Nonlinear Damping Term Research Articles

Long-time dynamics of a problem of strain gradient porous elastic theory with nonlinear damping and source terms

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of ‘consistency’ due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained.

Dynamics properties for a viscoelastic Kirchhoff-type equation with nonlinear boundary damping and source terms

This work is devoted to studying a viscoelastic Kirchhoff-type equation with nonlinear boundary damping-source interactions in a bounded domain. Under suitable assumptions on the kernel function g, density function, and M, the global existence and general decay rate of solution are established. Moreover, we prove the finite time blow-up result of solution with negative initial energy.

Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions

Abstract In the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.

Nonlinear Radiation Damping: A New Method for Dissipating Energy in Dynamic Earthquake Rupture Simulations

Abstract Dynamic earthquake rupture simulations are used to understand earthquake mechanics and the ground shaking that earthquakes produce. These simulations can help diagnose past earthquake behavior and are also used to generate scenarios of possible future earthquakes. Traditional dynamic rupture models generally assume elastic rock response, but this can lead to peak on-fault slip rates and ground shaking that are higher than those inferred from seismological observations. Some have approached this challenge using inelastic off-fault rock behavior to dissipate energy, but the addition of inelasticity can make it difficult to select parameters and establish suitable initial conditions, and increases the model’s complexity and computational cost. We propose a new method that works by adding a nonlinear radiation damping term to the friction law, with the surrounding rocks remaining linear elastic. Our new method results in lower peak slip rates, reduced seismic radiation, and an increasing slip-weakening critical distance with increasing rupture propagation distance, all within a linear elastic model. In addition, it is easy to implement.

Adaptive fuzzy finite-time tracking control of nonlinear systems with unmodeled dynamics

This paper considers an adaptive fuzzy finite-time tracking control issue for uncertain nonlinear systems in the presence of unmodeled dynamics via the backstepping technique. In the design procedure, the traditional dynamic signal structure is improved firstly such that it is suitable for the controller design within finite-time interval universally for the controlled systems and it can be utilized to dominate the unmodeled dynamics. The dynamic disturbances are compensated by nonlinear damping terms. The fuzzy logic systems (FLSs) are used to package the unknown nonlinearities. By constructing the appropriate Lyapunov functions in recursive step and employing the finite-time stability theorem, the developed FLS-based adaptive tracking control strategy within finite-time interval, which ensures the boundedness of all the closed-loop signals and the convergence of tracking error. In the end, simulation results are provided to test the availability of the developed strategy.

Finite time extinction for a critically damped Schrödinger equation with a sublinear nonlinearity

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schrödinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{{\varepsilon}+(|u|^2)^\alpha}u,$ with $a\in{\mathbb{C}},$ ${\varepsilon}\ge0,$ $2\alpha=(1-m)$ and $m\in[0,1).$ Here, we consider the sublinear case $0 < m < 1$ with a critical damped coefficient: $a\in{\mathbb{C}}$ is assumed to be in the set $ D(m)=\big\{z\in{\mathbb{C}}; \; {\mathrm{Im}}(z) > 0 \text{ and } 2\sqrt m{\mathrm{Im}}(z)=(1-m){\mathrm{Re}}(z)\big\} . $ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropriate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.

Optimal design and assessment of tuned mass damper inerter with nonlinear viscous damper in seismically excited multi-storey buildings

