Nonlinear dynamics in the flexible shaft rotating–lifting system of silicon crystal puller using Czochralski method

Silicon crystal puller (SCP) is key equipment in silicon wafer manufacture, which is, in turn, the base material for the most currently used integrated circuit chips. With the development of the techniques, the demand for longer mono-silicon crystal rod with larger diameter is continuously increasing in order to reduce the manufacture time and the price of the wafer. This demand calls for larger SCP with an increasing height, though it causes serious swing phenomenon of the crystal seed. The strong swing of the seed increases the possibility of defects in the mono-silicon rod and the risk of mono-silicon growth failure. The main aim of this paper is to analyze the nonlinear dynamics in flexible shaft rotating–lifting (FSRL) system of the SCP. A mathematical model for the swing motion of the FSRL system is derived. The influence of relevant parameters, such as system damping, excitation amplitude, and rotation speed, on the stability and the responses of the system is analyzed. The stability of the equilibrium, bifurcation, and chaotic motion is demonstrated, which have been observed in practical situations. Melnikov method is used to derive the possible parameter region which leads to chaotic motion. Three routes to chaos are identified in the FSRL system, including period doubling, symmetry-breaking bifurcation, and crisis. The work in this paper analyzes and explains the complex dynamics in FSRL system of the SCP, which will be helpful for the designers in the designing process in order to avoid the swing phenomenon in the SCP.

     

一种改进的冲量测试装置及其数据处理方法

对摆锤式片炸药比冲量测试装置及其数据处理方法进行了改进。首先,在对机械结构进行了改进设计的基础上,利用光栅测试系统测量摆动角度,并解决了测试时波形失真的问题,提高了装置的精密性和稳定性。然后,采用该装置研究了阻尼随摆动幅度的变化规律,并用实测数据进行了验证;提出了片炸药比冲量测试时的最大摆角修正方法,并进行了试验对比研究。结果表明:改进后的装置及其数据处理方法能够更加准确地测量片炸药的比冲量。     

微载荷含油轴承摩擦性能研究Ⅱ.摩擦试验分析

采用自行研制的径向轴承微载荷摩擦试验机研究了微载荷下径向含油轴承的摩擦性能,采用小波方法对试验数据进行滤波降噪.结果表明,含油轴承在稳定微载荷状态下的瞬态摩擦系数不固定,且随着润滑剂、轴承转速、径向载荷与混合润滑状态等因素而变化.在微载荷状态下,当600 r/min、采用液晶添加剂(5CB)与46号机械油以体积比1%～2%配制的润滑剂润滑时,平均摩擦系数达到最小值;当转速从600 r/min增至3 000 r/min时,平均摩擦系数由小变大、再变小;瞬态摩擦系数随着载荷从0.5 N增至10.0 N呈现高-低-稳定的变化趋势.与混合润滑状态相比,充分润滑下的摩擦系数较大;当混合润滑时,在主轴转速为1 500～2 400 r/min下的摩擦系数出现不稳定.     

Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports

Cracks appearing in the shaft of a rotary system are one of the main causes of accidents for large rotary machine systems. This research focuses on investigating the bifurcation and chaotic behavior of a rotating system with considerations of various crack depth and rotating speed of the system’s shaft. An equivalent linear-spring model is utilized to describe the cracks on the shaft. The breathing of the cracks due to the rotation of the shaft is represented with a series truncated time-varying cosine series. The geometric nonlinearity of the shaft, the masses of the shaft and a disc mounted on the shaft, and the viscoelasticity of the supports are taken into account in modeling the nonlinear dynamic rotor system. Numerical simulations are performed to study the bifurcation and chaos of the system. Effects of the shaft’s rotational speed, various crack depths and viscosity coefficients on the nonlinear dynamic properties of the system are investigated in detail. The system shows the existence of rich bifurcation and chaos characteristics with various system parameters. The results of this research may provide guidance for rotary machine design, machining on rotary machines, and monitoring or diagnosing of rotor system cracks.     

