NON-LINEAR DYNAMIC MODEL OF AN INEXTENSIBLE ROTATING FLEXIBLE ARM SUPPORTED ON A FLEXIBLE BASE

A mathematical model for a flexible arm undergoing large planar flexural deformations, continuously rotating under the effect of a hub torque and supported by a flexible base is developed. The position of a typical material point along the span of the arm is described using the inertial reference frame via a transformation matrix from the body co-ordinate system, which is attached to the flexible root of the rotating arm. The condition of inextensibility is employed to relate the axial and transverse deflections of the material point, within the beam body co-ordinate system. The position and velocity vectors obtained, after imposing the inextensibility conditions, are used in the kinetic energy expression while the exact curvature is used in the potential energy. Lagrangian dynamics in conjunction with the assumed modes method is utilized to derive, directly, the non-linear equivalent temporal equations of motion. The resulting non-linear model, which is composed of four coupled non-linear ordinary differential equations, is discussed, simulated and the results of this simulation are presented. The effects of the base flexibility are explored by comparing the resulting simulation results, for various flexibility coefficients, with previously published works of the authors. Moreover, the numerical results show that the base flexibility has a very important effect on the stability of rotating flexible arms that should be accounted for when simulating such systems.     

DYNAMIC RESPONSE OF A ROTATING FLEXIBLE ARM CARRYING A MOVING MASS

A clamped-free flexible arm rotating in a horizontal plane and carrying a moving mass is studied in this paper. The arm is modelled by the Euler-Bernoulli beam theory in which rotatory inertia and shear deformation effects are ignored. The assumed mode method in conjunction with Hamilton's principle is used to derive the equation of motion of the system which takes into account the effect of centrifugal stiffening due to the rotation of the beam. The eigenfunctions of a cantilever beam which satisfy the prescribed geometric boundary conditions are used as basis functions in the assumed mode method. The equation of motion is expressed in non-dimensional matrix form. Pre-designed transformed cosine profiles are used as trajectory inputs for the hub angle and the moving mass. The equation of motion is solved numerically using the fourth order Runge-Kutta method. Graphical results are presented to show the influence of centrifugal stiffening effect, moving mass values, mass travelling time, hub angle and mass trajectory profile on the deflection of the beam.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

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     

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.     

辐射对称金字塔型剪纸的力学行为

通过在弹性薄板上引入切口,构建了多边形辐射对称金字塔型剪纸结构.利用伽辽金法求解的悬臂梁形变公式和悬臂梁组合的方法,创建了用于解释形变过程的"梁模型",得到n个模块的正N边形金字塔结构的弹性系数与结构参数的关系公式,并求出弹性系数线性阈值的表达式,解释了该结构产生平面外扭曲的原因.利用推导的"梁模型"公式,并通过有限元仿真和实验的方法,系统研究了辐射对称金字塔型剪纸结构的力学响应特征,验证模型的准确性,并用于已有报道的石墨烯剪纸结构的力学特征分析.这项工作系统解释了竖直拉伸的金字塔型剪纸结构的力学响应.     

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.     

非惯性系下考虑剪切变形的柔性梁的动力学建模

研究非惯性坐标系下考虑剪切变形的柔性梁的动力学建模. 首先借鉴Euler-Bernoulli梁的几何非线性变形模式,考虑了Timoshenko梁弯曲以及剪切变形产生的几何非线性效应对纵向、横向变形位移的影响,在考虑两个方向的变形耦合项后,利用有限元法对柔性梁进行了离散,采用Lagrange方程建立了柔性梁的动力学模型,首次建立了包含变形二次耦合量的Timoshenko梁的动力学方程. 关键词： 非惯性坐标系 剪切变形 柔性梁 动力学建模     

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.     

