热环境下混凝土矩形薄板在弹性地基上的振动

基于刚性板的小挠度理论，考虑混凝土的材料非线性，推导了双参数弹性地基上混凝土矩形薄板热弹性问题的动力方程。采用级数法，导出了热环境下双参数弹性地基上四边简支混凝土矩形薄板的固有频率计算公式和强迫振动下的挠度函数。为便于工程应用，给出了双参数弹性地基上四边简支混凝土矩形薄板在恒向变温和温度均匀变化时的固有频率和均布荷载作用下的挠度函数。针对Winlder弹性地基的情况，讨论了板的材料弹性常数、几何尺寸（长宽比）、相对厚度、刚度系数k和温度对薄板固有频率和挠度函数的影响，从而为工程结构中热环境下弹性地基上混凝土矩形薄板的振动计算提供了理论依据。     

正交各向异性矩形板的自由振动特性分析

采用改进Fourier级数方法,建立了正交各向异性矩形薄板的弯曲振动模型,推导出与振动控制方程等价的矩阵方程,得到控制方程在任意边界条件下的解析解。弯曲振动的位移函数表示为标准的二维Fourier余弦级数和辅助Fourier级数之和,通过辅助级数的引入,解决了振动位移函数的偏导数在各边界处潜在不连续的问题。矩形板的振动模态信息能够通过求解一个标准的矩阵特征值而得到。最后进行数值计算并与现有的文献结果进行比较,验证了该方法的快速收敛性和计算精确性。     

车辆变速器弯扭振动特性分析

为研究车辆变速器弯扭振动特性，以某车辆变速器为研究对象，根据振动弯曲理论，建立变速器齿轮轴系弯扭振动数学模型；将不规则的变速器侧面箱体简化为质量均匀的薄板，基于薄板理论推导考虑传动轴系弯扭振动激励下的箱体振动方程；对传动轴弹性化处理后进行模态分析得到其振型函数，采用数值方法求解齿轮轴系和箱体的振动响应；分析箱体和齿轮轴系振动特性，并设计台架试验进行验证。结果表明：测点试验结果与计算结果的均方根值最大误差为32%，且振动趋势一致。研究结果可为探明车辆变速器弯扭振动传递机理与变速器动态可靠性研究提供理论基础。     

弹性地基上四边自由矩形薄板的自由振动

将弹性地基用Winkler模型来代替。首先把弹性地基上矩形薄板的动力学方程表示成为Hamilton正则方程，然后采用辛几何方法对全状态相变量进行分离变量，并利用得到的共扼辛正交归一关系，求出弹性地基上四边自由矩形薄板的固有频率和振型的解析解表达式。由于在求解过程中不需要事先人为的选取挠度函数，而是从弹性地基上矩形薄板的动力学基本方程出发，直接利用数学的方法求出可以满足四边自由边界条件的固有频率和振型的解析解表达式，使得问题的求解更加合理化。文中的最后还给出了计算实例来验证本文所采用的方法以及所推导出公式的正确性。     

矩形悬壁板自由振动精确解法

本文提出矩形薄板振动主振方向的概念。根据振形正交性的思想,在主振方向上矩形板振动波形是唯一的,由此建立了符合矩形悬臂板边界条件所激发出的振动曲线形态的振形函数表达式,并推导出矩形板的频率方程及相应的振形曲线。计算表明,这种方法符合板实际振动的规律性。     

均匀热环境下四边固支矩形PCB薄板的自由振动

表面贴装形式中PCB板可简化为四边固支矩形薄板。基于刚性板的小挠度理论,推导了热载下四边固支矩形PCB薄板的自由振动微分方程。从微分方程中得出,热载下的PCB薄板等效于面内受均布张力的薄板,进而通过结构力学方法将热载下四边固支薄板振动问题转换为受面内均布张力固支薄板振动问题。利用虚位移理论,得出了温度沿厚度均匀线性变化的热载下四边固支矩形PCB薄板固有频率和自由振动的挠度值的计算方法。讨论了热载下温度、薄板的几何尺寸对矩形PCB薄板自由振动固有频率的影响。结论可为矩形PCB薄板在热载下的振动分析以及固有频率计算提供方法上的参考。     

