Nonlinear Energy Sink Parameters Research Articles

Gust response alleviation of the aircraft wing aeroelastic systems in supersonic flows using nonlinear energy sink

In order to reduce the gust response level of the aircraft wing aeroelastic systems in supersonic flows, the nonlinear energy sink (NES) is introduced into the aeroelastic system, and the effect of the nonlinear energy sink on the aeroelastic gust response is studied. The gust response dynamic model without and with NES is established by Lagrange equation, considering the cubic nonlinear characteristics of both plunge and pitch stiffness. The piston theory model as well as the '1-cos' model are employed to describe the unsteady aerodynamic load and the gust velocity, respectively. Besides, the comparison study of the gust response amplitude and energy characteristics of the wing aeroelastic systems with and without NES are analyzed. For systems with NES, the amplitude of gust dynamic response decreases rapidly, while vibration can be transmitted to the NES mechanism and causing its vibration. From an energy perspective, about 72.15 % of the original vibration energy has been transferred into NES, and further consumed through its damping. The influence of NES parameters, such as frequency ratio, mass ratio, damping ratio, and installation position on gust response, is discussed. Numerical simulations demonstrate that: as the frequency ratio, damping ratio and mass ratio increase, the dynamic response amplitude of the aeroelastic system decays faster, and the energy transferred from the wing system also increases. In addition, an optimal relative distance for the installation of NES can be found with best gust alleviation effect. The influence of nonlinear energy sink on the targeted transfer of vibration energy in aeroelastic systems are analyzed under different working conditions. The NES shows great gust alleviation effect and good robustness under different flight speeds, gust intensities, and gust lengths.

Vibration suppression in SDOF systems coupled to a nonlinear energy sink under colored noise

This study analyzes the stochastic vibration suppression and optimization of the single-degree-of-freedom (SDOF) system equipped with the cubic stiffness nonlinear energy sink (NES) under colored noise excitation. Two theoretical methods are proposed: an integrated method (denoted as EL-ELM) that combines evolutionary Lyapunov theory with the equivalent linearization method, and the other is an empirical formula. Using EL-ELM, the coupled nonlinear system is simplified to an equivalent linear stochastic system, allowing for a theoretical analysis of the impact of NES structural parameters on suppression performance and the precise determination of optimal parameter configurations for the best suppression effects. Subsequently, based on the results from the EL-ELM and the response data, an empirical formula has been developed that clearly describes the comprehensive laws governing the optimal NES parameters as they vary with vibration system parameters and stochastic excitation. Through error analysis and comparison of the two methods, it is found that the empirical formula significantly outperforms EL-ELM in terms of accuracy and computational cost, but it is contingent on solid prior knowledge. This study explores the influence of NES structural parameters on the system’s dynamic response and energy, further validating the effectiveness of the proposed methods in identifying optimal structural parameters. The phenomenon of targeted energy transfer (TET) under different NES structural parameters is also explained. The methodologies introduced in this study have strengthened the theory of vibration suppression. Specifically, the empirical formula excels in accuracy and computational efficiency by effectively using prior knowledge. The EL-ELM method, owing to its theoretical insights, is vital for analyzing complex stochastic nonlinear models. Combining these approaches offers guidance for advancing vibration control in theoretical and practical domains.

Nonlinear Dynamic Response of Galfenol Cantilever Energy Harvester Considering Geometric Nonlinear with a Nonlinear Energy Sink

