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Nonlinear Bresse-Timoshenko beam model with thermal, mass diffusion and theormoelastic effects is studied. We state and prove the well-posedness of problem. The global existence and uniqueness of solution is proved by using the classical Faedo-Galerkin approximations along with two a priori estimates. We prove an exponential stability estimate under assumption $ (2.3)_{1} $ and polynomial decay rate for solution under $ (2.3)_{2} $, by using a multiplier technique combined with an appropriate Lyapuniv functions.

In this paper, the long time behavior of a quasilinear viscoelastic equation with nonlinear damping is considered. Under suitable assumptions, the existence of global attractors is established.

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

In the present paper, we consider an important problem from the application perspective in science and engineering, namely, one-dimensional porous–elastic systems with nonlinear damping, infinite memory and distributed delay terms. A new minimal conditions, placed on the nonlinear term and the relationship between the weights of the different damping mechanisms, are used to show the well-posedness of the solution using the semigroup theory. The solution energy has an explicit and optimal decay for the cases of equal and nonequal speeds of wave propagation.

In the literature, we investigate the decentralized control of multiple strict feedback nonlinear systems with parameter uncertainties. A decentralized controller is designed recursively to realize output synchronization and guarantee the boundedness of the overall system. The decentralized controller of each agent has four parts: state feedback of itself, information of its neighbors, adaptive parameter updaters and nonlinear damping items. A series of distributed z-passive identifiers are proposed for parameter updating and the boundedness of the networked system is guaranteed by the nonlinear damping items. It is proved that when the undirected graph is connected, the control objective can be achieved, i.e., output synchronization is realized while the overall system maintains bounded. Simulation results are presented to verify the effectiveness of the proposed controller.

We study existence of solutions, and in particular well-posedness, for a class of inhomogeneous, nonlinear partial differential equations (PDE's). The main idea is to use system theory to write the nonlinear PDE as a well-posed infinite-dimensional linear system interconnected with a static nonlinearity. By a simple example, it is shown that in general well-posedness of the closed-loop system is not guaranteed. We show that well-posedness of the closed-loop system is guaranteed for linear systems whose input to output map is coercive for small times interconnected to monotone nonlinearities. This work generalizes the results presented in [1], where only globally Lipschitz continuous nonlinearities were considered. Furthermore, it is shown that a general class of linear port-Hamiltonian systems satisfies the conditions asked on the open-loop system. The result is applied to show well-posedness of a system consisting of a vibrating string with nonlinear damping at the boundary.

The nonlinear modes of a non-conservative nonlinear system are sometimes referred to as damped nonlinear normal modes (dNNMs). Because of the non-conservative characteristics, the dNNMs are no longer periodic. To compute non-periodic dNNMs using classic methods for periodic problems, two concepts have been developed in the last two decades: complex nonlinear mode (CNM) and extended periodic motion concept (EPMC). A critical assessment of these two concepts applied to different types of non-conservative nonlinearities and industrial full-scale structures has not been thoroughly investigated yet. Furthermore, there exist two emerging techniques which aim at predicting the resonant solutions of a nonlinear forced response using the dNNMs: extended energy balance method (E-EBM) and nonlinear modal synthesis (NMS). A detailed assessment between these two techniques has been rarely attempted in the literature. Therefore, in this work, a comprehensive comparison between CNM and EPMC is provided through two illustrative systems and one engineering application. The EPMC with an alternative damping assumption is also derived and compared with the original EPMC and CNM. The advantages and limitations of the CNM and EPMC are critically discussed. In addition, the resonant solutions are predicted based on the dNNMs using both E-EBM and NMS. The accuracies of the predicted resonances are also discussed in detail.

As a composite material, the damping energy consumption mechanism of fiber reinforced polymer-confined reinforced concrete (FRP-C RC) structure is very complex. In previous dynamic calculation models, the bond-slip effect for steel bars was often ignored, which would lead to a considerable error in the response of the FRP-C RC structures. In this paper, a new numerical model of FRP-C RC columns considering the bond-slip for steel bars is established using a zero-length element and nonlinear beam-column elements in the OpenSees software, and the results of the model are verified by experimental results. Based on the complex damping theory, the loss factor expression and nonlinear damping model of FRP-C RC columns with the bond slip effect are proposed. Finally, the dynamic response of FRP-C RC columns with nonlinear damping under harmonic load is calculated and compared with the results available in literature. The results show that the proposed model considering steel bars’ bond-slip can provide better prediction for dynamic response of FRP-C RC columns and make the future seismic design of FRP-C RC columns safer.