三边固支一边自由混凝土矩形薄板的热弯曲

为了得到温度作用下三边固支一边自由混凝土矩形薄板的挠度和弯矩解析解。该文基于薄板的小挠度理论和叠加原理,考虑横向变温情况,将温度作用下的三边固支一边自由矩形薄板看作是面内温差作用下的三边简支一边自由的矩形薄板和相邻三边作用弯矩的三边简支一边自由矩形薄板的叠加。首先,通过在自由边界上试设具有待定参数的挠度函数,采用李维解法,推导出三边简支一边自由矩形薄板在自由边界挠度函数作用下的解析解;其次,推导出温度作用下三边简支一边自由矩形薄板的解析解;再之,利用相邻三边弯矩作用下四边简支矩形薄板的解答,推导出相邻三边弯矩作用下三边简支一边自由矩形薄板的解答;最后,采用叠加原理得出横向变温作用下三边固支一边自由矩形薄板的挠度和弯矩解析解,并利用MATLAB编制程序得到横向变温作用下三边固支一边自由矩形薄板的计算系数用表,从而为以后工程结构中三边固支一边自由混凝土矩形薄板在热环境下的设计计算提供了理论依据。     

采用面积坐标方法和形函数谱方法构造四边形薄板元

构造一个四边形薄板元,其构造方法有两个特点:1)采用四边形面积坐标方法以代替等参元坐标方法,从而使单元对网格畸变不敏感。2)采用形函数谱方法从已知的膜元推导出新的薄板元,换言之,利用已知的膜元形函数(低阶形函数)来导出待求的薄板元形函数(高阶形函数),此法的要点是:形函数谱是由低阶和高阶形函数所组成,而高阶形函数则是对低阶形函数加以升阶而导出。此法的优点是:使新单元的推导过程大为简化,而且导出的高阶形函数也非常简洁。     

基于波函数法的附加弹簧阻尼薄板的振动研究

基于Kirchhoff薄板弯曲振动理论和波函数法Wave Based Method(WBM)理论,推导了运用WBM将附加弹簧阻尼结构转化为点激励的方法,构建了基于WBM计算含弹簧阻尼支承薄板振动响应的系统矩阵,得到了含弹簧阻尼支承的薄板弯曲振动响应。以四边简支矩形板为例,计算了50~600 Hz频段内参考点的振动响应,并与解析法和有限元法的计算结果进行了对比。运用该方法对比计算了添加不同弹簧阻尼结构数与无弹簧阻尼结构时薄板在120 Hz的弯曲振动响应。结果表明:通过将弹簧阻尼结构转换成点激励的方法,能有效的将WBM应用于附加弹簧阻尼支承薄板弯曲振动响应的仿真计算,与有限元法相比,有着更高精度和收敛速度。     

任意边界条件下矩形板薄板自由振动特性分析

提出一种基于改进傅里叶级数的方法,对矩形薄板在任意边界条件下自由振动特性进行求解。通过将薄板振动的位移函数表示成二维傅里叶余弦级数和辅助级数的线性组合,克服传统傅里叶级数法中薄板位移函数边界处不连续的缺陷;基于位移函数列出矩形薄板拉格朗日方程,然后通过Hamilton原理求解得到矩形薄板自由振动频率与相应位移函数的系数。计算结果与文献及有限元解吻合良好,方法准确可靠;此外,通过改变边界约束弹簧刚度模拟任意边界条件;大量计算表明,固支边界条件与弹性边界条件组合中,随着固支边条界范围增大,矩形薄板无量纲频率参数呈增大趋势;简支及自由边界条件与弹性边界条件组合中,随着弹性边条界的增多,矩形薄板无量纲频率参数呈增大趋势。     

On the Use of a Lattice Model for Analyzing of In-Plane Vibration of Thin Plates

In this paper, a novel approach for simulating in-plane vibration of thin plates is proposed. It is based on the spectral element method (SEM) used within a lattice modeling framework. First, derivation of a frequency dependent dynamic stiffness matrix for a spectral beam element is presented. Then, the lattice modeling concept is introduced. In the model, the two-dimensional plate is discretized as a set of (one dimensional) spectral beam elements connected at the ends. The proposed approach is then used for modal analysis of rectangular plates of different aspect ratios (1 and 2) and boundary conditions (completely free and clamped). Simulated natural frequencies and modal shapes are compared to results available in the literature. It was found that the proposed model can reasonably reproduce low natural frequencies (in most cases within 10%) and modal shapes. Future work will focus on the use of the model as an aid in non-destructive testing of structures.     

