时滞耦合质量摆动力吸振器减振系统的等峰优化理论与实验

摆式调谐质量阻尼器因其便于安装、维修、更换,且经济实用,广泛应用于结构减振.它通过将摆的自振频率调谐到接近主系统的控制频率,使摆产生与主系统相反的振动,从而抑制或消除主系统的振动.本文通过对主系统无阻尼的被动减振系统和主系统有阻尼的时滞反馈主动减振系统进行多目标优化设计,实现了对主系统幅频响应曲线的等峰控制和共振峰与反共振峰差值的有效控制.首先,建立了时滞耦合质量摆动力吸振器减振系统的力学模型和振动微分方程,通过对主系统无阻尼的被动减振系统进行等峰优化,获得了减振系统的最优频率比和质量摆的最优阻尼比.对于主系统存在阻尼的被动减振系统,在该优化参数下主系统的幅频响应曲线等峰优化失效.其次,对于主系统存在阻尼的时滞反馈优化控制系统,采用CTCR方法得到了反馈增益系数和时滞的稳定区域.在保证系统稳定的前提下,通过调节反馈增益系数和时滞量两个控制参数能够实现对主系统幅频响应曲线的等峰控制.再次,对共振点处主系统振幅放大因子时滞敏感度和反馈增益系数敏感度进行分析,表明共振点幅值对反馈增益系数比对时滞更为敏感.最后,通过实验分别在频域和时域内对理论结果进行了验证.研究表明,通过采用时滞反馈对摆式调...     

OPTIMAL CONTROL OF NONLINEAR VIBRATION RESONANCES OF SINGLE-WALLED NANOTUBE BEAMS

An optimal time-delay feedback control method is provided to mitigate the primary resonance of a single-walled carbon nanotube （SWCNT） subjected to a Lorentz force excited by a longitudinal magnetic field. The nonlinear governing equations of motion for the SWCNT under longitudinal magnetic field are derived and the modulation equations are obtained by using the method of multiple scales. The regions of the stable feedback gain are worked out by using the stability conditions of eigenvalue equation. Taking the attenuation ratio as the objective function and the stable vibration regions as constrained conditions, the optimal control parameters are worked out by using minimum optimal method. The optimal controllers are designed to control the dynamic behaviors of tile nonlinear vibration systems. It is found that the optimal feedback gain obtained by the optimal method can enhance the control performance of the primary resonance of SWCNT devices.     

考虑间隙反馈控制时滞的磁浮车辆稳定性研究

常导磁吸型(EMS)磁悬浮列车在悬浮控制中的每个环节,时滞是不可避免的,当时滞超过一定程度后,系统有可能失稳.本文针对EMS磁浮列车控制环节的临界时滞与车辆参数(如运行速度、反馈控制增益、导轨参数和悬挂参数)的关系开展研究.建立了磁浮车辆/导轨耦合动力学模型,车辆包含1节车辆和4个磁浮架,考虑车辆的10个自由度,每个磁浮架上包含4个悬浮电磁铁.导轨模拟为一系列简支Bernoulli-Euler梁,采用模态叠加法对导轨振动方程进行求解.采用传统线性电磁力模型实现车辆和轨道的耦合.采用比例-微分控制算法对电磁铁电流进行反馈控制,实现车辆稳定悬浮,并假设时滞均发生在控制环节,且只考虑间隙反馈控制环节的时滞.采用四阶龙格库塔法对耦合系统动力学方程进行求解,编写了数值仿真程序,计算得到车辆导轨耦合系统在考虑间隙反馈控制时滞时的响应.将系统运动发散时的时滞大小视为临界时滞,开展了参数规律影响分析.通过分析,给出了提高时滞条件下车辆稳定性的方法,包括增大导轨的弯曲刚度和阻尼比,减小间隙反馈控制增益并增大速度反馈控制增益,以及增大二系悬挂阻尼.      

