直线运动柔性梁非线性动力学--主参数共振与内共振联合激励

针对基础直线运动柔性梁，基于Kane方程建立了相应的非线性动力学方程。采用多尺度法并结合笛卡尔坐标变换，导出了系统受前两阶模态间3：1内共振及第二阶模态主参激共振时的非线性调制方程组．数值求解了该方程组的定常解及相应的稳定性问题。研究表明，系统的平凡、单模态、双模态稳态解共存，超临界及亚临界叉形分岔只发生在单模态状态下，相反，鞍结分岔及Hopf分岔只在双模态状态下产生，一些稳定的极限环随参数变化经一系列倍周期分岔后导致运动的突然跳跃。     

A geodesic beam finite element for use in the-static analysis of reinforced circular cylindrical shells

The development of a geodesic beam finite element for use with a specific circular cylindrical shell finite element in the analysis of reinforced circular cylinders is described. The basic strain and curvature change equations are given and, from these, three versions of the geodesic beam element are developed. Two of the beams have nodal degress-of-freedom identical with the shell element. They differ in the treatment of the terms relating to rotation about the principal normal. The first version ignores this parameter but, under certain circumstances, the stiffness matrix contains terms which can contribute to the strain energy under arbitrary rigid body movement of the beam. This deficiency is removed by applying an aspect of Koiter's theory which is used to transform the curvature change equations. The introduction of additional rotational degress-of-freedom, at the end nodes of the beam element, produces a variation which is capable of accurately representing and transmitting in-surface bending effects to an adjoining beam element. Numerical evaluation successfully compares finite element solutions to basic problems for straight, circular and helical beams with theoretical strain energy solutions. Finally the beam is used in conjunction with a shell element to analyse an infinitely long circular cylinder, reinforced with equispaced rings, subjected to internal pressure.     

外激励力作用下的轴向运动梁非线性振动的联合共振

研究轴向运动梁在外激励力作用下非线性振动的联合共振问题.利用哈密顿原理建立横向振动的轴向运动梁的振动微分方程,采用分离变量法分离时间变量和空间变量并利用Galerkin方法离散运动方程.采用IHB法进行非线性振动求解,分析在内共振条件且外激励作用下的联合共振问题,对周期解进行稳定性的判定.典型算例获得了不同外激励力振幅时系统非线性振动的复杂频幅响应曲线.     

直线运动柔性梁非线性动力学--组合参数共振与内共振联合激励

针对基础直线运动柔性梁，基于Kane方程建立了相应的非线性动力学方程。采用多尺度法并结合笛卡尔坐标变换，导出了系统受前两阶模态间3：1内共振及其组合参激共振时的非线性调制方程组，数值求解了该方程组的定常解及相应的稳定性问题。研究表明，系统的平凡响应与双模态非平凡响应共存，由内共振所产生的非平凡响应皆为不稳定的鞍点，平凡及非平凡解分支都存在Hopf分岔现象，一些稳定的极限环随参数变化最终经倍周期分岔后产生混沌运动。     

Diffraction Efficiency and Angular Selectivity of Volume Phase Holograms Recorded in Photorefractive Materials

The recording and replay of volume phase gratings in photorefractive crystals is investigated for both transmission and reflection geometries. Differential equations are derived and solved for a range of parameters including the length of the crystal, the magnitude, spatial distribution and phase angle of the refractive index modulation, the beam ratio at recording, and the angular range at replay. The recording process is assumed to reach a steady-state limit before replay with a weak probe beam. Solution of the repaly equations is mainly by numerical integration, although analytic solutions are derived for special cases. It is found that in certain cases the diffraction efficiency can be greatly increased by replaying the hologram at an angle different from the recording angle.     

Acousto-optic interaction of a Gaussian laser beam with an ultrasonic wave of cylindrical symmetry

We present experimental studies of the interaction between a narrow Gaussian laser beam and a standing cylindrical ultrasonic wave. As a theoretical approach, a Fourier-optics-based successive diffraction model is used. Depending on the ratio of the Gaussian laser beam diameter to the first nodal diameter of the cylindrical ultrasound, light refraction or diffraction is observed. We experimentally investigate the time-averaged light intensity as well as the modulation of light in the far field of light refraction-diffraction by a cylindrical ultrasound. It is revealed that significant focusing appears if the phase front of the incident light is curved. The focusing effects of the acousto-optic system depend on the width of the laser beam and curvature of the phase front. Finally, possible applications are discussed.     

