Nonlinear Damping Coefficients Research Articles

Complex Multi-nonlinearity for piezoelectric energy harvesting systems based on an accurate higher-order perturbation method

A theoretical model of a complex nonlinear coupling energy harvester under hybrid excitations is established with analytical and numerical solutions. Different orders of the method of multiple scales are derived to approximate and analyze the equation of motion for the electromechanical coupling system, thus obtaining higher-order approximate analytical solutions for its operating status and output performance indicators. The significant effects of external excitation, linear and nonlinear damping coefficients, as well as impedance on the transverse displacement, voltage, and power of higher-order and lower-order methods are analyzed. The results show that higher-order solution methodology are more sensitive to parameter excitation. For a large damping, the higher-order analysis has more excellent performance in capturing energy. The higher order method shows more pronounced nonlinear softening phenomenon and it is more suitable for low frequency environments. The higher order methods improves solution accuracy of piezoelectric energy harvesting systems. Consequently, it is easier to derive the nonlinear performance of the harvesting systems, and thus improves the average output power.

Nonlinear roll damping coefficients and natural frequency estimation via augmented extended Kalman filtering for floating body

ABSTRACT The nonlinear roll damping coefficients and natural frequency of a floating body are discussed in this article using an augmented Extended Kalman Filtering (EKF) method. This paper presents a non-linear model-based observer for the combined estimation of free roll decay motion states and damping coefficients. The identification framework uses an improved extended Kalman filter (EKF) to deal with model parameter variability and noisy measurement input. The estimated error was compared to the theoretical results obtained from the covariance matrix for filtering correctness. It is found that the nonlinear damping coefficients and natural frequency obtained from the EKF approach are a more realistic representation of the roll dynamics from the general estimation procedure given by the literature. The proposed method also has the built-in capability to remove random noise from the measured roll. The influence of noise levels in the measurements on the identification accuracy was investigated and discussed.

Analysis of the effects of nonlinear damping on a multiple-degree-of-freedom quasi-zero-stiffness vibration isolator

In the present paper the effect of nonlinear damping on a multi-directional, Quasi-Zero-Stiffness (QZS) vibration isolator is assessed. A two degree-of-freedom vibration isolator is considered, the design consisting of an X arrangement of two struts comprising concentric magnets known to exhibit quasi-zero-stiffness. Nonlinear damping was accounted for in the system through rotational friction at the joints and damping in line with the struts themselves. The equations of motion were derived for each degree of freedom separately, namely transverse and twist motions, using the Energy method and Lagrange-Euler equations. The equations of motion were solved analytically using the Harmonic Balance method and numerically via Simulink. The numerical results validated the analytical predictions for both degrees of freedom. Absolute transmissibility was used to compare the performance of the system with both linear and nonlinear damping. Damping coefficients were found for optimal performance in each degree of freedom of the isolator. The transmissibility curves were optimised with no linear damping, and low strut and rotational damping coefficients. Complex attributes of the response were identified and discussed including a shift in the time averaged response and non-periodic motion in the response under certain conditions. These complexities can be reduced through a careful selection of linear and nonlinear damping coefficients. The research supports the hypothesis that nonlinear damping can improve the performance of quasi-zero-stiffness vibration isolators and identifies complex behaviours that must be considered in the isolator design.

Nonlinear Stiffness and Nonlinear Damping in Atomically Thin MoS2 Nanomechanical Resonators

We report on experimental measurements and quantitative analyses of nonlinear dynamic characteristics in ultimately thin nanomechanical resonators built upon single-layer, bilayer, and trilayer (1L, 2L, and 3L) molybdenum disulfide (MoS2) vibrating drumhead membranes. This synergistic study with calibrated measurements and analytical modeling on observed nonlinear responses has led to the determination of nonlinear damping and stiffness coefficients at cubic and quintic orders for these two-dimensional (2D) resonators operating in the very high frequency (VHF) band (up to ∼90 MHz). We find that the quintic force can be ∼20% of the Duffing force at larger amplitudes, and thus, it generally cannot be ignored in a nonlinear dynamics analysis. This study provides the first quantification of nonlinear damping and frequency detuning characteristics in 2D semiconductor nanomechanical resonators and elucidates their origins and dependency on engineerable parameters, setting a foundation for future exploration and utilization of the rich nonlinear dynamics in 2D nanomechanical systems.

