Stabilization of a type III thermoelastic Timoshenko system in the presence of a time‐distributed delay

In this paper, we investigate a Timoshenko‐type system of thermoelasticity of type III in the presence of a distributed delay. We prove the well‐posedness and two exponential stability results in the presence as well as in the absence of an extra frictional damping. In case of absence of the frictional damping, our stability result is obtained under the equal‐speed of propagation and a smallness condition on the weight of delay.     

Multidimensional thermoelasticity for nonsimple materials – well-posedness and long-time behavior

An initial-boundary value problem for the multidimensional type III thermoelaticity for a nonsimple material with a center of symmetry is considered. In the linear case, the well-posedness with and without a (second-order in space) Kelvin–Voigt and/or frictional damping in the elastic part as well as the lack of exponential stability in the elastically undamped case are proved. Further, a frictional damping for the elastic component is shown to lead to exponential stability. A Cattaneo-type hyperbolic relaxation for the thermal part is introduced and the well-posedness and uniform stability under a nonlinear frictional damping are obtained using a compactness-uniqueness-type argument. Additionally, a connection between exponential stability and exact observability of unitary strongly continuous groups is established.     

UNIFORM DECAY IN WEAKLY DISSIPATIVE TIMOSHENKO SYSTEM WITH INTERNAL DISTRIBUTED DELAY FEEDBACKS

In this paper we consider one-dimensional Timoshenko system with linear frictional damping and a distributed delay acting on the displacement equation.Under suitable assumptions on the weight of the delay and the wave speeds,we establish the well-posedness of the system and show that the dissipation through the frictional damping is strong enough to uniformly stabilize the system even in the presence of delay.     

A new stability number of the Bresse‐Cattaneo system

In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L²‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results     

SOME STABILITY RESULTS FOR TIMOSHENKO SYSTEMS WITH COOPERATIVE FRICTIONAL AND INFINITE-MEMORY DAMPINGS IN THE DISPLACEMENT

In this paper, we consider a vibrating system of Timoshenko-type in a onedimensional bounded domain with complementary frictional damping and infinite memory acting on the transversal displacement. We show that the dissipation generated by these two complementary controls guarantees the stability of the system in case of the equal-speed propagation as well as in the opposite case. We establish in each case a general decay estimate of the solutions. In the particular case when the wave propagation speeds are different and the frictional damping is linear, we give a relationship between the smoothness of the initial data and the decay rate of the solutions. By the end of the paper, we discuss some applications to other Timoshenko-type systems.     

On the control of n-dimensional thermoelastic system

In this article, we consider n-dimensional thermoelastic system with a nonlinear weak frictional damping. We establish an explicit and general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term.     

A general stability result for the semilinear viscoelastic wave model under localized effects

Our main goal in the present work is to address an integro-differential model under localized viscoelastic and frictional effects arising in the Boltzmann theory of viscoelasticity. More precisely, we consider a general version in the history context of the pioneer localized viscoelastic problem approached by Cavalcanti and Oquendo (2003) in the null history scenario, and more recently by Cavalcanti et al. (2018) in the history framework. By means of a new observability inequality, we prove a general stability result to the model under a weaker assumption on the localized frictional damping and a slower condition on the decreasing memory kernel (of polynomial type) than the previously mentioned works. To achieve such stability results, we still work in a general setting by removing the assumption on complementary damping mechanisms and show, in some reasonable situations concerning the density coefficient, that the localized viscoelastic effect is enough to ensure the general stability (of polynomial type) to the problem.     

The optimal decay rate for a weak dissipative Bresse system

In this work we consider the Bresse system with frictional damping operating only on the angle displacement and we show that under a certain assertion the solution decays polynomially and the decay rate is optimal.     

Frictional versus Kelvin‐Voigt damping in a transmission problem

We consider 2 transmission problems. The first problem has 2 damping mechanisms acting in the same part of the body, one of frictional type and other of Kelvin‐Voigt type. In this case, we show that, even though it has too much dissipation, the semigroup is not exponentially stable. The second problem also has those damping terms but they act in complementary parts of the body. For this case, we show that the semigroup is exponentially stable and it is not analytic.     

