On the stabilizability of a structural acoustic model which incorporates shear effects in the structural component

In this paper we consider a linear three-dimensional structural acoustic model which takes account of displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the deflections of the structural component of the structure are governed by the Reissner–Mindlin plate equations. We show strong stabilization of the coupled model without incorporating viscous or boundary damping in the equations for the gas dynamics and without imposing geometric conditions. It turns out that damping is needed in the interior of the plate, to which end Kelvin–Voigt damping is introduced in the plate equations. As our main tool we use a resolvent criterion for strong stability due to Tomilov.     

Global dynamics and control of a nonlinear body equation with strong structural damping

A nonlinear hinged extensible elastic body equation with strong structural damping and Balakrishnan-Taylor damping of full exponent is studied as a general model for large space structures of higher dimensions. In this paper, the absorbing sets and flat inertial manifold are obtained for this nonlinear body equation. The control spillover problem associated with the stabilization of this equation is resolved by constructing a linear finite dimensional feedback, control based on the existence of inertial manifolds of the uncontrolled equation. Moreover, the results obtained are robust with respect to the uncertainty in structural parameters. Supported by the National Natural Science Foundation of China (No. 19701023)     

On a structural acoustic model which incorporates shear and thermal effects in the structural component

In this paper we consider a structural acoustic model which takes account of thermal effects over and above displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the structural medium is a Reissner-Mindlin plate into which an additional degree of freedom, viz. temperature variation in the plate, has been introduced and the constitutive equations for the structural acoustic model couple parabolic dynamics with hyperbolic dynamics. We show unique solvability of the mathematical model and investigate the effect of the presence of thermal effects on the mechanical dissipation devices needed to attain uniform stabilization of the two-dimensional model in which the structural component is a Timoshenko beam. It turns out that, as in linear structural acoustic models which use the Euler-Bernoulli equation or the Kirchoff equation to describe the deflections of the thermo-elastic structural medium, uniform stabilization of the energy associated with the model can be attained without introducing mechanical dissipation at the free edge of the beam. Open problems with regard to the stabilization of the three-dimensional model are outlined.     

Boundary stabilization of a 3-dimensional structural acoustic model

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).     

On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping

We study the well-posedness and decay properties of a onedimensional thermoelastic laminated beam system either with or without structural damping, of which the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is wellposed by using the Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method. When the speeds are not equal, whether the system without structural damping may has polynomial stability is left as an open problem.     

Exponential decay for elastic systems with structural damping and infinite delay

ABSTRACT

In this paper, we consider nonlinear evolution equations of second order in Banach spaces involving unbounded delay, which can model an elastic system with structural damping involving infinite delays. By using fixed point for condensing maps, we prove the existence and exponential decay of mild solutions. The obtained results can be applied to the nonlinear vibration equation of elastic beams with structural damping and infinite delay.     

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

A rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time‐evolving displacements along the flexible portion Γ₀ of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter η∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ₀. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth‐order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of . By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.     

Mechatronic system with shunted piezoelectric damper modelled with structural damping

Paper presents analysis of an one-dimension flexural vibrating mechatronic system. The considered system is a cantilever beam with a piezoelectric transducer bonded to the beam's surface. An external electric circuit is adjoined to the transducer's clamps in order to damp vibrations. System was analyzed on the basis of an approximate Galerkin method. Verification and assumptions of the approximate method were described in the previous papers where analysis of the mechatronic system with piezoelectric shunt damper was presented. Structural damping of all system's components was being taken into consideration. Rheological properties were introduced using Kelvin-Voigt model of materials. Influences of component's structural damping coefficients values on the system's dynamic flexibility were defined. Obtained results were presented on 3D graphs as dynamic flexibility dependence on the structural damping coefficient and frequency of an external force that was applied to the system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有强结构阻尼的非线性梁方程的整体动力学和控制

在本文中，我们考虑了高维具有强结构阻尼和全指数Ｂａｌａｋｒｉｓｈｎａｎ－Ｔａｙｌｏｒ阻尼的非线性固定边界可伸展的弹性梁方程，得到它的吸收集和平坦惯性流形的存在性．基于无控制方程的惯性流形的存在性，得到了相应的溢出问题的有限维反馈镇定控制．进而，此结果关于结构参数的不确定性是鲁棒的．     

Uniform stability in structural acoustic models with flexible curved walls

The aim of this paper is twofold. First, we develop an explicit extension of the Kirchhoff model for thin shells, based on the model developed by Michel Delfour and Jean-Paul Zolésio. This model relies heavily on the oriented distance function which describes the geometry. Once this model is established, we investigate the uniform stability of a structural acoustic model with structural damping. The result no longer requires that the active wall be a plate. It can be virtually any shell, provided that the shell is thin enough to accommodate the curvatures.     

A classification of structural dissipations for evolution operators

In this paper, we study the asymptotic profile of the solution for a σ‐evolution equation with a time‐dependent structural damping. We introduce a classification of the damping term, which clarifies whether the solution behaves like the solution to an anomalous diffusion problem. We call this damping effective, whereas we say that the damping is noneffective when the solution shows oscillations in its asymptotic profile that cannot be neglected. Our classification shows a completely new interplay between the strength of the damping and the long time behavior of its coefficient. Copyright © 2015 John Wiley & Sons, Ltd.     

