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.     

非线性转子的低频振动失稳机理分析

包括两部分内容：１）材料内阻作用下转子自激振动的局部分岔分析；２）考虑湍流因素的滑动轴承油膜力作用下转子轴承系统油膜失稳机理的全面分析。结果表明，非线性转子的自激振动表现出复杂的动力学现象，这些低频振动现象的揭示，为工程上转子故障的识别和预防提供了理论依据。     

Modeling and analyzing of nonlinear dynamics for linear guide slide platform considering assembly error

It is industry tendency to accurately predict the dynamics of the mechanical systems with full consideration of errors. Based on the Hertz contact theory and general bearing modeling methods, this study proposed a more practical model using numerical method to investigate the influence of assembly error on the dynamics of linear guide slide platform. First, the modeling methods of five types of assembly errors are established, based on which, a nonlinear dynamic model is developed to investigate the influence of assembly error. In consideration of assembly error, the modeling method enables the sum of restoring forces and restoring moments equal to zero when no external load applied to the platform. Second, the simulation results indicate that the assembly error can cause uneven load distribution, change the dynamics of the system. In addition, different from previous research results, the stability of the system cannot be improved by simply increasing the preload. Last, in order to validate the proposed method, the proposed model is compared with previous fewer degrees-of-freedom model, and a series of experiments are conducted on a specialized platform to estimate the parameters of the system and verify the proposed model.

     

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.      

UKF在深组合GPS/INS导航系统中的应用

采用联邦滤波的深组合GPS/INS导航系统预滤波器量测模型具有很强的非线性,导致扩展卡尔曼滤波（EKF）的预滤波器估计精度不高。Unscented卡尔曼滤波（UKF）方法是一种非线性分布近似方法,它使用有限数量的sigma点去逼近整个非线性动态系统的分布可能,从而避免了对非线性测量模型进行线性化,具有较高的精度和较好的鲁棒性。在分析深组合导航系统预滤波器模型和UKF原理的基础上,设计了基于UKF滤波算法的预滤波器,对码相位误差、载波相位误差、载波频率误差、载波频率变化率等参数进行估计,同时将UKF和EKF算法进行了仿真比较。结果表明,在深组合导航系统中使用UKF滤波比EKF有更高的导航定位精度。     

A paradigmatic system to study the transition from zero/Hopf to double-zero/Hopf bifurcation

A two-d.o.f.?system experiencing codimension-three double-zero/Hopf bifurcation is considered. This is a special bifurcation which simultaneously involves a defective and a nondefective pair of critical eigenvalues, therefore, requiring a perturbation method specifically tailored on it. A nonstandard version of the multiple scale method is implemented, in which fractional power expansions, both for state-variables and time are used, and high-order arbitrary amplitudes are introduced. Bifurcation equations are obtained, governing the slow flow on the center manifold, which turns out to be tangent to the space spanned by the four critical eigenvectors. These are used to analyze the transition from codimension-three to codimension-two single-zero/Hopf bifurcations, occurring when the modulus of the damping is increased from small to order-one values. Bifurcation charts are obtained, displaying the role of quasi-periodic motions in the transition.     

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.     

Reduced model of a rotor-shaft system using modified SEREP

A modified System Equivalent Reduction Expansion Process (SEREP) is reported to reduce large linear system-equations of motion, which also include gyroscopic effect, internal and external damping. The process is employed on equations of motion written in state-space and utilizes both right and left eigenvectors of the original system unlike only the former in conventional SEREP approach. Numerical example with a finite element model representing multi-disc rotor-shaft-bearing-dynamics shows effectiveness of the process. Nice agreement of Campbell diagrams, FRF plots and unbalance response amplitude between reduced and original systems show usefulness of the reduction process.     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

