Dependent Damping Research Articles

Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation

In this paper, we consider the 1D Euler equation with time and space dependent damping term −a(t,x)v. It has long been known that when a(t,x) is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient a satisfies the following condition |a(t,x)−μ0|≤a1(t)+a2(x),where μ0≥0 and a1 and a2 are integrable functions with t and x. Under this condition, we show the global existence and the blow-up with small initial data, when μ0>0 and μ0=0 respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.

DMA study of recrystallization kinetic in non-ferrous alloys

Effect of recrystallization in Al-, Cu-, Mg-, Ti- and Ni-based alloys on temperature and amplitude dependent damping of forced mechanical vibration measured by DMA Q800 TA Instruments is analyzed. Different annealing regimes such as instant and interrupted heating, isothermal annealing are proposed. Formation of pseudo recrystallization internal friction peak is discussed.

Analytical solutions for thermo-elastic damping of rotational ring resonators incorporating thermal relaxations and elastic small scales

At the micro and nanoscales, thermo-elastic damping is regarded as one of the most crucial factors for promoting energy dissipation in vibrating structures. The thermo-elastic damping of a thin rotational ring resonator is explored in this paper based on three phase lag heat conduction model for thermal relaxations and nonclassical elasticity for small scale effects. Solving the heat conduction equation for radial direction heat flow and zero heat flux boundary conditions yields the temperature spread across ring cross-section. The equation takes into account only the rotating narrow ring’s in-plane vibrations. For the frequency dependent thermo-elastic damping, the combined nonlocal thermo-elasto-dynamic equations are solved. The present model is validated with the experimental and finite element data. The thermo-elastic damping of rotating rings has been found to be over predicted by classical elasticity. For a particular small scale factor, as rotational speed rises by 80%, the equivalent thermo-elastic damping of the ring resonator reduces by 43%. In addition, the effects of cross-sectional shape, nonlocal scale factor, rotating speed, modenumber, ambient temperature, and substance on thermo-elastic damping are thoroughly explored in this work. The results presented here are useful to build and construct future nanoscale resonant sensors and radio-frequency devices.

Increased velocity feedback gains in the presence of sensory noise can explain paradoxical changes in trunk motor control related to back pain

Literature reports paradoxical findings regarding effects of low-back pain (LBP) on trunk motor control. Compared to healthy individuals, patients with LBP, especially those with high pain-related anxiety, showed stronger trunk extensor reflexes and more resistance against perturbations. On the other hand, LBP patients and especially those with high pain-related anxiety showed decreased precision in unperturbed trunk movement and posture. These paradoxical effects might be explained by arousal potentially increasing average and variance of muscle spindle firing rates. Increased average firing rates could increase resistance against perturbations, but increased variance could decrease precision. We performed a simulation study to test this hypothesis.We modeled the trunk as a 2D inverted pendulum, stabilized by two antagonistic Hill-type muscles, based on their open-loop muscle activation dependent intrinsic stiffness and damping and through 25 ms-delayed, noisy contractile element length and velocity feedback. Reference feedback gains and sensory noise levels were tuned based on previously reported experimental data. We assessed the effect of increasing feedback gains on precision of trunk orientation at different perturbation magnitudes and assessed sensitivity of the effects to open-loop muscle stimulation and noise levels.At low perturbation magnitudes, increasing reflex gains consistently caused an increase in the variance of trunk orientation. At larger perturbation magnitudes, increasing reflex gains consistently caused a decrease in the variance of trunk orientation.Our results support the notion that LBP and related anxiety may increase reflex gains, resulting in an increase in the average and variance of spindle afference, which in turn increase resistance against perturbations and decrease movement precision.

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.

Extended statistical linearization approach for estimating non-stationary response statistics of systems comprising fractional derivative elements

An efficient technique to analyze non-stationary nonlinear random vibrations of dynamic systems endowed with a fractional derivative term is presented. The technique itself represents an extension of the concept of statistical linearization to this kind of systems, and it is applicable for both analytic and hysteretic system nonlinearities. The technique first resorts to harmonic balancing in deriving response-amplitude dependent equivalent damping and stiffness. This enables representation of the fractional derivative term as a linear combination of the system response displacement and velocity with amplitude dependent coefficients. Then, the expected values of these parameters are considered in proceeding to formulate a statistical linearization solution scheme. In this context, the solution procedure is completed by integrating in time the covariance Lyapunov equation associated with the derived equivalent linear system. The reliability of the proposed technique is tested by a series of germane Monte Carlo studies. This juxtaposition is also used to elucidate salient features of the technique, by varying the order of the fractional derivative term, and of the degree of the nonlinearity in the system. It also points out the versatility of the technique in determining the non-stationary values of auto-correlation and cross-correlations response parameters involving even the fractional derivative term.

