 , which models small transverse deflections u(x, t) of an extensible beam with moving ends. Without restriction on the initial configuration u0 and considering the initial velocity u1 with a bounded gradient, we succeed to prove that, given T an arbitrary positive real number, there exists a unique solution for the unilateral problem defined for all t ∈ [0,T ]. © 2006 Elsevier Inc. All rights reserved