How far can relaxation functions be increasing in viscoelastic problems?

It is known that when we add a viscoelastic damping to a frictional damping acting in the domain we might lose the property of exponential stability of the system. Moreover, a necessary condition for a system to be sub-exponentially stable is that the kernel itself must be sub-exponentially decaying to zero. Having this in mind, a natural question to be asked is that of when this necessary condition is also sufficient. We prove that this is the case for a fairly large class of kernels.     

A general decay result of a viscoelastic equation with infinite history and nonlinear damping

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and in the presence of an infinite-memory term. We prove an explicit and general decay result using some properties of the convex functions. Our approach allows a wider class of kernels, from which those of exponential decay type are only special cases.     

Investigation of Damping of Vibrations in a System With Two‐Disc Inseparable Clutch

This paper presents the results of tests on free and forced harmonic torsional vibrations in a system with a two‐disc inseparable clutch, with structural friction taken into account. Nonlinear histeresis loop describing the frictional‐elastic properties of the system was introduced into the model. The mathematical model of the vibrating system containing two disks inseparable clutch was built. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude torsional displacement was determined. When vibrations were harmonically forced, the amplitude ‐ frequency characteristics of the system were determined numerically. The system was used as a nonlinear torsional vibration damper in a linear system with a harmonic force. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors

The asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave.     

Exponential stability for laminated beams with a frictional damping

We study a laminated beam which consists of two identical layers of uniform thickness, taking into account that an adhesive of small thickness is bonding the two surfaces thereby producing an interfacial slip. We show that when the frictional damping acts on the effective rotation angle there is no need for any other kind of internal or boundary control to produce exponential stability for the system.     

Asymptotics for 2‐D wave equations with Wentzell boundary conditions in the square

This paper concerns the asymptotics of the linear wave equation with frictional damping only on Wentzell boundary in the square. After reformulating the model into an abstract Cauchy problem, we show that the spectrum for the corresponding operator matrix has no purely imaginary values. Moreover, by analyzing a family of eigenvalues for the operator matrix, we prove that there exists a solution of the system, whose energy decay rate can be arbitrarily slow.     

Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one‐dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright © 2008 John Wiley & Sons, Ltd.     

The effect of tailpipe friction on pressure and velocity oscillations in a nonlinear,lumped parameter,pulse combustor model

A lumped parameter model of pulse combustor behavior is formulated which takes into account the effect of tailpipe friction. The model is analyzed by employing asymptotic expansions of the pressure in the chamber and the velocity of the combustion products in the tailpipe and invoking the Poincarè–Lindstedt methodology. It is shown that the presence of frictional damping in the tailpipe can lead to situations where periodic pressure and velocity oscillations occur with periods so large that they would, in effect, not be physically observable.     

Methode zur Bestimmung von Dämpfungsgrössen in Transonic- und Überschallkanälen

Summary Test difficulties obviate the application of the methods for the determination of damping coefficients of wind tunnel models normally used, namely dying-out tests or tests with constrained oscillation respectively.To prevent these difficulties the rotor method has been developed and successfully tested in a transonic wind tunnel.With this method the model carries out an oscillation with constant amplitude, which consumes the energy of a rotor wheel. The damping quantity can be determined by measuring the oscillation frequency as a function of time without influence of the oscillation. By running a test without model but with an equivalent mass, damping and frictional components can be separated.     

General energy decay of solutions for a weakly dissipative Kirchhoff equation with nonlinear boundary damping

In this article, we study the weak dissipative Kirchhoff equation \({u_{tt}} - M\left( {\left\| {\nabla u} \right\|_2^2} \right)\Delta u + b\left( x \right){u_t} + f\left( u \right) = 0\), under nonlinear damping on the boundary \(\frac{{\partial u}}{{\partial v}} + \alpha \left( t \right)g\left( {{u_t}} \right) = 0\). We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping. Our result extends and improves some results in the literature such as the work by Zhang and Miao (2010) in which only exponential energy decay is considered and the work by Zhang and Huang (2014) where the energy decay has been not considered.