VIBRATION OF A COMPLIANT TOWER IN THREE-DIMENSIONS

The three-dimensional motion of an offshore compliant tower using both rigid and flexible beam models is studied in this paper. The tower is modelled as a beam supported by a torsional spring at the base with a point mass at the free end. The torsional spring constant is the same in all directions. When the beam is considered rigid, the two-degree-of-freedom model is employed. The two degrees constitute the two angular degrees of spherical co-ordinates, and the resulting equations are coupled and non-linear. When the beam is considered as elastic, three displacements are obtained as functions of the axial co-ordinate and time; again with coupled and non-linear equations of motion. The free and the forced responses due to deterministic loads are presented. The free responses of the rigid and elastic beams show rotating elliptical paths when viewed from above. The rate at which the path rotates depends on the initial conditions. When a harmonic transverse loading is applied in one direction, the displacement in that direction shows subharmonic resonance of order 1/2 and 1/3 while the displacement in the perpendicular direction is affected minimally. Next, in addition to the harmonic load in one direction, a transverse load is applied in the perpendicular direction. The transverse load varies exponentially with depth but is constant with time. It is found that the transverse load affects the transverse displacements in the perpendicular direction minimally.     

Multi-objective optimal design and experimental validation of tracking control of a rotating flexible beam

This paper investigates the tracking control problem for a rotating flexible beam. A proportional-integral-derivative control is designed to meet multiple objectives including overshoot, peak time, tracking error of the rigid movement of the rotating base, and the defectional angle of the flexible beam. A multi-objective optimization problem is then formulated for the control design and is solved with the cell mapping method. Numerical simulations and experiments are carried out to demonstrate the effect of the different control gains in the Pareto set, and to study the difference between a linear and a nonlinear model of the flexible beam.     

基于旋转场曲率的二维剪切梁单元建模

对二维剪切梁单元进行研究,利用平面旋转场理论推导了精确曲率模型.采用几何精确梁理论构建了剪切梁单元弹性力矩阵.通过绝对节点坐标方法建立了系统的非线性动力学方程,提出基于旋转场曲率的二维剪切梁单元,并分别引入经典二维剪切梁单元和基于位移场曲率的二维剪切梁单元进行比较研究.首先,静力学分析证明了所提模型的正确性;其次,特征频率分析验证了模型可与理论解符合,收敛精度高,并且能准确地预测单元固有频率对应的振型;最后,在非线性动力学问题上,通过与ANSYS结果对比分析,证明了该模型可有效处理柔性大变形问题,并且与经典二维剪切梁单元相比具有缓解剪切闭锁的优势.因此,本文提出的基于旋转场曲率的二维剪切梁单元在处理几何非线性问题中具有较大的应用潜力.     

激光干涉任意转角测量信号的获得及误差补偿技术

采用信号处理的方法对激光干涉测角系统的特性进行线性化处理,需要产生两路正交信号。采用空间互相垂直的两套干涉系统获得信号难免会使系统复杂、调整困难,因此提出了一种插值处理方法,可以利用测量得到的一路信号获得线性输出信号。输出信号的非线性误差较处理前大大减小了,但是无法满足高精度测量的要求。还提出了针对误差产生的原因进行的误差补偿技术。实验结果表明,提出的方法可以实现任意转角的高精度测量,光程差的测量误差小于±0 2μm,对应的转角测量误差为0 63″。     

A time-of-scan laser triangulation technique for distance measurements

This paper reports the development of an electro-optic device for relative distance measurement. The time-of-scan triangulation technique has been used as measurement principle and a rotating mirror employed as beam deflection system. A calibration technique is needed to calculate the geometrical parameters of the system. The device has an accuracy of 100 μm, a working distance of 20 cm and a range of 10 mm. The accuracy obtained depends on the instability of the rotation speed of the mechanical scanner that affects the measurement of the scanning time.     

Design and analysis of a laser micrometer for undulator gap measurement applications

In this paper we report the design of a laser micrometer for undulator gap measurement studies. In this scheme, the parallel light emerging from the rotating polygon mirror passes through an F-theta lens makes a vertical line on the image plane. When an object intercepts the line, the interrupted and the uninterrupted light is detected in a DSO through a detector. The beam spot size and the linear velocity of the beam spot on the scan line are the two important parameters of the laser micrometer. The beam spot size measurements show that it is constant along the scan line for a number of RPMs. It measures a constant value at ±3 mm to the effective focal length of the F-theta lens. The measurements are verified for two types of objects i.e. solid object and a circular hole and also for an electromagnet undulator. The electromagnet undulator is measured with an accuracy of 3–5 µm.     