Insight into the intrinsic time-varying aerodynamic properties of a truss girder undergoing a flutter with subcritical Hopf bifurcation

In this study, the investigation on the intrinsic time-varying nonlinear aerodynamic properties and actual energy feedback mechanism of limit cycle oscillation (LCO) and subcritical Hopf bifurcation, which is of great importance of the flutter design of bridges, but once rarely be conducted, was comprehensively carried out. The response characteristics of the nonlinear flutter were experimentally investigated. The dynamic mechanism of the subcritical Hopf bifurcation was firstly introduced in terms of the equivalent modal damping ratio as a function of amplitude. Further, a modified nonlinear self-excited force model was proposed to investigate the intrinsic time-varying characteristics of the aerodynamic properties and the real energy feedback mechanism of the subcritical Hopf bifurcation. Based on the model, the characteristics of nonlinear self-excited forces and nonlinear aerodynamic damping coefficients, as well as their contribution to the energy exchange behaviors, were studied. Subsequently, the energy feedback mechanisms of LCO and subcritical Hopf bifurcation were qualitatively discussed in detail considering the hysteresis loop of nonlinear self-excited forces (NSEF), which highlighted the important role of the linear self-excited-moment (including damping and stiffness) in the generation of a subcritical Hopf bifurcation, and the higher-order force components on the generation of stable LCO. Finally, the driving mechanisms of LCO and subcritical Hopf bifurcation were also quantitatively explained via an energy budget analysis, wherein the total work applied by structural and aerodynamic forces was presented as a function of amplitude and time.

Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System and Biomechanical Behaviour Prediction

The biomechanical responses of the driver-truck system in a dynamic environment have become a significant concern in the design and control of trucks. When evaluating the riding comfort of the seat-cab system, it is necessary to predict the biomechanical responses of the driver’s different parts and directions. However, there is no reliable model and method for effectively predicting the response characteristics of the driver-seat-cab system. Aiming at such problem, firstly, based on the 7 DOF (degree-of-freedom) seated human biodynamic model established previously, a 10 DOF non-linear dynamic model of the driver-seat-cab system was created, and its vibration differential equations were established. Secondly, the vibration signals for simulation and verification were collected through the road test using a truck. Thirdly, based on the Newmark-β integration method, the specific solution process of the model was given. The non-linear damping coefficients of the front and rear dampers for the cab suspensions were measured with a bench test. Moreover, the simulations were conducted based on the measured model parameters, taking the collected frame vibration signals as the inputs. The results show that the simulation results agree with the test results, proving that the dynamic model can effectively predict the driver’s biomechanical responses. Finally, some useful conclusions were obtained through the simulation analysis. The established model and conclusions can provide technical support for comfort evaluation, optimization design, and control of the seat-cab suspension system.

Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination

The widely cited Haken–Kelso–Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

Nonlinear roll damping parameter identification using free-decay data

A common practice for evaluating the nonlinear damping in ship roll motion is to perform a free-decay experiment. Although various methods have been developed to estimate the coefficients of a nonlinear damping model, these methods are limited to certain and relatively simple models. In addition, the performances of these methods have been greatly affected by the random noise embedded in the measured data. In this article, an efficient and versatile parameter identification method, applicable to rather general nonlinear damping and restoring models, to estimate nonlinear damping coefficients from contaminated free-decay data is proposed. The proposed method approximates the roll motion as complex exponentials by using the Prony-SS method. Because the Prony-SS method has a build-in noise rejection mechanism via the usage of truncated singular value decomposition, random noise in the measured data could be largely removed. Based on the analytical complex exponential representation of free-decay data, the damping coefficients can be easily estimated from the nonlinear equation of roll motion by using a least-square technique. Numerical examples demonstrate the effectiveness of the proposed method, even when the simulated data are significantly corrupted by random noise. Furthermore, the developed method performs well when nonlinear restoring parameters are also treated as unknown.