In the identification of mechanical systems through inverse receptance methods, a linear model is usually assumed, at least on a first approximation. However, most of the systems have a degree of nonlinearity in their elastic or dissipative properties. Usually, stiffness nonlinearities are identified, while damping nonlinearities are neglected or even not detected. Indeed, an equivalent linear damping model, correctly representing the nonlinear dissipation of the system under testing condition, can always be found, but it is test-specific and not generally valid. This paper addresses the identification of dissipative models for multi degrees of freedom mechanical systems with a single damping nonlinearity. Based on the direct damping matrix identification through inverse receptance methods, this research proposes an extension of the Stabilised Layers Method, valid for linear systems, to damping nonlinearities, resulting in the identification of the test-independent linear and nonlinear damping matrices of the system. The proposed method is theoretically derived for the identification of nonlinear damping forces depending on powers of displacement and velocity. The sinusoidal input describing function approximation of the nonlinear damping is exploited to identify the coefficients of the nonlinear damping force from the system receptances, measured under sinusoidal sweeps at constant amplitudes of oscillation across the nonlinearity. The presented method is applied to identify the linear and nonlinear damping matrices of a multi degrees of freedom system with a localised nonlinear magnetic damper. The coefficients of the nonlinear magnetic damping force are identified for two configurations of the magnetic damper. The proposed approach is a parametric method to identify the nonlinear damping models of mechanical systems using standard experimental techniques usually adopted for linear systems. The identified model is valid under general excitation forces, predicting the system behaviour in a broader range of operation than the test-specific linear equivalent model.

In this paper, a single-degree-of-freedom dynamic model is described, with displacement- and velocity-dependent nonlinearities represented by power laws. The model is intended to support the dynamic identification of structural components subjected to harmonic excitation. In comparison to other analytical expressions, the data-driven estimation of the nonlinear exponents provides a large versatility, making the generalised model adaptable for a wide number of different nonlinearities in both stiffness and damping. For instance, the proposed damping formulation can naturally accommodate air drag (quadratic) damping as well as dry friction. Differently to purely data-driven methods (e.g. black boxes), the obtained model is fully inspectable. The proposed formulation is here applied to the large oscillations of a prototype highly flexible wing and fitted on its steady state response in the frequency domain. These large-amplitude flap-wise bending oscillations are known to be affected by nonlinearities in both the stiffness (nonlinear hardening) and the velocity-dependent damping terms. The model is validated against experiments for different structural configurations and input amplitudes, as both these nonlinearities are energy-dependent.

Based on the method of contractive functions, the existence of the time-dependent global attractor of the plate equation with nonlinear damping and linear memory is proved.

In this paper, we prove that the existence of global attractors for a Kirchhoff wave equation with nonlinear damping and memory.

In the present work, we study a one-dimensional laminated Timoshenko beam with a single nonlinear structural damping due to interfacial slip. We use the multiplier method and some properties of convex functions to establish an explicit and general decay result. Interestingly, the result is established without any additional internal or boundary damping term and without imposing any restrictive growth assumption on the nonlinear term, provided the wave speeds of the first equations of the system are equal.

This paper is concerned with the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with additive white noise, for which the nonlinear damping has a critical cubic growth rate. By showing the pullback asymptotic compactness of the stochastic dynamic systems, we prove the existence of a random attractor in $H_{0}^{1}\times L^{2}$H01×L2.

In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, nonlinear weak damping, and a superlinear source: $$ u_{tt}+\Delta ^{2} u- \int _{0}^{t} g(t-\tau )\Delta ^{2} u(\tau )\,\mathrm{d} \tau + \vert u_{t} \vert ^{m-2}u_{t}= \vert u \vert ^{p-2}u,\quad \text{in }\varOmega \times (0,T). $$ When the source is stronger than dissipations, we obtain the existence of certain weak solutions which blow up in finite time with initial energy $E(0)=R$ for any given $R\geq 0$ .