Vibration control in fluid conveying pipes using NES with nonlinear damping

Control of vibrations in fluid transmitting pipes has emerged as a major engineering challenge. For higher fluid flow rate, support vibration, and excitation load, the vibrations in pipes that convey fluid can be complex and, at times, catastrophic. Recent works have demonstrated that Nonlinear Energy Sink (NES) can be effectively employed for passive vibration control of fluid-conveying pipes. The present work explores the possibility of using NES with nonlinear damping to overcome the limitations of conventional NES-based vibration control in fluid-transmitting pipes. Geometric nonlinear damping (velocity-displacement dependent damping), whose physical realization is possible by a suitable geometric configuration of the linear viscous damper, is considered in this study. Euler–Bernoulli beam theory models the fluid conveying pipe under an external harmonic load. The reduced order model obtained through the Galerkin procedure is investigated analytically using the complex averaging method. The occurrence of strongly modulated and weakly modulated responses are demonstrated, and their effect on vibration mitigation is shown. The frequency range exhibiting strongly modulated response in the system is significantly increased with the addition of nonlinear damping to NES. This assists in mitigating pipe vibrations by initiating targeted energy transfer from pipe to NES. The Saddle–Node and Hopf bifurcation boundaries in the model are identified using the slow-flow dynamics obtained by complex averaging. The effect of nonlinear damping, external excitation, the location of NES, and the flow rate on the bifurcation boundaries are investigated in detail. It is shown that the external excitation needed to trigger Saddle–Node bifurcation is largely reduced for NES having nonlinear damping. Furthermore, a study of Hopf bifurcation in the frequency domain shows that using NES with nonlinear damping significantly decreases the frequency range in which high-amplitude isolated solutions occur. The method of multiple scales is used to derive the analytical expression for the slow invariant manifold of the pipe-NES system. The vibration suppression mechanism and the occurrence of strongly modulated responses are explained with the help of flow on slow invariant manifolds. A comparison of the topology of the slow invariant manifolds shows that using nonlinear damping with NES substantially reduces the vibration amplitude of the pipe structure. The percentage of energy transferred to the NES is quantified to show the effectiveness of the proposed absorber over normal NES for different flow speeds.

Spectral element formulation for rock-socketed mono-pile under horizontal dynamic loads

The current study entails a thorough investigation of the dynamic response of a mono-pile foundation in homogeneous soil and varied layered soil–rock conditions under horizontal dynamic loads. A sequence of visco-elastic springs linked along its length represents the mono-pile immersed in elastic half space. The frequency dependent stiffness and damping of a mono-pile foundation embedded in layered elastic half space are developed using spectral element formulation. The variations in stiffness and damping with frequency for various soil–rock systems are demonstrated. The dynamic response characteristics of the pile were found to be influenced by the depth of the rock-socketed pile. The presence of the underlying layer of stiffer layer of rock enhances the overall stiffness of the system. The frequency versus amplitude responses of a mono-pile supported heavy mass block for both horizontal translation and rocking motion are calculated using the stiffness and damping of the mono-pile under various soil rock conditions. The horizontal amplitude of the pile decreases with increasing pile rock socketing depth. The resonant amplitudes for rocking motion increases as the pile’s rock socketing depth increases. Finally, the changes in natural frequency and peak amplitudes with varied pile slenderness ratios are investigated for diverse pile–soil–rock conditions.

Seismic design of hybrid braced frames with self-centering braces and fluid viscous damping braces

Combining self-centering braces (SCBs) and fluid viscous damping braces (FVDBs) in parallel emerges as a promising seismic-resistant strategy, because it has the potential of simultaneously controlling peak deformation, peak acceleration, peak base shear and nearly eliminating residual deformation. However, the corresponding seismic design method is still under development. To this end, this paper extended the performance-based plastic design (PBPD) method to hybrid braced frames (HBFs) equipped with SCBs and FVDBs. The PBPD method selected target drift and yield mechanism at the very beginning of the design procedure, and then it derived the design base shear using energy equivalent concept. The key step was to obtain the constant-ductility spectra of the equivalent single-degree-of-freedom systems through nonlinear time history analysis (NLTHA). The regression functions of the spectra were then incorporated with the PBPD procedure. A six-story steel frame was selected for demonstrating the design method. A total of 4 HBFs were designed by defining 4 levels of viscous damping ratio (ξv) supplemented by FVDBs. The seismic responses of the designed HBFs were examined by subjecting them to 20 earthquake ground motions. The NLTHA results indicated that the designed HBFs can well satisfy the performance targets. The design method was robust, although the ξv values were varied in a wide range. The design method may also shed light to the other types of hybrid systems with displacement- and velocity- dependent damping braces.

Can transient simulation efficiently reproduce well known nonlinear effects of jointed structures?