Nonlinear equations for a thin beam

Summary. Finite oscillations of a thin beam are analyzed, taking into account large deflections and rotations with small strains. It is shown that, depending on the assumptions, problems for an elastic and viscoelastic beam lead to solving hyperbolic or parabolic nonlinear partial differential equations. As an example, a cantilever beam is considered. The solutions for nonlinear vibrations are obtained by means of the Galerkin method, and numerical results are presented graphically for four material models.     

柔性机械臂的非线性振动分析

建立了由两个柔性杆和一个柔性饺链组成的柔性机械臂的动力学方程,利用非线性振动的平均法进行了摄动分析,求得了系统在主共振和多重共振下的定常解,发现系统在第二阶主共振和1∶2 内共振的情况下有内共振阈值和双拟饱和现象存在,增大低阶模态阻尼是抑制内共振、防止产生大幅低频振动的有效手段。     

曲梁弯扭屈曲分析

本文基于有限变形理论导出了薄壁曲梁位移和应变的一般表达式。从探讨薄壁曲梁弯扭屈曲的变形模态出发,考虑其弯矩作用平面内变形的影响,利用变分原理导出了曲梁稳定分析的基本微分方程。稳定分析表明,即使绕弱轴弯曲,曲梁也可能发生弯扭屈曲。对于曲率甚小的情况,本文把曲梁看作为具有初曲率的直梁,较精确地分析了初曲率对直梁弯扭屈曲临界弯矩的影响。此外,本文给出的闭合解与Timoshenko,Vlasov及Chai Hong Yoo等人的结果进行了比较,指出了其间差异的根源所在。     

三缸发动机式增程器扭转振动模糊PID主动抑制

针对增程器系统在发动机转矩波动激励下的多谐波扭转振动问题,提出一种模糊PID控制算法,利用发电机输出扭矩主动抑制扭转振动。建立8自由度轴系强迫振动模型,然后以角加速度以及转速为控制目标,采用模糊PID控制器主动抑制扭转振动。以均平方角加速度为扭转振动的评价指标。分析稳态和瞬态工况下扭转振动主动抑制效果。仿真结果表明,稳态工况下相比只以转速为控制目标的PID控制方式,均平方角加速度降低68.3%;相比传统的PID控制方法,均平方角加速度降低29.0%。所提出的方法可以有效抑制增程器轴系的扭转振动。     

Forced vibration of a beam system with autoparametric coupling effects

Small non-linear interactions of the type referred to as autoparametric may have a considerable effect on the pattern of forced vibration response of certain categories of structures exhibiting low damping characteristics. Under conditions of internal resonance these may bring about a complete neutralisation of the normal forced vibration resonance curve with a transfer of energy to an indirectly excited mode. The paper examines non-linear interactions between a pair of beam modes in forced vibration of a simple structure. Steady state solutions are obtained for the non-linear system equations. Measurements of vibratory acceleration and strain on a laboratory model are shown to give good confirmation of the analytical results and show clearly the substantial effects of internal resonance.     

Exact Solutions and Mode Transition for Out-of-Plane Vibrations of Nonuniform Beams with Variable Curvature

The two coupled governing differential equations for the out-of-plane vibrations of non-uniform beams with variable curvature are derived via the Hamilton's principle. These equations are expressed in terms of flexural and torsional displacements simultaneously. In this study, the analytical method is proposed. Firstly, two physical parameters are introduced to simplify the analysis. One derives the explicit relations between the flexural and the torsional displacements which can also be used to reduce the difficulty in experimental measurements. Based on the relation, the two governing characteristic differential equations with variable coefficients can be uncoupled into a sixth-order ordinary differential equation in terms of the flexural displacement only. When the material and geometric properties of the beam are in arbitrary polynomial forms, the exact solutions with regard to the outof- plane vibrations of non-uniform beams with variable curvature can be obtained by the recurrence formula. In addition, the mode transition mechanism is revealed and the influence of several parameters on the vibration of the non-uniform beam with variable curvature is explored.     

Locking-free curved beam element based on curvature

The formulation of a curved beam element with 3 nodes for curvature to eliminate the shear/membrane locking phenomenon is presented. The element is based on curvature so that it may represent the bending energy fully, and the shear/membrane strain energy is incorporated into the formulation by the equilibrium equations. To deal with general boundary conditions, a transformation matrix between nodal curvature and nodal displacement vector is introduced. Several examples are presented in order to verify the element formulation and its analytical capability. The solutions obtained reveal that the element describes the curved beam behaviour quite correctly and efficiently, showing no locking phenomena, and that it is also applicable to the analysis of both thin and thick curved beams.     

