Active control and experiment study of a flexible hub-beam system

The first-order approximation coupling （FOAC） model was proposed recently for dynamics and control of flexible hub-beam systems. This model may deal with system dynamics for both low and high rotation speed, while the classical zeroth-order approximation coupling （ZOAC） model is only available for low rotation speed. This paper assumes the FOAC model to present experimental study of active positioning control of a flexible hub-beam system. Linearization and nonlinear control strategies are both considered. An experiment system based on a DSP TMS320F2812 board is introduced. The difference between linearization and nonlinear control strategies are studied both numerically and experimentally. Simulation and experimental results indicate that, linearized controller can make the system reach an expected position with suppressed vibration of flexible beam, but the time taken to position is longer than expected, whereas nonlinear controller works well with precise positioning, suppression of vibration and time control.     

中心刚体-柔性梁系统的最优跟踪控制

对考虑阻尼影响的中心刚体-柔性梁系统的动力特性和主动控制进行研究. 研究 中考虑了3种动力学模型：一次近似耦合模型、一次近似简化模型和线性化模型. 一次近 似模型中同时考虑了柔性梁的轴向变形和横向变形. 若在一次近似耦合模型中忽略轴向变 形的影响,则可得出一次近似简化模型. 线性化模型是对一次近似简化模型的线性化处理. 另外研究中考虑了3种阻尼因素：结构阻尼、风阻、中心刚体轴承处的阻尼. 控制设计采 用最优跟踪控制方法. 给出了从物理测量中提取模态坐标的滤波器方法. 研究结果显 示,一次近似简化模型能够有效地对系统的动力学行为进行描述;阻尼对系统的动力学特 性有着重要影响;当系统大范围运动为低速时,模态滤波器能够较好地提取出控制律所需 的模态坐标,最优跟踪控制方法能够使得系统跟踪所期望的运动轨迹,并且柔性梁的弹性 振动可得到抑制.     

旋转运动柔性梁的假设模态方法研究

采用假设模态法对旋转运动柔性梁的动力特性进行研究，给出简化的控制模型. 首先采用Hamilton原理和假设模态离散化方法，在计入柔性梁由于横向变形而引起的轴向变形的二阶耦合量的条件下，推导出基于柔性梁变形位移场一阶完备的一次近似耦合模型，然后对该模型进行简化，忽略柔性梁纵向变形的影响，给出一次近似简化模型，最后将采用假设模态离散化方法的结果与采用有限元离散化方法的结果进行了对比研究. 研究中考虑了两种情况：非惯性系下的动力特性研究和系统大范围运动为未知的动力特性研究. 研究结果显示，当系统大范围运动为高速时，在假设模态离散化方法中应增加模态数目，较少的模态数目将导致较大误差. 一次近似简化模型能够较好地反映出系统的动力学行为，可用于主动控制设计的研究.     

Dynamic analysis of a flexible slider–crank mechanism with clearance

In this paper we are interested in the study of the dynamic behavior of a planar flexible slider–crank mechanism with clearance. Simulation and experimental tests were carried out for this goal. For the simulation tests, we have built the model under the software ADAMS. We used a contact model based on the so called Impact-function. An experimental set-up was designed and built to achieve some experimental validations. The presented results show that, in the presence of clearance, the mechanism responses were greatly influenced. The motion is characterized by the occurrence of three phases: a free motion, a continuous contact motion and an impact motion. In this paper, both the case of the mechanism with rigid link and the case with flexible link were studied. It is shown that in the presence of clearance, the coupler flexibility has a role of suspension for the mechanism.     

Stability and oscillations in a slow-fast flexible joint system with transformation delay

Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a linkrotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slowfast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.     

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation. The project supported by the National Natural Science Foundation of China (10472066, 10372057). The English text was polished by Yunming Chen.     

中心刚体-楔形梁-质点刚柔耦合系统动力学分析

研究了中心刚体-楔形梁-质点系统的固有特性和动力学响应.楔形梁为Euler-Bernoulli梁,高度和宽度都沿着梁的长度方向线性变化.利用广义Hamilton原理和一阶近似耦合模型得到了含有楔形梁完全耦合且时变的微分/代数控制方程.考虑了离心刚化效应,利用有限元得到了系统完全耦合的有限维方程.忽略轴向与横向位移的相互作用,得到了系统的一致质量、阻尼和刚度矩阵.最后对楔形梁和等截面梁在有无端部质点的四种结构进行仿真,结果表明存在显著差异,重点比较了同等条件下楔形梁与等截面梁的差异指数,说明均匀梁和楔形梁的截面细微的差别能够导致系统频率和动力学响应的明显差别.指出实际系统中使用楔形梁模型能够得到更为精确的仿真结果.     