The nonlinear energy sink (NES) and Galfenol material can achieve vibration suppression and energy harvesting of the structure, respectively. Compared with a linear structure, the geometric nonlinearity can affect the output performances of the cantilever beam structure. This investigation aims to present a coupled system consisting of a nonlinear energy sink (NES) and a cantilever Galfenol energy harvesting beam with geometric nonlinearity. Based on Hamilton’s principle, linear constitutive equations of magnetostrictive material, and Faraday’s law of electromagnetic induction, the theoretical dynamic model of the coupled system is proposed. Utilizing the Galliakin decomposition method and Runge–Kutta method, the harvested power of the external load resistance, and tip vibration displacements of the Galfenol energy harvesting model are analyzed. The influences of the external excitation, external resistance, and NES parameters on the output characteristic of the proposed coupling system have been investigated. Results reveal that introducing NES can reduce the cantilever beam’s vibration while considering the geometric nonlinearity of the cantilever beam can induce a nonlinear softening phenomenon for the output behaviors. Compared to the linear system without NES, the coupling model proposed in this work can achieve dual efficacy goals over a wide range of excitation frequencies when selecting appropriate parameters. In general, large excitation amplitude and NES stiffness, small external resistance, and small or large NES damping values can achieve the effect of broadband energy harvesting.

Vibration mitigation of spar-buoy floating wind turbines using a nonlinear energy sink

This work investigates the possibility of mitigating the wind-wave exerted vibration of floating offshore wind turbines using a nonlinear energy sink (NES). The vibration mitigation of floating wind turbines using passive linear control strategies have been intensively investigated, however, the effectiveness of the linear absorbers only works in a narrow frequency band, resulting in limited vibration suppression performance. The NES comprising a mass, a damper and a cubic stiffness element has the ability to mitigate vibrations in a broadband frequency range, which has been adopted in multiple engineering structures with significant performance benefits identified. In this paper, taking spar-buoy floating wind turbines as prototype structure, a simplified model with a NES mounted in nacelle subjected to wind and wave loads is presented. The harmonic balance method is then employed to analyse its nonlinear responses to the influence of NES parameters, together with the pseudo-arc-length method. The optimum NES parameter values are subsequently identified. Results show that the performance improvement can be up to 74.2% and 14.0%, respectively, comparing to that with no absorber and the TMD. Furthermore, to demonstrate the vibration suppression ability of the optimised NES in more realistic spar-buoy wind turbines, Orcaflex-the world’s leading package for the dynamic analysis of offshore systems, is adopted as the simulation tool. The spar-buoy wind turbine as well as the NES are established in Orcaflex and it is demonstrated that the performance benefits with the NES preserve under various loading conditions.

The effect of a nonlinear energy sink on the gust response of a wing

In this paper, the potential effectiveness of a nonlinear energy sink (NES) to absorb the energy from a wing that is vibrating as a result of flying in a gusty environment is investigated. The structural dynamics of the wing is simulated using a rigid airfoil mounted on two linear/nonlinear springs to represent the bending and torsional stiffness of the wing. The wing is subjected to a combination of gust and aerodynamic loads. The unsteady aerodynamic lift and moment are modelled using Wagner's theory. Furthermore, the gust loads are obtained by assuming two different gust profiles, e.g. sharp-edged and 1-cosine gust profiles. A nonlinear energy sink, which comprises of a concentrated mass, damper and a nonlinear spring, is attached to the wing, and its effectiveness to absorb the gust energy is investigated. The coupled nonlinear aeroelastic equations are integrated numerically to determine the response of the wing. To verify the developed aeroelastic model, the obtained results are compared with the available results in the literature and an excellent agreement is observed. The results highlight that adding the NES to the wing is capable of reducing the gust oscillation amplitude of the wing significantly when the NES parameters are chosen accordingly.

Influence of parameters of a piezoelectric nonlinear energy sink on targeted energy transfer characteristic

In the coupling system of elastic plate and piezoelectric nonlinear energy sink, the shunt circuit of piezoelectric material as a nonlinear energy sink (NES) is coupled with the main system, where the nonlinear stiffness and damping are realized by the shunt circuit of piezoelectric material. The targeted energy transfer (TET) characteristics of the system are analyzed by complexification-averaging method and harmonic balance method, the variation range of NES parameters is determined, and the slow invariant manifold, platform amplitude of the frequency responses, nonlinear normal modes and threshold interval of the system under different parameters are analyzed. The influences of piezoelectric NES parameters, such as mass ratio, damping and nonlinear stiffness, on the TET characteristics are obtained, and a general criterion for design of piezoelectric NES parameters is proposed.