On stabilization of an elastic system by fast-oscillating time-variant feedback

We study a class of elastic systems described by a (hyperbolic) second-order partial differential equation. Our working example is the equation of a vibrating string subject to a destabilizing linear disturbance. Our main goal is to establish conditions for stabilization and asymptotic stabilization of the equilibrium configuration of the string by applying to it fast oscillating controlled force. In the first situation studied we assume that the string is subject to damping; after that we consider the same system without damping. We extend the tools of high-order averaging and of chronological calculus for studying the stability of this distributed parameter system.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupled parabolic–hyperbolic Stokes–Lamé PDE system: limit behaviour of the resolvent operator on the imaginary axis

This article is focused on an established, genuinely physical fluid-structure interaction model, whereby the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic–hyperbolic system of two partial differential equations in three dimensions with non-standard coupling at the boundary interface: the (dynamic) Stokes system (parabolic, modelling the fluid) and the Lamé system (hyperbolic, modelling the structure). This system generates a contraction semigroup on the natural energy space [G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: explicit semigroup generator and its spectral properties, Fluids and Waves, Amer. Math. Soc. Contemp. Math. 440 (2007), pp. 15–59] (canonical model) and [G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Series S, 2(3) (2009), pp. 417–447]. The boundary interface may or may not include a ‘damping’ (or dissipative) term. If damping is active on the entire interface, then uniform (exponential) stabilization is ensured, regardless of the geometry of the structure [G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst. 22(4) 2008, pp. 817–835, special issue, invited paper] (canonical model) and [G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system, J. Evol. Eqns 9(2009), pp. 341–370]. This article emphasizes the case of, at most, partial damping. At any rate, the main result is a precise uniform-operator limit behaviour of the resolvent operator of the semigroup generator on the imaginary axis of interest in itself, which holds true with or without damping. It, in turn, then implies a fortiori strong stability results: most notably, on the whole state space, under at least partial damping at the interface; and, in the absence of damping, on the whole state space, after factoring out an explicit one-dimensional null eigenspace, at least for a large class of geometries of the structure: these are characterized by a uniqueness property of a special over-determined elliptic problem.     

Boundary stabilization of structural acoustic interactions with interface on a Reissner–Mindlin plate

Boundary stabilization of a structural acoustic model comprised of a wave and a Reissner–Mindlin plate is addressed. Both the components of the dynamics are subject to localized nonlinear boundary damping: the acoustic dissipative feedback is restricted to the flexible boundary and only a portion of the rigid wall; the plate is damped only on a segment of its edge.Derivation of stabilization/observability inequalities for a coupled system requires weighted energy multipliers dependent on the geometry of the domain, and special microlocal trace estimates for the Reissner–Mindlin plate. The behavior of the energy at infinity can be quantified by a solution to an explicitly constructed nonlinear ODE. The nonlinearities in the feedbacks may include sub- and superlinear growth at infinity, in which case the decay scheme presents a trade-off between the regularity of trajectories and attainable uniform dissipation rates of the finite energy.     

Adaptive stabilization of the sine‐Gordon equation by boundary control

This paper is concerned with adaptive global stabilization of the sine‐Gordon equation without damping by boundary control. An adaptive stabilizer is constructed by the concept of high‐gain output feedback. The closed‐loop system is shown to be locally well‐posed by the Banach fixed point theorem and then to be globally well‐posed by the Lyapunov method. Moreover, using a multiplier method global exponential stabilization of the system is proved. Copyright © 2004 John Wiley & Sons, Ltd.     

Analyticity and exponential stability of semigroups for the elastic systems with structural damping in Banach spaces

In this paper, we consider a second order evolution equation in a Banach space, which can model an elastic system with structural damping. New forms of the corresponding first order evolution equation are introduced, and their well-posed property is proved by means of the operator semigroup theory. We give sufficient conditions for analyticity and exponential stability of the associated semigroups.     

Spectral analysis of a high-order Hermitian algorithm for structural dynamics

The spectral analysis of an efficient step-by-step direct integration algorithm for the structural dynamic equation is presented. The proposed algorithm is formulated in terms of two Hermitian finite difference operators of fifth-order local truncation error and it is unconditionally stable with no numerical damping presenting a fourth-order truncation error for period dispersion (global error). In addition, although it is in competition with higher-order algorithms presented in the literature, the computational effort is similar to that of the classical second-order Newmark’s method. The numerical application for nonlinear structural dynamic problems is also considered.     

Boundary feedback stabilization of coupled sine-Gordon equations

^** Email: koba{at}cntl.kyutech.ac.jp^*** Email: sakamoto{at}cntl.kyutech.ac.jp This paper is concerned with global stabilization of the systemgoverned by coupled sine-Gordon equations without damping. Astabilizer is constructed by boundary velocity feedback. Theclosed-loop system is shown to be well posed by the non-linearsemigroup approach. Moreover, using a multiplier method, globalexponential stabilization of the closed-loop system is proved.