Dynamic analysis of horizontal axis wind turbine by thin-walled beam theory

A mixed flexible-rigid multi-body mathematical model is applied to predict the dynamic performance of a wind turbine system. Since the tower and rotor are both flexible thin-walled structures, a consistent expression for their deformations is applied, which employs a successive series of transformations to locate any point on the blade and tower relative to an inertial coordinate system. The kinetic and potential energy terms of each flexible body and rigid body are derived for use in the Lagrange approach to formulate the wind turbine system's governing equation. The mode shapes are then obtained from the free vibration solution, while the distributions of dynamic stress and displacement of the tower and rotor are computed from the forced vibration response analysis. Using this dynamic model, the influence of the tower's stiffness on the blade tip deformation is studied. From the analysis, it is evident that the proposed model not only inherits the simplicity of the traditional 1-D beam element, but also able to provide detailed information about the tower and rotor response due to the incorporation of the flexible thin-walled beam theory.     

高准确度光束偏转装置的设计与分析

基于矢量折射定理推导了光束经过正交双棱镜后的偏转表达式.给出了装置的主要设计参量;用数值模拟的方法分析了主要误差项,求出了总误差和实际准确度指标.结果表明,光束在水平张角及垂直张角500 μrad内可实现准确度优于0.8 μrad的偏转,偏离准确度主要受随机误差影响;反映到棱镜转角上的总误差为12.72 arcsec,引起的光束偏离误差为0.365 μrad,大于系统读数分辨率0.0387 μrad,且小于光束偏离准确度指标0.8μrad.     

Vibrational response of a moving suspension-slider loading system exciting a rotating flexible disk

To investigate the vibrational response of the magnetic read/write head in hard disk drives this paper models a rotating flexible disk excited by a moving suspension-slider system which is considered to be a mass-dashpot-spring loading system, with the initial unstressed transverse runout integrated into the rotating disk dynamic model. The slider motion on the disk surface is driven by the suspension rotating at a constant speed. By subtracting the steady-state deflection component from the instantaneous deflection response of the rotating disk system, the relative vibration transverse deflection of the slider caused by the motion of the suspension-slider loading system is obtained. The effects of the slider initial and final positions, speed of movement, the disk rotational speed, and the disk mode of the initial transverse runout on the maximum amplitude of the relative vibration deflection are analyzed.     

A worst balance analysis of flexible rotating shafts

A worst balance analysis is formulated for a rotating shaft. The analysis provides the maximum deflection of the shaft resulting from the worst combination of errors in placing balance weights along the axis of the rotor. The analysis is formulated in general as a non-linear programming problem and in particular cases as a linear programming problem. Both formulations require computational implementation for multi-mass rotors. Curves illustrating the trade-off between the error in balance weight placement and the corresponding worst disturbance are presented.     

Calibration of centre-of-mass energies at LEP 2for a precise measurement of the W boson mass

The determination of the centre-of-mass energies for all LEP 2 running is presented. Accurate knowledge of these energies is of primary importance to set the absolute energy scale for the measurement of the W boson mass. The beam energy between 80 and 104 GeV is derived from continuous measurements of the magnetic bending field by 16 NMR probes situated in a number of the LEP dipoles. The relationship between the fields measured by the probes and the beam energy is defined in the NMR model, which is calibrated against precise measurements of the average beam energy between 41 and 61 GeV made using the resonant depolarisation technique. The validity of the NMR model is verified by three independent methods: the flux-loop, which is sensitive to the bending field of all the dipoles of LEP; the spectrometer, which determines the energy through measurements of the deflection of the beam in a magnet of known integrated field; and an analysis of the variation of the synchrotron tune with the total RF voltage. To obtain the centre-of-mass energies, corrections are then applied to account for sources of bending field external to the dipoles, and variations in the local beam energy at each interaction point. The relative error on the centre-of-mass energy determination for the majority of LEP 2 running is 1.2 x 10^-4, which is sufficiently precise so as not to introduce a dominant uncertainty on the W mass measurement. Received: 4 August 2004, Revised: 15 December 2004, Published online: 21 January 2005