Effects of a non-linear energy sink (NES) on vortex-induced vibrations of a circular cylinder

We investigate in detail the passive control of vortex-induced vibrations of a freely oscillating circular cylinder using a non-linear energy sink consisting of a secondary system having linear damping and an essential non-linear cubic stiffness. The loads on the cylinder are calculated using a direct numerical simulation of the incompressible flow over the cylinder using a parallel computational fluid dynamics code. A strongly coupled fluid structure control numerical model is used to determine the responses of the cylinder and the sink as well as the flow. We vary the sink parameters (mass and damping) and determine their effects on the response of the coupled system. We find multiple stable responses of the coupled system for different mass ratios and damping coefficient of the sink, depending on the initial conditions.     

Limit cycle oscillation suppression of 2-DOF airfoil using nonlinear energy sink

This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.     

Power harvesting from transverse galloping of square cylinder

The concept of exploiting galloping of square cylinders to harvest energy is investigated. The energy is harvested by attaching a piezoelectric transducer to the transverse degree of freedom. A representative model that accounts for the coupled cylinder displacement and harvested voltage is used to determine the levels of the harvested power. The focus is on the effect of the Reynolds number on the aerodynamic force, the onset of galloping, and the level of the harvested power. The quasi steady approximation is used to model the aerodynamic loads. A linear analysis is performed to determine the effects of the electrical load resistance and the Reynolds number on the onset of galloping, which is due to a Hopf bifurcation. We derive the normal form of the dynamic system near the onset of galloping to characterize the type of the instability and to determine the effects of the system parameters on its outputs near the bifurcation. The results show that the electrical load resistance and the Reynolds number play an important role in determining the level of the harvested power and the onset of galloping. The results also show that the maximum levels of harvested power are accompanied with minimum transverse displacements for both low- and high-Reynolds number configurations.     

Control of cross-flow-induced vibrations of square cylinders using linear and nonlinear delayed feedbacks

We investigate the effectiveness of linear and nonlinear time-delay feedback controls to suppress high amplitude oscillations of an elastically mounted square cylinder undergoing galloping oscillations. A representative model that couples the transverse displacement and the aerodynamic force is used. The quasi-steady approximation is used to model the galloping force. A linear analysis is performed to investigate the effect of linear time-delay controls on the onset speed of galloping and natural frequencies. It is demonstrated that a linear time-delay control can be used to delay the onset speed of galloping. The normal form of the Hopf bifurcation is then derived to characterize the type of the instability (supercritical or subcritical) and to determine the effects of the linear and nonlinear time-delay parameters on their outputs near the bifurcation. The results show that the nonlinear time-delay control can be efficiently implemented to significantly reduce the galloping amplitude and suppress any dangerous behavior by converting any subcritical Hopf bifurcation into a supercritical one.     

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.     

Experimental and theoretical investigation of galloping of transversely inclined slender prisms

Galloping is characterized by large and periodical oscillations which may lead to collapse of slender structures. This study is the first attempt of a comprehensive experimental and theoretical investigation of galloping of transversely inclined prisms. A modified quasi-steady model is proposed with a constant term to estimate the galloping of a transversely inclined prism, which is later experimentally investigated by conducting a static Synchronous Multi-Pressure Sensing System (SMPSS) test and an aeroelastic test in a boundary layer wind tunnel. The galloping responses of the prisms were measured in the aeroelastic test, while the aerodynamic force coefficients were determined from the SMPSS test. These experimental results were subsequently utilized to validate the quasi-steady model. Based on the proposed model, the galloping responses of the prisms were predicted and compared with the experimental results. The experimentally measured and theoretically predicted galloping responses are discussed with respect to aerodynamic damping ratios, onset galloping wind speeds, distributed pressure coefficients, point pressure spectra and vortex shedding frequencies. Interesting findings are summarized.     

Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay

The active control approach generally requires power input to suppress vibrations of structures, while the conventional passive manner often causes waste of energy after transferring vibrations of the primary structure to the auxiliary system. In this work, an innovative control strategy based on energy harvesting for efficiently suppressing the cross-flow-induced vibrations such as galloping is proposed. The novel design facilitates the harvester of not only alleviating the oscillation of the primary structure but also seizing the transferred vibrational energy. An analytical model for the coupled nonlinear dynamical system is established by utilizing the Euler–Lagrange principle and implementing the Galerkin discretization. The impacts of the electrical load resistance and tip mass of the energy harvester on the coupled frequency, damping, and the onset speed of instability of the coupled multi-mode system are investigated in details. The results show that there exists an optimal load resistance for each tip mass which maximizes the onset speed of galloping. For control purposes, it is found that there is a well-defined tip mass of the energy harvester at which the coupled system has the highest onset speed of instability, and hence, the bluff body has the lowest vibration amplitude for all considered load resistances. However, to efficiently harvest energy and control the bluff body, both the tip mass of the energy harvester and electrical load resistance can be accurately determined.     

Autorotation dynamics of a low aspect-ratio rectangular prism

This paper presents two previously unreported aspects of the autorotation dynamics of low aspect ratio rectangular prisms, observed during an experimental study of the dynamics of helicopter underslung loads. Low-speed wind tunnel tests of a simplified container model free to rotate on a fixed axis demonstrated (a) that autorotation rate can lock-in to a structural mode and (b) that static hysteresis in autorotation rate can occur at low speeds. Autorotation lock-in behaves in a similar manner to vortex-shedding lock-in, suggesting that a similar feedback flow process between vortex wake dynamics and body motion is operating, and may provide a partial explanation for the complex changes in behaviour of rotating slung loads at high airspeeds. Static hysteresis at low speeds results in a bifurcation diagram for autorotation which is similar to that for cross-wind galloping of a square prism, including the effects of friction and inertia. The similarity in bifurcation behaviour seems likely to indicate similar dynamics rather than flow physics, suggesting that it may be possible to apply techniques developed to model the effect of non-linear damping characteristics in galloping to the modelling of autorotation.     

Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping

The dynamics of a system of coupled oscillators possessing strongly nonlinear stiffness and damping is examined. The system consists of a linear oscillator coupled to a strongly nonlinear, light attachment, where the nonlinear terms of the system are realized due to geometric effects. We show that the effects of nonlinear damping are far from being purely parasitic and introduce new dynamics when compared to the corresponding systems with linear damping. The dynamics is analyzed by performing a slow/fast decomposition leading to slow flows, which in turn are used to study transient instability caused by a bifurcation to 1:3 resonance capture. In addition, a new dynamical phenomenon of continuous resonance scattering is observed that is both persistent and prevalent for the case of the nonlinearly damped system: For certain moderate excitations, the transient dynamics “tracks” a manifold of impulsive orbits, in effect transitioning between multiple resonance captures over definitive frequency and energy ranges. Eventual bifurcation to 1:3 resonance capture generates the dynamic instability, which is manifested as a sudden burst of the response of the light attachment. Such instabilities that result in strong energy transfer indicate potential for various applications of nonlinear damping such as in vibration suppression and energy harvesting.     

