应变能中耦合项对旋转悬臂梁振动频率的影响

大范围运动悬臂梁的动力学建模问题对动力学特性分析及控制系统设计具有极其重要的作用. 当前研究多采用一次近似模型,其忽略了由轴向和横向变形所产生的应变能中的耦合项,然而这些项对动力学特性会产生影响. 通过讨论应变能的选取方式,计入了应变能中的耦合项;利用哈密尔顿原理建立结构的耦合振动模型;再借助瑞利—里兹法,以无大范围运动时的振型函数作为基本解组,得到了结构振动广义特征方程并求解. 通过数值算例对比分析,指出考虑应变能耦合项得到的频率与不考虑应变能耦合项得到的频率存在明显差别.      

基于初应力法的作大范围运动柔性梁动力学建模理论研究

研究了初应力法的作大范围运动柔性梁的建模理论.根据连续介质理论,考虑应变-位移中的非线性项,用一致质量有限元法对柔性梁进行离散,基于Jourdain速度变分原理导出定轴转动下大范围运动为自由的柔性梁刚-柔耦合动力学方程.从其刚柔耦合动力学方程出发,考虑在大范围运动已知情况下的结构动力学方程.通过引入准静态概念,把其结构动力学方程转化为准静态方程.对纵向和横向变形节点坐标进行坐标分离,解出与纵向变形相关的准静态方程,得到准静态时的纵向应力表达式,从而获得附加刚度项.并对此非惯性系下作大范围运动柔性梁的结构动力学方程进行数值仿真,对零次近似模型、一次近似模型、初应力法动力学模型的仿真结果进行分析,揭示三种模型的动力学性质的差异.     

作大范围运动矩形板的动力学建模理论研究

研究了初应力法的作大范围运动矩形板的建模理论。根据连续介质理论,考虑应变-位移中的非线性项,用一致质量有限元法对柔性板进行离散,基于Jourdain速度变分原理导出定轴转动下大范围运动为自由的柔性板刚-柔耦合动力学方程。从其刚柔耦合动力学方程出发,考虑在大范围运动已知情况下的结构动力学方程。通过引入准静态概念,把其结构动力学方程转化为准静态方程。对纵向和横向变形节点坐标进行坐标分离,解出与纵向变形相关的准静态方程,得到准静态时的纵向应力表达式,从而获得附加刚度项;并对此非惯性系下作大范围运动柔性板的结构动力学方程进行数值仿真,验证了采用初应力法柔性板的动力学建模方法来计算经历大范围运动的不规则柔性板的动力学响应是可行的,体现了初应力法对柔性板建模的优越性。     

水下环向双周期加肋圆柱壳体的自由振动

以浸没于水中的弹性环向双周期加肋薄圆柱壳为研究对象，考虑介质与结构振动的耦合效应，研究流固耦合系统的自由振动。基于Kennard薄壳理论、Helmholtz方程以及壳壁外表面的运动协调条件，并借助Dirac-δ函数引进肋骨对壳体的作用，从而建立耦合系统的运动方程。通过富氏积分变换、引进算子，并利用算子的周期性，得到系统的频率方程。采用沿实波数轴搜索求根的方法，重点计算了水下无限长环向双周期加肋柱壳的自由传播波频率系数，并进一步研究了流场和肋参数对壳体固有频率的影响。     

磁耦合悬臂梁双稳动力学分析与实验

研究了一个自由端附加小磁铁的悬臂梁在磁力作用下的双稳态动力学行为．首先,利用Hamilton原理和Euler-Bernoulli梁的基本方程建立了系统在非零平衡点处做微幅振动的动力学方程．其次,利用多尺度法对建立的模型进行理论分析,得到悬臂梁在非零平衡点处振动的幅频方程和位移解,并对解进行了稳定性分析．最后,通过建立实验装置,得到悬臂梁不同运动形式下的参数平面分类和悬臂梁在非零平衡点处振动的幅频关系,通过观察系统在非零平衡点处振动的理论预测,实验结果验证了非零平衡点处振动的理论分析的正确性．对照理论、实验和数值结果得到：在不同的外激励幅值和频率作用下,悬臂梁有三种不同的运动形式：在非零平衡点处的微幅振动;大范围往返运动;在两个非零平衡点之间的无规律运动．     