Very large amplitude vibrations of flexible structures: Experimental identification and validation of a quadratic drag damping model

This article presents a theoretical, numerical and experimental study of resonant structures undergoing very large amplitude vibrations. The purpose of this work is to validate a model for the damping due to the action of the air on a structure’s single-mode response in the steady-state. Experiments are performed on cantilever beams and beam assemblies of various sizes, from centimetric to micrometric, under harmonic base excitation. Dimensionless linear and nonlinear modal damping coefficients are simultaneously identified by means of frequency-domain identification techniques. These measurements demonstrate the pertinence of the presented model.

Parameter estimation for a ship's roll response model in shallow water using an intelligent machine learning method

In order to accurately identify the ship's roll model parameters in shallow water, and solve the problems of difficult estimating nonlinear damping coefficients by traditional methods, a novel Nonlinear Least Squares - Support Vector Machine (NLS-SVM) is introduced. To illustrate the validity and applicability of the proposed method, simulation and decay tests data are combined and utilized to estimate unknown parameters and predict the roll motions. Firstly, simulation data is applied in the NLS-SVM model to obtain estimated damping parameters, compared with pre-defined parameters to verify the validity of the proposed method. Subsequently, decay tests data are used in identifying unknown parameters by utilizing traditional models and the new NLS-SVM model, the results illustrate that the intelligent method can improve the accuracy of parametric estimation, and overcome the conventional algorithms' weakness of difficult identification of the nonlinear damping parameter in the roll model. Finally, to show the wide applicability of the proposed model in shallow water, experimental data from various speeds and Under Keel Clearances (UKCs) are applied to identify the damping coefficients. Results reveal the potential of using the NLS-SVM for the problem of the roll motion in shallow water, and the effectiveness and accuracy are verified as well.

Estimation of Nonlinear Roll Damping by Analytical Approximation of Experimental Free-Decay Amplitudes

Damping is critical for the roll motion response of a ship in waves. A common method for the assessment of damping in a ship’s rolling motion is to perform a free-decay experiment in calm water. In this paper, we propose an approach for estimating nonlinear damping that involves a linear exponential analytical approximation of the experimental roll free-decay amplitudes, followed by parametric identification based on the asymptotic method. The restoring moment can be strongly nonlinear. To validate this method, we first analyzed numerically simulated roll free-decay data using rolling equations with two alternative parametric forms: linear-plus-quadratic and linear-plus-cubic damping. By doing so, we obtained accurate estimates of nonlinear damping coefficients, even for large initial roll amplitudes. Then, we applied the proposed method to real free-decay data obtained from a scale model of a bulk barrier, and found the simulated results to be in good agreement with the experimental data. Using only free-decay peak data, the proposed method can be used to estimate nonlinear roll-damping coefficients for conditions with a strongly nonlinear restoring moment and large initial roll amplitudes.

Нелинейное затухание и нелинейный фазовый набег интенсивных спиновых волн в экранированных ферритовых пленках

AbstractWe analyze for the first time nonlinear damping and nonlinear phase shift of intense surface spin waves in screened yttrium–iron garnet films. Nonlinear damping coefficients are determined using the phenomenological model of spin wave propagation in such films. It is found that nonlinear phase shift in screened films is a weak effect as compared with free films and amounts to tens of degrees. At the same time, nonlinear damping in screened films is comparable with the corresponding value for free films. It is shown that for micrometer-thick films, the losses introduced upon an increase of power to 15 dBm become larger by 2–5 dB.

Commercial Vehicle Ride Comfort Optimization Based on Intelligent Algorithms and Nonlinear Damping

The method chosen to conduct vehicle dynamic modeling has a significant impact on the evaluation and optimization of ride comfort. This paper summarizes the current modeling methods of ride comfort and their limitations. Then, models based on nonlinear damping and equivalent damping and the multibody dynamic model are developed and simulated in Matlab/Simulink and Adams/Car. The driver seat responses from these models are compared, showing that the accuracy of the ride comfort model based on nonlinear damping is higher than the one based on equivalent damping. To improve the reliability of ride comfort optimization and analysis, a ride comfort optimization method based on nonlinear damping and intelligent algorithms is proposed. The sum of the frequency‐weighted RMS of the driver seat acceleration, the RMS of dynamic tyre load, and suspension working space is taken as the objective function in this article, using nonlinear damping coefficients and stiffness of suspension as design variables. By applying the particle swarm optimization (PSO), cuckoo search (CS), dividing rectangles (DIRECT), and genetic algorithm (GA), a set of optimal solutions are obtained. The method efficiency is verified through a comparison between frequency‐weighted RMS before and after optimization. Results show that the frequency‐weighted RMS of driver seat acceleration, RMS values of the suspension working space of the front and rear axles, and RMS values of the dynamic tyre load of front and rear wheels are decreased by an average of 27.4%, 21.6%, 25.0%, 19.3%, and 22.3%, respectively. The developed model is studied in a pilot commercial vehicle, and the results show that the optimization method proposed in this paper is more practical and features improvement over previous models.