Vibration suppression of a cable under harmonic excitation by a nonlinear energy sink

The application research of nonlinear energy sink (NES) on vibration suppression in many fields is a hot topic. In this paper, the vibration suppression mechanism of a nonlinear energy sink on a cable under a harmonic load is studied novely. First, the dynamic model of the cable-NES system under harmonic excitation is established. The partial differential Equations (PDEs) of motion governing the in-plane vibration of the cable-NES system are derived by Hamilton’s principle. Then, by considering the coupling effect among the first three in-plane modes of the cable, the ordinary differential equations (ODEs) of the coupled system are obtained applying Galerkin integral. The incremental harmonic balance (IHB) method is used to solve the nonlinear ordinary differential equations. Meanwhile, the stability of the solution is determined using Floquet theory. Finally, the primary resonances of the first and third in-plane modes of the cable are explored respectively, and the vibration mitigation performance of the NES and the wavelet transform of the cable displacement response are analyzed. The research results show that the NES has a significant effect on the dynamic behavior and vibration mitigation of the cable, especially, when the NES parameters are the same, the vibration mitigation effect of the NES on the primary resonance of the 3rd mode of the cable is better than that of the 1st mode.

Effects of weights on vibration suppression via a nonlinear energy sink under vertical stochastic excitations

A primary structure attached by a nonlinear energy sink (NES) moving vertically under a Gaussian white noise excitation is investigated. This paper demonstrates the stochastic responses and vibration suppression with emphasis on the effects of the weights. Four-dimensional state space equations which consider or ignore the weights in the vertical direction are given. Numerical probability density functions computed via the path integration method based on the Gauss-Legendre scheme are confirmed by Monte Carlo simulations. Probability density functions of the structure’s responses are compared between two cases under various random excitation intensities. The root-mean-square (RMS) displacement of the primary structure or random vibration suppression is estimated through the path integration method. Results reveal that the NES is able to suppress broadband random vibration of the structure. As the stochastic excitation decreases, the NES weight has significant effects on both probability density functions of the structure’s responses and RMS displacements of the primary structure. The NES parameters are discussed to explore random vibration suppression. For purpose of more accurate predictions, the effects of weights should be taken into consideration when the primary structure is excited by small random excitations or coupled with a large and reasonable NES mass. Simulations and discussions in this work provide theorical meanings to predict vibration suppression via a NES in stochastic environment.

Vibration suppression and dynamic behavior analysis of an axially loaded beam with NES and nonlinear elastic supports

Recently, dynamic analysis of a beam structure with nonlinear energy sink (NES) and various supports is attracting great attention. Most of the existing studies are about the beam structure with NES or nonlinear boundary supports with zero rotational restraint, respectively. However, there is little research accounting for such two types of complex factors simultaneously. In this work, the dynamic behavior of an axially loaded beam with both NES and general boundary supports is modeled and studied. The Galerkin truncated method (GTM) is employed to make the prediction of dynamic behavior of such a beam system, in which the mode functions of axially loaded Euler–Bernoulli beam with linear elastic boundary conditions are selected as the trail and weight functions. Then, the Galerkin condition is used to discretize the nonlinear governing equation of the beam system and establish the residual equations. The Runge–Kutta method is used to solve the residual matrix which consists of residual equations directly, and the harmonic balance method is also used to verify the results from the GTM. The influence of NES on vibration suppression and dynamic behavior of the beam structure is investigated and discussed. Results show that the vibration states of the beam structure can be transformed effectively through the change of NES parameters. On the other hand, the NES with suitable parameters has a beneficial effect on the vibration suppression at both ends of the beam structure.