漂浮基柔性关节空间机器人自适应滑模运动控制及振动主动抑制

讨论了关节柔性且系统参数不确定的漂浮基空间机器人系统的动力学建模过程、运动轨迹跟踪控制算法设计及系统柔性振动的主动抑制问题。利用系统动量、动量矩守恒关系和拉格朗日法对系统动力学进行分析,并建立系统动力学方程。基于奇异摄动法将系统分解为表示系统刚性运动部分的慢变子系统和表示系统柔性运动部分的快变子系统。针对慢变子系统提出了一种自适应滑模控制算法。该控制算法是由基于滑模面的等效控制项、自适应控制项和PID反馈控制项组成。因此,它集合了滑模控制、自适应算法和PID技术的优点,且弥补了三种算法各自的缺点。该控制算法能够有效地补偿系统的转动误差和不确定参数,提高控制系统的精度。针对快变子系统,提出基于速度差值的反馈控制算法来抑制柔性关节引起的系统柔性振动,保证系统的稳定性。最后,通过仿真实验证明了提出的混合控制算法的有效性。     

Locating order-chaos-order transition in elastic pendulum

An undamped elastic pendulum being a nonintegrable Hamiltonian system always has some chaotic trajectories observable on choosing appropriate initial conditions. This is true even if the pendulum is in libration with small amplitude; in this situation, the pendulum may be seen as a nearly integrable system. Since the measure of the set of the local chaotic trajectories in the phase space may be very small, the trajectories are hard to locate. However, the emergence of widespread chaos when the elastic pendulum is at autoparametric resonance is well-documented. The transition from the local and the widespread chaos is typically established through the Chirikov overlap criterion that approximates the phase portrait around a resonance using a one degree-of-freedom pendulum Hamiltonian. We argue in this paper that the aforementioned transition in the elastic pendulum is due to interaction between two resonances of same kind and their coexistence can be analytically located using perturbation methods, like the method of averaging, whereas the technique of the pendulum Hamiltonian is inapplicable. Furthermore, in the course of validating the result numerically, we also showcase the order-chaos-order transition in the elastic pendulum using the fast Lyapunov indicator.

     

Direct adaptive neural control of chaos in the permanent magnet synchronous motor

In this paper, a direct adaptive neural speed tracking control is addressed for the chaotic permanent magnet synchronous motor (PMSM) drive systems via backstepping. Neural networks are directly used to approximate unknown and desired control signals and a novel direct adaptive tracking controller is constructed via backstepping. The proposed adaptive neural controllers guarantee that the tracking error converges to a small neighborhood of the origin. Compared with the conventional backstepping method, the designed neural controller??s structure is very simple. Simulation results show that the proposed control scheme can suppress the chaos of PMSM and guarantees the perfect tracking performance even with the existence of unknown parameters.     

Rotor whirl damping by dry friction suspension systems

An efficient and automatic attenuation technique for the whirling motion of rotating machinery can be achieved by supporting the journal boxes elastically and providing them with suitable rubbing surfaces subject to dry friction normal to the shaft axis. The critical flexural speeds are easily cut off and the whirl amplitude is minimized throughout the frequency range. Confining the usual operative angular speed of the rotor in the range of adhesive contact between the dry friction surfaces, there is no significant increase of power dissipation or heat production as a whole due to this type of suspension system, whose task is just to suppress the resonant peaks when passing the critical speeds. Moreover, the wear of the rubbing surfaces can be easily compensated by use of suitable spring loading systems for the friction contact. The dry friction damping is also compared with an equivalent viscous damping, where the equivalence has to be understood in terms of work dissipated per single revolution of the rotor. As for other conventional cases, the shaft hysteresis is found to exert a destabilizing effect above the first critical speed, which however can be compensated by the other dissipation sources. The system stability is here studied perturbing the periodic motion and applying the Floquet theory.     

