Vibration suppression of a time-varying stiffness AMB bearing to multi-parametric excitations via time delay controller

The applications of active magnetic bearings are growing in industry due to its amazing advantages in reducing friction losses. In this research, the vibration of a two-degree-of-freedom rotor, active magnetic bearings system is suppressed via a nonlinear time delay controller at the confirmed worst resonance case. The selected resonance case is the simultaneous primary and sub-harmonic resonance case. The main aim of this paper was to study the effects of the nonlinear, time delay controller on the behavior of the vibrating system. The multiple time scale perturbation technique is applied to obtain an approximate solution to the second-order approximation. The steady-state solution is obtained around the worst resonance case. The stability of the system is studied applying both frequency response equations and phase-plane method. The worst resonance case is confirmed applying numerical technique. The effects of the different parameters on the steady-state response of the vibrating system are investigated. The obtained approximate solution is validated numerically. Some recommendations are given regarding the design of such system. At the end of the work, a comparison is made with the available published work.     

基于唯象相变理论的磁致伸缩材料耦合滞回动力学模拟

本文基于改进的Landau唯象相变理论,构造一个耦合的非线性常微分方程模型来模拟一维磁致伸缩材料的磁滞动态特性。模型的构造通过引入一个非凸的自由能函数来模拟磁致伸缩磁材料中不可逆的磁极化翻转与磁致应变,该自由能函数的每一个局部极小值都对应材料的一个磁化方向。通过热力学平衡条件建立能刻画磁致伸缩效应的非线性本构关系。所构造的模型成功地模拟出了磁场与弹性场之间的磁滞曲线和蝶形曲线,并采用实验结果对模型进行了验证。     

Field analysis and complex resonance frequency of the quasi-TE0,1,1-mode in an inhomogeneously filled resonator with losses

The object of the present investigation is the determination of the field intensity and the resonance frequency of the quasi-TE_0,1,1-mode (with rotational symmetry) in an inhomogeneously filled resonator. The resonator is filled with a number of concentrically located tubes consisting of different materials, which may be lossy. The top and the bottom walls of the resonator are electrically perfectly conducting; at the side wall an impedance boundary condition is employed. Numerical results for a case of practical interest in magnetic resonance experiments are obtained.     

Global stabilization of the oscillating eccentric rotor

The oscillating eccentric rotor has been widely studied to model resonance capture phenomena occurring in dual-spin spacecraft and rotating machinery. This phenomenon arises during spin-up as a resonance condition is encountered. We consider the related problem of rotor despin. Specifically, we determine nonlinear feedback control laws that not only despin the rotor but also bring its translational motion to rest. These globally asymptotically stabilizing control laws are derived using partial feedback linearization and integrator backstepping schemes. For the case in which the oscillating eccentric rotor is excited by a translational sinusoidal forcing function, the control law is shown to attenuate the amplitude of the translational oscillation.Research supported in part by the Air Force Office of Scientific Research under Grant F49620-92-J-0127.Research supported in part by NSF Grant MSS-9309165.     

Parametric resonance of a single-degree-of-freedom system with double bilinear hysteresis

A simple pendulum with a hinge of double bilinear hysteretic restraining moment-rotation characteristic under parametric excitation is studied. In contrast with a linear system with viscous damping, a double bilinear hysteretic system leads, in general, to finite response under parametric resonance. The response curves and the conditions under which unbounded response results are given. Further, it is shown that unlike the bilinear hysteretic system, a double bilinear hysteretic system may be shock excited into parametric resonance even when the exciting frequency is outside the parametrically resonant frequency range.     

Reduction of subsynchronous vibrations in a single-disk rotor using an active magnetic damper

Rotor instabilities in turbomachinery often manifest themselves as a re-excitation of the first rotor critical speed resulting in lateral rotor vibrations at a frequency below the rotor operating frequency. Considerable work exists in the literature involving the analysis of destabilizing mechanisms and passive solutions for reducing subsynchronous vibrations. The authors propose here a novel active control solution utilizing active magnetic bearing (AMB) technology in conjunction with conventional support bearings. The AMB is utilized as an active magnetic damper (AMD) at rotor locations inboard of conventional support bearings. Presented here are initial proof-of-concept experimental results using an AMD for vibration control of subsynchronous rotor vibrations in a high-speed single-disk laboratory rotor. The study shows that subsynchronous vibrations are reducible with an AMD and up to a 93% reduction in the amplitude of subsynchronous vibrations is demonstrated. The study also shows that the AMD can significantly increase synchronous vibration response (up to 218% in one case) by increasing system stiffness and pushing a critical speed closer to an operating speed. The overall results from this work demonstrate that reduction in subsynchronous response is feasible and that full rotor dynamic analysis and design is critical for the successful application of this approach.     