磁场中轴向运动导电梁弯曲自由振动分析

研究磁场环境下轴向运动导电梁的弯曲自由振动．首先给出系统的动能、势能以及电磁力表达式,进而应用哈密顿变分原理,推得磁场中轴向运动导电梁的磁弹性弯曲振动方程．在位移函数设定基础上,应用伽辽金积分法分别推出三种不同边界约束条件下,轴向运动梁的磁弹性自由振动微分方程和频率方程,得到固有频率表达式．通过算例,得到了弹性梁固有振动频率的变化规律曲线图,分析了轴向运动速度、磁感应强度和边界条件对固有振动频率和临界值的影响．     

纤维曲线铺放的复合材料层合板的自由振动分析

假设曲线纤维的方向角沿板的长度方向按照线性规律变化,导出了纤维曲线铺设时的参考路径,将参考路径沿板的宽度方向平移可得一种曲线纤维增强复合材料单层板,当这种纤维曲线铺设的单层板对称铺放时即可得相应的曲线纤维增强复合材料层合板．基于弹性薄板的小挠度理论,建立了曲线纤维增强复合材料层合板自由振动问题的基本方程,采用微分求积法进行数值求解,得到了层合板的自振频率及相应的振型．与已有文献计算结果的比较,验证了本文计算结果的正确性．通过数值算例分析了微分求积法求解本问题时的收敛性,研究了纤维铺放路径和边界条件的不同对曲线纤维增强复合材料层合板频率及振型的影响,研究结果可为该种结构的设计提供一定的参考．     

一类非均质楔形直杆的振动分析

本文从一类变系数二阶微分方程的一个通解入手,分析了一类非均质楔形直杆的纵向自由振动,得到了这类直杆在一种边界条件下的频率方程和振型函数.     

CFRP索斜拉桥的索-浅拱模型及面内自由振动分析

本文针对斜拉桥的受力特点,基于索和浅拱的经典动力学运动方程,结合拉索与浅拱之间的耦合边界条件,并且考虑两者的几何非线性,建立了斜拉桥的多索-浅拱面内自由振动模型。将浅拱分段处理,结合索、浅拱连接处的动态平衡条件,应用分离变量法,建立多索-浅拱模型的面内自由振动理论。以双索浅拱模型为例,求解其特征值问题。同时,建立了相应的有限元模型,有限元计算结果与本文理论分析吻合良好。最后针对CFRP索斜拉桥的关键参数,基于本文的索-浅拱理论,对面内自由振动进一步研究。研究表明：浅拱的矢高在一定范围内变化,仅对某一阶频率产生影响,而其他各阶频率几乎没影响;CFRP拉索能显著改善索-浅拱组合结构的基本动力学特性。     

大跨度斜拉桥的动力特性及其颤振临界风速的计算

基于有限单元法理论,确定了斜拉桥合乎其构造特点的窨计算模型,建立了斜拉桥的自由振动的三维运动方程,对斜拉桥的动力特性进行了求解和分析,在此基础上,建立了在气动荷载作用下斜拉桥的三维运动方程,提出了用于分析斜拉桥颤振临界状态的子空间复特征值方法,推导出计算颤动临界风速的理论公式,并编写了用于计算斜拉桥颤振频率的多振型参与双参数自动搜索的计算机程序。     

