Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

SENSITIVITY ANALYSIS AS A METHOD OF ABSORBER TUNING FOR REDUCTION OF STEADY STATE RESPONSE OF LINEAR PARAMETRIC SYSTEMS

The aim of this paper is to determine whether one dynamic absorber can reduce the amplitude of the steady state vibration of a parametric system for natural and parametric resonance frequencies simultaneously. The efficiency of both the conventional dynamic absorber and the parametric absorber is analyzed. The first order sensitivity analysis of parametric periodic systems in the time domain is applied to obtain logarithmic sensitivity functions in the frequency domain. The first order sensitivity logarithmic functions are used to tune the conventional absorber and the parametric absorber.     

Autoparametric vibration absorber effect to reduce the first symmetric mode vibration of a curved beam/panel

This paper presents the implementation of autoparametric phenomena to reduce the symmetrical vibration of a curved beam/panel under external harmonic excitation. The internal energy transfer of a first symmetric mode into first anti-symmetric mode in a curved panel is one example of autoparametric vibration absorber effect. This is similar to the vibration energy transfer from the resonance of a primary structure to the resonance of a secondary spring–mass (tuned mass damper). The nonlinear response of a curved beam is analyzed using an equation with two modes, and a shaker test. The effect of different configurations of the curve beam/panel, including damping ratios and excitation levels, on the energy transfer of the first symmetric mode to the first anti-symmetric mode was studied.The conventional tuned mass damper (TMD) can reduce the resonance response by energy transfer using damping dissipation, whereas an autoparametric vibration absorber (AVA) can reduce the resonance response by energy transfer using parametric interaction. The results indicate that there is a non-absorption region in which vibration is amplified. For the AVA, the non-absorption region can be minimized by tuning the resonance frequency of the first anti-symmetric mode to half of the first symmetric mode resonance frequency using additional mass. No additional damping material is required for achieving sufficient vibration reduction. The AVA can maintain reliable performance in hot and corrosive environments where damping material cannot perform effectively. This paper presents the first successful experimental results of an autoparametric vibration absorption mechanism in a curved beam.     

Analysis of Coulomb friction vibration dampers

An analysis of the steady state response of a rotational Coulomb friction vibration damper has been carried out. Such dampers are sometimes used in various industrial applications. Analysis of the steady state phase plane is used to determine various response quantities such as amplitude ratio, phase lag, energy dissipated per cycle, and rms power loss. The analysis shows that the response can be categorized into one of three main types. Expressions are developed to predict effect of the addition of a damper on the power and the disturbance amplitude transmitted to the load. It is shown that simple one parameter relationships exist for predicting the maximum vibration reduction achievable with this type of damper.     

双稳态结构中的1/2次谐波共振及其对隔振特性的影响

以典型的双稳态系统——屈曲梁结构为例,基于等效模型,结合解析、数值和实验手段,研究了双稳态结构中的1/2次谐波共振特性、演化过程、参数调节规律及其对隔振特性的影响.研究发现,当非线性刚度系数或激励幅值增加到一定程度时,系统会在一定带宽下产生显著的1/2次谐波共振;随着激励幅值增加,阻尼系统的1/2次谐波遵循“产生-增强-衰退-消失”的过程,该过程对峰值频率和峰值传递率有重要影响;适当提高非线性强度能有效改善双稳态结构隔振特性.针对双稳态屈曲梁结构开展的实验验证了1/2次谐波特性和隔振特性变化规律.     

On the efficacy of an active absorber with internal state feedback for controlling self-excited oscillations

An active absorber, utilizing the state feedback of the absorber mass, is proposed for controlling the self-excited vibration of a single degree-of-freedom extended Rayleigh oscillator. The control strategy renders the design standalone. The process of optimizing the control gains is discussed. The analysis reveals that by selecting a suitably high value of the absorber frequency, the overall damping of the system can be made as high as the critical damping irrespective of the amount of negative linear damping present in the primary self-excited system. It is shown that a higher value of the absorber frequency is profitable in almost all respects related to the performance as well as the robustness of the system under parametric uncertainty. The nonlinear analysis of the system reveals that the proposed absorber can control the amplitude of oscillation even in case detuning (up to some limit) of the absorber frequency from its nominal value. The region of global stability increases with the increase in the value of the absorber frequency. However some aspects, like higher absorber deflection, reduced lower bound of the admissible detuning and the lower range of the tolerance on the mass ratio limit using a very high value of absorber frequency. The results of numerical simulations confirm the analytical results.     