具有组合非线性阻尼的非线性能量阱振动抑制响应分析

非线性能量阱是一种振动能量吸收装置,其在结构振动抑制中具有十分重要的作用.文章对具有组合非线性阻尼非线性能量阱的系统进行振动抑制相关的分析.首先对具有组合非线性阻尼非线性能量阱的系统进行理论模型的描述,对系统模型的运动方程利用复变量平均法进行推导,得到系统的慢变方程.其次对系统的慢变方程运用多尺度法进行强调制响应的分析,通过对系统进行慢不变流形和相轨迹的研究,描述系统强调制响应发生的条件基础.此外,还利用一维映射对系统进行分析,揭示外激励幅值对强调制响应存在时频率失谐系数取值区间的影响规律.最后利用能量谱、时间响应和庞加莱映射对耦合组合非线性阻尼非线性能量阱系统进行了振动抑制的相关研究,揭示组合非线性阻尼的非线性能量阱不同阻尼比、阻尼和刚度对其振动抑制效果的影响规律,得出组合非线性阻尼非线性能量阱和主结构响应存在一致性的现象,并验证所提出的组合非线性阻尼非线性能量阱模型具有较好的振动抑制能力.     

Nonlinear Dynamics and Stability of a Two D.O.F. Elastic/Elasto-Plastic Model System

The local and global nonlinear dynamics of a two-degree-of-freedom model system is studied. The undeflected model consists of an inverted T formed by three rigid bars, with the tips of the two horizontal bars supported on springs. The springs exhibit an elasto-plastic response, including the Bauschinger effect. The vertical rigid bar is subjected to a conservative (dead) or non-conservative (follower) force having static and periodic components. First, the method of multiple scales is used for the analysis of the local dynamics of the system with elastic springs. The attention is focused at modal interaction phenomena in weak excitation at primary resonance and in hard sub-harmonic excitation. Three different asymptotic expansions are utilised to get a structural response for typical ranges of excitation parameters. Numerical integration of the governing equations is then performed to validate results of asymptotic analysis in each case. A full global nonlinear dynamics analysis of the elasto-plastic system is performed to reveal the role of plastic deformations in the stability of this system. Static 'force-displacement' curves are plotted and the role of plastic deformations in the destabilisation of the system is discussed. Large-amplitude non-linear oscillations of the elasto-plastic system are studied, including the influence of material hardening and of static and sinusoidal components of the applied force. A practical method is proposed for the study of a non-conservative elasto-plastic system as a non-conservative elastic system with an 'equivalent' viscous damping. This revised version was published online in July 2006 with corrections to the Cover Date.     

高精度惯导系统重力扰动的阻尼抑制方法

重力扰动已经成为高精度长航时惯导系统的主要误差源之一。针对船用高精度惯导系统的重力扰动抑制问题,从舰船INS误差模型出发,推导了重力扰动在惯导系统中的传播特性。仿真结果表明垂线偏差将引起系统较大的舒拉振荡误差。为抑制重力扰动对系统的影响,引入常速度误差反馈阻尼网络和相位超前串联阻尼网络。分析了重力扰动在水平阻尼网络中的传递特性,实现了相应滤波器的设计。在此基础上完成了实验验证,海上试验结果表明,所引入的两种阻尼网络都能够阻尼掉重力扰动引起的舒拉振荡型导航误差,其中,相位超前串联阻尼网络效果更优,抑制率达到70%以上。     

Study on Dynamics, Stability and Control of Multi-Body Flexible Structure System in Functional Space

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｔｈｅｍｏｄｅｒｎｓｃｉｅｎｃｅａｎｄｅｎｇｉｎｅｅｒｉｎｇｔｅｃｈｎｏｌｏｇｙ ,ｅｓｐｅｃｉａｌｌｙａｓｔｒｏｎ ａｖｉｇａｔｉｏｎｈａｖｉｎｇｂｅｃｏｍｅｐｒａｃｔｉｃａｌｌｙ ,ｔｈｅｐｒｏｂｌｅｍｄｉｓｃｕｓｓｉｎｇｄｙｎａｍｉｃｓ,ｓｔａｂｉｌｉｔｙａｎｄｃｏｎｔｒｏｌｏｆｓｙｓｔｅｍｏｆｔｈｅｌａｒｇｅｆｌｅｘｉｂｌｅｓｐａｃｅｓｔｒｕｃｔｕｒｅ (ＬＦＳＳ)ｈａｓｂｅｅｎａａｃｔｉｖｅｒｅｓｅａｒｃｈｆｉｅｌｄａｔｈｏｍｅａ…     