Vibration absorption of parallel-coupled nonlinear energy sink under shock and harmonic excitations

Nonlinear energy sink (NES) can passively absorb broadband energy from primary oscillators. Proper multiple NESs connected in parallel exhibit superior performance to single-degree-of-freedom (SDOF) NESs. In this work, a linear coupling spring is installed between two parallel NESs so as to expand the application scope of such vibration absorbers. The vibration absorption of the parallel and parallel-coupled NESs and the system response induced by the coupling spring are studied. The results show that the responses of the system exhibit a significant difference when the heavier cubic oscillators in the NESs have lower stiffness and the lighter cubic oscillators have higher stiffness. Moreover, the e±ciency of the parallel-coupled NES is higher for medium shocks but lower for small and large shocks than that of the parallel NESs. The parallel-coupled NES also shows superior performance for medium harmonic excitations until higher response branches are induced. The performance of the parallel-coupled NES and the SDOF NES is compared. It is found that, regardless of the chosen SDOF NES parameters, the performance of the parallel-coupled NES is similar or superior to that of the SDOF NES in the entire force range.

Robust design optimization of nonlinear energy sink under random system parameters

Generally, in designing nonlinear energy sink (NES), only uncertainties in the ground motion parameters are considered and the unconditional expected mean of the performance metric is minimized. However, such an approach has two major limitations. First, ignoring the uncertainties in the system parameters can result in an inefficient design of the NES. Second, only minimizing the unconditional mean of the performance metric may result in large variance of the response because of the uncertainties in the system parameters. To address these issues, we focus on robust design optimization (RDO) of NES under uncertain system and hazard parameters. The RDO is solved as a bi-objective optimization problem where the mean and the standard deviation of the performance metric are simultaneously minimized. This bi-objective optimization problem has been converted into a single objective problem by using the weighted sum method. However, solving an RDO problem can be computationally expensive. We thus used a novel machine learning technique, referred to as the hybrid polynomial correlated function expansion (H-PCFE), for solving the RDO problem in an efficient manner. Moreover, we adopt an adaptive framework where H-PCFE models trained at previous iterations are reused and hence, the computational cost is less. We illustrate that H-PCFE is computationally efficient and accurate as compared to other similar methods available in the literature. A numerical study showcasing the importance of incorporating the uncertain system parameters into the optimization procedure is shown. Using the same example, we also illustrate the importance of solving an RDO problem for NES design. Overall, considering the uncertainties in the parameters have resulted in a more efficient design. Determining NES parameters by solving an RDO problem results in a less sensitive design.

Nonlinear energy sink to enhance the landing gear shimmy performance

The shimmy phenomenon is a significant concern in aircraft landing gear dynamics. The prediction of the shimmy instability is an essential issue in landing gear design to develop a passive or active suppression method. This work investigates the application of the nonlinear energy sink (NES) concept to mitigate the effects of shimmy in landing gears. The NES concept has been used in recent research on mechanical vibrations. It comprises a passive target energy transfer method that refers to a one-way energy transfer from a primary to a nonlinear subsystem. The landing gear model is based on torsional displacement coupled with the tyre classical elastic string analogy model. The NES device connects to the wheel shaft, and it comprises a mass, a linear damper, and a pure cubic spring. The numerical integration in time was used to assess the shimmy onset speed and the post-shimmy limit cycle oscillations. A parametric analysis of the landing gear nonlinear dynamics without the NES is presented. The design space of possible NES parameters is given, obeying design constraints, and inspected to assess adequate NES designs. The best NES samples are included in the landing gear dynamics to study their influence. Results have shown that the NES can adequately expand the operational speed range for no shimmy and lead to lower LCO amplitudes in the post-shimmy for a reasonable range of speeds. The NES concept’s successful employment for the landing gear dynamics suggests an enormous potential form of passive shimmy control.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

The suppression of nonlinear panel flutter response at elevated temperatures using a nonlinear energy sink