简谐激励下黏弹性非线性能量阱的研究

黏弹性材料作为一种良好的减振材料,广泛应用于机械、航空和土木等领域.本文用黏弹性Maxwell器件代替传统非线性能量阱中的阻尼元件,提出一种新型的黏弹性非线性能量阱,并对该模型在简谐激励下的减振性能进行分析.首先,根据牛顿第二定律建立系统的动力学方程,采用谐波平衡法求解系统的幅频响应曲线,并利用MATLAB中的Runge-Kutta数值方法验证解析解的正确性,结果吻合良好.然后,分析黏弹性非线性能量阱的减振性能和参数的影响.最后,分析了不同质量比下非线性刚度比和阻尼比同时变化时减振效果的变化趋势,并讨论了黏弹性非线性能量阱的最佳取值范围.研究结果表明:主系统的最大振幅随着非线性刚度的增加先减小后增大;当参数选取恰当时,黏弹性非线性能量阱比传统非线性能量阱的减振效果更优;另外,随着质量比的增加,主系统最大振幅的最小值出现先减小后趋于不变的现象,且非线性刚度比和阻尼比的最佳取值范围有所增大.以上结论对黏弹性非线性能量阱的实际应用提供了一定的理论依据.     

Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

Non-linear dynamic analysis of a HSFD mounted gear-bearing system

An investigation is carried out on the systematic analysis of the dynamic behavior of the hybrid squeeze-film damper (HSFD) mounted a gear-bearing system with strongly non-linear oil-film force and gear meshing force in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless unbalance coefficient, damping coefficient and the dimensionless rotating speed ratio as control parameters. The non-dimensional equations of the gear-bearing system are solved using the fourth order Runge-Kutta method. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents and fractal dimension of the gear-bearing system. The results presented in this study provide some useful insights into the design and development of a gear-bearing system for rotating machinery that operates in highly rotating speed and highly non-linear regimes.     

Nonlinear Analysis of Store-Induced Limit Cycle Oscillations

Flight tests of modern high-performance fighter aircraft reveal the presence of limit cycle oscillation (LCO) responses for aircraft with certain external store configurations. Conventional linear aeroelastic analysis predicts flutter for conditions well beyond the operational envelope, yet these store-induced LCO responses occur at flight conditions within the flight envelope. Several nonlinear sources may be present, including aerodynamic effects such as flow separation and shock-boundary layer interaction and structural effects such as stiffening, damping, and system kinematics. No complete theory has been forwarded to accurately explain the mechanisms responsible. This research examines a two degree-of-freedom aeroelastic system which possesses kinematic nonlinearities and a strong nonlinearity in pitch stiffness. Nonlinear analysis techniques are used to gain insight into the characteristics of the behavior of the system. Numerical simulation is used to verify and validate the analysis. It is found that when system damping is low, the system clearly exhibits nonlinear interaction between aeroelastic modes. It is also shown that although certain applied forcing conditions may appear negligible, these same forces produce large amplitude LCOs under specific realizable circumstances.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

Numerical optimisation of a classical stochastic system for targeted energy transfer

The paper studies stochastic dynamics of a two-degree-of-freedom system, where a primary linear system is connected to a nonlinear energy sink with cubic stiffness nonlinearity and viscous damping. While the primary mass is subjected to a zero-mean Gaussian white noise excitation, the main objective of this study is to maximise the efficiency of the targeted energy transfer in the system. A surrogate optimisation algorithm is proposed for this purpose and adopted for the stochastic framework. The optimisations are conducted separately for the nonlinear stiffness coefficient alone as well as for both the nonlinear stiffness and damping coefficients together. Three different optimisation cost functions, based on either energy of the system’s components or the dissipated energy, are considered. The results demonstrate some clear trends in values of the nonlinear energy sink coefficients and show the effect of different cost functions on the optimal values of the nonlinear system’s coefficients.     

Random vibration of viscoelastic system under broad-band excitations

An analytical method is proposed to study the response of a viscoelastic system with strongly non-linear stiffness force and under broad-band random excitations. The random excitations can be additive, or multiplicative, or both, and they can be stationary or non-stationary with evolutionary spectra. With the proposed method, contributions of the viscoelastic force to both damping and stiffness are taken into account separately, and then the extended version of the stochastic averaging, called the quasi-conservative averaging, is applied to the system to derive the averaged equation of energy envelope. Probability density functions of system responses, such as the total energy, the amplitude, and the state variables, can then be obtained analytically. The accuracy of the method is substantiated by comparing the analytical results with those from Monte Carlo simulations. Effects of parameters in the viscoelastic force and in the non-linear stiffness force on the system responses are also investigated.     