Lagrange-Maxwell equation and magnetic saturation parametric resonance of generator set

Lagrange-Maxwell's equation is extended firstly.With the theory of elec- tromechanical analytical dynamics,the magnetic complement energy in air gap of gener- ator is acquired.The torsional vibration differential equations with periodic coefficients of rotor shafting of generator which is in the state of magnetic saturation are established. It is shown that the magnetic saturation may cause double frequency electromagnetic moment.By means of the averaging method,the first approximate solution and corre- sponding solution of the primary parametric resonance is obtained.The characteristics and laws of the primary parametric resonance excited by the electromagnetism are ana- lyzed and some of new phenomena are revealed.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.     

Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution

This work investigates the nonlinear longitudinal forced vibrations of a bar with hysteretic dynamics typical of rocks and man-made geomaterials. The material properties are described by a constitutive relation that includes energy dissipation effects consistently with hysteretic dynamics. The harmonic balance method and a perturbation technique that uses the material's modulus defect as perturbation parameter are employed to solve the equation of motion. The spatial dependence of the modulus defect, which is disregarded in earlier solutions of this problem, is duly accounted for. According to this model, the resonance frequency shift is proportional to the amplitude of the excitation, while the inverse of the quality factor increases as the square root of the latter. Further, the spatial variation of the modulus defect is shown to affect the modulation of the fundamental component along the bar. The nonlinear spectral components are of odd order, with amplitude proportional to the maximum value of the modulus defect and to the square of the excitation's amplitude, and decreases non-monotonically with increasing harmonic order. The motivation for this work is twofold. Analytical models may improve our understanding of the dynamics of hysteretic materials in general, and of the mutual interaction of material defects in particular. Secondly, this work establishes a bench-mark result on a mathematically simple hysteretic system against which alternative mathematical approaches, possibly concerning material with complex constitutive relations, could be tested.     

Nonlinear normal modes and primary resonance of horizontally supported Jeffcott rotor

The nonlinear normal modes of a horizontally supported Jeffcott rotor are investigated. In contrast with a vertically supported rotor, there are localized and nonlocalized nonlinear normal modes because the linear natural frequencies in the horizontal and vertical directions are slightly different due to both gravity and the nonlinearity of restoring force. Reflecting such nonlinear normal modes, the frequency response curves are characterized in the primary resonance. In the case where the eccentricity is small, i.e., the response amplitude is small, the whirling motion is localized in the horizontal or vertical direction in the resonance. On the other hand, when the eccentricity is large, two kinds of whirling motion, which are localized in the vertical direction and nonlocalized in any direction, coexist simultaneously in a region of rotational speed. Experiments are conducted, and the theoretically predicted nonlinear responses based on localized and nonlocalized nonlinear normal modes are observed.     

磁悬浮挠性转子系统模型复合辨识方法

针对磁悬浮挠性转子的纯有限元分析模型精度低的问题,提出一种磁悬浮挠性转子系统模型复合辨识方法。该方法首先采用有限元方法把磁悬浮挠性转子划分为多个Timoshenko梁单元,建立磁悬浮挠性转子系统模型,并对其挠性临界频率、阻尼、刚度和振型等关键参数进行分析,进而得到磁悬浮挠性转子的等效降阶数学模型;然后采用鲁棒自适应方法分析磁悬浮挠性转子系统的动态特性;最后,采用变LEVY方法从动态特性分析数据中对磁悬浮挠性转子进行系统辨识,校正有限元分析得到的降阶数学模型。实验结果表明,本方法可以得到磁悬浮挠性转子较为准确的系统模型。     

Investigation of the whirling motion and rub/impact occurrence in a 16-pole rotor active magnetic bearings system with constant stiffness

Nonlinear Dynamics - In this paper, the weight of the rotor is considered in a 16-pole rotor active magnetic bearing system with constant stiffness. The equations of motion are derived to show the...     

Dynamics of a balanced rotor under the action of an elastic force with a hysteresis characteristic

Recently, the constructions of rotors rotating on passive-type magnetic bearings designed with the use of high-temperature superconductors have been developed [1, 2]. One still open problem in the stability analysis of rotors on such bearings is the problem on the influence of the bearing rigidity characteristic on the system dynamics. The stability of a steady-state motion of a rotor with arbitrary eccentricity is studied in [3]. The present paper deals with the dynamics of a steady-state motion of a balanced rotor rotated by an infinite-power motor; in this case, the elastic force acting on the rotor has a hysteresis characteristic [4]. The equations of motion are described by a nonautonomous system of differential equations which is reduced with prescribed accuracy to an autonomous system, and the latter is then analyzed [5].     

