Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping

2021 ◽  
Vol 496 (1) ◽  
pp. 124785
Author(s):  
Pengyan Ding ◽  
Zhijian Yang
Keyword(s):  
Nonlinear Damping ◽  
Beam Equation ◽  
Rotational Inertia ◽  
Longtime Behavior

scholarly journals Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

2020 ◽  
Vol 4 (1) ◽  
pp. 400-416
Author(s):  
Abdelmajid Ali Dafallah ◽  
◽  
Fadlallah Mustafa Mosa ◽  
Mohamed Y. A. Bakhet ◽  
Eshag Mohamed Ahmed ◽  
...  
Keyword(s):  
Nonlinear Damping ◽  
Random Attractor ◽  
Beam Equation ◽  
Stochastic Dynamic ◽  
Uniqueness Of Solutions ◽  
Stochastic Dynamical System ◽  
Autonomous Case ◽  
Bounded Absorbing Set ◽  
Linear Damping ◽  
Longtime Behavior

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

scholarly journals Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Danxia Wang ◽  
Jianwen Zhang ◽  
Yinzhu Wang ◽  
Sufang Zhang
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Solutions ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Rotational Inertia ◽  
Absorbing Set ◽  
Viscous Effect ◽  
Structural Damping ◽  
Longtime Behavior

Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equationutt-uxxtt+uxxxx-σ(∫0l‍(ux)2dx)uxx-ϕ(∫0l‍(ux)2dx)uxxt=q(x), in [0,L]×R+with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.

Pull-In Dynamics of an Elastic Beam Actuated by Distributed Electrostatic Force

2003 ◽  
Author(s):  
Slava Krylov ◽  
Ronen Maimon
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Instability ◽  
Electrostatic Force ◽  
Normal Modes ◽  
Elastic Beam ◽  
Nonlinear Damping ◽  
Rotational Inertia ◽  
Electrostatic Forces ◽  
Spatial Region ◽  
Electrically Actuated

A detailed study of the transient nonlinear dynamics of an electrically actuated micron scale beam is presented. A model developed using the Galerkin procedure with normal modes as a basis accounts for the distributed nonlinear electrostatic forces, nonlinear distributed squeezed film damping forces, and rotational inertia of a mass carried by the beam. Special attention is paid to the dynamics of the beam near instability points. Results generated by the model and confirmed experimentally show that nonlinear damping leads to shrinkage of the spatial region where stable motion is realizable. The voltage that causes dynamic instability, in turn, approaches the static pull-in value.

Uniform decay rate estimates for the beam equation with locally distributed nonlinear damping

2021 ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Leonel G. Delatorre ◽  
Valéria N. Domingos Cavalcanti ◽  
Victor H. Gonzalez Martinez ◽  
Daiane C. Soares
Keyword(s):  
Decay Rate ◽  
Nonlinear Damping ◽  
Beam Equation ◽  
Uniform Decay

scholarly journals Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Erhan Pişkin
Keyword(s):  
Contraction Mapping ◽  
Blow Up ◽  
Nonlinear Damping ◽  
Contraction Mapping Principle ◽  
Beam Equation ◽  
Source Terms ◽  
Banach Contraction ◽  
Damping And Source Terms ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

AbstractWe consider the existence, both locally and globally in time, the decay and the blow up of the solution for the extensible beam equation with nonlinear damping and source terms. We prove the existence of the solution by Banach contraction mapping principle. The decay estimates of the solution are proved by using Nakao’s inequality. Moreover, under suitable conditions on the initial datum, we prove that the solution blow up in finite time.

Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force

2004 ◽  
Vol 126 (3) ◽  
pp. 332-342 ◽  
Author(s):  
Slava Krylov ◽  
Ronen Maimon
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Instability ◽  
Electrostatic Force ◽  
Normal Modes ◽  
Elastic Beam ◽  
Nonlinear Damping ◽  
Rotational Inertia ◽  
Electrostatic Forces ◽  
Spatial Region ◽  
Electrically Actuated

A detailed study of the transient nonlinear dynamics of an electrically actuated micron scale beam is presented. A model developed using the Galerkin procedure with normal modes as a basis accounts for the distributed nonlinear electrostatic forces, nonlinear squeezed film damping, and rotational inertia of a mass carried by the beam. Special attention is paid to the dynamics of the beam near instability points. Results generated by the model and confirmed experimentally show that nonlinear damping leads to shrinkage of the spatial region where stable motion is realizable. The voltage that causes dynamic instability, in turn, approaches the static pull-in value.

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