This work is concerned with the Dirichlet initial boundary problem for a semilinear viscoelastic wave system with nonlinear weak damping and source terms. For nonincreasing positive functions g and h, we show the finite time blow-up of some solutions whose initial data have arbitrarily high initial energy.

A general vibration model of a flexible rotor system is established to investigate the influences of the damping characteristics on vibration behaviors. Based on the multi-scale method, the analytical solutions of steady-state and transient-state are derived under the positive and negative conditions of nonlinear damping. The physical significance of the coefficients and their influences on rotor behaviors are analyzed through theoretical analysis and numerical calculation. The experimental results elaborate the damping effect and verify the rationality of the model.

A homogeneous continuous visco-elastic shear-beam, describing the dynamics of base-isolated tall buildings exposed to a uniformly distributed steady wind flow, is studied. The shear beam is constrained at the bottom end by a nonlinear visco-elastic device and free at the top end. The aeroelastic effects, responsible for self-excitation, are evaluated via the quasi-static theory. The occurrence of Hopf bifurcation is detected. Critical and post-critical behavior is analyzed by applying a perturbation scheme. The critical wind velocity and the associated complex galloping mode are found. The steady value of the oscillation amplitude on the stable limit-cycle and its stability are evaluated as function of the mean wind velocity. The mechanical performances of the structure are investigated in according to the effectiveness of the visco-elastic isolation system. A passive controller is proposed to increase the galloping wind velocity and to reduce the amplitude of the limit-cycle.

In this work, we develop a mesoscopic lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model to solve (2 + 1)-dimensional wave equation with the nonlinear damping and source terms. Through the Chapman-Enskog multiscale expansion, the macroscopic governing evolution equation can be obtained accurately by choosing appropriate local equilibrium distribution functions. We validate the present mesoscopic model by some related issues where the exact solution is known. It turned out that the numerical solution is in very good agreement with exact one, which shows that the present mesoscopic model is pretty valid, and can be used to solve more similar nonlinear wave equations with nonlinear damping and source terms, and predict and enrich the internal mechanism of nonlinearity and complexity in nonlinear dynamic phenomenon.

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in \cite{AB01, ACK01, AB02, AL12} for the linear damping case, and in \cite{AB07} for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of \cite{AB05, AB10}., 32 pages, no figures

The hydraulic realization of the inerter, called fluid inerter, is a piston-cylinder device designed to convey a fluid through an external helical channel, thus producing rotational inertia of a fluid mass. This paper proposes a novel base-isolation scheme that combines friction pendulum isolators with a Tuned Fluid Inerter (TFI), wherein a grounded fluid inerter benefits from the interaction with an auxiliary set of low-damping rubber isolators as tuning elements. Based on experiments on a small-scale prototype, the fluid inerter is modeled as a linear inerter in parallel with a nonlinear dashpot, whose power-law damping term is related to the pressure drops in the helical channel. Instead, the hysteretic behavior of friction pendulum isolators is described by a standard Coulomb-type frictional model. Optimal design of the fluid inerter is performed within a probabilistic framework, by modeling the base acceleration as a Kanai-Tajimi filtered random process and handling the nonlinearities of fluid inerter and friction isolators through statistical linearization. An extensive parametric study is conducted to assess the influence of different characteristics of isolators and earthquake excitation and, above all, of the nonlinearity of the fluid inerter. The effectiveness of the TFI is demonstrated through nonlinear response history analysis under a suite of 100 ground-motion records. The proposed isolation scheme not only significantly reduces the displacement demand of the isolators, but also mitigates the acceleration and interstory drift response of the superstructure. Moreover, the benefits of the inherent nonlinear damping effect produced by the fluid inerter proves to be particularly effective to reduce the peak response under pulse-like ground motions that may occur in near-field earthquake events, which indeed represent critical excitations for structures with friction isolators.

