Boundary Damping Research Articles

The Exponential Stability Result of an Euler-Bernoulli Beam Equation with Interior Delays and Boundary Damping

We study the exponential stability of Euler-Bernoulli beam with interior time delays and boundary damping. At first, we prove the well-posedness of the system by the C0 semigroup theory. Next we study the exponential stability of the system by constructing appropriate Lyapunov functionals. We transform the exponential stability issue into the solvability of inequality equations. By analyzing the relationship between delays parameters α and damping parameters β, we describe (β,α)-region for which the system is exponentially stable. Furthermore, we obtain an estimation of the decay rate λ⁎.

On behavior of solutions to a class of nonlinear hyperbolic inverse source problem

This article is concerned with a class of hyperbolic inverse source problem with memory term and nonlinear boundary damping. Under appropriate assumptions on the initial data and parameters in the equation, we establish two results on behavior of solutions. At first we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity and finally a blow-up result is established for certain solution with positive initial energy.

Well-posedness of Mindlin–Timoshenko Plate with Nonlinear Boundary Damping and Sources

This study addresses well-posedness of a Mindlin---Timoshenko (MT) plate model that incorporates nonlinear viscous damping and nonlinear source term in Neumann boundary conditions. The main results verify local and global existence of solutions as well as their continuous dependence on the initial data in appropriate function spaces. Along with (Pei et al. in J Math Anal Appl 418(2):535---568, 2014, in Nonlinear Anal 105:62---85, 2014) this work completes the fundamental well-posedness theory for MT plates under the interplay of damping and source terms acting either in the interior or on the boundary of the plate.

Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback

We study the stabilization problem for the wave equation with localized Kelvin--Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the result of [19] where, in a more general setting, the case of distributed structural damping is considered.

Asymptotics for wave equations with Wentzell boundary conditions and boundary damping

We are concerned with linear wave equations with Wentzell boundary conditions of dynamical type, where only one velocity feedback force acts on the Wentzell boundary. By using the theory of strongly continuous semigroups of linear operators, we prove that the energies of the solutions are strongly stable. Moreover, we show in the one dimensional case that there are solutions decaying at arbitrarily slow rates.

Asymptotic stability for a class of boundary control systems with non-linear damping

The asymptotic stability of boundary controlled port-Hamiltonian systems defined on a 1D spatial domain interconnected to a class of non-linear boundary damping is addressed. It is shown that if the port-Hamiltonian system is approximately observable, then any boundary damping which behaves linear for small velocities asymptotically stabilizes the system.

On the Transverse Vibrations of Strings and Beams on Semi-Infinite Domains

In this paper, we study the transverse vibrations of a string and of a beam which are infinitely long in one direction. These vibration problems can be used as a toy model for rain-wind induced oscillations of cables. In order to suppress undesired vibrations in the string (or beam), dampers are used at the boundary. The main aim of this paper is to show how solutions for these string and beam problems on a semi-infinite domain can be computed. We derive explicit solutions for a linear string problem which is attached to a mass-spring-dashpot system at x = 0 by using the D’Alembert method, and for a transversally vibrating beam problem which has a pinned, sliding, clamped or damping boundary, respectively, at x = 0 by using the method of Laplace transforms. It will be shown how waves are reflected for different types of boundaries.

Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition

The paper studies the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary conditionutt-Δu+Δ2u-Δut-Δg(u)=f(x). The main result is concerned with nonlinearitiesg(u)with supercritical growth. In that case, we construct a bounded absorbing set with further regularity and obtain quasi-stability estimates. Then the exponential attractor is established in natural energy spaceV2×H.

General Decay Rate for Nonlinear Thermoviscoelastic System with a Weak Damping and Nonlinear Source Term

In this paper, we consider a nonlinear system of thermoviscoelastic with a nonlinear boundary damping and nonlinear source, in a bounded domain Ω, under appropriate assumptions imposed on relaxation function and with certain initial data, we establish the general decay rate of the solution energy, from which the usual exponential and polynomial decay are only special cases. This work generalizes and improves earlier results in the literature.

Uniform stabilization of wave equation with localized damping and acoustic boundary condition

In this paper we deal with stability of the nonlinear wave equation with acoustic boundary conditions to non-locally reacting boundary.

Universal Characteristic Frequency Equation for Cable Transverse Component System and Its Universal Numerical Solution

AbstractIt is meaningful to develop an accurate universal approach to solve the dynamical characteristic problem of a cable system combined with additional components used in real engineering. A segmented dynamic governing equation is presented for a taut shallow cable supported in the middle by a transverse component. Then, a universal frequency characteristic equation is proposed that considers all the influence factors, including the flexural rigidity, sagging, inclination angle, finite rigidity or damped boundary condition, and the intermediate supporting component. A general numerical solution is presented by giving an analytical derivative expression for the dynamic stiffness of the cable system. As a result, the modal parameters of an arbitrary complicated cable system can be obtained via the root of the equation. Finally, the accuracy of the proposed approach is verified using numerical cases, and its validity is also proved by applying it to case studies of several typical cable systems.

Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

We consider the problems of trajectory generation and tracking for general 2×2 systems of first-order linear hyperbolic PDEs with anti-collocated boundary input and output. We solve the trajectory generation problem via backstepping. The reference input, which generates the desired output, incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by Vazquez, Krstic, and Coron for the backstepping stabilization of 2×2 linear hyperbolic systems. We apply our approach to a wave PDE with indefinite in-domain and boundary damping. For tracking the desired trajectory we employ a PI control law on the tracking error of the output. We prove exponential stability of the closed-loop system, under the proposed PI control law, when the parameters of the plant and the controller satisfy certain conditions, by constructing a novel “non-diagonal” Lyapunov functional. We demonstrate that the proposed PI control law compensates in the output the effect of in-domain and boundary disturbances. We illustrate our results with numerical examples.

Experiment inspired numerical modeling of sediment concentration over sand–silt mixtures

A series of flume experiments has been conducted to investigate sediment transport of sand–silt mixtures in both wave-only and wave-with-current conditions. Two types of sediments collected from a typical silty tidal flat were used: a silt-sized mixture with median grain size of 46μm, and a very fine sand-sized mixture with median grain size of 88μm. A high concentration layer (HCL) was observed near the bottom together with ripples under wave-only conditions. Sediment concentrations inside the HCL are quasi-stationary with the bulk Richardson number approaching a constant value. The thickness of the HCL can be scaled with approximately two times the damped wave boundary layer thickness. For the concentration profiles, we find that the vertical profile of the silt concentration appears different from the profile of the sand concentration, since the silt concentration decreases logarithmically within HCL, while homogeneously distributes outside the HCL. Finally, the reference concentration formulation of van Rijn (2007b) was recalibrated for the silt classes and applied in a multi-fraction model to predict the vertical concentration profile for silt and sand classes. The results show a promising agreement with the measurements, for both wave-only and wave-with-current conditions.

Problem on vibration of a bar with nonlinearsecond-order boundary damping

In this paper, we study the initial-boundary problem with nonlinear dynam- ical boundary condition for the pseudohyperbolic equation. This problem repre- sents a mathematical model of longitudinal vibration in a thick short bar with dynamic nonlinear second-order boundary damping. The existence and unique- ness of a generalized solution are proved. The proof is based on a priori estimates and Galerkin procedure. This approach allows to construct approximation in the suitable for practical application form.

Acoustic multi-stopband metamaterial plates design for broadband elastic wave absorption and vibration suppression

This paper presents the modeling technique, working mechanism and design guidelines for acoustic multi-stopband metamaterial plates for broadband elastic wave absorption and vibration suppression. The metamaterial plate is designed by integrating two-degree of freedom (DOF) mass–spring subsystems with an isotropic plate to act as vibration absorbers. For an infinite metamaterial plate without damping, a working unit is modeled using the extended Hamilton׳s principle, and two stopbands are observed through dispersion analysis on the averaged three-DOF model. For a finite metamaterial plate with boundary conditions and damping, shear-deformable conforming plate elements are used to model the whole plate, and stopbands are investigated by frequency response analysis and transient analysis. Influences of absorbers׳ resonant frequencies and damping ratios, plate׳s boundary conditions and dimensions, and effective plate-absorber vibration modes are thoroughly investigated. Results show that the metamaterial plate is essentially based on the concept of conventional vibration absorbers. The local resonance of the two-DOF subsystems generates two stopbands, and the inertial forces generated by the resonant vibrations of absorbers straighten the plate and attenuate/stop wave propagation. Each stopband׳s bandwidth can be increased by increasing the absorber mass and/or reducing the average mass of isotropic plate in each working unit. Moreover, while a low damping ratio for the primary absorber can guarantee absorbers׳ quick response to transient excitations, a high damping ratio for the secondary absorber can combine the two stopbands into a wide one. In the end, a sensitivity analysis on absorber resonant frequencies is conducted and relatively large sensitivity is found at the two stopband regions.

Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as ``thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.

On the Transverse, Low Frequency Vibrations of a Traveling String With Boundary Damping

In this paper, we study the free transverse vibrations of an axially moving (gyroscopic) material represented by a perfectly flexible string. The problem can be used as a simple model to describe the low frequency oscillations of elastic structures such as conveyor belts. In order to suppress these oscillations, a spring–mass–dashpot system is attached at the nonfixed end of the string. In this paper, it is assumed that the damping in the dashpot is small and that the axial velocity of the string is small compared to the wave speed of the string. This paper has two main objectives. The first aim is to give explicit approximations of the solution on long timescales by using a multiple-timescales perturbation method. The other goal is to construct accurate approximations of the lower eigenvalues of the problem, which describe the oscillation and the damping properties of the problem. The eigenvalues follow from a so-called characteristic equation obtained by the direct application of the Laplace transform method to the initial-boundary value problem. Both approaches give a complete and accurate picture of the damping and the low frequency oscillatory behavior of the traveling string.

General decay of solutions for a viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary damping-source interactions

A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions imposed on the source and the damping, we establish uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.

Blow-up for wave equation with weak boundary damping and source terms

In this paper, we consider the wave equation with nonlinear boundary damping and source terms. This work is devoted to prove a finite time blow-up result under suitable condition on the initial data and positive initial energy. The main goal of the present paper is to generalize our previous result in Ha (2012) treating the boundary damping term in a more general setting.

Exponential stabilization of a viscoelastic wave equation with dynamic boundary conditions

In this paper, we study the stabilization of a semi-linear viscoelastic wave equation subject to semi-linear and dynamical boundary conditions. The kernels used are of strongly positive definite type. We prove that internal and boundary memory damping are strong enough, via transmission process (\({u|_{\Gamma} = v}\)), to stabilize the whole system.