Polynomial stability of the Rao-Nakra beam with a single internal viscous damping

Zhuangyi Liu; Bopeng Rao; Qiong Zhang

doi:10.1016/j.jde.2020.04.030

Polynomial stability of the Rao-Nakra beam with a single internal viscous damping

Zhuangyi Liu, Bopeng Rao, Qiong Zhang

Mathematics & Statistics

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18 Scopus citations

Abstract

In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations.

Original language	English (US)
Pages (from-to)	6125-6162
Number of pages	38
Journal	Journal of Differential Equations
Volume	269
Issue number	7
DOIs	https://doi.org/10.1016/j.jde.2020.04.030
State	Published - Sep 15 2020

Bibliographical note

Funding Information:
This work was supported by the Beijing Natural Science Foundation (grant 4182059 ) and the National Natural Science Foundation of China (grant 61873036 ).

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

Beam
Riesz basis
Semigroup
Stability
Viscous damping

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title = "Polynomial stability of the Rao-Nakra beam with a single internal viscous damping",

abstract = "In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations.",

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author = "Zhuangyi Liu and Bopeng Rao and Qiong Zhang",

note = "Funding Information: This work was supported by the Beijing Natural Science Foundation (grant 4182059 ) and the National Natural Science Foundation of China (grant 61873036 ). Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

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AU - Liu, Zhuangyi

AU - Rao, Bopeng

AU - Zhang, Qiong

N1 - Funding Information: This work was supported by the Beijing Natural Science Foundation (grant 4182059 ) and the National Natural Science Foundation of China (grant 61873036 ). Publisher Copyright: © 2020 Elsevier Inc.

PY - 2020/9/15

Y1 - 2020/9/15

N2 - In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations.

AB - In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations.

KW - Beam

KW - Riesz basis

KW - Semigroup

KW - Stability

KW - Viscous damping

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Polynomial stability of the Rao-Nakra beam with a single internal viscous damping

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Keywords

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