Dynamical Analysis of an Axially Vibrating Nanorod

Heidari, Hanif

doi:10.12732/ijam.v29i2.9

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…The first example consists in the following probem (see [, Chap. 8], [] and [, (32)])

({, \begin{matrix} u_{t t} - a u_{x x t t} - u_{x x} + α u_{t} = 0 & in (0, L) \times (0, \infty), \\ u (0, t) = u (L, t) = 0, & in (0, \infty), \\ u false(\cdot, 0 false) = u_{0}, u_{t} false(\cdot, 0 false) = u_{1}, & in (0, L), \end{matrix})

where a and L are positive constants and α is a non‐negative function that belongs to

L^{\infty} false(0, L false)

. Such a problem enters into our framework by taking

V = H_{0}^{1} false(0, L false)

A u = - u_{x x}, A_{0} u = - a u_{x x}, \forall u \in V,

that are clearly linear self‐adjoint and positive operators from

H_{0}^{1} false(0, L false)

into its dual

H^{- 1} false(0, L false)

.…”

Section: Motivationmentioning

confidence: 99%

“…Remark If B is bounded from U into H and if

D false(scriptA false) = D false(A_{0} false)

, then the operator

scriptA

is bounded from

D (A) \times D (A)

consequently for an initial datum

U_{0} \in D false(scriptA false) \times D false(scriptA false)

, problem has a strong solution

U \in C^{1} false([0, \infty), D (A) \times D (A)

, see Theorem 2 of [] in the case of problem without damping (

α = 0

).…”

Section: Well‐posedness Of the Systemmentioning

confidence: 99%

“…Let us further mention that all references mentioned above are devoted to modelling or to some numerical illustrations. The mathematical analysis of model can be found in [], where the well‐posedness of the problem is considered and a spectral analysis is performed. Due to the attrativity of nanostructures and their potential applications, the understanding of their dynamics is crucial.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stabilization and asymptotic behavior of non local damped wave equations

Nicaise

2019

Z Angew Math Mech

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“…The first example consists in the following probem (see [, Chap. 8], [] and [, (32)])

({, \begin{matrix} u_{t t} - a u_{x x t t} - u_{x x} + α u_{t} = 0 & in (0, L) \times (0, \infty), \\ u (0, t) = u (L, t) = 0, & in (0, \infty), \\ u false(\cdot, 0 false) = u_{0}, u_{t} false(\cdot, 0 false) = u_{1}, & in (0, L), \end{matrix})

where a and L are positive constants and α is a non‐negative function that belongs to

L^{\infty} false(0, L false)

. Such a problem enters into our framework by taking

V = H_{0}^{1} false(0, L false)

A u = - u_{x x}, A_{0} u = - a u_{x x}, \forall u \in V,

that are clearly linear self‐adjoint and positive operators from

H_{0}^{1} false(0, L false)

into its dual

H^{- 1} false(0, L false)

.…”

Section: Motivationmentioning

confidence: 99%

“…Remark If B is bounded from U into H and if

D false(scriptA false) = D false(A_{0} false)

, then the operator

scriptA

is bounded from

D (A) \times D (A)

consequently for an initial datum

U_{0} \in D false(scriptA false) \times D false(scriptA false)

, problem has a strong solution

U \in C^{1} false([0, \infty), D (A) \times D (A)

, see Theorem 2 of [] in the case of problem without damping (

α = 0

).…”

Section: Well‐posedness Of the Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stabilization and asymptotic behavior of non local damped wave equations

Nicaise

2019

Z Angew Math Mech

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show abstract

“…The micro and nanoscale physical phenomena have different properties from macroscale [1][2][3]. Carbon nanotubes (CNTs) are allotropes of carbon.…”

Section: Introductionmentioning

confidence: 99%

“…Nonlocal longitudinal vibration of viscoelastic-coupled doublenanorod systems is studied by Karlicic et al [7]. Heidari investigates controllability and stability analysis of a nanorod [2].…”

Section: Introductionmentioning

confidence: 99%

Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod

Heidari

Zwart

2019

Mathematical and Computer Modelling of Dynamical Systems

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Analysis of nonlocal axial vibration in a nanorod is a crucial subject in science and engineering because of its wide applications in nanoelectromechanical systems. The aim of this paper is to show how these vibrations can be modelled within the framework of port-Hamiltonian systems. It turns out that two port-Hamiltonian descriptions in physical variables are possible. The first one is in descriptor form, whereas the second one has a non-local Hamiltonian density. In addition, it is shown that under appropriate boundary conditions these models possess a unique solution which is non-increasing in the corresponding 'energy', i.e., the associated infinitesimal generator generates a contraction semigroup on a Hilbert space, whose norm is directly linked to the Hamiltonian.

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Nonlocal longitudinal vibration in a nanorod, a system theoretic analysis

Heidari¹,

Zwart²

2022

Math. Model. Nat. Phenom.

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Analysis of longitudinal vibration in a nanorod is an important subject in science and engineering due to its vast application in nanotechnology. This paper introduces a port-Hamiltonian formulation for the longitudinal vibrations in a nanorod, which shows that this model is essentially hyperbolic. Furthermore, it investigates the spectral properties of the associated system operator. Standard distributed control and feedback are shown not to be controllable nor stabilizing.

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Dynamical Analysis of an Axially Vibrating Nanorod

Cited by 3 publications

References 15 publications

Stabilization and asymptotic behavior of non local damped wave equations

Stabilization and asymptotic behavior of non local damped wave equations

Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod

Nonlocal longitudinal vibration in a nanorod, a system theoretic analysis

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