Nonlinear adaptive task space control for a 2-DOF redundantly actuated parallel manipulator

A nonlinear adaptive (NA) controller in the task space is developed for the trajectory tracking of a 2-DOF redundantly actuated parallel manipulator. The dynamic model with nonlinear friction is established in the task space for the parallel manipulator, and the linear parameterization expression of the dynamic model is formulated. Based on the dynamic model, a new control law including adaptive dynamics compensation, adaptive friction compensation and error elimination items is designed. After defining a quadratic performance index, the parameter update law is derived with the gradient descent algorithm. The stability of the parallel manipulator system is proved by the Lyapunov theorem, and the convergence of the tracking error and the error rate is proved by the Barbalat’s lemma. The NA controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated parallel manipulator, and the experiment results are compared with the APD controller.     

捷联惯组减振系统角振动、线振动共振频率理论分析

当前实际工程中,角振动试验技术和设备尚不成熟。以指导捷联惯组减振系统设计过程为目的,结合捷联惯性组合减振系统对角共振频率的特殊要求,对线振动、角振动共振频率的关系进行了分析与讨论。建立了捷联惯性组合减振系统的动力学模型,得到了捷联惯组减振系统数学模型,利用刚体动力学方法分析捷联惯组的转动惯量和回转半径,得出理想情况下捷联惯性组合减振系统线共振、角共振频率存在一定的比例关系,并与减振器安装中心到转动中心线的平均距离和惯组回转半径有关的结论。最后讨论了改变减振系统刚度和减振器布置方式两种改变共振频率的方法,并对两种方法进行了比较。     

主动约束层阻尼梁的数值模型

为对主动约束层阻尼结构建立精确完善的数学模型，采用有限元建模，并考虑到压电材料的机电耦合效应和粘弹性材料的本构关系随温度、频率的变化而变化的特点，将有限元方法与粘弹性材料的GHA模型相结合，从而避免因粘弹性材料导致的非线性微分方程，能直接求解模态频率、模态阻尼及结构响应。为进一步设计控制器，先在物理空间进行动力缩聚，将系统降至适当的维数，然后在状态空间用鲁棒防阶的方法进一步降阶。这样既能大大降低系统维数，又能保证降阶后系统稳定、可控、可观。这对于重量轻、柔度大、低频密集的大型空间柔性结构尤其重要。     

Optimal solution error quantification in variational data assimilation involving imperfect models

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliary control problem. If the model is not perfect, the optimal solution error includes an additional component because of the presence of the model error. In this paper, we study the influence of the model error on the optimal solution error covariance, considering strong and weak constraint data assimilation approaches. For the latter, an additional equation describing the model error dynamics is involved. Numerical experiments for the 1D Burgers equation illustrate the presented theory. Copyright © 2016 John Wiley & Sons, Ltd.     

光纤捷联惯导系统高阶误差模型的建立与分析

针对多位置标定算法中利用的陀螺和加表的误差模型,在捷联惯导系统误差传播方程中,考虑陀螺和加表的标度因数误差和安装误差,建立了一种高阶误差模型。为了评价该模型的准确性,将其与不考虑标度因数误差和安装误差的模型比较,设计了系统静态和动态仿真实验。在系统静态仿真中,分别加入陀螺漂移和加表安装误差,而在动态仿真中同时加入各项误差项,求取以这些误差项为初值的模型微分方程的解,使其与惯导系统输出误差进行比较。仿真结果发现,建立的高阶误差模型比不考虑标度因数误差和安装误差的模型精度高出约三个数量级。     

Forced Vibration of Continuous System with Unsymmetrical Collision Characteristics

This paper deals with steady-state response of a continuous system with nonlinear boundary conditions which are motion-limiting constraint. An analytical method of approximate solution for the continuous system with unsymmetrical collision characteristics in which the beam end collides with a stop once in one period of its vibration is presented. Some numerical results of the approximate solution are shown. Contrary to the case of continuous system with symmetrical collision characteristics, the resonance curves of nonlinear response of approximate solution are shown as discontinuous line. Some numerical results of a continuous system with no hysteresis damping are compared with those of a continuous system with hysteresis damping and a single-degree-of-freedom system.