Chaotic oscillations in pipes conveying pulsating fluid

Chaotic motions of a simply supported nonlinear pipe conveying fluid with harmonie velocity fluetuations are investigated. The motions are investigated in two flow velocity regimes, one below and above the critical velocity for divergence. Analyses are carried out taking into account single mode and two mode approximations in the neighbourhood of fundamental resonance. The amplitude of the harmonic velocity perturbation is considered as the control parameter. Both period doubling sequence and a sudden transition to chaos of an asymmetric period 2 motion are observed. Above the critical velocity chaos is explained in terms of periodic motion about the equilibrium point shifting to another equilibrium point through a saddle point. Phase plane trajectories, Poincaré maps and time histories are plotted giving the nature of motion. Both single and two mode approximations essentially give the same qualitative behaviour. The stability limits of trivial and nontrivial solutions are obtained by the multiple time scale method and harmonic balance method which are in very good agreement with the numerical results.     

Analysis on the nonlinear response of cracked rotor in hover flight

The motion equations for a Jeffcott rotor in hover flight are derived. A periodically sampled peak-to-peak value diagram is used for characterizing and distinguishing different types of nonlinear responses in hovering state. The nonlinear responses become more apparent when the rotor is running above the critical speed in flat flight. There are three ways for rotor responses going to chaos, namely through quasi-periodic, intermittence, or period-3 bifurcation to chaos. The hover flight might suppress some nonlinear responses. However, the position of axis center might obviously deflect, leading to either nonlinear response or peak-to-peak value jump near the fraction frequency of swing critical speed.     

Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spin-down operation. In this article, initially the equations of motion for a spinning shaft with non-constant speed are derived, then the system is discretised, and finally a nonlinear dynamic analysis is performed using multiple scales perturbation method. The system in first-order approximation takes the form of two coupled sets of paired equations. The first pair describes the torsional and the rigid body rotation, whilst the second consists of the equations describing the two lateral bending motions. Notably, equations of the lateral bending motions of first-order approximation coincide with the system in case of constant rotating speed, and considering the amplitude modulation equations, as it is shown, there are detuning frequencies from the Campbell diagram. The nonlinear normal modes of the system have been determined analytically up to the second-order approximation. The comparison of the analytical solutions with direct numerical simulations shows good agreement up to the validity of the performed analysis. Finally, it is shown that the Campbell diagram in the case of spin-up or spin-down operation cannot describe the critical situations of the shaft. This work paves the way, for new safe operational ‘modes’ of rotating structures bypassing critical situations, and also it is essential to identify the validity of the tools for defining critical situations in rotating structures with non-constant rotating speeds, which can be applied not only in spinning shafts but in all rotating structures.     

非稳态非线性油膜力作用下柔性轴裂纹转子的动力特性分析

分析了在动载轴承非稳态非线性油膜力作用下，具有横向裂纹柔性轴Jeffcott转子在非线性涡动影响下的动力特性。通过数值计算表明，在油膜失稳转速前，随着裂纹轴刚度变化比的增大，系统在低转速区域内具有丰富的非线性动力行为，出现倍周期分叉及混沌现象，涡动振幅随转速升高而减小，直到非稳态非线性油膜失稳，在无裂纹转子油膜临界失稳点处发现了类Hopf分叉现象，系统运动由平衡变为拟周期运动；裂纹转子在油膜临界失稳时的系统运动亦为拟周期运动，裂纹转子轴刚度变化对油膜失稳点及油膜失稳之后转子的运动影响不大，转子系统作拟周期运动。     

水平地震时斜面水坝受到的非线性动水压力的全过程的解析解

本文给出在非线性Ｒａｙｌｅｉｇｈ阻尼作用下，由水平地震激发的水库内动水压力变化全过程的解析解：在激发初期其幅值较小，可用小参数法使控制方程线性化，从而求得各阶渐近控制方程的解析解；当动水压力的幅值增大到小参数展开不适用时，用ｖａｎ ｄｅｒｐｏｌ法求得非线性控制方程的渐近解，从而显示了动水压力的幅值从有序变化到发生突然跳跃的多值性的非线性本质。     