A nonlinear energy sink (NES) is used to suppress the nonlinear aerothermoelastic response of panels at elevated temperatures. A nonlinear aeroelastic model for a two-dimensional panel with an NES in a supersonic airflow is established using the Galerkin method, with the aerodynamic load being based on first-order piston aerodynamic theory. The model is then used to study the nonlinear dynamic response behavior of a heated panel with an NES to obtain the NES suppression region. The effects of different NES parameters and temperature elevations on the NES suppression region are examined and a design technique is developed to facilitate finding the most suitable combination of parameter values to suppress the nonlinear panel flutter response. The results show that the NES is effective at reducing the aeroelastic response amplitude within a specific range. Varying the installed position, damping, mass ratio and stiffness of the NES will lead to variations in its controlling effect upon the suppression region and the dynamic behavior of the heated panel. An increase in the temperature elevation results in a decrease in the NES suppression region. When the temperature reaches a certain point, the NES suppression region disappears and the NES no longer has any effect on the panel flutter response. Overall, it is found that an NES provides an effect and robust solution, and a wide range of different parameter combinations can be found, that will meet the requirements identified by the proposed design method.

Wind tunnel demonstration of galloping mitigation with a purely nonlinear energy sink

Galloping is a critical type of flow-induced vibration (FIV) arising on power transmission lines, high rise buildings, pipe and cables bundles in the oil and gas industry. In this paper, we present a purely nonlinear energy sink (NES) that mitigates the galloping of a square prism. The NES is composed of a ball rotating freely in a circular track attached to the prism. The ball’s dynamics is coupled to that of the prism in a purely nonlinear way by inertia. We experimentally assess how this simple NES reduces the prism vibration by comparing the prism amplitude responses with and without the NES. A supplementary video presents these experiments, during which the NES ball exhibits different dynamics in three regimes; oscillatory, intermittent, and rotational. We characterize the ball behaviour and its effect on the prism response in each regime. The oscillatory regime appears at low flow speeds at which both the prism and the ball oscillate with small amplitude. The intermittent regime represents a transition mode within a small range of flow speeds and corresponds to a small jump in the vibration amplitude of the prism. The rotational regime appears at higher flow speeds, where the ball oscillates with relatively high angular speeds resulting in a strong modulated response of the prism. The design of the NES allows to easily vary its track dimensions to use a ball of different sizes and masses. Accordingly, we demonstrate the influence of the main NES parameters, which are the ball mass, NES track radius, ball friction, and radial clearance between NES track walls and the rotating ball, on both the prism response and the ball behaviour. The NES we present is directly amenable to mitigate other types of FIV.

Suppressing nonlinear aeroelastic response of laminated composite panels in supersonic airflows using a nonlinear energy sink

This paper proposes using a Nonlinear Energy Sink (NES) to suppress the nonlinear aeroelastic response of laminated composite panels in supersonic airflows. Relevant aeroelastic equations are established using Hamilton’s principle and a finite element approach, drawing upon Von Karman’s large deflection theory and first order piston theory. The idea of the NES suppression region is proposed and the effects of NES parameters on the NES suppression region are studied in detail. The results show that the nonlinear aeroelastic responses of the panel can be completely suppressed by the Transient Resonance Capture (TRC); the appropriate NES parameter values can increase the critical dynamic pressure for flutter and suppress the nonlinear aeroelastic response effectively. Increasing the mass ratio of the NES can improve the NES suppression region; the nonlinear stiffness coefficient and damping of the NES within a specific range can suppress the nonlinear aeroelastic response. The most effective installation position for a NES is in a specific region behind the center-line of the panel in the direction of the airflow.

Vortex-induced vibration on a low mass ratio cylinder with a nonlinear dissipative oscillator at moderate Reynolds number

