旋转柔性悬臂梁动力学的Bezier插值离散方法研究

在旋转柔性梁变形场描述中,引入Bezier插值离散方法.首先构建旋转运动悬臂梁物理模型,接着采用第二类Lagrange动力学方程和Bezier插值离散方法,在计入柔性梁横向弯曲变形引起的纵向缩短的情况下,推导了旋转柔性梁的刚柔耦合动力学方程,并编制旋转柔性梁的动力学仿真软件,然后通过仿真算例对系统的动力学问题进行研究.最后将仿真结果与有限元法、假设模态法进行分析比较,验证了提出的Bezier插值离散方法的正确性,并得出Bezier插值离散法的计算效率较高;计算精度符合工程实际需要,高速时计算精度大于假设模态法;Bezier插值离散方法在处理大柔性问题时比假设模态法合理.因此在多体系统动力学领域具有优良性能和应用价值的Bezier插值离散方法将具有推广价值.     

基于绝对节点坐标法的大变形柔性梁几种动力学模型研究

对在平面内大范围转动的大变形柔性梁动力学进行了研究, 基于绝对节点坐标法建立了一种新的大变形柔性梁的非线性动力学模型. 该动力学模型中考虑了柔性梁的轴向拉伸变形和横向弯曲变形, 利用Green-Lagrangian应变张量计算柔性梁的轴向应变及应变能, 利用曲率的精确表达式计算柔性梁的横向弯曲变形能. 运用拉格朗日恒等式给出了柔性梁横向弯曲变形能新的表达式, 该变形能表达式更加简洁, 通过新的变形能表达式得到了新的弹性力模型, 由此得到的动力学方程可以精确地描述柔性梁的几何大变形问题. 通过与高次耦合模型以及ANSYS中BEAM188非线性梁单元模型的比较, 验证了本模型在计算大变形时的正确性以及高次耦合模型在处理大变形问题时的不足. 进一步研究发现, 新的广义弹性力模型可以适当地简化, 给出了两种简化模型, 根据不同模型的计算效率以及计算精度的比较确定了不同模型的适用范围.     

大口径、高精度二维快速控制平面反射镜系统设计北大核心CSCD

为解决光电跟踪仪跟踪精度测量过程中光束大范围指向的模拟问题,设计了一种大口径、高精度二维快速控制反射镜(fast steering mirror,FSM)。采用微晶材料设计了长、短轴分别为230 mm和160 mm的椭圆形平面反射镜,面形精度优于λ/30。采用音圈电机驱动,通过柔性支撑铰链设计及DSP嵌入式控制系统,运动行程达到±30 mrad,运动控制精度达到5μrad,运动控制线性度优于±0.20%,角分辨率优于1μrad。通过软件控制,实现对入射光束圆形轨迹运动、直线轨迹运动、随机运动等形式的运动模拟。最后,对设计指标进行实际测试,可以满足跟踪精度光束动态模拟的测试需求。     

AN ESTIMATOR-BASED SLIDING-MODE CONTROL FOR MANEUVERING A FLEXIBLE SPACECRAFT

An estimator-based sliding-mode controller (ESMC) is discussed for a linear stochastic system with a known disturbance and is utilized in a flexible spacecraft for the reduction of residual vibration while allowing natural deflection during operation. By converting the tracking problem into a regulator problem, the ESMC minimizes the expected value of the guadratic objective function composed of errors which always remain in the intersection of sliding hypersurfaces. For the numerical evaluation to take place in a flexible with a flexible spacecraft, a large slewing maneuver strategy is devised, with a tracking model for the nominal trajectory. A start-coast-stop strategy for an economical maneuver is employed in conjunction with the input shaping technique. The performance and efficacy of the proposed control scheme are illustrated with a comparison of different maneuvering strategies.     

On the finite settling time and residual vibration control of flexible structures

Rapid end-point positioning of a structure with minimal residual vibration under a single bounded control input is considered. The structure between the actuator and the end point is assumed to be flexible, and is the main cause of positioning error from residual vibration. A standard solution to this problem is offered by the well known bang-bang control. Implementation of such a bang-bang control strategy leads to decreased response time but introduces unwanted vibrations from repeated overshooting of the switching contour, caused by errors in the implementation of the discontinuous control law. An appropriately shaped input function based on minimum energy and bounded control for such end-point positioning systems totally eliminates the residual vibration while making the response time small. The proposed forcing input has two discontinuities at most and therefore does not suffer from the undesirable intermediate discontinuities present in the bang-bang control. This novel control force compares favorably with the bang-bang solution with respect to response time. It also compares favorably with the ramped sinusoidal input force proposed by Meckl and Seering [1] with respect to residual vibration. They shaped the input function by eliminating the resonant frequencies where as the resonant frequencies are retained in the function proposed. As an illustration, the newly shaped forcing input is applied to a rotating thin flexible beam to suppress residual vibration.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

