Local Damping Research Articles

Polynomial stability of an abstract system with local damping

In this paper, we consider the polynomial stability for an abstract system of the typeutt+Lu+But=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results.

Finite element formulation for the dynamic analysis of shear deformable thin-walled beams

The present study develops a finite element formulation for the natural vibration and dynamic analysis of thin-walled members with asymmetric cross-sections. The formulation captures shear deformation within the middle surface, global and local warping, viscous damping, translational and rotatory inertial effects, and inertial coupling. The formulation starts with the extended variational form of Hamilton’s principle to account for non-conservative damping effects, in conjunction with a special interpolation scheme aimed at circumventing shear locking. The resulting finite element has two nodes with seven degrees of freedom per node. The discretized equations of motion are integrated using the unconditionally stable Newmark integration scheme with constant average acceleration. For members with doubly symmetric sections, comparisons with shell solutions show that the formulation reliably predicts the natural frequencies and vibration modes associated with flexure/shear and torsion/warping responses. Further comparisons with other non-shear deformable thin-walled beam solutions show that shear deformation effects gain significance in shorter spans, higher vibration modes, and in flexural vibration about the major axis. For monosymmetric and asymmetric sections, comparisons with benchmark solutions showcase the ability of the formulation to capture the inertial coupling arising between the flexure/shear and torsion/warping responses, while keeping the computational cost to a minimum. Comparisons to shell-based solutions demonstrate the ability of the solution to accurately capture the damped time history response of members with doubly symmetric, monosymmetric, and asymmetric sections subjected to harmonic, ramped, and blast-like forces. A separate treatment is provided for degenerate cross-sections with a vanishing global warping function.

Stability of Transmission Wave-Plate Equations with Local Indirect Damping

Stability of Transmission Wave-Plate Equations with Local Indirect Damping

Distinct Surface Warming Response Over the Western and Eastern Equatorial Pacific to Radiative Forcing

AbstractThis study examines regional characteristics of atmospheric and oceanic feedback processes in the western and eastern equatorial Pacific, by applying a localized surface heating in the respective region in a hierarchy of climate models. A western Pacific forcing is largely offset by a negative shortwave cloud radiative effect and damping via wind‐evaporation‐SST feedback. In contrast, an eastern Pacific forcing, while being partially compensated for by ocean dynamical adjustments, induces an amplified warming extending to the central Pacific due to weak local damping mechanisms. As for the inter‐model spread of the future tropical Pacific surface warming pattern, the ocean heat uptake response in the east can explain much of the spread both on fast (<5 years) and slow (>100 years) timescales. Our results suggest that an “El Niño‐like” warming pattern is probable in response to increasing greenhouse gases owing to the strong negative feedback intrinsic to the western Pacific.

Energy transport in one-dimensional oscillator arrays with hysteretic damping

Energy transport in 1-dimensional oscillator arrays has been extensively studied to date in the conservative case, as well as under weak viscous damping. When driven at one end by a sinusoidal force, such arrays are known to exhibit the phenomenon of supratransmission, i.e. a sudden energy surge above a critical driving amplitude. In this paper, we study 1-dimensional oscillator chains in the presence of hysteretic damping, and include nonlinear stiffness forces that are important for many materials at high energies. We first employ Reid's model of local hysteretic damping, and then study a new model of nearest neighbor dependent hysteretic damping to compare their supratransmission and wave packet spreading properties in a deterministic as well as stochastic setting. The results have important quantitative differences, which should be helpful when comparing the merits of the two models in specific engineering applications.

Stability of a suspension bridge with a localized structural damping

<p style='text-indent:20px;'>Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes, winds, ..., features and modifications such as dampers are used to stabilize these bridges. In this work, we use a minimum amount of dissipation to establish exponential decay- rate estimates to the following nonlocal evolution equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}(x,y,t)+\Delta^2 u(x,y,t) - \phi(u) u_{xx}- \left(\alpha(x, y) u_{xt}(x,y,t)\right)_x = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>which models the deformation of the deck of either a footbridge or a suspension bridge.</p>

The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization

<p style='text-indent:20px;'>In this paper, we investigate the stabilization of a linear Bresse system with one singular local frictional damping acting in the longitudinal displacement, under fully Dirichlet boundary conditions. First, we prove the strong stability of our system. Next, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if the three waves have the same speed of propagation. On the contrary, we prove that the energy of our system decays polynomially with rates <inline-formula><tex-math id="M1">\begin{document}$ t^{-1} $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M2">\begin{document}$ t^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula>.</p>

Eventual differentiability of coupled wave equations with local Kelvin-Voigt damping

