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Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam

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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.

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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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Authors and Affiliations

  1. Faculté des Mathématiques, Université des Sciences et de la Technologie Houari Boumediene, BP 32, EL-Alia, 16111, Bab Ezzouar, Alger, Algeria

    Abdelkarim Kelleche

  2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Nasser-eddine Tatar

Authors
  1. Abdelkarim Kelleche
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  2. Nasser-eddine Tatar
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Correspondence to Abdelkarim Kelleche.

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Kelleche, A., Tatar, Ne. Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam. Appl Math Optim 75, 343–364 (2017). https://doi.org/10.1007/s00245-016-9334-8

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  • DOI: https://doi.org/10.1007/s00245-016-9334-8

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