Damping Term Research Articles

New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new classes of equations that were not covered by the previous criteria. According to symmetry between the negative and positive solutions of the studied equation, we present some new criteria that ensure the oscillation of all solutions by using the Riccati transformation and comparison method. We provide an example to illustrate our main results.

Modelling of vortex-induced force and prediction of vortex-induced vibration of a bridge deck using method of multiple scales

With the increase of span length, long-span bridges become more flexible and susceptible to vortex-induced vibrations (VIV). Several models for vortex-induced force (VIF) and various methods for predicting VIV have been developed accordingly but without consensus. This paper uses the method of multiple scales, from a different angle of view, to analyze the general polynomial model for VIF and present a concise polynomial model for VIF and a simple method for predicting VIV of a bridge deck. The aeroelastic vibration of a deck section is simulated by the computational fluid dynamics (CFD) and the results are used to verify the concise polynomial model of VIF. The forced vibration of a deck section is also simulated by CFD to yield the simple method for predicting VIV together with the frequency response equation and the critical equation of VIV derived by the method of multiple scales. The results show that the even and cross terms of aerodynamic damping and stiffness in the general polynomial VIF model are small compared with the odd terms and can be thus omitted. The lock-in region and the amplitude of VIV of the deck section can be accurately predicted by the proposed simple method.

Review of Wind-Induced Effects Estimation through Nonlinear Analysis of Tall Buildings, High-Rise Structures, Flexible Bridges and Transmission Lines

The nonlinear effects exhibited by structures under the action of wind loads have gradually stepped into the vision of wind-resistant researchers. By summarizing the prominent wind-induced nonlinear problems of four types of wind-sensitive structures, namely tall buildings, high-rise structures, flexible bridges, and transmission lines, the occurrence mechanism of their nonlinear effects is revealed, providing cutting-edge research progress in theoretical studies, experimental methods and vibration control. Aerodynamic admittance provides insights into the aerodynamic nonlinearity (AN) between the wind pressure spectrum and wind speed spectrum of tall building surfaces. The equivalent nonlinear equation method is used to solve nonlinear vibration equations with generalized van-der-Pol-type aerodynamic damping terms. The elastic–plastic finite element method and multiscale modeling method are widely employed to analyze the effects of geometric nonlinearity (GN) and material nonlinearity (MN) at local nodes on the wind-induced response of latticed tall structures. The AN in blunt sections of bridges arises from the amplitude dependence of the aerodynamic derivative and the higher-order term of the self-excited force. Volterra series aerodynamic models are more suitable for the nonlinear aerodynamic modeling of bridges than the polynomial models studied more in the past. The improved Lindstedt–Poincare perturbation method, which considers the strong GN in the response of ice-covered transmission lines, offers high accuracy. The complex numerical calculations and nonlinear analyses involved in wind-induced nonlinear effects continue to consume significant computational resources and time, especially for complex wind field conditions and flexible and variable structural forms. It is necessary to further develop analytical, modeling and identification tools to facilitate the modeling of nonlinear features in the future.

The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

In the present investigation, a quadratically forced damped pendulum-type equation is solved analytically using several effective and highly accurate approaches. Some different types of pendulum oscillators linked to the forced and damped terms, in addition to the power of the damping term, are analyzed. In the first part, the Krylov–Bogoliubov–Mitropolsky (KBM) technique and the ansatz method are carried out for analyzing the quadratically damped pendulum oscillator. In the second part, the two variants of the KBM technique are implemented for investigating the quadratically forced damped pendulum oscillator. Some symmetric approximations are derived and compared with the fourth-order Runge–Kutta numerical approximation. In addition, the maximum distance error is estimated in the whole study domain for ensuring that all obtained approximations are accurate. The obtained results are illustrated through concrete examples.

A Non-Local Non-Homogeneous Fractional Timoshenko System with Frictional and Viscoelastic Damping Terms

We are devoted to the study of a non-local non-homogeneous time fractional Timoshenko system with frictional and viscoelastic damping terms. We are concerned with the well-posedness of the given problem. The approach relies on some functional analysis tools, operator theory, a priori estimates and density arguments. This work can be considered as a contribution to the development of energy inequality methods, the so-called a priori estimate method inspired from functional analyses and used to prove the well-posedness of mixed problems with integral boundary conditions.

Dynamics of nonlinear Reissner–Mindlin–Timoshenko plate systems

AbstractThe problem of Reissner–Mindlin–Timoshenko plate systems with nonlinear damping terms is considered. The main result is the existence of global attractors. By showing the system is gradient and asymptotic smoothness via a stabilizability inequality, we establish the existence of global attractors with finite fractal dimension. The continuity of global attractors regarding the parameter in a residual dense set is also proved. The above results allow the damping terms with polynomial growth.

