# Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam

## 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.

- Authors:

- Université des Sciences et de la Technologie Houari Boumediene, Faculté des Mathématiques (Algeria)
- King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics (Saudi Arabia)

- Publication Date:

- OSTI Identifier:
- 22617044

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 75; Journal Issue: 3; Other Information: Copyright (c) 2017 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BEAMS; DECAY; ELASTICITY; KERNELS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; SIMULATION; VELOCITY; VISCOSITY

### Citation Formats

```
Kelleche, Abdelkarim, E-mail: kellecheabdelkarim@gmail.com, and Tatar, Nasser-eddine, E-mail: tatarn@Kfupm.edu.sa.
```*Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam*. United States: N. p., 2017.
Web. doi:10.1007/S00245-016-9334-8.

```
Kelleche, Abdelkarim, E-mail: kellecheabdelkarim@gmail.com, & Tatar, Nasser-eddine, E-mail: tatarn@Kfupm.edu.sa.
```*Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam*. United States. doi:10.1007/S00245-016-9334-8.

```
Kelleche, Abdelkarim, E-mail: kellecheabdelkarim@gmail.com, and Tatar, Nasser-eddine, E-mail: tatarn@Kfupm.edu.sa. Thu .
"Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam". United States.
doi:10.1007/S00245-016-9334-8.
```

```
@article{osti_22617044,
```

title = {Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam},

author = {Kelleche, Abdelkarim, E-mail: kellecheabdelkarim@gmail.com and Tatar, Nasser-eddine, E-mail: tatarn@Kfupm.edu.sa},

abstractNote = {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.},

doi = {10.1007/S00245-016-9334-8},

journal = {Applied Mathematics and Optimization},

number = 3,

volume = 75,

place = {United States},

year = {Thu Jun 15 00:00:00 EDT 2017},

month = {Thu Jun 15 00:00:00 EDT 2017}

}