Dynamic behaviour of a metamaterial system with negative mass and modulus

Different models of metamaterials have been developed to generate negative mass and/or negative modulus. The resulting mass and modulus in existing works, however, cannot be independently controlled. The current study presents a new representative cell of elastic metamaterials in an effort to provide a comprehensive model for generating negative mass and/or negative modulus. The current model consists of a series of properly arranged rigid bodies and linear springs. By introducing both translational and rotational motions in the representative cell, negative mass and negative modulus can be obtained in a controlled manner. The mechanisms and conditions under which negative mass and/or negative modulus can be achieved are studied in detail. Numerical examples indicate that by varying the design of the representative cell, different properties of the material system can be reliably generated, i.e., double negative (mass and modulus) or single negative (mass or modulus). The dynamic behaviour of the developed material system under different loading frequencies is evaluated and the longitudinal elastic wave propagation in such metamaterials is studied.     

Dymamics of a rotating flexible manipulator with tip load

The dynamics of a flexible manipulator is investigated in this paper. From the point of view of dynamic blance, the motion equations of a rotating beam with tip load are established by using Hamilton's principle. By taking into account the effects of dynamic stiffening and dynamic softening, the stability of the system is proved by employing Lyapunov's approach. Furthermore, the method of power series is proposed to find the exact solution of the eigenvalue problem. The effects of rotating speed and tip load on the vibration behavior of the flexible manipulator are shown in numerical results. Supported by National Natural Science Foundation. of China     

Dynamic analysis of a tethered satellite system with a moving mass

This paper presents a dynamic analysis of a tethered satellite system with a moving mass. A dynamic model with four degrees of freedom, i.e., a two-piece dumbbell model, is established for tethered satellites conveying a mass between them along the tether length. This model includes two satellites and a moving mass, treated as particles in a single orbital plane, which are connected by massless, straight tethers. The equations of motion are derived by using Lagrange’s equations. From the equations of motion, the dynamic response of the system when the moving mass travels along the tether connecting the two satellites is computed and analyzed. We investigate the global tendencies of the libration angle difference (between the two sections of tether) with respect to the changes in the system parameters, such as the initial libration angle, size (i.e. mass) of the moving mass, velocity of the moving mass, and tether length. We also present an elliptic orbit case and show that the libration angles and their difference increase as orbital eccentricity increases. Finally, our results show that a one-piece dumbbell model is qualitatively valid for studying the system under certain conditions, such as when the initial libration angles, moving mass velocity, and moving mass size are small, the tether length is large, and the mass ratio of the two satellites is large.     

Dynamic analysis of a disc brake under frictional and thermomechanical internal loading

The study of the vibratory response of a mechanical system as complex as a disc brake needs to consider the complexity of the problem induced by the coupling of tribological, thermomechanical and dynamical effects. Experimental consideration are discussed here for two set ups at the full scale of the disc brake and at a local scale focussed on the third body interface. A numerical model with thermomechanical and dynamical couplings is then presented, followed by a substantial discussion.     

Modal test and analysis of cantilever beam with tip mass

The phenomenon of dynamic stiffening is a research field of general interest for flexible multi-body systems. In fact, there are not only dynamic stiffening but also dynamic softening phenomenon in the flexible multi-body systems. In this paper, a non-linear dynamic model and its linearization characteristic equations of a cantilever beam with tip mass in the centrifugal field are established by adopting the general Hamilton Variational Principle. Then, the problems of the dynamic stiffening and the dynamic softening are studied by using numerical simulations. Meanwhile, the modal test is carried out on our centrifuge. The numerical results show that the system stiffness will be strengthened when the centrifugal tension force acts on the beam (i.e. the dynamic stiffening). However, the system stiffness will be weakened when the centrifugal compression force acts on the beam (i.e. the dynamic softening). Furthermore, the equilibrium position of the system will lose its stability when the inertial force reaches a critical value. Through theoretical analysis, we find that this phenomenon comes from the effect of dynamic softening resulting from the centrifugal compression force. Our test results verify the above conclusions and confirm that both dynamic stiffening and softening phenomena exist in flexible multi-body systems. The project supported by the National Natural Science Foundation of China (19972002) and the Doctoral Programme from The State Education Commission China (20010001011)     

带裂纹旋转柔性梁系统的刚柔耦合动力学研究

对具有刚柔耦合效应的带裂纹旋转柔性梁进行建模和动力学特性分析研究。采用晶格弹簧离散模型,利用无质量弹簧模拟梁上裂纹,通过考虑梁变形的二阶耦合项建立了带裂纹旋转柔性梁系统的一次近似耦合动力学控制方程。数值计算结果表明,裂纹的存在会使旋转柔性梁的固有频率降低,并且随着梁转速的增大,这种降低效应呈减弱趋势;值得注意的是,裂纹梁的固有频率与裂纹处的弯矩具有正相关关系。此外,裂纹的存在不仅会使转速变化阶段梁的末端位移响应增大,还会对转速稳定后梁的末端振荡产生显著的影响。     

Dynamic properties of a body with discontinual mass variation

In this paper the procedure for the dynamic analysis of body separation is introduced. Based on the general laws of classical dynamics, the method for obtaining the velocity and the angular velocity of the remainder body during separation is developed. Due to the discontinual mass variation, the jump-like change of the velocity and the angular velocity of the body is evident. Various types of motion of the separated body are considered. Depending on the type of motion of the separated body the dynamic properties of the remainder body are obtained. As a special case the in-plane motion of the body before and after separation is considered. The theoretical considerations are applied for the separation analysis of a rotor (a shaft-disc system). The transient motion of the body after separation is investigated. To prove the correctness of the procedure suggested in the paper, the case when the mass and the moment of inertia of the separated body are infinitesimal is analyzed. The obtained differential equations are the same as those previously obtained.     