Dynamic design of a nonlinear energy sink with NiTiNOL-steel wire ropes based on nonlinear output frequency response functions

A novel vibration isolation device called the nonlinear energy sink(NES)with Ni Ti NOL-steel wire ropes(Ni Ti-ST) is applied to a whole-spacecraft system. The Ni Ti-ST is used to describe the damping of the NES, which is coupled with the modified Bouc-Wen model of hysteresis. The NES with Ni Ti-ST vibration reduction principle uses the irreversibility of targeted energy transfer(TET) to concentrate the energy locally on the nonlinear oscillator, and then dissipates it through damping in the NES with Ni Ti-ST.The generalized vibration transmissibility, obtained by the root mean square treatment of the harmonic response of the nonlinear output frequency response functions(NOFRFs),is first used as the evaluation index to analyze the whole-spacecraft system in the future.An optimization analysis of the impact of system responses is performed using different parameters of NES with Ni Ti-ST based on the transmissibility of NOFRFs. Finally, the effects of vibration suppression by varying the parameters of Ni Ti-ST are analyzed from the perspective of energy absorption. The results indicate that NES with Ni Ti-ST can reduce excessive vibration of the whole-spacecraft system, without changing its natural frequency. Moreover, the NES with Ni Ti-ST can be directly used in practical engineering applications.     

Nonlinear dynamics of an aeroelastic airfoil with free-play in transonic flow

Nonlinear dynamic behaviors of an aeroelastic airfoil with free-play in transonic air flow are studied. The aeroelastic response is obtained by using time-marching approach with computational fluid dynamics (CFD) and reduced order model (ROM) techniques. Several standardized tests of transonic flutter are presented to validate numerical approaches. It is found that in time-marching approach with CFD technique, the time-step size has a significant effect on the calculated aeroelastic response, especially for cases considering both structural and aerodynamic nonlinearities. The nonlinear dynamic behavior for the present model in transonic air flow is greatly different from that in subsonic regime where only simple harmonic oscillations are observed. Major features of the responses in transonic air flow at different flow speeds can be summarized as follows. The aeroelastic responses with the amplitude near the free-play are dominated by single degree of freedom flutter mechanism, and snap-though phenomenon can be observed when the air speed is low. The bifurcation diagram can be captured by using ROM technique, and it is observed that the route to chaos for the present model is via period-doubling, which is essentially caused by the free-play nonlinearity. When the flow speed approaches the linear flutter speed, the aeroelastic system vibrates with large amplitude, which is dominated by the aerodynamic nonlinearity. Effects of boundary layer and airfoil profile on the nonlinear responses of the aeroelastic system are also discussed.     

Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation

The nonlinear behaviors and vibration reduction of a linear system with a nonlinear energy sink(NES) are investigated. The linear system is excited by a harmonic and random base excitation, consisting of a mass block, a linear spring, and a linear viscous damper. The NES is composed of a mass block, a linear viscous damper, and a spring with ideal cubic nonlinear stiffness. Based on the generalized harmonic function method,the steady-state Fokker-Planck-Kolmogorov equation is presented to reveal...     

Galloping instabilities of two-dimensional triangular cross-section bodies

Galloping is a type of aeroelastic instability characterized by large amplitude, low frequency, normal to wind oscillations. It normally appears in bodies with small stiffness and structural damping when they are placed in a flow and the incident velocity is high enough. In this paper a systematic approach for the analysis of galloping of triangular cross-section bodies is reported. Wind tunnel experiments have been conducted aiming at establishing the unstable characteristics of isosceles triangular cross-section bodies when subjected to a uniform flow with angles of attack ranging from 0 to 180°. The results have been summarized in a stability map, where galloping instability zones in the angle of attack—main vertex angle plane—are identified.