非线性电容RLC串联电路的主共振研究

研究非线性电容RLC串联电路,应用多尺度法,得到非线性振动系统主共振的一次近似解并进行数值计算,分析电阻、电感、电容和电动势对主共振幅频响应的影响.结果表明:RLC串联电路的主共振响应有跳跃和滞后现象;随着电动势的增加,主共振的振幅和共振区增大;随着电阻的增大,主共振的振幅和共振区减小.     

Nonequilibrium dynamics of parametrically excited cold atoms via magnetic field-gradient modulation

We propose that the driven cold atomic system whose trap potential is periodically perturbed via parametric modulation of the magnetic field-gradien is a novel system to investigate the complex dynamics in nonlinear dynamical systems. We calculate the atomic trajectories and basins of attraction by varying the modulation amplitude and the modulation frequency. The calculation results show parametric resonance similar to those in Kim et. al.'s work [Opt. Commun. 236 (2004) 349] on cold atoms under parametric modulation of trap laser intensities in the case of small modulation amplitude or low modulation frequency. As the modulation amplitude or the modulation frequency is increased, the nonlinear effects are enhanced so that the dynamics of the system shows a wide variety of nonlinear behaviors, such as period doubling and chaos. We experimentally demonstrate the parametric resonance when the magnetic field gradient is parametrically modulated, which evidences the realization of the proposed system. We expect that this system is useful for understanding the stochastic phenomena occurring between complex basins of attraction, such as fluctuation induced transitions across the complex basin boundaries.     

Geometrically nonlinear dynamic response of stiffened plates with moving boundary conditions

An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The first approximation solution of the non-resonance is obtained by means of the method of multiple scales. The primary parametric resonance and primary resonance of the stiffened plate are studied by using the same method. The accuracy of the method is validated by comparing the results with those of finite element analysis via ANSYS. Numerical examples for different stiffened plates are presented to discuss the steady response of the non-resonance and the amplitude-frequency relationship of the primary parametric resonance and primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude-frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.     

时滞反馈控制扭转振动系统的振动

研究了采用时滞反馈来控制扭转振动系统的振动问题.在一个带有非线性动力吸振器的扭转振动系统中,采用时滞反馈来控制主系统的振动.研究了反馈增益系数和时滞对主系统振动的影响.研究结果表明,对某一固定的反馈增益系数,存在时滞的某段调节区间,可以通过调节时滞来抑制主系统的振动.在时滞的调节区间内存在一个最佳点,主系统的振动被抑制到最小值.可以同时调节反馈增益系数和时滞两参数,当反馈增益系数和时滞都调节到最佳值时,主系统振动的振幅由0.24减小到0.03,取得了很好的减振效果. 关键词： 时滞反馈 扭转振动 减振     

窄带随机噪声作用下单自由度非线性干摩擦系统的响应

研究了单自由度非线性干摩擦系统在窄带随机噪声参数激励下的主共振响应问题.用Krylov-Bogoliubov平均法得到了关于慢变量的随机微分方程.在没有随机扰动情形,得到了系统响应幅值满足的代数方程.在有随机扰动情形,用线性化方法和矩方法给出了系统响应稳态矩计算的近似计算公式.讨论了系统阻尼项、非线性项、随机扰动项和干摩擦项等参数对于系统响应的影响.理论计算和数值模拟表明,当非线性强度增大时系统的响应显著变小,系统分岔点滞后;随着激励频率的增大系统响应变大,而当激励频率小于一定的值时,系统响应为零;增加干 关键词： 单自由度非线性干摩擦系统 主共振响应 Krylov-Bogoliubov平均法     

