Linear Damping Term Research Articles

Conformal structure-preserving SVM methods for the nonlinear Schrödinger equation with weakly linear damping term

In this paper, by applying the supplementary variable method (SVM), some high-order, conformal structure-preserving, linearized algorithms are developed for the damped nonlinear Schrödinger equation. We derive the well-determined SVM systems with the conformal properties and they are then equivalent to nonlinear equality constrained optimization problems for computation. The deduced optimization models are discretized by using the Gauss type Runge-Kutta method and the prediction-correction technique in time as well as the Fourier pseudo-spectral method in space. Numerical results and some comparisons between this method and other reported methods are given to favor the suggested method in the overall performance. It is worthwhile to emphasize that the numerical strategy in this work could be extended to other conservative or dissipative system for designing high-order structure-preserving algorithms.

Terahertz plasma oscillations and dissipating solitary pulses in electron–hole semiconductor plasmas

Abstract Linearized modes of oscillation in terahertz range and weakly dissipating electrostatic solitary pulses are studied in semiconductor plasmas in the framework of quantum hydrodynamics. Many-particle effect is expressed in the local density approximation with the help of exchange-correlation potential and the species space charge effect is included through Poisson’s equation. Using reductive perturbation technique, damped Korteweg de-Vries equations is derived with a linear damping term arising due to electron (hole)–phonon collisions. Time evolution of solitary acoustic pulses is presented analytically and numerically. Typical values corresponding to GaAs, GaSb, InP and GaN semiconductors are used for parametric analysis and pulse profile with collision-induced dissipation and quantum effects of statistical pressure, Bohm potential and exchange-correlation potential. The pulses are stable and can withstand perturbations for a considerable time before damping.

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.

Frequency Characteristics of Sloshing Resonance in a Three-Dimensional Shallow-Water Rectangular Tank

The frequency characteristics of free surface elevation time histories of shallow-water sloshing in a three-dimensional rectangular tank is presented. The numerical model for sloshing motion uses an accurate velocity potential Boussinesq-type equation. A particular solution is adopted to express the external excitations and a linear damping term is introduced to replace a friction effect produced by tank walls. The excitation amplitude of the forced movement of the tank is tiny, while the excitation frequency varies around the first resonance frequency. The numerical results show that the wave energy in the rectangular tank under resonance is concentrated in the integer multiple of excitation frequencies, the sum excitation frequencies, and the difference excitation frequencies.

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.

Nonlinear response of passively flapping foils

This study numerically investigates two-dimensional incompressible flows over elastically mounted foils, undergoing semi-passive and fully passive motion. In our strongly coupled numerical models, we employ linear and cubic stiffness and damping terms in order to examine their highly nonlinear response. The undamped model of the fully passive system exhibits various responses from periodic to chaotic and then to flip-over for the reduced velocity, ranging from 1 to 10. However, introducing the cubic damping terms causes a significant decrease in the magnitude of plunging and pitching amplitudes without affecting the onset point of bifurcation. Also, plunging and pitching amplitudes are altered significantly after the point of onset. Furthermore, the performance metrics of each passive system are computed for power generation applications to demonstrate that semi-passive system attain efficiency up to 20% for a pitching amplitude of 50° with the excitation frequency in the narrow range of 0.15 to 0.20. On the other hand for a fully passive system, the efficiency of around 34% is obtained near the onset point of a bifurcation with a low mass ratio and linear damping terms. However, introducing cubic damping terms causes degradation in efficiency to bring it down to 14−20% for a wide range of reduced velocity.

Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Liénard nonlinear oscillator

We study the dynamics of a parametrically and externally driven Rayleigh-Liénard hybrid model and report the emergence of extreme bursting events due to a novel pulse-shaped explosion mechanism. The system exhibits complex periodic and chaotic bursting patterns amid small oscillations as a function of excitation frequencies. In particular, the advent of rare and recurrent chaotic bursts that emerged for certain parameter regions is characterized as extreme events. We have identified that the appearance of a sharp pulse-like transition that occurred in the equilibrium points of the system is the underlying mechanism for the development of bursting events. Further, the controlling aspect of extreme events is attempted by incorporating a linear damping term, and we show that for sufficiently strong damping strength, the extreme events are eliminated from the system, and only periodic bursting is feasible.

Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids

We characterize the $$L^2$$ decay rate of solutions to the 3D magneto-micropolar system in terms of the decay character of the initial datum. Due to a linear damping term, the microrotational field has a faster decay rate. We also address the asymptotic behaviour of solutions by comparing them to solutions to the linear part. As a result of the linear damping, the difference between the microrotational field and its linear part also decays faster. As part of the proofs of these results, we prove estimates for the derivatives of solutions which might be of independent interest.

