Bernoulli-Euler梁横向振动固有频率的轴力影响系数

给出了考虑轴力对于Bernoulli-Euler梁横向振动固有频率影响系数的高精度表达式。与动力刚度法推导等截面梁自由振动分析的动态刚度阵不同,首先获得承受常轴力的Bernoulli-Euler梁横向自由振动微分方程的通解,并通过位移边界条件消去待定常数,得到精确形函数;使用有限元方法,建立了使用精确形函数表达等截面Bernoulli-Euler梁动态刚度阵的微分格式,该微分格式精确刚度阵与动力刚度法得到的刚度阵完全一致。仿照Timoshenko对压弯梁静态挠度表达中取用轴力影响因子的方法,提出了Bernoulli-Euler梁横向振动固有频率的轴力影响系数表达式,结合Wittrick-Williams算法和动态刚度阵证明了当轴力在±0.5倍第1阶欧拉临界力之间变化时,轴力影响系数表达式最大误差不超过2%,且随固有频率阶次的提高,误差越来越小。     

非对称Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀Bernoulli-Euler薄壁梁单元自由振动的控制运动微分方程,推导了其精确的动态传递矩阵。采用Bernoulli-Euler弯扭耦合梁理论,假定梁横截面没有任何对称性,考虑了薄壁梁在两个方向的弯曲振动及翘曲刚度的影响。动态传递矩阵可以用于计算非对称薄壁梁及其集合体的精确固有频率和模态形状。针对具体的算例,给出了各种边界条件下固有频率的数值结果并与文献中已有的结果进行了比较,还讨论了翘曲刚度对固有频率和模态形状的影响,结果表明如果忽略翘曲刚度的影响,可能得到毫无意义的结果。     

用动态刚度法分析旋转变截面梁横向振动特性

通过引入动态刚度法分析旋转变截面梁的振动特性。首先基于欧拉-伯努利梁理论给出旋转变截面梁自由振动方程,然后通过动态刚度法推导该旋转梁的动态刚度矩阵,最后运用MATLAB中的fzero函数求解特征值方程得到旋转梁横向振动的固有频率和模态振型。数值计算结果证明了动态刚度法的精度和有效性,同时分析了轮毂半径、转速以及渐变系数对固有频率的影响。     

磁场作用下轴向运动功能梯度Timoshenko梁的振动特性

轴向运动结构的工程振动问题一直是动力学领域中的重要课题之一。为了更全面地分析工程中的振动,针对磁场作用下轴向运动功能梯度Timoshenko 梁的振动特性展开论述。基于梁的动力学方程组和相应的简支边界条件,应用复模态方法,得到不同参数时固有频率和衰减系数与轴向运动速度的对应关系。采用微分求积法分析磁场作用下前四阶固有频率和衰减系数随轴向运动速度的变化,并与复模态方法的结果进行对比验证。数据结果表明复模态方法得到的结果是精确解析解。衰减系数呈现不对称性,耦合固有频率呈现分离性。随着轴速、磁场强度和功能梯度指数的增大,梁的固有频率减小;随着支撑刚度参数的增大,梁的固有频率增大。     

U型波纹管的扭转振动固有频率的计算

采用等效的薄壁圆管模型来分析U型波纹管的扭转振动固有频率,为此要确定U型波纹管的等效半径和扭转刚度。等效半径采用两种计算方法。扭转刚度的计算即采用了钱伟长关于旋转壳的扭转刚度积分公式,也采用了等效的圆管模型方法,由此得到的相应的U型波纹管扭转固有频率的计算方法分别称为积分法和简化法。没有关于波纹管扭转固有频率的实验见诸文献。采用有限元法来验证上述计算方法,取得了很好的一致。积分法更为精确,简化法更适于工程计算。     

某车辆动力总成异常振动分析与改进

针对某SUV车柴油机动力总成产生异常振动导致操纵手柄剧烈振动这一实际问题,将振动试验测试与有限元仿真分析结合起来研究动力总成的振动动态特性。通过整车道路试验及转鼓试验,发现柴油机工作转速的二阶激励激起了动力总成系统共振,通过采用有限元方法建立了一种柴油机动力总成振动分析模型,计算分析了此动力总成的振动固有频率和固有振型,找到了导致动力总成弯曲刚度变差的原因—飞轮壳结构刚度不足。在验证了有限元仿真模型计算合理性基础上,通过改进设计飞轮壳结构,提高了动力总成的固有频率,使其避开了共振频率区间,最终消除了操纵手柄的异常振动。该试验与仿真分析方法对解决同类工程问题具有参考指导和应用价值。     