Nonlinear free vibration of a microscale beam based on modified couple stress theory

This paper presents a nonlinear free vibration analysis of the microbeams based on the modified couple stress Euler–Bernoulli beam theory and von Kármán geometrically nonlinear theory. The governing differential equations are established in variational form from Hamilton principle, with a material length scale parameter to interpret the size effect. These partial differential equations are reduced to corresponding ordinary ones by eliminating the time variable with the Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the size-dependent characteristic relations of nonlinear vibration frequency vs. central amplitude of the microbeams are obtained successfully. The comparisons with available published results show that the current approach and algorithm are of good practicability. A parametric study is conducted involving the dependency of the frequency on the length scale parameter along with Poisson ratio, which shows that the nonlinear vibration frequency predicted by the current model is higher than that by the classical one.     

Squeal and chatter phenomena generated in a mountain bike disc brake

This paper examines squeal and chatter phenomena generated experimentally in mountain bike disc brakes. There are two kinds of frictional self-excited vibrations in the bike disc brakes, called squeal with frequency of 1 kHz and chatter with frequency of 500 Hz. In order to reproduce the squeal and chatter, a bench test apparatus using an actual bike was set up to determine the associated frequency characteristics experimentally. The results show the frequencies to be independent of pad temperature and disc rotating speed. Squeal is shown to be in-plane vibration in the direction of the disc surface which is caused by the frictional characteristics having negative slope with respect to the relative velocity in the vibrating system, which includes brake unit, spokes and hub. Chatter is generated within a limited high temperature region. Again, it is frictional vibration in which the squeal and out-of-plane vibration of the disc due to Coulomb friction combine through the internal resonance relation between in-plane and out-of-plane nonlinear vibration caused by the temperature increase of the disc during braking.     

Stability and transient dynamics of a propeller–shaft system as induced by nonlinear friction acting on bearing–shaft contact interface

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.     

柴油机轴系扭振系统的强非线性主共振

研究柴油机轴系扭振系统强非线性问题.根据拉格朗日方程建立柴油机轴系扭振系统的动力学模型,通过参数变换,应用Modified Lindstedt-Poincaré方法得到柴油机轴系扭振系统强非线性主共振的幅频响应方程,分析系统不同参数对主共振幅频响应的影响.结果表明,系统的幅频响应曲线存在跳跃,随着简谐力矩的减小和阻尼的增大,系统的非线性跳跃减弱,系统的振幅减小,系统主共振的区域也随之减小;随着调谐参数的变化,系统的主共振力幅响应曲线存在两种拓扑结构.MLP方法得出的近似解析解与龙格库塔法得出的数值解吻合.     

Forced vibrations of a torsional oscillator with Coulomb friction under a periodically varying normal load

The forced vibration response of a single degree of freedom torsional system with Coulomb friction, under a periodically varying normal load, is studied. First, an enhanced multi-term harmonic balance method is developed to calculate nonlinear responses directly in frequency domain; this should be in general applicable to periodically varying nonlinear systems. Second, a pulse width modulated normal load is approximated by a truncated Fourier series with a reasonable number of harmonic components and utilized for case studies. Finally, the effects of duty ratio on nonlinear frequency responses are examined.     

Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling

The response of a two-degree-of-freedom system with autoparametric coupling under the action of broad band random excitation is investigated. The system corresponds to the autoparametric vibration absorber and is also typical of many common structural configurations. A method based upon the Markov vector approach, together with an approximate treatment of third and higher statistical moments, is used to derive a set of fourteen coupled non-linear equations for the first and second moments of the system responses. A numerical integration procedure is used to obtain quantitative results for the system mean and mean square responses over a range of system parameters.The results show that large random motions of the coupled system may occur when the internal detuning parameter is close to the principal internal resonance, and that these motions may give rise to a suppression effect on the random motions of the main system. A feature of the results is that under conditions of internal resonance the random motions are found to be quasi-stationary, with steady oscillatory terms in the response moments. This suggests the possibility of entrainment of regular harmonic responses by the system random motions.     