基于转镜同步跟踪系统的研究

为了研究高速飞行弹丸的运动姿态问题,提出转镜同步跟踪技术。在高速CCD相机主光轴方向放置一面转镜,将弹道线位置上飞行弹丸的运动姿态反射到高速CCD相机内实现同步跟踪。设计了基于高速CCD相机视场中点的转镜跟踪系统,建立了弹丸和转镜的运动模型,并利用MATLAB软件得到了其随时间变化的曲线,分析了相机和转镜空间位置对成像质量的影响。针对参数H=200 m,V=100 m/s,对系统存在的误差进行了分析,结果表明该系统可以实现对高速弹丸的同步跟踪。     

旋转内接悬臂梁的刚柔耦合动力学特性分析

对固结于旋转刚环上内接柔性梁的刚柔耦合动力学特性进行了研究. 在精确描述柔性梁非线性变形基础上, 利用Hamilton变分原理和假设模态法, 在计入柔性梁由于横向变形而引起的轴向变形二阶耦合量的条件下, 推导出一次近似耦合模型. 忽略柔性梁纵向变形的影响,给出一次近似简化模型,引入无量纲变量, 对简化模型做无量纲化处理. 首先分析在非惯性系下内接悬臂梁的动力学响应, 并与外接悬臂梁进行比较; 其次研究内接悬臂梁的稳定性;最后分析内接悬臂梁失稳临界转速的收敛性. 研究发现, 与外接悬臂梁存在动力刚化效应不同,内接悬臂梁存在着动力柔化效应; 给出了内接悬臂梁无条件稳定的临界径长比以及失稳的临界转速的计算方法; 若第一阶固有频率随转速增大而减小,则该内接悬臂梁处于有条件稳定; 随着模态截断数的增加,内接悬臂梁失稳的临界转速减小且有收敛值. 关键词： 内接悬臂梁 一次近似简化模型 动力柔化 临界转速     

Dynamic analysis of an axially moving finite-length beam with intermediate spring supports

This paper presents the analysis for the transverse vibration of an axially moving finite-length beam inside which two points are supported by rotating rollers. In this study, the rollers are modeled as uniaxial springs in the transverse direction. Hamilton?s principle is applied to derive the equations of motion and boundary conditions of the system. The equations of motion include translational and rotational motions as well as flexible motion. These equations are discretized using Galerkin?s method, and then the dynamic characteristics of a flexible beam with spring supports are studied by solving an eigenvalue problem. The veering phenomenon of natural frequency loci and mode exchanges are investigated for different positions of the springs and various values of the spring stiffness. In addition, the mode localization is also analyzed using the peak amplitude ratio. It is found in this study that the first mode is localized in one of the beam spans if an appropriate value of the spring constant is selected. Furthermore, it is shown that mode localization can be used to reduce the vibration transferred from one span to the other span while a beam moves axially.     

A FINITE ELEMENT FORMULATION FOR COUPLING RIGID AND FLEXIBLE BODY DYNAMICS OF ROTATING BEAMS

The work presented in this paper is based on an existing comprehensive formulation for rotating flexible systems. In the existing formulation the flexible degrees of freedom (d.o.f.) are represented by an analytically computed modal basis and the coupling matrices between the rigid- and the flexible-body d.o.f. are developed based on the analytical modal representation of the flexible d.o.f. In this paper, the existing formulation is generalized for rotating beams by representing the flexible d.o.f. either as physical d.o.f. of a finite element formulation or as a set of retained and internal d.o.f. of a Craig-Bampton formulation. The coupling matrices between the rigid-body rotation and the flexible d.o.f. are developed accordingly. The non-linear effects from the work done by the centrifugal forces are included in the formulation. Finite element shape functions of a beam element in a three-dimensional space and finite element shape functions for solid elements are employed for deriving the coupling terms between the rigid-body d.o.f. and the physical d.o.f. An additional transformation is required and performed when the right-body d.o.f. are coupled with the internal and the retained d.o.f. of a Craig-Bampton formulation. The coupled system of equations is solved in the time domain by combining the Newmark method for time integration and the Newton-Raphson method for solving the non-linear system of equations within each time step. Analyses are performed for a flexible rotating beam in order to validate the development. An analytical solution is compared with the new formulations that represent the rotating beam flexibility with the physical d.o.f. of beam or solid elements. The analytical solution is also compared to the formulation that represents the flexible d.o.f. in terms of retained and internal d.o.f. of a Craig-Bampton formulation. Very good correlation between the analytical and numerical results is observed.     

The role of non-linear theories in transient dynamic analysis of flexible structures

Transient dynamic analysis of flexible structures undergoing large motions is considered. For rotating structures, it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non-linear (at least second-order) beam theory. Use of a first-order (linearized) linear beam theory results in a spurious loss of bending stiffness. For a rotating plane beam, a set of linear partial differential equations of motion—that includes all inertia effects (Coriolis, centrifugal, acceleration of revolution) and coupling between extensional and flexural deformations—is derived from the fully non-linear beam theory by consistent linearization. The analysis is subsequently extended to the more general case of a plate, accomodating shear deformation, and undergoing a general three-dimensional rotating motion. The discretization process of the resulting linear equations of motion for the beam and the plate is also discussed.     