This paper describes a stochastic approach to the vibration behavior of a nonlinear damping structure. The deterministic approach is based on motion equations including the non-linearity of damping. For the stochastic calculation, the deviations were assigned to the input parameters of the mathematical model. The probabilistic distribution of these parameters must reflect real operational situations. Numerical simulations were performed with MATLAB using the Monte Carlo and perturbation method. The resulting curves for the parameters studied can find a practical application in the optimization of rotating machines (washing machine), where the parameters of mass, position of the center of gravity, etc. change during operation.

Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddot{u}(t) + Au(t) + g(\dot{u}(t))=h(t),\quad t\in\mathbb{R}^+ ,$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^\infty$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ and this result is optimal. If $h$ is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.

We consider a class of nonlinear lattices with nonlinear damping (0.1) u ¨ n ( t ) + ( − 1 ) p Δ p u n ( t ) + α u n ( t ) + h ( u n ( t ) ) + g ( n , u ˙ n ( t ) ) = f n , where n ∈ Z , t ∈ R + , α is a real positive constant, p is any positive integer and Δ is the discrete one-dimensional Laplace operator. Under suitable conditions on h and g we prove the existence of a global attractor for the continuous semigroup associated with (0.1) . Our proofs are based on a difference inequality due to M. Nakao [M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations 227 (2006) 204–229].

This paper is concerned with the p -system of hyperbolic conservation laws with nonlinear damping. When the constant states are small, the solutions of the Cauchy problem for the damped p -system globally exist and converge to their corresponding nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation given by the Darcy's law. The optimal convergence rates are also obtained. In order to overcome the difficulty caused by the nonlinear damping, a couple of correction functions have been technically constructed. The approach adopted is the elementary energy method together with the technique of approximating Green function. On the other hand, when the constant states are large, the solutions of the Cauchy problem for the p -system will blow up at a finite time.

This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping. Several new oscillation criteria are established under quite general assumptions, which improve and extend the known results in the literature.

The paper is devoted to discuss the effects of nonlinear damping on the post-critical behavior of the Ziegler’s column. The classical Ziegler’s double-pendulum is considered in regime of finite displacements, in which, moreover, nonlinear damping of Van der Pol-type is introduced at the hinges. A second-order Multiple Scale analysis is carried out on the equations of motion expanded up to the fifth-order terms. The nature of the Hopf bifurcation, namely, supercritical or subcritical, as well as the occurrence of the ‘hard loss of stability’ phenomenon, are investigated. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally studied for different linearly damped columns.

We prove the existence of a compact, finite dimensional, global attractor for a system of strongly coupled wave and plate equations with nonlinear dissipation and forces. This kind of models describes fluid-structure interactions. Though our main focus is on the composite system of two partial differential equations, the result achieved yields as well a new contribution to the asymptotic analysis of either (uncoupled) equation.

In this paper, we consider the nonlinear viscoelastic equation u t t − Δ u + ∫ 0 t g ( t − τ ) Δ u ( τ ) d τ + u t | u t | m − 2 = u | u | p − 2 with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g and for p > m , we prove that there are solutions with positive initial energy that blow up in finite time.

Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R3 with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C1(|s|−C2)⩽|g(s)|⩽C3(1+|s|p), where 1⩽p0) is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f′(s)|⩽C4(1+|s|q), where q⩽2. It is remarkable that when 2

We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.

In this paper a time domain output-only Dynamic Identification approach for Civil Structures (DICS) first formulated some years ago is reviewed and presented in a more generalized form. The approach in question, suitable for multi- and single-degrees-of-freedom systems, is based on the statistical moments and on the correlation functions of the response to base random excitations. The solving equations are obtained by applying the Itô differential stochastic calculus to some functions of the response. In the previous version ([21] Cavaleri, 2006; [22] Benfratello et al., 2009), the DICS method was based on the use of two classes of models (Restricted Potential Models and Linear Mass Proportional Damping Models) while its generalization for use with different models from the ones mentioned above is discussed. In the paper the new class of models to which the DICS method is applicable are described. Further, the advantages and disadvantages of the approach in question are examined, also by a comparison with some techniques available in the literature. © 2013 Elsevier Ltd.