Resonance behaviour of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field

The oscillatory flow of a viscoelastic fluid in a circular pipe under the influence of a transversal magnetic field is studied. Exact solutions for the axial velocity and flow rate are presented. The velocity enhancement and the resonance behaviour are analysed both numerically and asymptotically in the case of small pipe radii. Approximations for the resonance frequencies and the achievable velocity enhancements are derived. Copyright © 2005 John Wiley & Sons, Ltd.     

Theoretical rotor dynamic analysis of two-pole induction motors regarding excitation due to static rotor eccentricity

The paper shows a theoretical rotor dynamic analysis of two-pole induction motors regarding excitation due to static rotor eccentricity, which causes an oscillating magnetic force in the air gap of the machine. The hereby caused rotor vibrations are mathematically described, showing the orbit movements of the rotor centre, the shaft journals and the bearing housings. The influence of the rotor speed as well as influence of the direction of the magnetic force on the vibration behavior is investigated, based on a numerical example. On the one hand the aim of the paper is to derive the mathematical coherences—based on a simplified rotor dynamic model—between rotor dynamics, electromagnetic and the oil film characteristics of the sleeve bearings for two-pole induction machines, considering electromagnetic excitation due to a static rotor eccentricity. On the other hand the aim is to derive a method for calculating the maximum height of the shaft vibrations—as a worst case—caused by a static rotor eccentricity. Therefore the paper shall prepare the basis for implementing this method into more detailed numerical programs, e.g., finite element programs.     

直升机旋翼/机体动稳定性研究进展

首先对直升机旋翼/机体动不稳定性问题的种类进行了简要概述,包括旋翼挥舞/变距、变距/摆振、挥舞/摆振和挥舞/摆振/变距耦合等孤立旋翼动不稳定性问题,以及直升机地面共振和空中共振等旋翼/机体耦合动不稳定性问题,然后分别从气动力与结构的高精度数值模型、动稳定性的计算分析方法和实验模型测试3 个方面详细介绍了直升机旋翼/机体动不稳定性问题的研究现状,并着重讨论了直升机旋翼/机体动稳定性分析技术最近的主要研究方向:耦合CFD(computational fluid dynamics)/CSD(computational structuraldynamics) 的直升机旋翼气弹动稳定性分析、复合材料旋翼动稳定性分析及其材料不确定性影响、带减摆器的旋翼/机体动稳定性分析和先进直升机构型的旋翼/机体动稳定性分析,最后对直升机旋翼/机体动稳定性研究的发展趋势进行了展望.     

非线性碰摩力对碰摩转子分叉与混沌行为的影响

研究了具有非线性碰摩力的转子局部碰摩的分叉与混沌运动,利用计算机仿真对某发动机转子的碰摩故障进行了数值模拟,讨论了转子系统参数的变化对转子混池运动状态的影响,并与碰摩实验结果进行了比较,发现了具有非线性碰摩力的转子局部碰摩转子系统的各种多周期运动和混沌运动及其演变过程。     

Effect of elasticity on the dynamics of a superconducting rotor rotating in a magnetic field

A spherical shape of the outer surface of rotors of some types of noncontact gyroscopes gives rise to conditions, where the force field ensures the stability of the center of mass relative to the base and has an insignificant effect on the angular motion of the rotor. However, there are some effects (for instance, the Barnett—London effect), which lead to emergence of moments of mechanical forces even for spherical bodies. The effect of rotor elasticity on the motion of a superconducting deformable spherical solid body in a magnetic field is studies. It is shown that the moment of mechanical forces acting on the body in the magnetic field is proportional in the first approximation to the angular velocity squared. The effect of this moment on the dynamics of angular motion of the rotor is studied.     

Hysteretic deteriorating model for quasi-brittle materials based on micromechanical damage approach

A damage model, which is based on the stochastic modeling of the microstructures, is developed for the quasi-brittle materials subjected to repeated loading. According to this model, the overall response of the material is represented with a series of micro-elements joined in parallel. A combined model is proposed for the micro-element considering the fracture as well as the hysteretic energy dissipation. To account for the progressive failure, the random fracture strains are assigned to the micro-elements. Therefore the overall parallel bundle is considered as a stationary random field. Then by averaging the microscopic random field, the overall loading, unloading and reloading curves are derived analytically. Two hysteretic rules are derived from the proposed model, and the overall hysteretic deteriorating behaviors could be well reproduced. To demonstrate the validity of the present model, the numerical results are shown against the stochastic simulated curves as well as the experimental data. The present model provides an alternative approach for the efficient modeling of the hysteretic deteriorating behaviors for the quasi-brittle materials.