Calculation of the problem on a rotating flexible thread

A solution to the problem of a flexible thread rotating around a horizontal axis is presented. Dependences calculated in elementary functions with graphic applications are obtained for determining the parameters of the rotating thread, such as the thread-shape outline, the largest deviation from the axis, the values of curvature angles, and the thread tension, among others.     

Dynamic analysis of very flexible beams

The dynamic analysis of flexible beams with large deformations is difficult and few studies have been performed. In this paper, the vibration analysis of several very flexible beams with large deflections using the finite element approach is studied. The examples were a cantilever beam and rotating flexible robot arms. The results were compared with the results available in the published literature. Several successful checks on the finite element results were performed to ensure the accuracy of the solutions. Due to the geometrical nonlinearity, several static equilibrium shapes can exist for large deflections of a cantilever beam for a given load. Nonlinear dynamic finite element analysis was implemented to investigate the stability of these shapes.     

Vibration suppression control of smart piezoelectric rotating truss structure by parallel neuro-fuzzy control with genetic algorithm tuning

The main goal of this paper is to develop a novel approach for vibration control on a piezoelectric rotating truss structure. This study will analyze the dynamics and control of a flexible structure system with multiple degrees of freedom, represented in this research as a clamped–free–free–free truss type plate rotated by motors. The controller has two separate feedback loops for tracking and damping, and the vibration suppression controller is independent of position tracking control. In addition to stabilizing the actual system, the proposed proportional–derivative (PD) control, based on genetic algorithm (GA) to seek the primary optimal control gain, must supplement a fuzzy control law to ensure a stable nonlinear system. This is done by using an intelligent fuzzy controller based on adaptive neuro-fuzzy inference system (ANFIS) with GA tuning to increase the efficiency of fuzzy control. The PD controller, in its assisting role, easily stabilized the linear system. The fuzzy controller rule base was then constructed based on PD performance-related knowledge. Experimental validation for such a structure demonstrates the effectiveness of the proposed controller. The broad range of problems discussed in this research will be found useful in civil, mechanical, and aerospace engineering, for flexible structures with multiple degree-of-freedom motion.     

OPTIMAL VIBRATION ESTIMATION OF A NON-LINEAR FLEXIBLE BEAM MOUNTED ON A ROTATING COMPLIANT HUB

To eliminate the need for sensor placement on rotating flexible beams such as turbine blades, helicopter rotors and like applications, a new approach has been developed based on the linear quadratic estimator (LQE) technique for estimating the vibration of any point on the span of a rotating flexible beam mounted on a compliant hub (plant) in the presence of process and measurements noise. A non-linear model of the plant is utilized in this study to mimic the actual plant behavior. The corresponding plant dynamics of the LQE are in the form of a reduced order linear model constructed from the eigenvalues and eigenfuctions of a finite element dynamic model of the plant formulated in the state space. A virtual hub deflection (that mimics the actual measurement of the vertical hub deflection needed by the estimation process) is generated by the non-linear model of the plant. The LQE reconstructs the states of the plant, including transverse deflection of the beam at any point, from the measurements of the vertical deflection of the hub, assuming that it is the most accessible state for measurement. Estimated beam tip deflection obtained by the proposed technique is then compared to the tip deflection generated by the non-linear model and the results show good agreement.     

Surface excitations of a bose-einstein condensate

Surface modes in a Bose-Einstein condensate of sodium atoms have been studied. We observed excitations of standing and rotating quadrupolar and hexadecapolar modes. The modes were excited with high spatial and temporal resolution using the optical dipole force of a rapidly scanning laser beam. This novel technique is very flexible and should be useful for the study of rotating Bose-Einstein condensates and vortices.     

Theoretical model of ball joint squeak

The theoretical model for the ball joint squeak problem is proposed. The ball joint is modeled as a sphere attached to a rotating flexible beam and it is in contact with a semi-spherical rigid socket. The relative rotation and vibration between the ball and socket produces friction stresses causing the dynamic instability of the system. The squeak propensity of the ball joint system is numerically investigated through the eigenvalue sensitivity analysis at the sliding equilibrium. Stability analysis shows that the bending modes of the beam can produce the dynamic instability under the negative friction-velocity slope. It is also highlighted that the squeak propensity under the poor friction characteristics can be controlled by the system parameters such as the sliding speed, the contact stiffness, the radius of the ball, the axial load and the tilting angle.     

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.     

Linear parameter varying control design for rotating systems supported by journal bearings

Linear parameter varying (LPV) control is a model-based control technique that takes into account time-varying parameters of the plant. In the case of rotating systems supported by lubricated bearings, the dynamic characteristics of the bearings change in time as a function of the rotating speed. Hence, LPV control can tackle the problem of run-up and run-down operational conditions when dynamic characteristics of the rotating system change significantly in time due to the bearings and high vibration levels occur. In this work, the LPV control design for a flexible shaft supported by plain journal bearings is presented. The model used in the LPV control design is updated from unbalance response experimental results and dynamic coefficients for the entire range of rotating speeds are obtained by numerical optimization. Experimental implementation of the designed LPV control resulted in strong reduction of vibration amplitudes when crossing the critical speed, without affecting system behavior in sub- or super-critical speeds.