Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

Guiver, Christopher; Opmeer, Mark R.

doi:10.1016/j.sysconle.2010.07.002

Cited by 8 publications

(21 citation statements)

References 6 publications

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“…(1) It is technically convenient to start with the proof of statement (2). To this end, we show that, for the case when κ 1 = κ 2 = 0, the operator L is dissipative.…”

Section: Proof Of Theorem 24mentioning

confidence: 97%

“…(iii) The situation when the combination of two positive control parameters involves one diagonal element of the control matrix (α or β) and one co-diagonal element (κ 1 or κ 2 ) is more complicated. It is proven in statements (2) and (3) of theorem 2.4 that, for two cases when either α > 0, κ 1 > 0 and β = κ 2 = 0 or β > 0, κ 2 > 0 and α = κ 1 = 0, the stability result holds, i.e. the entire spectra of the corresponding operators are located in the open upper half-plane.…”

Section: Introductionmentioning

confidence: 94%

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Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

Shubov

2019

Proc. R. Soc. A.

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The Euler–Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped–free self-adjoint problem have been established.

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“…(1) It is technically convenient to start with the proof of statement (2). To this end, we show that, for the case when κ 1 = κ 2 = 0, the operator L is dissipative.…”

Section: Proof Of Theorem 24mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 94%

Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

Shubov

2019

Proc. R. Soc. A.

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“…Now we replace the free right-end conditions from Eq. (3) with the following boundary feedback control law [2,4]. Define the input and the output as…”

Section: The Initial-boundary Value Problem For the Euler-bernoulli Bmentioning

confidence: 99%

“…Based on the results of [1,2], the dynamics generator has a purely discrete spectrum, whose location on the complex plane is determined by the controls k 1 and k 2 . Having in mind the practical applications of the asymptotic formulas [3][4][5], we discuss the case of k 1 ≥ 0 and k 2 ≥ 0, such that |k 1 | þ |k 2 | . 0 (see Proposition 2).…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we mention that the feedback control of beams is a well-studied area [6], with multiple applications to the control of robotic manipulators, long and slender aircraft wings, propeller blades, large space structure [7,8], and the dynamics of carbon nanotubes [9]. The analysis of a classical beam model with nonstandard feedback control law that originated in engineering literature [4,[10][11][12] may be of interest for both analysts and practitioners. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Analysis and Numerical Investigation of a Flexible Structure with Nonconservative Boundary Data

Shubov¹,

Kindrat²

2020

Functional Calculus

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Analytic and numerical results of the Euler-Bernoulli beam model with a twoparameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters (k 1 and k 2 ) relates a two-dimensional input vector (the shear and the moment at the right end) and the observation vector (the time derivatives of displacement and the slope at the right end). The following results are contained in the paper. First, high accuracy numerical approximations for the eigenvalues of the discretized differential operator (the dynamics generator of the model) have been obtained. Second, the formula for the number of the deadbeat modes has been derived for the case when one control parameter, k 1 , is positive and another one, k 2 , is zero. It has been shown that the number of the deadbeat modes tends to infinity, as k 1 ! 1 þ and k 2 ¼ 0. Third, the existence of double deadbeat modes and the asymptotic formula for such modes have been proven. Fourth, numerical results corroborating all analytic findings have been produced by using Chebyshev polynomial approximations for the continuous problem.

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Stabilisation of Timoshenko beam system with delay in the boundary control

Wang

2013

International Journal of Control

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In this paper, we consider the exponential stabilisation problem of a Timoshenko beam with delay in boundary control. Suppose that the controller outputs of the forms α 1 u 1 (t) + β 1 u 1 (t − τ ) and α 2 u 2 (t) + β 2 u 2 (t − τ ) where u 1 (t) and u 2 (t) are the inputs of boundary controllers. In the past, most of the stabilisation results are required α j > β j > 0, j = 1, 2. In the present paper, we shall give a new dynamic feedback control law that makes the system exponential stabilisation ∀τ > 0 provided that |α j | = |β j |(j = 1, 2). The exponential stabilisation result is proved via test of exact observability of the system.

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Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

Abstract: We show that a non-dissipative feedback that has been shown in the literature to exponentially stabilize an Euler-Bernoulli beam makes a Rayleigh beam and a Timoshenko beam unstable.

Cited by 8 publications

References 6 publications

Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

Spectral Analysis and Numerical Investigation of a Flexible Structure with Nonconservative Boundary Data

Stabilisation of Timoshenko beam system with delay in the boundary control

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