热环境中功能梯度圆柱壳内共振非线性模态

研究热环境中无限长功能梯度薄壁圆柱壳内共振非线性模态,给出系统发生内共振条件;采用多尺度法建立系统具有内共振的非线性调谐方程;讨论梯度指数、温度变化及振动能量对系统非线性模态频响特性影响。研究表明,随调谐参数的变化,系统非线性模态会发生分岔;调谐参数分岔值取决于梯度指数、温度及振动能量。     

定轴转动与基础激励下梁的非线性动力学

采用Kane方程，建立了含耦合的几何及惯性非线性项的定轴转动与轴向基础激励联合作用下柔性梁的非线性动力学控制方程组，该方程组不仅包含二次及三次非线性项，而且体现了参数激励与外激励的联合作用。运用多尺度法，研究了匀转速，顺臂安装下悬臂梁的一阶模态主参激共振与外激励1/2次亚谐共振同时作用时梁的一阶近似稳态响应。结果表明，梁的一阶模态幅频特性将受到转速，旋转半径和激励幅值等参数变化的显著影响。     

Localised formulations for thick “sandwich” laminated and composite structures

A “sandwich” panel under axial compressive loading is studied over its post-buckling range. Alternative formulations were developed to cover solutions other than the deflection shape given by the classical Rayleigh-Ritz analysis. General elasticity principles are used to derive the total potential energy function in terms of the displacement field. Given a constant modal amplitude displacement field, which is introduced in the total potential energy function, the results are discussed having as background models studied earlier. Techniques for the analysis of localized solutions are presented with reference to the study of the beam on nonlinear elastic foundation. The concepts of localization are introduced by using dynamical analogy and the double scale perturbation method and applied to the sandwich panel, to derive the governing amplitude modulation equations. The resulting formulations are discussed using some examples, and by solving analytical and numerically the amplitude differential equations.     

A non-discretized global method for free vibration of generally laminated fibre-reinforced pre-twisted cantilever plates

 The paper presents a computational method for characterizing the resonant frequency properties of cantilever pre-twisted plates composed of fibre-reinforced laminated composites. It aims to simulate a laminated turbomachinery blade or a fan blade with a relatively small aspect ratio for which the conventional beam model fails to provide accurate solutions. The strain and kinetic energy components are expressed in area integration equations with respect to the plate mid-surface. An energy functional is defined and a governing eigenvalue equation is derived. There is no domain mesh generation in the numerical analysis thus reducing accuracy loss due to discretization. New numerical solutions are presented and the effects of angle of pre-twist, aspect ratio, and symmetric and antisymmetric lamination for two different composite laminates are analyzed in detail. The limit of linear twisting curvature omitted in previous analysis is strictly adopted here and the consequences of exceeding the limit are discussed.     

Low-velocity impact response for laminated stepped beam

A numerical method is presented to evaluate the response of a laminated stepped beam under a transverse low velocity impact. The impactor and the laminated beam are considered as a single system and thus the impact force becomes the internal force of the system. The energies associated with each span of the beam and the impactor mass are derived in the form of a higher order shear deformation theory and Sun's modified contact law. These are substituted in the Lagrangian equation of motion to arrive at the governing equations of the system. The equations are uncoupled by principal coordinate transformations and then solved numerically. The solutions can be obtained for arbitrary boundary conditions. Natural frequency parameters are compared with the available literature results wherever possible and very good agreement is achieved. New case studies are discussed in detail to demonstrate the efficiency of the method.     

Effects of rf voltage modulation on particle motion

The effects of rf voltage modulation on synchrotron motion were studied experimentally. The experimental data revealed the resonance islands generated by the rf voltage modulation. With electron cooling, beam particles were observed to damp to the basins of these resonance islands or attractors, which were observed to rotate about the origin of the phase space at a half of the modulation frequency. The measured amplitude of the attractors as a function of the modulation frequency agreed very well with the theoretical prediction. The Poincaré maps in the resonance rotating frame were obtained from the experimental data and compared with tori of the Hamiltonian flow. Based on our theoretical formulation, slow beam extraction using rf voltage modulation and a bent crystal are also studied.