Towards linear modal analysis for an L-shaped beam: Equations of motion

We consider an L-shaped beam structure and derive all the equations of motion considering also the rotary inertia terms. We show that the equations are decoupled in two motions, namely the in-plane bending and out-of-plane bending with torsion. In neglecting the rotary inertia terms the torsional equation for the secondary beam is fully decoupled from the other equations for out-of-plane motion. A numerical modal analysis was undertaken for two models of the L-shaped beam, considering two different orientations of the secondary beam, and it was shown that the mode shapes can be grouped into these two motions: in-plane bending and out-of-plane motion. We compared the theoretical natural frequencies of the secondary beam in torsion with finite element results which showed some disagreement, and also it was shown that the torsional mode shapes of the secondary beam are coupled with the other out-of-plane motions. These findings confirm that it is necessary to take rotary inertia terms into account for out-of-plane bending. This work is essential in order to perform accurate linear modal analysis on the L-shaped beam structure.     

耦合变形对大范围运动柔性梁动力学建模的影响

柔性梁在作大范围空间运动时,产生弯曲和扭转变形,这些变形的相互耦合形成了梁在纵向以及横向位移的二次耦合变量。本文考虑了变形产生的几何非线性效应对运动柔性梁的影响,在其三个方向的变形中均考虑了二次耦合变量,利用弹性旋转矩阵建立了准确的几何非线性变形方程,通过Lagrange方程导出系统的动力学方程。仿真结果表明,在大范围运动情况下,仅在纵向变形中计及了变形二次耦合量的一次动力学模型,与考虑了完全几何非线性变形的模型具有一定的差异。     

柔性多体系统产生动力刚化原因的研究

传统的柔性多体系统建模理论由于对柔性体的变形及其与大范围运动产生惯性力之间的耦合处理得过于简单,所以在分析存在高速大范围运动柔性多体系统的动力学性态时会得到完全错误的结论。本文将通过对作大范围运动弹性薄板的讨论来揭示产生这种错误的及探讨对传统性多体系统建模理论作出改进的对策     

Analysis of Coupled Vibration Characteristics of Wind Turbine Blade Based on Green's Functions

This paper presents the analysis of dynamic characteristics of horizontal axis wind turbine blade, where the mode coupling among axial extension, flap vibration(out-of-plane bending), lead/lag vibration(in-plane bending) and torsion is emphasized. By using the Bernoulli-Euler beam to describe the slender blade which is mounted on rigid hub and subjected to unsteady aerodynamic force, the governing equation and characteristic equation of the coupled vibration of the blade are obtained. Due to the combined influences of mode coupling, centrifugal effect, and the non-uniform distribution of mass and stiffness, the explicit solution of characteristic equation is impossible to obtain. An equivalent transformation based on Green's functions is taken for the characteristic equation, and then a system of integrodifferential equations is derived. The numerical difference methods are adopted to solve the integrodifferential equations to get natural frequencies and mode shapes. The influences of mode coupling, centrifugal effect, and rotational speed on natural frequencies and mode shapes are analyzed. Results show that:(1) the influence of bending-torsion coupling on natural frequency is tiny;(2) rotation has dramatic influence on bending frequency but little influence on torsion frequency;(3) the influence of bending-bending coupling on dynamic characteristics is notable at high rotational speed;(4) the effect of rotational speed on bending mode is tiny.     

弹性连接旋转柔性梁动力学分析

采用Chebyshev 谱方法对考虑根部连接弹性的平面内旋转柔性梁动力学特性进行研究. 基于Gauss-Lobatto 节点与Chebyshev 多项式方法对柔性梁变形场进行离散,通过投影矩阵法施加固定及弹性连接边界条件. 利用Chebyshev 谱方法获得了系统固有频率和模态振型数值解,通过与有限元方法及加权残余法的比较,验证了方法的有效性. 分析了弹性连接刚度、角速度比率、系统径长比及梁的长细比等参数对系统固有频率及模态振型的影响. 研究发现:由于系统弯曲模态、拉伸模态的频率随各参数的变化规律不一致,将出现频率转向与振型转换现象;随着弹性连接刚度、角速度比率及系统径长比的增大,低阶弯曲模态频率增大并超过高阶拉伸模态频率,随着梁的长细比的增大,低阶拉伸模态频率增大并超过高阶弯曲模态频率.      