Dynamic Characteristic Analysis of a Toothed Electromagnetic Spring Based on the Improved Bouc—Wen Model

Electromagnetic spring active isolators have attracted extensive attention in recent years. The standard Bouc–Wen model is widely used to describe hysteretic behavior but cannot accurately describe asymmetric behavior. The standard Bouc–Wen model is improved to better describe the dynamic characteristic of a toothed electromagnetic spring. The hysteresis model of toothed electromagnetic spring is established by adding mass, damping, and asymmetric correction terms with direction. Subsequently, the particle swarm optimization algorithm is used to identify the parameters of the established model, and the results are compared with those obtained from the experiment. The results show that the current has a significant impact on the dynamic curve. When the current increases from 0.5 A to 2.0 A, the electromagnetic force sharply increases from 49 N to 534 N. Under different excitations and currents, the residual points predicted by the model proposed in this work fall basically in the horizontal band region of −20–20 N (for an applied current of 1.0 A) and −40–80 N (for an application of 4.5 mm/s). Furthermore, the maximum relative error of the model is 12.75%. The R2 of the model is higher than 0.98 and the highest value is 0.9993, proving the accuracy of the established model.

Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source term

We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity utt−Δut−Δpu+ut=|u|q−2uln|u|,x∈Ω,t>0 in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.

Dynamics of a class of extensible beams with degenerate and non-degenerate nonlocal damping

This work is concerned with new results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with three distinguished nonlocal nonlinear damping terms. In the first possibly degenerate case, the results feature the existence of a family of compact global attractors and a thickness estimate for their Kolmogorov’s $\varepsilon$-entropy. Then, in the non-degenerate context, the structure of the helpful nonlocal damping leads to the existence of finite-dimensional compact global and exponential attractors. Lastly, in a degenerate and critical framework, it is proved the existence of a bounded closed global attractor but not compact. To the proofs, we provide several new technical results by means of refined estimates that open up perspectives for a new branch of nonlinearly damped problems.

Electrical Response Estimation of Vibratory Energy Harvesters via Hilbert Transform Based Stochastic Averaging

Abstract Converting mechanical vibrations into electrical power with vibratory energy harvesters can ensure the portability, efficiency, and sustainability of electronic devices and batteries. Vibratory energy harvesters are typically modeled as nonlinear oscillators subject to random excitation, and their design requires a complete characterization of their probabilistic responses. However, simulation techniques such as Monte Carlo are computationally prohibitive when the accurate estimation of the response probability distribution is needed. Alternatively, approximate methods such as stochastic averaging can estimate the probabilistic response of such systems at a reduced computational cost. In this paper, the Hilbert transform based stochastic averaging is used to model the output voltage amplitude as a Markovian stochastic process with dynamics governed by a stochastic differential equation with nonlinear drift and diffusion terms. Moreover, the voltage amplitude dependent damping and stiffness terms are determined via an appropriate equivalent linearization, and the stationary probability distribution of the output voltage amplitude is obtained analytically by solving the corresponding Fokker–Plank equation. Two examples are used to demonstrate the accuracy of the obtained analytical probability distributions via comparisons with Monte Carlo simulation data.

A novel tuned mass damper inerter: optimal design, effectiveness comparison, and robustness investigation

This study proposes a novel tuned mass damper inerter (NTMDI) by altering the position of the damping element in tuned mass damper inerter (TMDI) or adding an inerter element to a variant tuned mass damper (VTMD). The proposed NTMDI can be regarded as a more general case of tuned viscous mass damper. A detailed mechanical model for a single-degree-of-freedom (SDOF) structure with NTMDI is introduced, and the optimal design of NTMDI is derived theoretically based on fixed-point theory. Analytical solutions for NTMDI optimal design, including the optimal frequency ratio and damping ratio, as well as the optimal dynamic amplification factor of the structure, are obtained. Then, considering three types of loads: including harmonic excitation, white noise excitation and natural seismic excitation, the effectiveness of NTMDI for vibration control was evaluated by comparing it with three other classical tuned mass dampers (TMD, VTMD, and TMDI). The results demonstrate that adding inertance to TMD and VTMD can significantly improve their effectiveness in mitigating structural responses. Additionally, it is found that NTMDI is more efficient in reducing structural responses than TMDI and the frequency range controlled by NTMDI is wider than that controlled by TMDI when the tuned physical mass and inertance of NTMDI are identical to those of TMDI. Finally, a series of robustness investigations are carried out, and NTMDI exhibits superior robustness to TMDI in situations where positive tuning occurs in the optimal frequency ratio. However, in cases where negative tuning is present, TMDI outperforms NTMDI. Fortunately, TMD, VTMD, TMDI, and NTMDI demonstrate excellent robustness in terms of damping.

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.