Hydrodynamic derivative determination based on CFD and motion simulation for a tow-fish

A new methodology to determine hydrodynamic derivatives of a tow-fish underwater vehicle using computational fluid dynamics (CFD) was presented. Hydrodynamic derivatives were derived from a second-order modulus expansion and consist of added mass, linear damping, and nonlinear damping coefficients. Added mass coefficients were analytically obtained using strip theory. All components of linear and nonlinear damping coefficients were determined by CFD simulations without any pre-elimination. The determination procedure for damping coefficients is as follows. First, a series of simulations was prepared for various situations reflecting the actual operating conditions of the tow-fish. Then, actual CFD simulations were performed for each of the cases, and 6 degree-of-freedom (6-DOF) forces and moments acting on the tow-fish were obtained. Finally, all linear and nonlinear damping coefficients were determined by optimized fitting of the CFD results using the least-squares procedure. The effectiveness of the determined hydrodynamic derivatives was demonstrated by reproducing the 6-DOF forces and moments and comparing them with those obtained from the CFD simulations. The relative significance of each coefficient was estimated by a global comparative view of normalized coefficients. To demonstrate the applicability of the current approach, 6-DOF simulations of the tow-fish for three different scenarios (L -, U -, and S -turn scenarios) at various towing speeds were conducted. The efficacy of the current methodology was strengthened by graphical and physical analyses of the virtual simulation results.

Power Takeoff Optimization to Maximize Wave Energy Conversions for Oscillating Water Column Devices

This paper presents an investigation on how to optimize the power takeoffs (PTOs) to maximize the mean wave energy conversion from seas for floating oscillating water column (OWC) devices. For this purpose, the linear air turbine PTOs are first analytically optimized to maximize the mean energy conversion for regular waves, based on which a simple and fast-turnaround assessment method is proposed to optimize the linear PTO damping coefficient for maximizing the mean energy conversion for the given sea states. Further on, the focus is on how we can reliably assess the maximized mean wave energy conversions if the OWCs are equipped with nonlinear air-turbine PTOs, as frequently seen in practical OWC plants. Conventionally, the time-consuming time-domain analysis is employed for searching the optimized nonlinear damping coefficients which may be both wave-height- and wave-period-dependent for the given sea state. This may not be suitable in the early design stages of the device when many different options may be compared and checked for optimizing the device itself. As such, a reliable fast-turnaround assessment is very desirable for the device performance and for the assessment of energy conversion for the sea states in the deployment site. From the examples in the study, it is shown that the OWC wave energy converters with the optimized linear PTOs have the same energy capture capacities as those with optimized nonlinear PTOs. As such, it can be suggested that the OWC wave energy capture capacity can be analyzed using the fast-turnaround frequency-domain analysis, regardless of whether the linear or nonlinear PTO is used in the wave energy converter.

Simulation-Based Determination of Hydrodynamic Derivatives and 6DOF Motion Analysis for Underwater Vehicle

This paper introduces a simulation-based determination method for hydrodynamic derivatives and 6DOF (degrees-of-freedom) motion analysis for an underwater vehicle. Hydrodynamic derivatives were derived from second-order modulus expansion and composed of the added mass, and linear and nonlinear damping coefficients. The added mass coefficients were analytically obtained using the potential theory. All of the linear and nonlinear damping coefficients were determined using CFD simulation, which were performed for various cases based on the actual operating condition. Then, the linear and nonlinear damping coefficients were determined by fitting the CFD results, which referred to 6DOF forces and moments acting on an underwater vehicle, with the least square method. To demonstrate the applicability of the current study, 6DOF simulations for three different scenarios (L-, U-, and S-turn) were carried out, and the results were validated on the basis of physical plausibility.