The vortex induced vibration (VIV) mitigation on a circular cylinder with low mass ratio (i.e. the ratio of structural to displaced fluid mass) parameter and moderate Reynolds numbers through a nonlinear energy sink (NES) is investigated numerically as basis for applications on dynamics of spar platform, marine tunnel or risers used in the ocean engineering industry. The Reynolds-Averaged-Navier–Stokes (RANS) equations and shear Stress transport (SST) k–ω turbulence model are used to calculate the flow field, while a fourth-order Runge–Kutta method is employed for evaluating the nonlinear structure dynamics of flow–cylinder–NES coupled system. The computational model includes an overset mesh solution which can avoid the negative volume grid problem as a dynamic mesh method is involved to deform the domain according to the motion of the fluid–structure interface. The numerical model is validated against experimental data of VIV of an isolated cylinder in uniform current. The study is aimed to investigate the effect of NES parameters, including mass, damping and nonlinear cubic stiffness, and various reduced velocities on the VIV response of the cylinder. The VIV amplitudes’ distribution along the various reduced velocities, trajectories of cylinder motion, hydrodynamic response, and temporal evolution of vortex shedding, are obtained by conducting numerical simulations. Subsequently, it is found that placing a NES with appropriate parameters inside the cylinder, can affect the distribution of the three-branch response and narrows the lock-in range. A good VIV amplitudes’ suppression can be achieved with a large NES mass, small nonlinear cubic stiffness, and the NES damping is higher in the critical range for this kind low mass ratio cylinder structures.

Passive Suppression of Panel Flutter Using a Nonlinear Energy Sink

A nonlinear energy sink (NES) is used to suppress panel flutter. A nonlinear aeroelastic model for a two-dimensional flat panel with an NES in supersonic flow is established using the Galerkin method. First-order piston aerodynamic theory is adopted to build the aerodynamic load. The effects of NES parameters on flutter boundaries of the panel are investigated using Lyapunov’s indirect method. The mechanism of the NES suppression of panel flutter is studied through energy analysis. Effects of NES parameters on aeroelastic responses of the panel are obtained, and a design technique is adopted to find a suitable combination of parameter values of the NES that suppresses the panel flutter effectively. Results show that the NES can increase or reduce the onset dynamic pressure of the panel flutter and it can reduce the aeroelastic response amplitude effectively within a certain range of dynamic pressure behind the onset dynamic pressure. The installation position of the NES depends on the direction of the airflow. The robust characteristics should be considered to find the suitable combination of parameter values of the NES.

Nonlinear vibration absorption of laminated composite beams in complex environment

Nonlinear vibration absorption of a laminated composite beam is investigated with the account of complex environment (moisture and temperature). A passive efficient nonlinear energy sink (NES) vibration absorber is used to control the transverse vibration. The generalized Hamilton principle is applied to derive a dynamic model of the laminated composite beam coupled with the NES. Numerical simulations reveal the effects of temperature, moisture, and laying angle on natural frequencies. It is numerically found that the NES can rapidly reduce the vibration amplitude. Then, approximate analytical solutions are sought via the harmonic balance method. The approximate analytical solutions are confirmed by the numerical solutions. Amplitude–frequency response curves show that the NES can reduce the amplitude to very low values for various temperatures, moisture levels, and laying angles. In a certain ranges of the NES parameters, different control effects are determined via an approximate analysis. It is demonstrated that the NES is a promising approach to control vibration of a laminated composite beam in complex environment.

Thermal Effect on Dynamics of Beam with Variable-Stiffness Nonlinear Energy Sink

Abstract In this study, we investigate the targeted energy transfer (TET) from a simply supported beam that is subjected to thermal variations and external excitations to a local nonlinear energy sink (NES). We derive the governing equation of motion for the beam with an NES device and study the influences of NES parameters on the vibration-suppressing effect. We obtain the optimized parameters of the NES under constant-amplitude harmonic excitation at room temperature. The optimized NES gradually loses its vibration absorption efficiency as the excitation amplitude and temperature increase. We change the nonlinear stiffness of the NES to mitigate the influence of temperature variation and show that NES efficiency can be enhanced by reducing the nonlinear stiffness. We propose a variable-stiffness NES, and the results demonstrate this NES is best for maintaining efficiency over the whole temperature range. We also analyze the transient responses of the system under impulse loads. Results indicate that, like the performance of the system subjected to harmonic excitation, an NES with relatively low stiffness can better suppress vibration with increasing impulse amplitude and temperature.