Blow-up and lifespan estimates for a damped wave equation in the Einstein–de Sitter spacetime with nonlinearity of derivative type

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.

Soliton-like solutions of general variable coefficient (2+1)-dimensional KdV equation with linear damping term

A general variable coefficient (2+1)-dimensional KdV equation with linear damping term (VC(2+1)d damping KdV equation for short) has been investigated by using the simplified homogeneous balance method. It has been proven firstly that if the coefficients of the equation satisfy a certain constraint condition, then the VC(2+1)d damping KdV equation has a nonlinear transformation that converts the VC(2+1)d damping KdV equation itself into a homogeneous quadratic equation which admits a series of exponential function solutions. By means of the nonlinear transformation derived here, one and two soliton-like solutions of the VC(2+1)d damping KdV equation are obtained successfully. As an illustrative example, a concrete VC(2+1)d damping KdV equation is studied, the nonlinear transformation, and one and two soliton-like solutions of the equation are all given.

The multiplication of distributions in the study of delta shock wave for a hyperbolic system of Temple class

The Riemann problem for a hyperbolic system of conservation laws belonging to Temple class is considered within frame of [Formula: see text]-solutions based on the concept of solutions defined in the setting of a distributional product. We obtain all the possible discontinuous solutions within a convenient space of distributions that contains Dirac delta measures and discontinuous functions. In addition, an interesting discontinuous solution is found for certain Riemann initial conditions, in which the Dirac delta measure is involved in both of the two state variables simultaneously. As an application, a detailed example for the mentioned system but involving a linear damping term is provided, in which the trajectories of discontinuous solutions including contact discontinuity, shock wave and delta shock wave change in exponential forms.

Dissipative dust acoustic solitons in magnetized nonextensive warm dusty plasma

A systematic theoretical investigation has been carried out in order to explore the propagation of both linear and nonlinear dust acoustic waves (DAWs) in a collisional magnetized warm dusty plasma in the presence of nonextensive electrons and ions. A linear dispersion relation is analyzed to show that the wave frequencies of both parallel and perpendicular modes are modified by the relevant plasma parameters. The standard reductive perturbative technique leads to a Korteweg–de Vries equation with a linear damping term for the propagation dynamics of the finite amplitude waves in warm dusty plasma. It is found that the dust-neutral collision induced dissipation is responsible for the linear damping. The analysis reveals that DAW propagates in the form of a weakly dissipative compressive/rarefactive solitons and the nonlinear excitations are significantly influenced by different plasma parameters, such as dust concentration, nonextensivity, external magnetic field strength, and dust-neutral collisionality. The present results can be relevant to the DAWs in some astrophysical plasma environments, such as Saturn’s rings, interstellar dusty clouds, etc.

Transformation of wave patterns in three-dimensional shallow-water tank due to vertical excitations

The wave patterns of sloshing motion in three-dimensional shallow-water rectangular tank were investigated based on Boussinesq-type equations with linear damping term. Different coupled excitations were studied when the vertical excitation frequency was equal to twice the second natural frequency. The transformation of wave patterns can be found in the tank due to the effect of vertical excitations. From linear standing waves to nonlinear traveling waves, two different traveling wave patterns can be found in the tank during the final steady-state. For the coupled surge–heave excitations with different vertical excitation frequencies and amplitudes, the transformations of wave patterns appeared when the excitation frequencies close to twice the second natural frequency.

Numerical and experimental investigations into fluid resonance in a gap between two side-by-side vessels

When two vessels in a side-by-side configuration are under wave actions, fluid resonance occurs in a narrow gap between two vessels, termed ‘gap resonance’, leading to large free-surface responses. Large free-surface responses due to gap resonance can induce large relative motions and drift forces by both vessels, which influences operational safety. The present study aims at investigating the free-surface response near resonance in a narrow gap between two fixed, identical barges with square corners numerically and experimentally. Gap resonance is driven by three typical incident wave conditions including beam sea, quartering sea, and head sea. In light of the fact that the potential-flow theory overestimates the response in the gap, a potential-flow model with energy dissipation effects is developed based on the boundary element method. A dissipation surface is devised at the bottom opening as well as two end openings of the gap. Both linear and quadratic damping terms associated with laminar boundary layer wall friction and flow separation from sharp corners are accounted for. Physical experiments of gap resonance excited by irregular waves with different significant wave heights and peak periods are carried out and used for calibration and validation of the potential-flow model. Nonlinear correlation is observed between the incident wave amplitude and resonant gap response, which is predicted with satisfactory agreement by the developed potential-flow model. In-depth discussion on the mode shapes is made. It is found that the first mode is predominant in the beam sea, and the natural modes exhibit standing wave behaviors. Under quartering sea and head sea excitations, the second and third modes dominate over other modes. Waves propagate slowly along the gap in the head sea condition. Under the quartering sea excitation, the first and second modes propagate along the gap, but the third mode exhibits standing wave behavior, indicating the quartering sea response is a transition between head sea and beam sea. The significance of the present study is threefold. Firstly, the experimental measurements provide reference results for numerical simulation. Secondly, a simple but effective damping model to suppress unrealistic free-surface response near resonance is developed. Lastly, this work gives an insight into the spatial structure and dominance of natural modes under different wave conditions.