《薄壁Timoshenko梁弯扭耦合振动的动态有限元法》

通过直接求解单对称均匀薄壁Timoshenko梁单元弯扭耦合振动的运动微分方程,推导了其精确的动态刚度矩阵。在本文研究中考虑了弯扭耦合、翘曲刚度、转动惯量和剪切变形的影响。针对某弯扭耦合的薄壁梁算例,应用本文推导的动态刚度矩阵,采用自动Muller法和结合频率扫描法的二分法求解频率特征方程,计算了该薄壁梁的固有特性,并讨论了翘曲刚度、剪切变形和转动惯量对该弯扭耦合薄壁梁的固有频率和模态形状的影响。数值结果验证了本文方法的精确性和有效性,并指出随着模态阶次的增加,剪切变形、转动惯量和翘曲刚度对薄壁梁的固有特性的影响更加显著。     

中厚圆柱壳自由振动的动力刚度法分析

该文阐述了将动力刚度法应用于中厚圆柱壳的自由振动分析。从考虑横向剪切变形和转动惯量的中厚壳理论出发,将圆柱壳的振动分解为一系列确定环向波数下的一维振动问题。用常微分方程求解器COLSYS求解该一维问题的动力刚度,通过Wittrick-Williams算法及导护型牛顿法求得该环向波数下结构的频率和振型。由于求解动力刚度时使用COLSYS对控制方程进行了精确求解,所以该文方法是精确方法。数值算例验证了中厚圆柱壳壳段固端频率计数J0计算方法的可靠性。综合表明:应用动力刚度法对中厚圆柱壳自由振动进行分析是可靠、精确的。     

基于小波有限元的悬臂梁裂纹识别

研究了悬臂梁裂纹识别中的正反问题．即通过裂纹位置和尺寸求解梁的固有频率以及利用梁的固有频率．识别裂纹位置和尺寸。以矩形截面裂纹悬臂梁为例，利用小波有限元方法建立了梁自由振动的有限元模型．其中裂纹被看作为一刚度已知的扭转线弹簧，求解出了系统的固有频率；通过行列式变换，将反问题求解简化为只含线弹簧刚度一个未知数的一元二次方程求根问题，分别做出了以不同固有频率作为输入值时裂纹位置与弹簧刚度之间的解曲线，曲线交点预测出裂纹的位置与尺寸。数值算例证实了算法的有效性，为工程结构裂纹故障预示与诊断提供了新的方法。     

基于液压变压器蓄能器变刚度机构的液压激振方法

提出了一种新的激振方法,即用液压变压器与蓄能器、作动缸组成变刚度弹性机构,该机构连接振体构成液压激振振动回路。调节变压器的排量可以改变弹性机构的刚度,进而改变振动回路的固有频率和振动特性。由于这种方法不再依赖液压阀控制振动,所以能够避免节流损失。构建了激振回路数学模型,进行了仿真分析,表明该回路的固有频率可通过液压变压器在一定范围内任意调节。     

Exact determinant for infinite order FEM representation of a Timoshenko beam–column via improved transcendental member stiffness matrices

Transcendental stiffness matrices for vibration (or buckling) have been derived from exact analytical solutions of the governing differential equations for many structural members without recourse to the discretization of conventional finite element methods (FEM). Their assembly into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or critical load factors) can be found with certainty using the Wittrick–Williams algorithm. A very recently discovered analytical property is the member stiffness determinant, which equals the FEM stiffness matrix determinant of a clamped ended member modelled by infinitely many elements, normalized by dividing by its value at zero frequency (or load factor). Curve following convergence methods for transcendental eigenproblems are greatly simplified by multiplying the transcendental overall stiffness matrix determinant by all the member stiffness determinants to remove its poles. In this paper, the transcendental stiffness matrix for a vibrating, axially loaded, Timoshenko member is expressed in a new, convenient notation, enabling the first ever derivation of its member stiffness determinant to be obtained. Further expressions are derived, also for the first time, for unloaded and for static, loaded Timoshenko members. These new expressions have the advantage that they readily reduce to corresponding expressions for Bernoulli–Euler members when shear rigidity and rotatory inertia are neglected. Additionally, the total equivalence of the normalized transcendental determinant with that of an infinite order FEM formulation aids understanding and evaluation of conventional FEM results. Examples are presented to illustrate the use of the member stiffness determinant. Copyright © 2004 John Wiley & Sons, Ltd.     

一系螺旋弹簧动刚度对车辆-轨道耦合振动影响分析

建立准确表征一系悬挂轴箱螺旋弹簧波动特性的力学模型,运用动刚度矩阵法求解,研究其对悬挂系统隔振性能影响。结合基于格林函数法的车辆-轨道耦合动力学模型,引入弹簧刚度频变特性,对比分析考虑一系螺旋弹簧频变刚度前后车辆动力学性能之间的差异。结果表明,动刚度矩阵法可以精确求解螺旋弹簧随频率变化的动刚度特性,在一阶模态振动频率后弹簧刚度值呈现103等级的剧烈变化,该结果与有限元模型结果一致;一系螺旋弹簧的动态频率特性导致轮轨激励由车轮至构架的振动位移传递率提高到接近于1,而对车体的振动传递率提高到了10-3左右;在整车车辆-轨道动力学计算中,其对轮轨振动影响较小,但车体与构架出现了较高的高频振动能量峰值。包含一系悬挂动刚度的车辆模型更接近实际,为了降低车辆振动,应尽量提高一系螺旋弹簧自振频率并降低动刚度变化幅值。       