<p style='text-indent:20px;'>In this work, we consider a coupled wave equations with partially and locally distributed Kelvin-Voigt damping, where only one equation is dissipative.</p><p style='text-indent:20px;'>Under the assumption that the damping coefficient changes smoothly near the interface of the damped and undamped regions, we investigate the effectiveness of the indirect control, and we prove that the associated semigroup is eventually differential.</p>

Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping

<p style='text-indent:20px;'>This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ b(x) = 0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M2">\begin{document}$ x\in (-1, 0] $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M3">\begin{document}$ b(x) = a(x)>0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ x\in (0, 1) $\end{document}</tex-math></inline-formula>. It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that <inline-formula><tex-math id="M5">\begin{document}$ a'(x) $\end{document}</tex-math></inline-formula> has a singularity at <inline-formula><tex-math id="M6">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>, we investigate the decay rate of the solution which depends on the order of the singularity.</p><p style='text-indent:20px;'>When <inline-formula><tex-math id="M7">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> behaves like <inline-formula><tex-math id="M8">\begin{document}$ x^{\alpha}(-\log x)^{-\beta} $\end{document}</tex-math></inline-formula> near <inline-formula><tex-math id="M9">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M10">\begin{document}$ 0\le{\alpha}<1, \;0\le\beta $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M11">\begin{document}$ 0<{\alpha}<1, \;\beta<0 $\end{document}</tex-math></inline-formula>, we show that the system can achieve a mixed polynomial-logarithmic decay rate.</p><p style='text-indent:20px;'>As a byproduct, when <inline-formula><tex-math id="M12">\begin{document}$ \beta = 0 $\end{document}</tex-math></inline-formula>, we obtain the decay rate <inline-formula><tex-math id="M13">\begin{document}$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} $\end{document}</tex-math></inline-formula> of solution for arbitrarily small <inline-formula><tex-math id="M14">\begin{document}$ \varepsilon>0 $\end{document}</tex-math></inline-formula>, which improves the rate <inline-formula><tex-math id="M15">\begin{document}$ t^{-\frac{1}{1-{\alpha}}} $\end{document}</tex-math></inline-formula> obtained in [<xref ref-type="bibr" rid="b14">14</xref>]. The new rate is again consistent with the exponential decay rate in the limit case <inline-formula><tex-math id="M16">\begin{document}$ \alpha\to 1^- $\end{document}</tex-math></inline-formula>. This is a step toward the goal of obtaining the optimal decay rate eventually.</p>

Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [0, 1], with p ∈ (1, ∞). The damping term is assumed to be linear and localized to an arbitrary open sub-interval of [0, 1]. We prove that the semi-group (Sp(t))t≥0 associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether p ≥ 2 or 1 < p < 2.

Coordinated Wide-Area Damping Control Using Deep Neural Networks and Reinforcement Learning

This paper proposes the design of two coordinated wide-area damping controllers (CWADCs) for damping low frequency oscillations (LFOs), while accounting for the uncertainties present in the power system. The controllers based on Deep Neural Network (DNN) and Deep Reinforcement Learning (DRL), respectively, coordinate the operation of different local damping controls such as power system stabilizers (PSSs), static VAr compensators (SVCs), and supplementary damping controllers for DC lines (DC-SDCs). The DNN-CWADC learns to make control decisions using supervised learning; the training dataset consisting of polytopic controllers designed with the help of linear matrix inequality (LMI)-based mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_2/H_\infty$</tex-math></inline-formula> optimization. The DRL-CWADC learns to adapt to the system uncertainties based on its continuous interaction with the power system environment by employing an advanced version of the state-of-the-art deep deterministic policy gradient (DDPG) algorithm referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bounded exploratory control</i> -based DDPG (BEC-DDPG). The studies performed on a 33 machine, 127 bus equivalent model of the Western Electricity Coordinating Council (WECC) system-embedded with different types of damping controls demonstrate the effectiveness of the proposed CWADCs.

Local indirect stabilization of same coupled evolution systems through resolvent estimates

&lt;p style='text-indent:20px;'&gt;In this paper, we consider same systems of two coupled equations (wave-wave, Schrödinger-Schrödinger) in a bounded domain. Only one of the two equations is directly damped by a localized damping term (indirect stabilization). Under geometric control conditions on both coupling and damping regions (internal or boundary), we establish the energy decay rate by means of a suitable resolvent estimate. The numerical contribution is interpreted to confirm the theoretical result of a wave-wave system.&lt;/p&gt;

A Numerical Investigation of the Characteristics of Seismic Signals Induced by Rockfalls

This study proposes a numerical coupling approach to simulate seismic signals of rockfalls and conducts a parametric analysis to explore the characteristics of the seismic signals generated by rockfalls. To validate the approach, three field rockfall tests were selected for comparison. The rockfall velocity, duration, seismic frequency, Husid plot, Arias intensity, and spectrogram of the seismic signals were compared. We found that friction between rocks and the ground affects rock falling behavior. In addition, the local damping and Rayleigh damping assignments in the numerical model have strong effects on the simulation results. The volume of the falling rock and the falling speed of the rock affect the Arias intensity. The coupling approach proposed could be extended and can potentially be used as a useful tool in rockfall hazard estimations.