We study, in the setting of a real Hilbert spaceH, the asymptotic behavior of trajectories of the second-order dissipative dynamical system with linear and gradient-driven nonlinear dampingwhereλ >0 andf, Φ:H → Rare two convex differentiable functions. It is proved that ifΦis coercive and bounded from below, then the trajectory converges weakly towards a minimizer ofΦ. In particular, we state that under suitable conditions, the trajectory strongly converges to the minimizer ofΦexponentially or polynomially.

WOS: 000328081500028 Using generalized Riccati transformations, we derive new interval oscillation criteria for a class of general type second-order nonlinear differential equations with nonlinear damping. Examples are also given to illustrate the results.

In the present study, the concept of the output frequency response function is applied to theoretically investigate the force transmissibility of multi-degree of freedom (MDOF) structures with a nonlinear anti-symmetric viscous damping. The results reveal that an anti-symmetric nonlinear viscous damping can significantly reduce the transmissibility over all resonance regions for MDOF structures while it has almost no effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to overcome the dilemma in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant region but increases the transmissibility over non-resonant regions.

We show that the nonlinear wave equation $\Box u+u_{t}^{3}=0$ is globally well-posed in radially symmetric Sobolev spaces $H^{k}_{\mathrm{rad}}(\mbi{R}^{3})\times H^{k-1}_{\mathrm{rad}}(\mbi{R}^{3})$ for all integers $k>2$ . This partially extends the well-posedness in $H^{k}(\mbi{R}^{3})\times H^{k-1}(\mbi{R}^{3})$ for all $k\in [1,2]$ , established by Lions and Strauss[12]. As a consequence we obtain the global existence of $C^{\infty}$ solutions with radial $C_{0}^{\infty}$ data. The regularity problem requires smoothing and non-concentration estimates in addition to standard energy estimates, since the cubic damping is critical when $k=2$ . We also establish scattering results for initial data $(u,u_{t})|_{t=0}$ in radially symmetric Sobolev spaces.

An adaptive approach of vibrating thin structures is proposed here. The method consists in applying an equivalent adimensional damping ratio to each potential resonance. This ratio is deduced from experimental data obtained in vacuum facility, in relation with frequencies, for several structural technologies. Consequently, it is possible to calculate the structure in a linear nondissipative context, valid out of resonance bands, and truncated in those bands. Thus, the equivalent damping ratio is directly used to define adimensional resonance truncation bandwith and level. The contribution consists in tested and applied modal methodology and algebraic representations of damping including several dissipations—viscous and internal microfrictions—inducing a nonmonotonous model. The here aim is to provide realistic recommendations for simple vibrational analysis of aerospace thin structures—panels and stiffeners.

We consider a one-dimensional weakly damped wave equation, with a damping coefficient depending on the displacement. We prove the existence of a regular connected global attractor of finite fractal dimension for the associated dynamical system, as well as the existence of an exponential attractor.

In this work we are concerned with the existence and uniqueness of strong global solutions and exponential decay of the total energy for an initial-boundary value problem associated with the Kirchhoff equation with variable coefficients on the action of a nonlinear internal damping.

In this work, we study the 1-D isentropic bipolar hydrodynamic model. This model takes the form of compressible Euler–Poisson system with nonlinear damping added to the momentum equations. Under some smallness conditions, the solutions to the Cauchy problem of the system globally exist and convergence to the nonlinear diffusion waves, which are the corresponding solutions of nonlinear parabolic equations given by the Darcy's law with a specified initial data. The optimal convergence rates are obtained by Green function method when the initial perturbation is in L 1 -space.

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered. We prove that, under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions, the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we obtain the blow-up of solutions in finite time when the initial energy is nonnegative. This improves earlier results in the literature.

In this paper we study the equation of viscoelasticity u t t − u x x t − F ( u x ) x = f ( x , t ) with the nonlinear damping term on the boundary. Making use of the so-called energy perturbation method and a comparison inequality we prove the relationship between decay rate of solutions and f ( x , t ) , that is, they have the same exponential decay or algebraic decay for the case F ( s ) = s + | s | α − 2 ( α ⩾ 2 ) , and for the case F ( s ) = | s | α − 2 ( α > 2 ) , they have the same algebraic decay.

We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $\gamma$ and $\kappa$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters., Comment: 38 pages. To appear in: Discrete Contin. Dyn. Syst. Series A