In recent years, the tuned mass damper inerter (TMDI) has been demonstrated in several theoretical studies to be an effective vibration absorber for the seismic protection of non-isolated buildings. Its effectiveness relies on careful tuning of the TMDI stiffness and damping properties, while its performance improves with the increase of the inertance property which is readily scalable. Nevertheless, in all previous studies, the energy dissipative TMDI element has been modelled by a linear viscous damper. Still, commercial viscous dampers display a nonlinear velocity-dependent power law behavior. In this regard, this paper investigates, for the first time in literature, the potential of the TMDI fitted with nonlinear viscous damper (NVD) for seismic response protection of multi-storey buildings. This is supported by an efficient optimal nonlinear TMDI (NTMDI) tuning approach which accounts for any absorber connectivity to the building structure and employs statistical linearization to treat the nonlinear damping term. For the special case of white-noise excited undamped buildings, optimal NTMDI tuning is derived analytically in closed-form which is shown to be sufficiently accurate for lightly damped structures. Comprehensive numerical data are presented to delineate trends of optimal NVD damping coefficient with the NVD power-law exponent and the inertance. Further, nonlinear response history analysis results pertaining to optimally tuned NTMDI application for a benchmark 9-storey steel structure demonstrate that reduced NTMDI stroke and inerter force can be achieved with negligible change in storey drifts and floor acceleration performance by adopting lower NVD exponent values, leading to practically beneficial NTMDI deployments.

Global existence, blow-up and optimal decay for a nonlinear viscoelastic equation with nonlinear damping and source term

In this paper, we are concerned with a viscoelastic wave equation with memory term, nonlinear damping and source term. Firstly, using the potential well method combined with Galerkin approximation procedure, the global weak solutions are obtained. Secondly, we investigate the blow-up of solutions with initial positive and negative energy, as well as our result improves the earlier ones in [29] and [36]. Finally, under some assumptions imposed on damping coefficient and the relaxation function, we establish the optimal decay of the solutions which conducted by perturbed energy method. Moreover, we obtain that the exponential form of relaxation function lead to better decay result and memory term can slow down the energy decay by displaying the energy decay graphically.

Energy decay and blow-up of solutions for a class of system of generalized nonlinear Klein-Gordon equations with source and damping terms

In this work, we investigate generalized coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms and initial-boundary value conditions, in a bounded domain. We obtain decay of solutions by use of Nakao inequality. The blow up of solutions with negative initial energy is also established.

Pullback attractors and upper semicontinuity for non-autonomous extensible two-beams

In this paper we study the long-time dynamics for non-autonomous extensible non-symmetric two-beams. The governing equations describe the equilibria of an elastically-coupled extensible two-beams system subject to damping terms and evenly compressive axial loads. Under quite general assumptions on nonlinear damping and sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We prove the existence of pullback attractors in the natural space energy. Finally, we prove the regularity of the family of pullback attractors and its upper semicontinuous with respect to the perturbed parameters.

The performance of nonlinear vibration control via NiTiNOL–Steel wire ropes

Conventional nonlinear vibration absorbers must provide additional mass and motion space to achieve the expected vibration control effect, which will significantly alter the original design of the target structure. In this study, the shape memory NiTiNOL–steel (NiTi–ST) wire rope and composite plate are coupled to investigate the vibration control effect of such a configuration. The vibration response of the model is studied using the Galerkin truncation method (GTM) and the harmonic balance method (HBM), and the stability of the frequency response curve is determined using the linearized stability theory. NiTi–ST wire rope is found to have a good damping effect for both low- and high-order resonances. Under a strong external excitation, the appearance of the closed detached response (CDR) may cause system amplitude mutation; however, it can also achieve effective inhibition. Finally, the effect of the structural parameters on the vibration control is discussed. The nonlinear damping term in NiTi–ST determines the soft and hard characteristics of the system, while the equivalent linear damping term is mainly responsible for vibration suppression. Therefore, as a novel shock absorber, NiTi–ST can be applied to engineering structures in order to prevent vibration damage.

LONG-TIME DECAY OF LERAY SOLUTION OF 3D-NSE WITH EXPONENTIAL DAMPING

We study the uniqueness, the continuity in [Formula: see text] and the large time decay for the Leray solutions of the 3D incompressible Navier–Stokes equations with nonlinear exponential damping term [Formula: see text], ([Formula: see text]).

Long time decay of solutions to the 3D incompressible Navier–Stokes equations with nonlinear damping

This paper is interested in the decay of solutions to 3D Navier–Stokes equations with a nonlinear damping term . We discuss the bounds of the global strong solutions in the negative Sobolev space and establish the optimal decay rates in of the global strong solutions. Moreover, the upper bound of the derivative is also obtained. Finally, we investigate the asymptotic stability of the solutions to the system.