FREE VIBRATION ANALYSIS OF A RECTANGULAR PLATE CARRYING ANY NUMBER OF POINT MASSES AND TRANSLATIONAL SPRINGS BY USING THE MODIFIED AND QUASI-ANALYTICAL AND NUMERICAL COMBINED METHODS

The natural frequencies and the corresponding mode shapes of a uniform rectangular plate carrying any number of rigidly attached (or elastically mounted) point masses and translational springs with various magnitudes and arbitrary locations are determined by using the modified Analytical and Numerical Combined Method (or modified ANCM) and the quasi-ANCM. Instead of seeking the closed-form solution analytically for the natural frequencies and normal mode shapes of the ‘unconstrained’ rectangular plate (without any concentrated elements attached) required for the pure ANCM, the normal mode shapes for the modified ANCM and the natural frequencies together with the normal mode shapes for the quasi-ANCM are obtained numerically, however. Then the characteristic equation of the ‘constrained’ rectangular plate (with any number of concentrated elements attached) is derived basing on the natural frequencies and normal mode shapes of the ‘unconstrained’ plate and applying the mode-superposition theory. Finally, the natural frequencies and mode shapes of the ‘constrained’ plate are obtained numerically. The pure ANCM is originally available only for the problems that the ‘closed-form’ solution for the natural frequencies and normal mode shapes of the ‘constrained’ system is obtainable. Now, the modified ANCM and quasi-ANCM presented in this paper break the limitation of the pure ANCM and extend the area of problems solvable by the ANCM. © 1997 John Wiley & Sons, Ltd.     

热环境对基础激励作用下纤维增强树脂基复合薄板振动特性的影响

采用理论与实际相结合的方式,研究了热环境对纤维增强树脂基复合薄板振动特性的影响。首先,建立了基础激励作用下纤维增强树脂基复合薄板在热环境中的解析模型,推导获得了固有频率、模态振型、振动响应及阻尼比的解析表达式,并给出了热环境下复合薄板振动特性的求解流程。然后,搭建了热环境下该类型复合薄板结构的振动测试系统,并以TC500碳纤维/树脂薄板为研究对象进行了实际测试。以相应的测试结果为例,研究发现,当温度从20℃上升到200℃时,复合薄板的模态振型基本不发生改变,固有频率降低幅度在2.3%~36.6%之间,振动响应幅值增大范围约为15.3%~58.4%,且阻尼性能也呈现增大的趋势,例如相对于常温下的阻尼结果,当温度上升到200℃时,其前6阶模态阻尼比的增大程度在13.9%~56.4%之间。另外,经过对比理论和测试获得的不同温度下的固有频率、模态振型、振动响应及阻尼结果可知,两者吻合较好,进而验证了解析分析方法及其结果的正确性。      

含焊接残余应力薄圆板结构自由振动近似解

研究焊接残余应力对薄圆板结构振动特性的影响,解决薄圆板结构振动中存在非均匀分布预应力问题。根据含预应力结构的应变-应力方程,建立含预应力薄圆板结构的运动控制方程。基于Rayleigh-Ritz法构造Lagrange能量泛函方程。将预应力和位移试函数展开成三角级数形式,对含预应力薄圆板结构的自由振动问题进行求解。以周边简支边界薄圆板结构为例,对比焊接残余应力的不同分布形式对薄圆板结构固有频率及振型的影响。数值计算结果验证了所提方法的有效性,可应用于解决任意分布预应力问题。     

Dynamic effective thickness in laminated-glass beams and plates

In recent years, several equations have been proposed to calculate deflections and stresses in laminated-glass beams and plates under static loading using the concept of effective thickness, which consists of calculating the thickness of a monolithic element with equivalent bending properties to a laminated element. Recently, an effective thickness for the dynamic behavior of laminated-glass beams has been proposed to enable the modal parameters (natural frequencies, loss factors and mode shapes) to be determined using an equivalent monolithic model. In the present paper, the technique has been extended to the two-dimensional case of rectangular laminated-glass plates and the steps needed to estimate the modal parameters of laminated-glass elements using this methodology are presented. The dynamic effective thickness concept has been validated by experimental tests made on a laminated-glass beam and a laminated-glass plate. The results show that good accuracy is achieved in the natural frequencies and mode shapes but high scatter is encountered in the loss factors.     