Characterization of the viscoelastic effects of thawed frozen soil on pile by measurement of free response

Most of the man-made structures in cold regions could expose in seasonally frozen conditions which could have significant impact on the soil-pile systems of structures. Little research has been conducted to investigate the thawed frozen soil effects on the dynamic performance of pile structures. An experimental investigation of the thawed frozen soil effects on the dynamic behavior of a pile structure is presented. Free-decay response approach is used in summer to estimate the dynamic properties of a pile partially embedded in Fairbanks silt one year ago. The frequency spectrum analyses are used to evaluate system's vibration properties. The empirical modal decomposition is used to decompose response signal for system parameter identification. Results show that two dominant modes can be used to characterize pile dynamic response, one is a component at 14Hz and the other one is at 920Hz. The former is a pile rocking mode in which pile behaves like a rigid body; the latter is the first order structural bending mode of the embedded pile. The rocking mode exhibits nonlinear characteristics in the time-frequency domain which is dominated by strong soil nonlinearity. The structural modes of embedded pile exhibit weak nonlinear characteristics. The damping ratios are identified, which can be used to characterize pile-thawed frozen soil interactions. A theoretical nonlinear model including equivalent nonlinear soil spring and nonlinear damping coefficients is developed to correlate with the experimental results.

A damping boundary condition for atomistic-continuum coupling

The minimization of spurious wave reflection is a challenge in multiscale coupling due to the difference of spatial resolution between atomistic and continuum regions. In this study, a new damping condition is presented for eliminating spurious wave reflection at the interface between atomistic and continuum regions. This damping method starts by a coarse–fine decomposition of the atomic velocity based on the bridging scale method. The fine scale velocity of the atoms in the damping region is reduced by applying nonlinear damping coefficients. The effectiveness of this damping method is verified by one- and two- dimensional simulations.

Parameter identification for nonlinear damping coefficient from large-amplitude ship roll motion using wavelets

In this paper, we have introduced an efficient Legendre wavelet spectral method (LWSM) to ship roll motion model for investigating the nonlinear damping coefficients. The accuracy of the proposed method is assessed by rolling with an initial amplitude using nonlinear type of equations. To the best of our knowledge until now there is no rigorous wavelet solution has been reported for the above model. Convergence analysis of the proposed method is discussed. The operational matrix of derivative is utilized to convert a ship roll motion equation into a set of algebraic equations. Accurate coefficients can be easily obtained for large rolling angles up to the amplitude at the maximum value of the restoring moment using wavelets. Some numerical examples are given to show the validity and applicability of the proposed wavelet method. Moreover the use of Legendre wavelets is found to be simple,flexible, accurate, efficient and small computation cost.

Hard-Spring Bistability and Effect of System Parameters in a Two-Degree-of-Freedom Vibration System with Damping Modeled by a Fractional Derivative

The harmonic balance coupled with the continuation algorithm is a well-known technique to follow the periodic response of dynamical system when a control parameter is varied. However, deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, therefore this paper introduces polynomial homotopy continuation technique to investigate the steady state bifurcation of a two-degree-of-freedom system including quadratic and cubic nonlinearities subjected to external and parametric excitation forces under a nonlinear absorber. The fractional derivative damping is considered to examine the effects of different fractional order, linear and nonlinear damping coefficients on the steady response. By means of polynomial homotopy continuation, all the possible steady state solutions are derived analytically, i.e. without numerical integration. Coexisting periodic solutions, saddle-node bifurcation and various effects of fractional damping on the steady state response are found and investigated. It is shown that the fractional derivative order and damping coefficient change the bifurcating curves qualitatively and eliminate the saddle-node bifurcation during resonance. Moreover, the system response depicts bigger and bigger region of hard-spring bistability with increasing fractional derivative order, but the region of hard-spring bistability of steady response becomes gradually small and then disappears when we increase the linear and nonlinear damping coefficients. In addition, the analytical results are verified by comparison with the numerical integration ones, it can be found that the present approximate resonance responses are in good agreement with numerical ones.