A three‐step Oseen‐linearized finite element method for incompressible flows with damping

AbstractThis paper introduces a three‐step Oseen‐linearized finite element method for the 2D/3D steady incompressible Navier–Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen‐linearized problems over a fine grid, which possess the same stiffness matrices with only various right‐hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two‐level method.

Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models

Abstract For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invariant manifold via Nehari manifold. To establish the results of global well-posedness of solution, we consider the problem at three different initial energy levels, i.e., subcritical initial energy level, critical initial energy level, and arbitrarily high initial energy level. We first obtain the local existence of the solution by using the contraction mapping principle. Then, in the framework of potential well, we obtain global existence, nonexistence, and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level. In the end, we establish the global nonexistence of solution for the problem with linear weak damping and strong damping at the arbitrarily high initial energy level.

Global well-posedness of fourth-order Petrovsky equation with weak and strong damping terms

In this paper, a semilinear fourth-order Petrovsky equation with nonlinear weak damping term and linear strong damping term is considered in the frame of the potential well. We prove the existence and uniqueness of the global strong solution with sub-critical initial energy. In addition, we obtain the global existence, asymptotic behaviour and finite time blowup for the weak (non-steady state) solution with a critical initial energy level.

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

Power characteristics of vibration-based piezoelectric energy harvesters: the effect of piezoelectric material nonlinearity

Piezoelectric energy harvesting has received tremendous interests in the past two decades as a viable solution to self-powered electronics and devices. Recently, significant emphasis has been given to nonlinear energy harvesters driven by the desire for broadband, high-performance energy harvesting. Numerous efforts have been devoted to the understanding and modeling of the electromechanical coupling and the effect of nonlinearities introduced by mechanical and electrical aspects of the system. However, very few works in the literature considered the effect of piezoelectric material nonlinearity on the system power performance. Nevertheless, it has been found that piezoelectric nonlinearity is significant even at low to moderate excitation level. This paper is motivated to study the power behavior of piezoelectric energy harvesters with piezoelectric nonlinearity, most importantly, the power limit and electromechanical coupling. For this purpose, an approximate model is developed from the nonlinear model in the literature to derive the closed-form expressions of important power characteristics. Analytical analysis shows that the effect of piezoelectric material nonlinearity results in a nonlinear damping term and a nonlinear stiffness term in the approximate model. The approximate solutions of optimal load resistance, maximum power, power limit, and critical electromechanical coupling are obtained and validated by numerical simulations first. The induced nonlinear damping reduces the power limit of the system compared to its linear counterpart. Interestingly, a harvester that exhibits strong electromechanical coupling under small excitation could become weakly coupled under large excitation. The analytical analysis and numerical results are validated by experiments.

General stability for piezoelectric beams with a nonlinear damping term

General stability for piezoelectric beams with a nonlinear damping term

Identification of parameters for same-order nonlinear damping terms in polynomial-type vortex-induced force models for bridge decks

Polynomial-type empirical models have been commonly adopted to describe the vortex-induced force (VIF) on bridge decks in the study of vortex-induced vibration. Previous studies chose different nonlinear damping terms like the Van der Pol or Rayleigh type in such models without explaining whether these same-order terms can be chosen arbitrarily. A novel parameter identification technique is developed in this paper to distinguish different same-order nonlinear terms in the VIF model with additional governing equations based on the orthogonality among motion-induced terms. The parameter identification results obtained from the wind-tunnel tests on a central-slotted box deck indicate that the Rayleigh-type damping terms are the only important ones in the vertical VIF, whilst none of the same-order nonlinear damping terms of the torsional VIF should be ignored. The finding is then theoretically explained assuming that the aerodynamic damping nonlinearity of VIF primarily comes from the continuous change of relative attack angle between the oscillating deck and the incoming wind.