Systematic design and development of a flexible wheel for low mass lunar rover

Low mass compact rovers provide cost effective means to explore extra-terrestrial terrains. Use of flexible wheels in such applications where the wheel size is restricted, improves traction at reduced slip and sinkage. Design of a flexible wheel for a given mission is a challenging task requiring consideration of stiffness of rim and spokes, stress induced in the wheel, chassis movement during wheel rotation and the operating mode of the wheel. Also, accurate mathematical models are required to save design and development time and reduce the number of prototypes for selection. It is observed that most of the research papers deal with performance testing of flexible wheels and information on analytical formulation is scarce. Therefore, in the present work, a methodology has been formulated to systematically design a flexible wheel for a low mass lunar rover. The prototype performance is tested and compared with analytical estimates and reasons for difference are investigated. Paper contains details of design criteria, mathematical modelling, realisation of wheel prototype, test fixture and analysis test comparison. Authors believe that this work provides a useful aid to the designer to systematically design flexible wheels for low mass lunar rovers.     

旋转悬臂梁的刚柔耦合动力学建模与频率分析

对固结于转动刚体上外接柔性梁的刚柔耦合动力学建模和频率特性进行了研究,在精确描述柔性梁非线性变形的基础上,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,推导出考虑"动力刚化"项的一次近似耦合模型。首先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,分析梁固有频率对模态截断数的依赖性;其次研究在一次近似简化模型和零次近似简化模型下,调谐角速度与共振现象的关系;最后分析一次近似耦合模型的动力特性。研究发现,为保证计算的精度,模态截断数应随无量纲角速度的增大而增加,合理的模态截断数具有收敛值;一次近似简化模型下悬臂梁横向弯曲振动不存在共振调谐角速度,一次耦合模型下柔性梁并没有出现屈曲失稳现象。现有典型文献的相关结论是值得商榷的。     

用具有负定矩阵的梯度系统构造稳定的变质量力学系统

随着科学技术的发展,对喷气飞机、火箭等变质量系统动力学的研究显得越来越重要, 并且总是希望变质量系统的解是稳定的或渐近稳定的. 而通用的研究稳定性的Lyapunov直接法有很大难度, 因为直接从微分方程出发构造Lyapunov函数往往很难实现. 本文给出一种研究稳定性的间接方法, 即梯度系统方法. 该方法不但能揭示动力学系统的内在结构, 而且有助于探索系统的稳定性、渐进性和分岔等动力学行为. 梯度系统的函数V通常取为Lyapunov函数, 因此梯度系统比较适合用Lyapunov函数来研究. 列写出变质量完整力学系统的运动方程,在系统非奇异情形下,求得所有广义加速度. 提出一类具有负定矩阵的梯度系统, 并研究该梯度系统解的稳定性. 把这类梯度系统和变质量力学系统有机结合,给出变质量力学系统的解可以是稳定的或渐近稳定的条件, 进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的变质量力学系统. 通过具体例子,研究了变质量系统的单自由度运动,在怎样的质量变化规律、微粒分离速度和加力下,其解是稳定的或渐近稳定的. 本文的构造方法也适合其它类型的动力学系统.      

考虑空气耦合的薄膜结构动力分析

假定空气是可压缩、无粘性和有势的,推导了薄膜结构的非线性刚度矩阵和空气与薄膜结构耦合作用的气动力表达式。根据薄膜结构的受力特点,建立了薄膜结构与空气耦合系统的运动方程。采用有限元方法对无限域的空气进行简化,建立了空气与薄膜结构的耦合有限元模型,模拟了空气对薄膜结构的附加质量和声致阻尼影响。考虑薄膜结构的几何非线性,对结构在空气中的自由振动和受迫简谐振动进行了系统分析,同时探讨了流体环境对薄膜结构动力响应的影响因素。     

An improved HSFR method for natural vibration analysis of an immersed cylinder pile with a tip mass

Immersed cylinder piles are usually modelled as immersed cantilever cylinder columns carrying a tip mass and rotary moment of inertia. In this paper, the equations of motion of an immersed cylinder pile along transversal modes of vibration are developed. Compressibility of water and structural damping are included in the formulation. Natural frequencies of the immersed pile are obtained from the developed equations using harmonic sweep frequency response analyses. The proposed method is applied to numerical examples, and the results obtained are shown satisfactory when compared to other numerical solutions in the literature, or to finite element solutions and experimental data.