Applied Mathematics & Optimization

, Volume 75, Issue 3, pp 343–364 | Cite as

Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam

Article

Abstract

The paper deals with an axially moving viscoelastic structure modeled as an Euler–Bernoulli beam. The aim is to suppress the transversal displacement (transversal vibrations) that occur during the axial motion of the beam. It is assumed that the beam is moving with a constant axial speed and it is subject to a nonlinear force at the right boundary. We prove that when the axial speed of the beam is smaller than a critical value, the dissipation produced by the viscoelastic material is sufficient to suppress the transversal vibrations. It is shown that the rate of decay of the energy depends on the kernel which arise in the viscoelastic term. We consider a general kernel and notice that solutions cannot decay faster than the kernel.

Keywords

Moving structure Euler–Bernoulli beam Viscoelasticity Nonlinear force Arbitrary decay 

Notes

Acknowledgments

The second author is grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Faculté des MathématiquesUniversité des Sciences et de la Technologie Houari BoumedieneBab Ezzouar, AlgerAlgeria
  2. 2.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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