Modeling and dynamic analysis of a magnetically actuated butterfly valve

In this work, a magnetically actuated butterfly valve is considered and a novel and accurate mathematical model is derived. The equilibrium of the system is investigated and the effects of the inlet velocity and direct current voltage (DC) on the stable rotation angle of the valve are presented. Considering a time periodic perturbation arising from electric circuit, the effects of the operating angle, inlet velocity, and driving parameters on the periodic and chaotic dynamics of the system are investigated. It is observed that, for an opening angle less than the cut-off angle, there exists a unique DC voltage for a stable equilibrium. The stability of this equilibrium depends nonlinearly on the inlet velocity and the seating torque. An expression is derived for the threshold value for the stability of the valve. Under periodic voltage, the inlet velocity and stable angle induce a backward shift on the resonant frequency, and jump phenomena and subharmonics are observed for some values of the driving amplitude. The highest amplitudes of vibration are detected for a fully open valve, for an almost closed valve, and for a valve with large inlet velocity. Using bifurcation diagrams and Lyapunov exponents, it is shown that the system exhibits a route to chaos with windows of period doubling and unbounded motion. Some guidance for design of magnetically actuated butterfly valves is proposed as well as recommendations for future work.     

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.     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

Single-mode response and control of a hinged–hinged flexible beam

The problem of suppressing the vibrations of a hinged–hinged flexible beam that is subjected to primary and principal parametric excitations is tackled. Different control laws are proposed, and saturation phenomenon is investigated to suppress the vibrations of the system. The dynamics of the beam are modeled with a second-order nonlinear ordinary-differential equation. The method of multiple scales is used to derive two-first ordinary differential equations that govern the time variation of the amplitude and phase of the response. These equations are used to determine the steady-state responses and their stability. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters on the steady-state amplitude and on saturation phenomena at resonance have been reported.     

深井提升动力学研究

随着浅部煤炭资源的日益枯竭, 我国煤炭开采朝着深部化和大型化方向发展, 新建和改扩建的大型立井年生产能力已达1000万吨, 开采最大深度已达1500 m. 千米深井在提升过程中, 造成提升钢丝绳、容器振荡, 特别是自由悬挂平衡尾绳大幅度摆动, 严重影响多绳摩擦提升系统向高速度、深度化发展. 在国家重点研发计划“深地资源勘查开采”重点专项“煤矿深井建设与提升基础理论及关键技术”支持下, 建立了单元数量自动调整的自由悬挂平衡尾绳提升系统动力学模型, 揭示了传统提升系统诱发平衡尾绳大摆动的机理; 提出了深井SAP提升新模式, 构建了多元耦合下的SAP提升系统动力学模型与非光滑动力学模型, 揭示了多参数影响下系统的非光滑动力学特性及非线性振动演化规律; 研发了适用于深部提升的SAP提升技术与装备, 开展了SAP提升技术与装备的现场研究, 解决了大强煤矿立井提升系统运行过程中尾绳大摆动、提升容器大振动等关键问题, 提高了提升系统高速运行的安全性, 消除了尾绳大摆动难控制的问题.      

Impulsive control induced effects on dynamics of single and coupled ODE systems

The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.     

Revisited chaotic vibrations in dielectric elastomer systems with stiffening

Dielectric elastomer is a type of soft materials which can deform under applied voltage. Here, irregular vibrations in a circular dielectric elastomer membrane with stiffening under periodic forcing are studied. The stiffening phenomenon can induce fast increases in the potential energies near the limiting stretches, which induces challenges to the numerical simulations. By comparing different numerical strategies, the adaptive step size method with allowable very small step sizes is used to simulate the system. For the system with or without damping, the existence of chaos is then verified through the positive maximum Lyapunov exponent and the fractal structures in the phase plane simultaneously. The local dynamic analysis shows the strong contribution of regions near the limiting stretches to the occurrence of chaos, revealing the important role of the stiffening. For the system with damping, the rich dynamical behaviors accompanying chaos such as the period-doubling route to chaos and the long chaotic transients also provide further consistent supports for the existence of chaos. For the system without damping, chaos region in a parameter plane is located by using different initial conditions, revealing the transitional behaviors from periodic states to chaos. Besides, the chaos is more easily to occur in the system without damping. Thus, the study here is useful to avoid or further handle such complex irregular dynamics.