Vibrations of extensible beams: Unilateral problem

Abstract

The present work is dedicated to study a unilateral problem relating to the operator L ˆ u ( x , t ) = ∂ 2 u ∂ t 2 − [ a ˆ ( t ) + b ˆ ( t ) ∫ α ( t ) β ( t ) ( ∂ u ∂ x ) 2 d x ] ∂ 2 u ∂ x 2 + q ∂ 4 u ∂ x 4 , which models small transverse deflections u ( x , t ) of an extensible beam with moving ends. Without restriction on the initial configuration u 0 and considering the initial velocity u 1 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 ] .