Refined models of elastic beams undergoing large in-plane motions: Theory and experiment

Accurate mechanical models of elastic beams undergoing large in-plane motions are discussed theoretically and experimentally. Employing the geometrically exact theory of rods with appropriate kinematic assumptions and asymptotic arguments, two approximate models are obtained—a relaxed model and its constrained version—that describe extensional and bending motions and neglect shear deformations. These models are shown to be suitable to predict, via an asymptotic approach, closed-form nonlinear motions of beams with general boundary conditions and, in particular, with boundary conditions that longitudinally constrain the motions. On the other hand, for axially unrestrained or weakly restrained beams, an inextensible and unshearable model is presented that describes bending motions only. The perturbations about the reference configuration up to third order are consistently derived for all beam models. Closed-form solutions of the responses to primary-resonance excitations are obtained via an asymptotic treatment of the governing equations of motion for two different beam configurations; namely, hinged–hinged (axially restrained) and simply supported (axially unrestrained) beams. In particular, considering the present theory and the existing theories, variations of the frequency–response curves with the beam slenderness or the relative boundary mass are investigated for the lowest modes. The fidelity of the proposed nonlinear models is ascertained comparing the theoretically obtained frequency–response curves of the first mode with those experimentally obtained.     

Vibration analysis of piezoelectric laminated slightly curved beams using distributed transfer function method

Piezoelectric laminated slightly curved beams (PLSCB) is currently one of the most popular actuators used in smart structure applications due to the fact that these actuators are small, lightweight, quick response and relatively high force output. This paper presents an analytical model of PLSCB, which includes the computation of natural frequencies, mode shapes and transfer function formulation using the distributed transfer function method (DTFM). By setting the radius of curvature of the proposed model to infinity, a piezoelectric laminated straight beams (PLSB) model can be obtained. The DTFM is applied and extended to carry out the transfer function formulation of the PLSCB and PLSB models. This method will be used to solve for the natural frequencies, mode shapes and transfer functions of the PLSCB and PLSB models in exact and closed form solution without using truncated series of particular comparison or admissible functions. The natural frequencies of the cantilevered PLSCB and PLSB are calculated by the DTFM and the Rayleigh–Ritz method. The analysis indicates that the stretching–bending coupling due to curvature has a considerable effect on the frequency parameters. Increasing the radius of curvature of the PLSCB has its largest effect on the natural frequencies. But the inhomogeneity of the boundary conditions does not have any effects on the natural frequencies or system spectrum due to the both receptance and boundary transfer functions have the same characteristic equations. The method can also be generalized to the vibration analysis of non-piezoelectric composite beams with arbitrary boundary conditions.     

Systematic modeling of a chain of N-flexible link manipulators connected by revolute–prismatic joints using recursive Gibbs-Appell formulation

The main intent of this paper is to represent a systematic algorithm capable of deriving the equations of motion of N-flexible link manipulators with revolute–prismatic joints. The existence of the prismatic joints together with the revolute ones makes the derivation of governing equations difficult. Also, the variations of the flexible parts of the links, with respect to time cause the associated mode shapes of the links to vary instantaneously. So, to derive the kinematic and dynamic equations of motion for such a complex system, the recursive Gibbs-Appell formulation is applied. For a comprehensive and accurate modeling of the system, the coupling effects due to the simultaneous rotating and reciprocating motions of the flexible arms as well as the dynamic interactions between large movements and small deflections are also included. In this study, the links are modeled based on the Euler–Bernoulli beam theory and the assumed mode method. Also, the effects of gravity as well as the longitudinal, transversal and torsional vibrations have been considered in the formulations. Moreover, a recursive algorithm based on 3 ×  3 rotational matrices has been applied in order to derive the system’s dynamic equations of motion, symbolically and systematically. Finally, a numerical simulation has been performed by means of a developed computer program in order to demonstrate the ability of this algorithm in deriving and solving the equations of motion related to such systems.