Forward and backward motion control of a vibro-impact capsule system

A capsule system driven by a harmonic force applied to its inner mass is considered in this study. Four various friction models are employed to describe motion of the capsule in different environments taking into account Coulomb friction, viscous damping, Stribeck effect, pre-sliding, and frictional memory. The non-linear dynamics analysis has been conducted to identify the optimal amplitude and frequency of the applied force in order to achieve the motion in the required direction and to maximize its speed. In addition, a position feedback control method suitable for dealing with chaos control and coexisting attractors is applied for enhancing the desirable forward and backward capsule motion. The evolution of basins of attraction under control gain variation is presented and it is shown that the basin of the desired attractors could be significantly enlarged by slight adjustment of the control gain improving the probability of reaching such an attractor.     

Control of the nonlinear oscillator bifurcation under a superharmonic resonance

A weakly nonlinear oscillator is modeled by a differential equation. A superharmonic resonance system can have a saddle-node bifurcation, with a jumping transition from one state to another. To control the jumping phenomena and the unstable region of the nonlinear oscillator, a combination of feedback controllers is designed. Bifurcation control equations are derived by using the method of multiple scales. Furthermore, by performing numerical simulations and by comparing the responses of the uncontrolled system and the controlled system, we clarify that a good controller can be obtained by changing the feedback control gain. Also, it is found that the linear feedback gain can delay the occurrence of saddle-node bifurcations, while the nonlinear feedback gain can eliminate saddle-node bifurcations. Feasible ways of further research of saddle-node bifurcations are provided. Finally, we show that an appropriate nonlinear feedback control gain can suppress the amplitude of the steady-state response.     

Study on the force transmissibility of vibration isolators with geometric nonlinear damping

By a special layout of the damper in a vibration isolation system, the velocity-nth power damping of the damper can be transformed into the velocity-displacement dependent damping in the moving direction due to geometric nonlinearity. This study is mainly concerned with the mechanism of this geometric nonlinear damping and tries to find some guidelines for designing isolators with high performance, regardless of the isolator is passive or active. The model used in this study is an unconstrained two-degree-of-freedom isolation system, which is suitable for investigating vibration isolation in space vehicles. The motion equation is solved by the averaging method to obtain the amplitude–frequency equation. The influence of damping coefficients on the force transmissibility is analyzed. Results show that this kind of damping has some advantages in improving isolation performance at both the resonance and higher frequencies. Meanwhile, it can also degrade the isolation performance if unreasonable parameters are chosen.     

主动控制压电旋转悬臂梁的参数振动稳定性分析

在工程实际中旋转机械由于制造和加工误差,装配的不均匀性等原因,往往会脉动运行,这将使得机械系统发生参数振动. 当脉动参数满足一定关系时,这种参数振动将会失稳,进而影响机械结构的正常运转. 本文针对这一问题,引入压电材料对 脉动旋转悬臂梁系统的振动进行控制,研究主动控制悬臂梁系统的参数振动优化设计问题,采用 Hamilton 变分原理与一阶 Galerkin 离散相结合的方法,建立了受速度反馈传感器主动控制的压电旋转悬臂梁的一阶近似线性控制方程. 运用多尺度方法,得到了压电旋转悬臂梁系统在发生1/2亚谐波参数共振时稳定性边界的控制方程,并利用直接分析方法验证了解析摄动解的正确性. 将摄动解中临界阻尼比和轮毂角速度脉动幅值的无量纲参数作为评价系统稳定性能的指标. 通过数值算例,分析了轮毂半径、轮毂角速度平均值和脉动幅值、梁长以及速度传感器的反馈增益系数对系统稳定性区域的影响. 研究结果表明,梁长、轮毂半径、脉动幅值会降低系统稳定性,反馈增益系数可以提高系统稳定性,而轮毂角速度平均值与系统稳定性之间有非单调的关系. 为进一步设计压电旋转机械结构提供了理论依据.      

