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.     

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.     

Uniform stability for the Timoshenko beam with tip load

In this paper we consider a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. We show that uniform stabilization of the model which includes the rotary inertia of the tip load can be obtained when feedback boundary moment and force controls are applied at the point of contact between the beam and the tip load. However, in the presence of the load stabilization is “slower” and subject to a restriction on the boundary data at the free end of the beam.     

Stabilization of the nonuniform Timoshenko beam

In this paper we consider a Timoshenko beam with variable physical parameters, we prove that the model can be stabilize by one control force for both internal and boundary cases.     

Dynamic boundary stabilization of a Reissner–Mindlin plate with Timoshenko beam

This paper is concerned with well‐posedness results for a mathematical model for the transversal vibrations of a two‐dimensional hybrid elastic structure consisting of a rectangular Reissner–Mindlin plate with a Timoshenko beam attached to its free edge. The model incorporates linear dynamic feedback controls along the interface between the plate and the beam. Classical semigroup methods are employed to show the unique solvability of the coupled initial‐boundary‐value problem. We also show that the energy associated with the system exhibits the property of strong stability. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Boundary stabilization of nonuniform Timoshenko beam

In this paper,the boundary stabiligstion of tbe Timoshenko equation of a nononiform beam,with clarrmped boundary condition at one end and witb bending moment and shear force controls at the other end, is considered. It is proved that the system is exponentially stahilizable when the bending moment and shear force controls are simultaneously appiied. The frequency domain method and the multiplier technique are used.     

Timoshenko梁的边界反馈一致镇定

该文研究在Timoshenko梁两端施加边界反馈控制的镇定问题.在某些线性边界反馈作用下,通过分析闭环系统算子的谱,并利用频域方法证明了相应的闭环系统的一致稳定性.     

Exponential decay of Timoshenko beam with locally distributed feedback

The problem of exponential stabilization of a nonuniform Timoshenkobeam with locally distributed controls is investigated. Withoutthe assumption of different wave speeds, it is shown that, undersome locally distributed controls, the vibration of the beamdecays exponentially. The proof is obtained by using a frequencymultiplier method.     

Buckling of a Timoshenko beam on elastic foundation with uncertain input data

A monotone dependence of the critical buckling load of a simplysupported Timoshenko–Mindlin beam both on the shear correctionfactor and on the stiffness of foundation is proved. Then theworst-scenario method is employed to find the ‘most dangerous’input data.     

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.     

The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels

In this paper we investigate the longtime behavior of the mathematical model of a homogeneous viscoelastic plate based on Reissner-Mindlin deformation shear assumptions. According to the approximation procedure due to Lagnese for the Kirchhoff viscoelastic plate, the resulting motion equations for the vertical displacement and the angle deflection of vertical fibers are derived in the framework of the theory of linear viscoelasticity. Assuming that in general both Lame's functions, λ and μ, depend on time, the coupling terms between the equations of displacement and deflection depend on hereditary contributions. We associate to the model a nonlinear semigroup and show the behavior of the energy when time goes on. In particular, assuming that the kernels λ and μ decay exponentially, and not too weakly with respect to the physical properties considered in the model, then the energy decays uniformly with respect to the initial conditions; i.e., we prove the existence of an absorbing set for the semigroup associated to the model.     

Boundary feedback stabilization of Timoshenko beam with boundary dissipation

Nonlinear boundary feedback control to a Timoshenko beam is studied. Under some nonlinear boundary feedback controls, the nonlinear semi-group theory is used to show the well-posedness for the correspnding closed loop system. Then by using the energy perturbed method, it is proved that the vibration of the beam under the proposed control actions decays asymptotically or exponentially as t→∞. Project supported by the National Natural Science Foundation of China.     

The Riesz basis property of a Timoshenko beam with boundary feedback and application

The Riesz basis property of the generalized eigenvector systemof a Timoshenko beam with boundary feedback is studied. Firstly,two auxiliary operators are introduced, and the Riesz basisproperty of their eigenvector systems is proved. This propertyis used to show that the generalized eigenvector system of aTimoshenko beam with some linear boundary feedback forms a Rieszbasis in the corresponding state space. Finally, it is concludedthat the closed loop system exhibits exponential stability.     

Solvability of a Reissner-Mindlin-Timoshenko plate-beam vibration model

In this paper, we consider the vibration of a plate–beamsystem consisting of a Reissner–Mindlin plate and a Timoshenkobeam. A variational form is derived directly from the equationsof motion and the constitutive equations. We show how an existenceresult for a general linear vibration problem in variationalform may be applied to the weak variational problem for thissystem.     

Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam

We introduce a set of conserved quantities of energy‐type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well‐posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd.     

A difference scheme for solving the Timoshenko beam equations with tip body

In this article, a Timoshenko beam with tip body and boundary damping is considered. A linearized three-level difference scheme of the Timoshenko beam equations on uniform meshes is derived by the method of reduction of order. The unique solvability, unconditional stability and convergence of the difference scheme are proved. The convergence order in maximum norm is of order two in both space and time. A numerical example is presented to demonstrate the theoretical results.     

Riesz basis property of the generalized eigenvector system of a Timoshenko beam

The Riesz basis property of the generalized eigenvector system of a Timoshenko beam with boundary feedback controls appliedto two ends is studied in this paper. The spectral property of the operator A determined by the closed loop system is investigated.It is shown that operator A has compact resolvent and generatesa C₀ semigroup, and its spectrum consistsof two branches and has two asymptotes under some conditions.Furthermore it is proved that the sequence of all generalizedeigenvectors of the system principal operator forms a Rieszbasis for the state Hilbert space.     

A locking-free scheme for the LQR control of a Timoshenko beam

In this paper we analyze a locking-free numerical scheme for the LQR control of a Timoshenko beam. We consider a non-conforming finite element discretization of the system dynamics and a control law constant in the spatial dimension. To solve the LQR problem we seek a feedback control which depends on the solution of an algebraic Riccati equation. An optimal error estimate for the feedback operator is proved in the framework of the approximation theory for control of infinite dimensional systems. This estimate is valid with constants that do not depend on the thickness of the beam, which leads to the conclusion that the method is locking-free. In order to assess the performance of the method, numerical tests are reported and discussed.