Damping formulations for finite difference linear dynamic analyses: Performance and practical recommendations

An in-depth study is presented on some numerical formulations used to reproduce damping of motion in solid continua. The focus is addressed in particular to those formulations that in the finite difference geotechnical code FLAC can be exploited to perform linear analyses. The numerical investigations are performed through a simple 1D model of a homogeneous soil deposit interested by a shear wave motion coming from the bedrock, and a gravity dam model subjected to seismic excitations in presence of dam-water-foundation interaction. The Rayleigh damping method is tested in particular to find an appropriate choice of the control frequency in order to minimize the problems of high-frequency overdamping. The analyses also investigate two other formulations (the local damping and its variant, the combined damping) that present the major advantages in providing a frequency independent action and in needing not significant computational resources. Limits and principal drawbacks of their possible use are highlighted.

Energy decay for a porous-elastic system with nonlinear localized damping

Energy decay for a porous-elastic system with nonlinear localized damping

Mechanical and Microstructural Assessment of Inhomogeneities in Oxide Ceramic Matrix Composites Detected by Air-Coupled Ultrasound Inspection

Ceramic Matrix Composites (CMC) are promising materials for high-temperature applications where damage tolerant failure behavior is required. Non-destructive testing is essential for process development, monitoring, and quality assessment of CMC parts. Air-coupled ultrasound (ACU) is a fast and cost-efficient tool for non-destructive inspections of large components with respect to the detection of material inhomogeneities. Even though ACU inspection is usually used for visual inspection, the interpretation of C-scan images is often ambiguous with regard to critical defects and their impact on local material properties. This paper reports on a new approach to link the local acoustic damping of an oxide CMC plate obtained from ACU analysis with subsequent destructive mechanical testing and microstructural analyses. Local damping values of bending bars are extracted from ACU maps and compared with the results of subsequent resonant frequency damping analysis and 3-point bending tests. To support data interpretation, the homogeneous and inhomogeneous CMC areas detected in the ACU map are further analyzed by X-ray computed tomography and scanning electron microscopy. The results provide strong evidence that specific material properties such as Young’s modulus are not predictable from ACU damping maps. However, ACU shows a high, beneficial sensitivity for narrow but large area matrix cracks or delaminations, i.e., local damping is significantly correlated with specific properties such as shear moduli and bending strengths.

Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

Importance of examining particle movements in modelling ballast bed via discrete element method

ABSTRACT The discrete element method is widely adopted to investigate the micro-mechanism of degradation and reinforcement of ballast tracks. Generally, numerical parameters of ballast particles are examined by macro characteristics obtained in field or indoor tests, while particle-scale responses are rarely examined due to the difficulties in practical measurement. In this study, we demonstrate the importance of examining particle movements in modelling ballast bed via discrete element method. A previous laboratory model test is repeated via Particle Flow Code, where 25 sets of damping coefficients are assigned. The results show that the variation of damping coefficients has more significant influence on particle movements than the macroscopic behaviours of ballast and ballast beds. According to the three-dimensional motion characteristics of ballast particles obtained using a SmartRock sensor, the appropriate local damping constant and critical contact damping ratio are suggested for typical loading frequencies.

Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface

In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depend on the speed of the degeneracy. This result improves the decay rate of the semigroup associated to the system on an earlier result of Z. Liu and Q. Zhang in [21] involving the wave equation with local Kelvin-Voigt damping and non-smooth coefficient at interface.

A micro study on centrifuge model test of dynamic compaction using discrete-element method

Using the discrete-element method (DEM), a numerical simulation of dynamic compaction (DC) under an acceleration field of 50g was used to study the microscopic behaviours and macroscopic responses of soils under DC on the basis of a centrifuge model test of DC. The crater depth of the numerical simulation was compared with that of the centrifuge model test and field test. The results showed that the crater depth of the numerical simulation was in reasonable agreement with that of the centrifuge model test and a field test. Microscopic behaviours and macroscopic responses of soils under DC can thus be simulated using DEM by applying a centrifugal force field. Variations of dynamic stress, porosity and coordination number under DC were analysed. The effect of local damping on crater depth was also assessed. This study sheds light on the microscopic mechanism of DC, which should be helpful for improving the effect of DC.