A new kind of energy transfer from high frequency mode to low frequency mode in a composite laminated plate

This paper brings to light a new type of nonlinear resonant motion in a fiber-reinforced composite laminated rectangular thin plate, which is not reported in other literature. The investigated system is a simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitation whose frequency is near to the first-order natural frequency of the plate. This new phenomenon demonstrates that the responses of a low-order frequency mode can be excited by those of a high-order frequency mode. The high-order frequency is the first-order natural frequency of the test plate, and the low-order frequency here is lower than the first-order nature frequency. Experimental research works on the nonlinear vibrations of the composite laminated rectangular thin plate have been carried out for the first time. It is found from the experimental results that the nonlinear dynamic responses consist of four modes, whose frequencies include a lower frequency than the first-order natural frequency, 1/3 sub-harmonic, 2/3 sub-harmonic and the first-order natural frequencies. In this case, the amplitude of the mode for lower frequency is larger than those of modes for the aforementioned frequencies. Moreover, the theoretical job goes to analyze this new phenomenon. An analytical mode is given to explain the interactions between the first-order mode and the lower-frequency mode observed in the experiment. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion are formulated for the test plate under parametric excitation. Galerkin’s method is utilized to discretize the partial differential governing equations of motion for the composite laminated rectangular thin plate to a two-degree-of-freedom nonlinear system. The results of numerical simulations qualitatively agree very well with the experimental results. In addition, the multi-pulse chaotic motions are also found in numerical simulations.     

弹性中厚板模态特性的近似计算方法

文中介绍了求解中厚板振动模态特性的一个近似方法一模态摄动法。这一方法是在薄板模态子空间内实施,最后将复杂的偏微分方程的求解转化为一个代数方程组的求解。对于四过简支板,得到一个十分简单的计算公式。文中还对剪切变形和转动惯量以及剪切折合系数对四过简支厚板固有频率的影响进行了分析。算例表明,这一方法简单且具有较好的精度。     

A Comparison of the Vibration Characteristics of Carbon Fiber Reinforced Plastic Plates with those of Magnesium Plates

An experimental and numerical investigation is conducted to evaluate vibration characteristics of an advanced composite material system, namely carbon fiber reinforced plastic (CFRP), relative to a magnesium alloy currently used in the vibration testing industry. Experimental test specimens for both materials, with varying thicknesses, are tested with two boundary conditions — the free condition, to evaluate the natural frequencies and damping of the two materials and, secondly, a general constrained condition, typical of vibration testing. Experimental modal analysis techniques are used to measure the vibration characteristics including natural frequencies, damping and mode shapes. The results from these tests show that the natural frequency and mode shape for relatively thin CFRP plates were comparable with those of magnesium plate. Although the mode shapes also compare well for thick CFRP and magnesium specimens, the natural frequencies were found to have significant differences between the two material systems. The largest difference between the two material systems, present for all thicknesses, is found to be the damping values for the respective vibration modes. This unique characteristic of the CFRP material presents an opportunity for a performance increase in the vibration testing system’s community.     

高温与脱粘对复合材料蜂窝板模态特性影响的试验

复合材料蜂窝板是目前航空航天领域中一类非常重要的结构,但此类结构在热环境下的模态特性研究鲜有公开报道。本文针对由碳纤维增强环氧树脂基复合材料层合板和Nomex蜂窝芯层复合而成的夹层板进行了不同温度下的模态试验,在加热过程中结构面板与芯层脱离,向外凸出。损伤发生前,结构固有频率随温度升高而下降,在识别出的前7阶模态中,弯曲振型的下降幅度较大,而扭转振型的降幅较小。对损伤后的结构再次进行常温模态试验,其模态特征发生显著变化,除了固有频率的变化外,模态阻尼比明显提高,而且其中两阶模态振型的顺序发生对调。此外,本文还对该损伤形式的成因进行了分析,所得结论对用于热环境的复合材料蜂窝结构设计具有一定的指导意义。     

具有中间支承的矩形板自由振动分析

应用一般解析解来求解具有中间支承矩形板的自由振动问题。一般解析解能求解任意边界条件矩形板的振动问题,求解过程是将整块板看成是沿中间支承分开的两块板,沿支承边两块板的挠度均等于零,斜度和弯矩均相等,再由全部边界条件和连续性条件可以求解各阶频率和振型。对几种具有简支边,平夹边或自由边的混合边界矩形板进行了计算。