参数激励下非线性受控系统的动力学和稳定性

研究两自由度参数激励系统的非线性动力学与控制问题．利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程，以多尺度方法获得一阶近似控制方程．然后，对系统受一阶摸态参激主共振与一、二阶模态间3：1内共振联合作用下的幅额响应及其稳定性，以及反馈参数对系统稳态行为的影响作了详细分析．结果表明，响应的稳定域位置和大小取决于位移反馈，位移立方反馈改变了系统的非线性程度，速度反馈类似于阻尼，可使系统呈现自激振动特性．     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

CHARACTERIZATION OF THE BEHAVIOUR OF A SIMPLE AEROSERVOELASTIC SYSTEM WITH CONTROL NONLINEARITIES

The characterization of the behaviour of nonlinear aeroelastic systems has become a very important research topic in the Aerospace Industry. However, most work carried to-date has concentrated upon systems containing structural or aerodynamic nonlinearities. The purpose of this paper is to study the stability of a simple aeroservoelastic system with nonlinearities in the control system and power control unit. The work considers both structural and control law nonlinearities and assesses the stability of the system response using bifurcation diagrams. It is shown that simple feedback systems designed to increase the stability of the linearized system also stabilize the nonlinear system, although their effects can be less pronounced. Additionally, a nonlinear control law designed to limit the control surface pitch response was found to increase the flutter speed considerably by forcing the system to undergo limit cycle oscillations instead of fluttering. Finally, friction was found to affect the damping of the system but not its stability, as long as the amplitude of the frictional force is low enough not to cause stoppages in the motion.     

A Simplified Optimal Control Method for Homoclinic Bifurcations

A simplified optimal control method is presented for controlling or suppressing homoclinic bifurcations of general nonlinear oscillators with one degree-of-freedom. The simplification is based on the addition of an adjustable parameter and a superharmonic excitation in the force term. By solving an optimization problem for the optimal amplitude coefficients of the harmonic and superharmonic excitations to be used as the controlled parameters, the force term as the controller can be designed. By doing so, the control gain and small optimal amplitude coefficients can be obtained at lowest cost. As the adjustable parameter decreases, a gain of some amplitude coefficient ratio is increased to the highest degree, which means that the region where homoclinic intersection does not occur will be enlarged as much as possible, leading to the best possible control performance. Finally, it is shown that the theoretical analysis is in agreement with the numerical simulations on several concerned issues including the identification of the stable and unstable manifolds and the basins of attraction.     

Vibration control by nonlocal feedback and jerk dynamics

Jerk dynamics is used for a new method for the suppression of self-excited vibrations in nonlinear oscillators. Two cases are considered, the van der Pol equation and nonlinear oscillator with quadratic and cubic nonlinearities. A nonlocal control force is introduced in such a way to obtain a third order nonlinear differential equation (jerk dynamics). Using the asymptotic perturbation method, two slow flow equations on the amplitude and phase of the response are obtained, and subsequently the performance of the control strategy is investigated. The feedback gains are connected with the stability and response of the system under control. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the self-excitations. Numerical simulation confirms the validity of the new method.     

Effects of damping on the speed of increase and amplitude of limit cycle for an aircraft braking system subjected to mode-coupling instability

A nonlinear model of an aircraft braking system is presented and used to investigate the effects of damping on the stability in Chevillot et al. (Arch Appl Mech 78(12):949–963, 2008). It has been shown that the addition of damping into the equations of motion does not lead systematically to the stabilization of the system. In the case of a mode-coupling instability, there is indeed an optimal ratio between the modal damping coefficients of the two modes in coalescence, that maximize the stable area. But the stable area is not a sufficient criterion. In dynamics, the amplitude of the vibrations and the transient behavior characterized by the speed of increase of the oscillations are best indicators. In this paper, the same nonlinear model of the aircraft braking system is used to compute time-history responses by integration of the full set of the nonlinear dynamic equations. The aim of the study is to evaluate the effects of damping on the nonlinear dynamics of the brake. It is shown that damping may be very efficient to significantly reduce and slow down the increase of the friction-induced vibrations. But, in the same way as for the stability area, there exists a value of the damping ratio that optimizes the effects of damping.     