考虑剪切变形影响的斜梁桥自振频率的解析方法

斜梁桥振动频率没有显式解,给使用《公路桥涵设计通用规范》方法计算冲击系数带来不便。考虑斜梁桥振动时的弯扭耦合效应,分别采用修正的Timoshenko梁理论建立其弯曲振动的动态刚度矩阵,采用Saint-Venant扭转理论建立其自由扭转振动的动态刚度矩阵,结合斜支承边界条件,导出斜支承坐标系下的动态刚度矩阵,提取弯矩-转角的刚度方程,根据其奇异条件建立关于斜梁桥自振频率的超越方程,采用二分法对超越方程进行求解以得到自振频率。该文分析了一座标准A型单跨斜箱梁桥考虑与不考虑剪切变形影响时的前5阶振动频率随斜交角的变化,比较了正交简支初等梁和正交简支深梁、斜支初等梁和斜支深梁的前5阶频率。结果显示:斜梁桥基频随斜交角的增大而增大、第2阶频率随斜交角的增大而减小;斜梁桥振动频率的计算应采用考虑剪切变形影响的深梁理论。     

Dynamic stiffness analysis of laminated beams using a first order shear deformation theory

In this paper the exact vibration frequencies of generally laminated beams are found using a new method, including the effect of rotary inertia and shear deformations. The effect of shear in laminated beams is more significant than in homogenous beams, due to the fact that the ratio of extensional stiffness to the transverse shear stiffness is high. The exact dynamic stiffness matrix is derived, and then any set of boundary conditions including elastic connections, and assembly of members, can be solved as in the classical direct stiffness method for framed structures. The natural frequencies of vibration of a structure are those values of frequency that cause the dynamic stiffness matrix to become singular, and one can find as many frequencies as needed from the same matrix. In the paper several examples are given, and compared with results from the literature.     

计及二阶效应的梁杆系统动力响应分析方法研究

基于动力刚度法和有限元理论提出了一种考虑二阶效应计算梁杆动力响应的新方法。通过求解轴向力作用下Bernoulli-Euler 梁横向和轴向挠度自由振动微分方程,利用位移边界条件反解出待定系数,得到了动态精确形函数;使用经典有限元方法推导了考虑截面自身旋转惯量的质量阵和考虑二阶效应的刚度阵,该质量阵和刚度阵各元素均为轴力和圆频率的超越函数;建立了杆系结构瞬态动力学分析的动力平衡方程,给出了稳定和高效的求解方案。对几个典型的算例进行了计算分析,并与通用软件ANSYS 的计算结果进行了比较。计算结果表明:该分析梁杆系统动力响应的新方法具有较高的计算精度和效率,特别是能够准确地计入轴力对于梁杆动力响应的影响。     

Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick–Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large‐scale structures. Copyright © 2003 John Wiley & Sons, Ltd.     

含裂纹损伤杆系结构的动态特性研究

运用动刚度有限元法,研究了含裂纹损伤杆系结构的动态特性。提出了一种含裂纹的杆单元,基于断裂力学的线弹簧模型,导出了相应的动刚度矩阵。在此基础上,对含裂纹的悬臂梁和平面框架进行了数值计算,并与已有的实验值和解析解进行了比较。结果表明:损伤位置和损伤程度的不同均会导致结构动态特性发生改变,因而在结构分析中应考虑损伤的影响;而该单元能够方便地用于含裂纹损伤杆系结构的动态特性分析,并具有很好的精度。     

High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method

In this paper, high-order free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness method. The governing partial differential equations of motion for one element are derived using Hamilton’s principle. This formulation leads to seven partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are divided into two ordinary differential equations by considering the symmetrical sandwich beam. Closed form analytical solutions of these equations are determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. The element dynamic stiffness matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by use of numerical techniques and the known Wittrick–Williams algorithm. Finally, some numerical examples are discussed using dynamic stiffness method.     

Eigensolution for large structural systems with substructures

A method for calculating natural frequencies and mode shapes of large structural systems with substructures and the subspace iteration is developed. The method uses only substructural stiffness matrices and the mass matrix for each finite element of the system. The mass matrix for the entire structure or any of its substructures need not be computed. However, efficiency of the method is improved when mass matrix for the entire structure is computed and saved in the computer core. No approximating assumptions are made. Thus, natural frequencies and mode shapes for the finite element model employed are the same with or without the substructuring algorithm. This is demonstrated by computing first ten natural frequencies and the corresponding mode shapes for an open truss helicopter tail-boom structure.