On the Analytical and Numerical Solutions of the Linear Damped NLSE for Modeling Dissipative Freak Waves and Breathers in Nonlinear and Dispersive Mediums: An Application to a Pair-Ion Plasma

In this work, two approaches are introduced to solve a linear damped nonlinear Schrödinger equation (NLSE) for modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). The linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution until now, due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in detail. The first solution is called a semi-analytical solution while the second is an approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as the initial solution to solve the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of the finite differences method (FDM) will be devoted to achieve this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. The numerical values of both maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated using the nonlinear polynomial based on the Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study helps us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.

Numerical model of energy dissipation and shallow-water sloshing motions in tank under different coupled excitations

Shallow-water sloshing motions in a three-dimensional rectangular tank are investigated. The Boussinesq-type equations in terms of velocity potential and the finite-difference scheme are applied for the solutions of numerical model. Through linking the rate of decay of the wave amplitudes to the energy dissipation due to the friction at the tank walls, a linear damping term is proposed and added into the free surface boundary condition. Taking the tank under excited frequencies near the lowest natural frequency, the maximum transient wave amplitudes and steady-state wave amplitudes of sloshing motions at the tank wall are presented and verified by the experimental results given in the literature. The characteristics of sloshing motions in tank under different coupled excitations are studied. The results indicate that coupled surge-sway excitations lead to the weaker nonlinear sloshing motions in tank than the single degree of freedom excitations. The intersection of sloshing wave crest lines finally tend to the diagonal line of the tank under the coupled surge-sway excitations with different amplitudes. And the irregular free surface oscillations appear at the corners of the tank excited by the coupled surge-sway-roll-pitch-yaw harmonic motions.

A Complex Stiffness Human Impedance Model With Customizable Exoskeleton Control.

The natural impedance, or dynamic relationship between force and motion, of a human operator can determine the stability of exoskeletons that use interaction-torque feedback to amplify human strength. While human impedance is typically modelled as a linear system, our experiments on a single-joint exoskeleton testbed involving 10 human subjects show evidence of nonlinear behavior: a low-frequency asymptotic phase for the dynamic stiffness of the human that is different than the expected zero, and an unexpectedly consistent damping ratio as the stiffness and inertia vary. To explain these observations, this article considers a new frequency-domain model of the human joint dynamics featuring complex value stiffness comprising a real stiffness term and a hysteretic damping term. Using a statistical F-test we show that the hysteretic damping term is not only significant but is even more significant than the linear damping term. Further analysis reveals a linear trend linking hysteretic damping and the real part of the stiffness, which allows us to simplify the complex stiffness model down to a 1-parameter system. Then, we introduce and demonstrate a customizable fractional-order controller that exploits this hysteretic damping behavior to improve strength amplification bandwidth while maintaining stability, and explore a tuning approach which ensures that this stability property is robust to muscle co-contraction for each individual.

Absence of Global Periodic Solutions for a Schrödinger-Type Nonlinear Evolution Equation

The problem of the absence of global periodic solutions for a Schrodinger-type nonlinear evolution equation with a linear damping term is investigated. It is proved that when the damping coefficient is nonnegative, the problem does not have global periodic solutions for any initial data, while when it is negative, the same is valid for “sufficiently large values” of the initial data.

New critical behaviors for semilinear wave equations and systems with linear damping terms

We consider the inhomogeneous semilinear wave equation with linear damping ∂ttu−Δu+∂tu=|u|p+w(t,x) in , where p>1 and w≢0. A general criterium for the nonexistence of global weak solutions is established. In the particular case w=ω(x), we obtain the critical exponent in the sense of Fujita (first critical exponent) and the critical exponent in the sense of Lee and Ni (second critical exponent) for the considered problem. Namely, we show that, if (p∗(N)=∞ if N∈{1,2}) and ω ≥ 0, then the problem admits no global weak solution; if p>p∗(N), then a global solution exists, for some ω>0 and suitable initial values. Moreover, when N ≥ 3 and p>p∗(N), we prove that for σ<N, if and ω(x) ≥ C|x|−σ for |x| large, then there is no global weak solution; if σ∗ ≤ σ<N, then a global solution exists for some ω>0 with ω(x) ≤ C|x|−σ for |x| large, and suitable initial values. Next, we extend our study to the case of systems.