Higher-order averaging procedure for performance analysis of a mono-stable Duffing piezoelectric energy harvesting system under white noise excitation

This work is concerned with the energy harvesting of mono-stable Duffing oscillators with piezoelectric coupling under white noise excitation. This is the first time that the higher-order averaging procedure is extended to find the first-, second- and third-order approximate stationary joint probability density functions of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. Then the mean-square electric voltage and mean output power are derived through the approximate relation between the electric voltage and the mechanical states. The analytical results are verified by Monte Carlo simulation. The effects of system parameters, such as excitation intensity, mechanical damping term, nonlinear stiffness coefficients, and the resistance ratio of the resistive–capacitive circuit on the mean-square displacement and mean output power are discussed in detail. It has been shown that the averaging procedure of the averaging method of coefficients in FPK equation can be considered as a particular case of the proposed higher-order averaging procedure and the higher-order approximate solutions are much more accurate than the one obtained by the averaging method of coefficients in FPK equation.© 2022 The Authors. Published by Elsevier Ltd.

Evaluation of Dynamic Properties of Trees Subjected to Induced Vibrations

The preservation of trees in urban and archeological areas is a theme of particular relevance. Modern systems of monitoring, together with approaches for deriving the main characteristics of trees influencing their response toward extreme events, are nowadays at the basis of a growing number of studies. The aim of the present paper is the dynamic identification of trees carried out by employing an approach which combines a simple data-acquisition system, direct and ambient sources of excitation, and different data-processing methods. In particular, using a single accelerometer placed at different sections of the trunk and considering excitations induced by either pulling tests or ambient vibrations, the derivation of the main frequencies and levels of modal damping characterizing the dynamic response of a sour cherry tree (Prunus cerasus) is carried out. A finite element model of the tree is also carried out to support the validation of the proposed approach and further analyze the derived outcomes. The obtained results underline the feasibility of the proposed approach in deriving information useful for assessing the behavior of trees toward dynamic actions and, consequently, of particular relevance for the identification of possible damages induced by variations in terms of dynamic characteristics (frequencies) and damping.

Exponential growth of solutions with L_p-norm of a nonlinear viscoelastic wave equation with strong damping and source and delay terms

" In this work, we are concerned with a problem for a viscoelastic wave equation with strong damping, nonlinear source and delay terms. We show the exponential growth of solutions with $L_{p}$-norm. i.e. $\displaystyle\lim_{t\rightarrow\infty}\Vert u\Vert_{p}^{p}\rightarrow\infty.$"

The Exponential Growth of Solution, Upper and Lower Bounds for the Blow-Up Time for a Viscoelastic Wave Equation with Variable- Exponent Nonlinearities

This paper aims to study the model of a nonlinear viscoelastic wave equation with damping and source terms involving variable-exponent nonlinearities. First, we prove that the energy grows exponentially, and thus in 𝐿p2 and 𝐿p1 norms. For the case 2 ≤ 𝑘(. ) < 𝑝(. ), we reach the exponential growth result of a blowup in finite time with positive initial energy and get the upper bound for the blow-up time. For the case 𝑘(. ) = 2, we use the concavity method to show a finite time blow-up result and get the upper bound for the blow-up time. Furthermore, for the case 𝑘(. ) ≥ 2, under some conditions on the data, we give a lower bound for the blow-up time when the blow-up occurs.

WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR FOR THE DISSIPATIVE P-BIHARMONIC WAVE EQUATION WITH LOGARITHMIC NONLINEARITY AND DAMPING TERMS

This paper concerns with the initial and boundary value problem for a p-biharmonic wave equation with logarithmic nonlinearity and damping terms. We establish the well-posedness of the global solution by combining Faedo–Galerkin approximation and the potential well method, and derive both the polynomial and exponential energy decay by introducing an appropriate Lyapunov functional. Moreover, we use the technique of differential inequality to obtain the blow-up conditions and deduce the life-span of the blow-up solution.

Energy Decay for a Type of Plate Equation with Degenerate Energy Damping and Source Term

Energy Decay for a Type of Plate Equation with Degenerate Energy Damping and Source Term

Exponential Growth of Solutions with L p -Norm of a Klein-Gordon Wave Equation with Strong Damping, Source and Delay Terms

Exponential Growth of Solutions with L p -Norm of a Klein-Gordon Wave Equation with Strong Damping, Source and Delay Terms

Active Second-Order Pole-Zero Cancellation Control for Speed Servo Systems With Current Sensor Fault Tolerance

This study presents an advanced output-feedback technique for speed regulation tasks in servo systems, which reduces the system model independence and is independent of the current feedback, leading to fault tolerance. Hence, the output position error drives the model-free order reduction filter for speed, and the acceleration feedback (as a replacement for the current loop) is admissible. Furthermore, the proposed control law includes specially structured active damping terms in the feed-forward loop to actively trigger second-order pole-zero cancellation, forming a simple single-loop proportional-integral derivative-type structure with compensation terms. Finally, a 420-W prototype DC servo system experimentally validates the effectiveness of the proposed technique.