Although joints are omnipresent in engineering structures, their proper consideration in simulations is still a challenging area of research. This is because joints introduce local nonlinearities which lead to load and preload-dependent resonance frequencies and damping, mode coupling, high variability of contact area and contact pressures. The knowledge about those phenomena has been gathered via systematic experimental testing. A meaningful digital or virtual replication of such tests would require a transient numerical time integration, where besides all dynamic effects, the local and fine discretized nonlinear contact and friction forces in the joint also need to be considered. The Finite Element Method can only provide this theoretically since the necessary computation times make practical use almost impossible. Newer methods, where the joint deformation is represented via so-called contact modes instead of nodal degrees of freedom, offer a perspective for transient virtual tribomechadynamics. This paper is devoted to the following question: Is it possible for the first time to numerically reproduce real experiments with low computing times so that the previously mentioned nonlinear effects can be observed? In this work the transient numerical response of a jointed structure due to different loads (sine sweep, sine, step and impulse) is evaluated and related to measurements. The application of the Hilbert Transform to the time data, as known from measurements, reveals load and preload dependent eigenfrequency and damping. Other measured phenomena like mode coupling, a high variability in the contact area and resulting contact force can also be observed numerically. The results are in good qualitative agreement with the experiments. This work proves the principal capability of virtual tribomechadynamics as an efficient complement to real testing for gaining deeper insight. All the powerful measurement methods can directly be applied to simulated data.

Influences of Attack Angles on Aerodynamic Derivatives and Flutter Characteristics of Flat Box Girders

The flutter response of Nanjing Fourth Bridge flat box girder under different wind attack angles was tested in detail through sectional model test. The evolution of unsteady and steady critical amplitude with wind speed was discussed. Based on the amplitude envelope of the flutter response and the Hilbert transform, the amplitude- dependent modal damping of the system is identified, and the mechanism of the flutter mode change in the wind angle of attack is initially explained. Secondly, the modal parameters of the system under different wind attack angles were extracted. Based on the bimodal coupled flutter analysis method, the nonlinear flutter derivatives of the section under different wind attack angles were identified, and the change law of the amplitude dependence of the key flutter derivatives on the wind attack angle and the potential influence on the flutter morphology and characteristics of the section were studied. Finally, the effect of wind attack angle on uncoupled and coupled aerodynamic damping is analyzed by deconstructing the modal damping one by one, and the dynamic mechanism of the differential flutter performance caused by fractional damping is illustrated.

Smoluchowski–Kramers approximation with state dependent damping and highly random oscillation

<p style='text-indent:20px;'>The small mass limit (Smoluchowski–Kramers approximation) of class systems of ordinary differential equations describing motions of small mass particle with state dependent friction and high oscillation is derived by a diffusion approximation approach. In the small mass limit, due to the state dependent damping, one additional term appears in the limit equation, which leads to a stochastic differential equation (SDE) as the highly random oscillation appears as a multiplicative white noise.</p>

Lifespan estimates for semilinear wave equations with space dependent damping and potential

In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are assumed to be critical or short range, spherically symmetric perturbation. The blow up results and the upper bound of lifespan estimates are obtained by the so-called test function method. The key ingredient is to construct special positive solutions to the linear dual problem with the desired asymptotic behavior, which is reduced, in turn, to constructing solutions to certain elliptic “eigenvalue” problems.

A thorough comparison between measurements and predictions of the amplitude dependent natural frequencies and damping of a bolted structure

Bolted joints are a significant source of damping and nonlinearity in built up structures. In the micro-slip regime (i.e. when the bolts do not slip completely) the log of the damping tends to increase linearly with the log of the vibration amplitude, the so-called power-law behavior. The natural frequency also tends to decrease slightly with vibration amplitude. While many works have successfully tuned phenomenological models to capture these behaviors, far fewer have sought to predict the nonlinearity in a bolted joint and validated the predictions with measurements over a range of bolt preloads and vibration amplitudes. This work presents a step towards such a prediction, using a detailed finite element model of a structure, including the preload forces in the bolts and Coulomb friction in the contact, to seek to predict the nonlinear damping and stiffness of the structure and how they vary with preload. The structure studied consists of two beams bolted together at their free ends, the so-called S4 Beam. While the contact interfaces were machined to be nominally flat, the micron-scale curvature in the contact interface is included in the model, approximated in two different ways, to seek to understand what fidelity is needed at the contact interface to successfully capture its dynamic behavior. The recently presented Quasi-Static Modal Analysis (QSMA) method is employed, where the amplitude dependent damping and natural frequency can be predicted from a single quasi-static simulation, avoiding the expense of numerical integration. The predicted damping and natural frequency are compared with measurements at various preloads, showing reasonable agreement if a Coulomb friction coefficient of 0.2 is used for all simulations. The simulations also revealed that, while the micron-scale curvature of the interfaces was important, the results were not very sensitive to how it was approximated.