几何非线性摩擦阻尼隔振系统动力学行为研究

非线性隔振系统由于具有较线性系统更优的隔振性能,因此在工程中应用广泛.本文通过配置与被隔振对象的运动方向相垂直的库伦摩擦阻尼器,构建了几何非线性摩擦阻尼模型.由于引入了几何非线性,因此其摩擦力与位移正相关,这与传统与位移无关摩擦力模型有显著不同.首先,建立了具有几何非线性摩擦阻尼的数学模型以及隔振系统的受迫振动方程;然后,使用谐波平衡法求解了动力学方程,并使用数值仿真方法验证了谐波平衡法求解的准确性;最后,研究了几何非线性摩擦阻尼隔振器的绝对位移传递率和相对位移传递率.研究结果表明,在库伦摩擦阻尼选择适当,非线性摩擦阻尼系统可以在保持高频振动衰减效果的前提下,显著降低系统共振峰,其性能优于传统的恒定摩擦阻尼隔振模型.同时,几何非线性摩擦阻尼系统能够避免传统摩擦阻尼系统中的“锁定”现象,从传递率角度来说,不利于共振峰控制;但从激励环境改变引发隔振系统失效的角度来看,几何非线性摩擦阻尼系统可以拓宽系统对激励幅值的适应范围,避免隔振系统失效.本文的研究结果对此类隔振系统的设计和摩擦阻尼参数的选择具有通用的指导意义.      

负刚度时滞反馈控制动力吸振器的等峰优化

相比于传统动力吸振器, 负刚度动力吸振器同时具有更好的减振能力和更宽的有效减振频带宽度, 为了进一步降低共振峰幅值, 在负刚度吸振器系统耦合时滞反馈控制. 对负刚度时滞反馈控制动力吸振器系统进行等峰优化设计, 优化设计的准则是:第一和第二共振峰的峰值相等; 同时兼顾两个目标, 一个目标是在优化时的最大共振峰幅值小于被动负刚度吸振器系统的反共振峰幅值, 另一目标是在优化时共振峰幅值与反共振峰幅值差小于被动吸振器系统. 接着, 通过设计和调节负刚度系数、吸振器阻尼系数和时滞反馈控制系数对控制系统进行等峰优化设计. 最后, 在降低幅值的同时, 分析结构参数对有效减振频带宽度的影响. 经过等峰优化之后, 选择本文的一组结构参数与两个典型的模型进行对比. 为了定量比较不同模型的降幅效果, 定义了减幅百分比, 研究发现在有效减振频带区间内减幅百分比超过40%以上. 结果表明, 通过等峰优化准则对结构参数进行优化设计和调节增益系数和时滞量, 共振峰幅值的减幅百分比也近似达到40%, 也可以调节增益系数和时滞量, 使得幅频响应曲线具有较宽的有效减振频带和较低的共振峰幅值与反共振峰幅值的差值.      

Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension

Non-linear vibrations of axially moving beam with time-dependent tension are investigated in this paper. The beam material is modelled as three-parameter Zener element. The Galerkin method and the fourth order Runge-Kutta method are used to solve the governing non-linear partial-differential equation. The effects of the transport speed, the tension perturbation amplitude and the internal damping on the dynamic behaviour of the system are numerically investigated. The Poincare maps and bifurcation diagrams are constructed to classify the vibrations. For small values of the transport speed and the amplitude of periodic perturbation the system is asymptotically stable with its response tending to zero. With the increase of parameters one can observe the coexistence of attractors. Regular and chaotic motion occur when the internal damping increases.     

Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers

This paper aims to study the nonlinear-forced vibrations of a viscoelastic cantilever with a piecewise piezoelectric actuator layer on its top surface using the method of Multiple Scales. The governing equation of motion is a second-order nonlinear ordinary differential equation with quadratic and cubic nonlinearities which appear in stiffness, inertia, and damping terms. The nonlinear terms are due to the piezoelectricity, viscoelasticity, and geometry of the system. Forced vibrations of the system are investigated in the cases of primary resonance and non-resonance hard excitation including subharmonic and superharmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including damping coefficient, thickness to width ratio of the beam, length and position of the piezoelectric layer, density of the beam, and the piezoelectric coefficient on the frequency-response curves are discussed for each case. It is shown that in all these cases, the response of the system follows a softening behavior due to the existence of the piezoelectric layer. The piezoelectric layer provides an effective tool for active control of vibration. In addition, the effect of the viscoelasticity of the beam on passive control of amplitude of vibration is illustrated.     

Non-linear analysis and quench control of chatter in plunge grinding

This paper aims at mitigating regenerative chatter in plunge grinding. To begin with, a dynamic model is proposed to investigate grinding dynamics, where eigenvalue and bifurcation analyses are adopted, respectively, for prediction of grinding stability and chatter. Generally, it is found that most grinding chatter is incurred by subcritical Hopf bifurcation. Compared with supercritical instability, the subcritical generates coexistence of stable and unstable grinding in the stable region and increases chatter amplitude in the chatter region. To avoid these adverse effects of the subcritical instability, bifurcation control is employed, where the cubic non-linearity of the relative velocity between grinding wheel and workpiece is used as feedback. With the increase of feedback gain, the subcritical instability is transformed to be supercritical not only locally but also globally. Finally, the conditionally stable region is completely removed and the chatter amplitude is decreased. After that, to further reduce the chatter amplitude, quench control is used as well. More specifically, an external sinusoid excitation is applied on the wheel to quench the existing grinding chatter, replacing the large-amplitude chatter by a small-amplitude forced vibration. Through the method of multiple scales, the condition for quenching the chatter is obtained.     

Periodic and quasiperiodic motion for complex non-linear systems

A damped complex non-linear system corresponding to two coupled non-linear oscillators with a periodic damping force is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Phase-locked solutions with period equal to the damping force period are possible only if the oscillators amplitudes are equal. On the contrary, if the oscillators amplitudes are different, periodic solutions exist only with a period different from the damping force period. These solutions are stable only for perturbations that conserve the phase difference and the square amplitude sum of the oscillators. Energy considerations are used in order to study existence and characteristics of quasiperiodic motion. We demonstrate that modulated motion can be also obtained for appropriate values of the detuning parameter and in this case an approximate analytic solution is easily constructed. If the detuning parameter decreases the modulation period increases and then diverges, an infinite-period bifurcation occurs and the resulting motion becomes unbounded. Analytic approximate solutions are checked by numerical integration.     

DYNAMICAL ANALYSIS ON A KIND OF SEMI-ACTIVE VIBRATION ISOLATION SYSTEMS WITH DAMPING CONTROL 1)

研究了一类基于相对速度反馈的含立方刚度的单自由度非线性半主动隔振系统.通过平均法得到了系统分别在基于加速度-相对速度反馈的加速度驱动阻尼控制策略、速度-相对速度反馈的天棚阻尼控制策略和位移-相对速度反馈的地棚阻尼控制策略下主共振响应的近似解析解,并利用数值解验证了近似解析解的准确性.通过 Lyapunov 理论对不同控制策略下系统的稳定性进行了分析,讨论了系统参数对控制效果的影响.分析结果表明,对 3 种基于相对速度反馈的控制策略进行解析研究时,切换条件中的控制参数具有相同的表达式;在抑制共振响应振幅方面,基于速度-相对速度反馈的天棚阻尼控制策略在低频时的减振效果最好,而基于加速度-相对速度反馈的加速度驱动阻尼控制策略在高频时的减振效果最优;在抑制瞬态响应振幅方面,基于速度-相对速度反馈的天棚阻尼控制策略的减振效果最好.此类解析研究方法可应用到其他半主动开关控制策略中,为半主动隔振系统的控制策略研究提供了有效的方法和手段.