Chatter suppression by parametric excitation: Model and experiments

Chatter vibration leads to challenges in precise machining due to its harmful effect on productivity and surface quality. In this study, a chatter suppression method based on parametric excitation was developed. The effect of parametric excitation on self-excited vibration was investigated based on a model of a van der Pol-Mathieu-Duffing oscillator with a time delay. It reveals that there can be a zero solution for the oscillator under the effect of parametric excitation, while it is impossible to have a stable zero equation without parametric excitation. The stability of a parametrically excited vibration system regarding the regenerative effect in the cutting processes was studied by the averaging method. The stability analysis shows that parametric excitation with an appropriate frequency and large amplitude has a chatter suppression effect no matter whether the waveform is a sinusoidal wave, square wave or triangular wave. To validate the effect of parametric excitation for chatter suppression, experiments were conducted based on a magnetorheological (MR) fluid-controlled boring bar, which can generate high-frequency parametric excitation based on the quick response of the MR fluid. Cutting experiments with an excitation current of different waveforms and diverse frequencies show that chatter can be significantly suppressed by the effect of parametric excitation.     

Effect of one-way clutch on the nonlinear vibration of belt-drive systems with a continuous belt model

This study focuses on the nonlinear steady-state response of a belt-drive system with a one-way clutch. A dynamic model is established to describe the rotations of the driving pulley, the driven pulley, and the accessory shaft. Moreover, the model considers the transverse vibration of the translating belt spans for the first time in belt-drive systems coupled with a one-way clutch. The excitation of the belt-drive system is derived from periodic fluctuation of the driving pulley. In automotive systems, this kind of fluctuation is induced by the engine firing harmonic pulsations. The derived coupled discrete–continuous nonlinear equations consist of integro-partial-differential equations and piece-wise ordinary differential equations. Using the Galerkin truncation, a set of nonlinear ordinary differential equations is obtained from the integro-partial-differential equations. Applying the Runge–Kutta time discretization, the time histories of the dynamic response are numerically solved for the driven pulley and the accessory shaft and the translating belt spans. The resonance areas of the coupled belt-drive system are determined using the frequency sweep. The effects of the one-way clutch on the belt-drive system are studied by comparing the frequency–response curves of the translating belt with and without one-way clutch device. Furthermore, the results of 2-term and 4-term Galerkin truncation are compared to determine the numerical convergence. Moreover, parametric studies are conducted to understand the effects of the system parameters on the nonlinear steady-state response. It is concluded that one-way clutch not only decreases the resonance amplitude of the driven pulley and shaft's rotational vibration, but also reduces the resonance region of the belt's transverse vibration.     

Flow-induced instability under bounded noise excitation in cross-flow

Flow-induced vibration of a single cylinder in a cross-flow is mainly due to vortex shedding, which is usually considered as a forced vibration problem. It is shown that flow-induced vibration of a cylinder in the lock-in region is a combination of forced resonant vibration and fluid-damping-induced instability, which leads to time-dependent-fluid-damping-induced parametric resonance and constant-negative-damping-induced instability. The time-dependent fluid damping can be modeled as a bounded noise. The dynamic stability of a two-dimensional system under bounded noise excitation with a narrow-band characteristic is studied through the determination of the moment Lyapunov exponent and the Lyapunov exponent. The case when the system is in primary parametric resonance in the absence of noise is considered and the effect of noise on the parametric resonance is investigated. For small amplitudes of the bounded noise, analytical expansions of the moment Lyapunov exponents and Lyapunov exponents are obtained, which are shown to be in excellent agreement with those obtained using Monte Carlo simulation. The theory of stochastic stability is applied to explore the stability of a cylinder in a cross-flow. The analytical and numerical results show that the time-dependent-fluid-damping-induced parametric resonance could occur, which suggests that parametric resonance also contributes to the vibration of the cylinder in the lock-in range.