A novel design of vibration isolator with high and frequency dependent damping characteristics based on a Large Negative Poisson’s Ratio (LNPR) structure

In this paper, a biconcave-lens shaped structure with a large value of negative Poisson’s ratio (LNPR) is proposed. By filling viscoelastic damping materials inside this LNPR structure, a novel vibration isolator with frequency dependent damping characteristics is designed. With this integrated design of negative Poisson’s ratio structural element and viscoelastic materials, the compression and tension of the filled viscoelastic materials will be enlarged by the LNPR structure during a vibration process, which can effectively dissipate the vibrational energy. This proposed novel vibration isolator can achieve large damping characteristics and low vibration transmissibility, particularly when applying in micro-level vibration isolation. Finite element simulation is performed to predict the static and dynamic characteristics of this proposed novel vibration isolator, and characterize the hyperelastic and viscoelastic properties of the damping rubber. Finally, a vibration isolation platform using four proposed vibration isolators is established to perform frequency-sweep testings, with which our proposed design concept of this novel vibration isolator is demonstrated.

Enhanced conductance response in radio frequency scanning tunnelling microscopy

Diverse spectroscopic methods operating at radio frequency depend on a reliable calibration to compensate for the frequency dependent damping of the transmission lines. Calibration may be impeded by the existence of a sensitive interdependence of two or more experimental parameters. Here, we show by combined scanning tunnelling microscopy measurements and numerical simulations how a frequency-dependent conductance response is affected by different DC conductance behaviours of the tunnel junction. Distinct and well-defined DC-conductance behaviour is provided by our experimental model systems, which include C60 molecules on Au(111), exhibiting electronic configurations distinct from the well-known dim and bright C60’s reported so far. We investigate specific combinations of experimental parameters. Variations of the modulation amplitude as small as only a few percent may result in systematic conductance deviations as large as one order of magnitude. We provide practical guidelines for calibrating respective measurements, which are relevant to RF spectroscopic measurements.

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.

Comment on the paper by R. Maniar and A. Bhattacharyay, [Random walk model for coordinate-dependent diffusion in a force field, Physica A 584 (2021) 126348

Paper (Maniar and Bhattacharyay, 2021) introduces a discrete random-walk (RW) model with a site dependent waiting time and an external force i.e. a site dependent drift. The RW is compared with a Smoluchowski equation in continuous space and time with space dependent diffusivity and space dependent damping. Using numerical simulations in Maniar and Bhattacharyay (2021) it is shown that the equilibrium distribution satisfies locally the Stokes–Einstein relation i.e. there is a constant temperature in the whole system. Here we show that the result can be recovered analytically with a continuous limit in the RW master equation. Moreover, we evidence that the RW model should be carefully defined in order to satisfy locally Stokes–Einstein relation, indeed a different definition of the discrete RW process describes a system that satisfies the Smoluchowski equation with constant damping, where the temperature is not uniform.

Nonclassical nonlinear elasticity of crystalline structures.

Hysteretic elastic nonlinearity has been shown to result in a dynamic nonlinear response which deviates from the known classical nonlinear response; hence this phenomenon was termed nonclassical nonlinearity. Metallic structures, which typically exhibit weak nonlinearity, are typically categorized as classical nonlinear materials. This article presents a material model which derives stress amplitude dependent nonlinearity and damping from the mesoscale dislocation pinning and breakaway to show that the lattice defects in crystalline structures can give rise to nonclassical nonlinearity. The dynamic nonlinearity arising from dislocations was evaluated using resonant frequency shift and higher order harmonic scaling. The results show that the model can capture the nonlinear dynamic response across the three stress ranges: linear, classical nonlinear, and nonclassical nonlinear. Additionally, the model also predicts that the amplitude dependent damping can introduce a softening-hardening nonlinear response. The present model can be generalized to accommodate a wide range of lattice defects to further explain nonclassical nonlinearity of crystalline structures.

Analytical model for damping behaviour of an orthotropic cantilever hollow member with polygonal perforations

This work presents a mathematical framework for analysing damping behavior of an orthotropic cantilever hollow member containing polygonal perforations while subjected to base excitation. Amplitude and strain rate depending damping models are considered in the framework. A generalized formulation of stress functions for different shapes of the polygonal perforations is derived using the complex variable approach. For this study, five different shapes of the perforations i.e., circular, triangular, diamond, pentagonal and hexagonal are considered. The final expression of the damping ratio is evaluated in terms of the excitation amplitude, perforation-based stress functions, number of perforations and material dependent damping coefficients. The damping ratio obtained through the proposed approach is validated by comparing with experimental results available in previous study for circular perforations. From results, the amplitude dependent damping behaviour is observed. Later, the damping ratio is obtained for different shapes of perforations. The damping ratio is significantly enhanced when the tube is perforated with triangular perforations along its length, which is increased for more number of perforations in the tube.