【作者】 郭强；

【导师】 钟宏志；

【作者基本信息】 清华大学 ， 土木工程， 2004， 硕士

【摘要】 样条微分求积法是一种新型的数值计算方法,其与传统微分求法的主要区别在于:它是基于B样条函数来构造基函数,进而获得权系数。目前,样条微分求积法的开发刚刚起步,应用范围还比较有限。本文的工作重点之一是进一步完善样条微分求积法并推广其在结构分析中的应用。文中详细阐述了四次与六次样条微分求积法的建立过程,得到了用于实际计算的权系数,并对奇数次与偶数次样条微分求积法做了归纳总结。通过实例计算,样条微分求积法体现出了稳定性强、灵活性好的特点,并且易于编程实现。本文另一个工作重点是梁的非线性振动分析。这里的非线性指的是几何非线性(大挠度),考虑了两种梁的理论,即Bernoulli-Euler梁理论和Timoshenko梁理论。梁的几何形式包括等截面与不等截面,后者分别为线性变宽度和线性变高度的楔形梁。对于Bernoulli-Euler梁,本文采用六次样条微分求积法求解,讨论了边界条件对求解过程的影响。对于Timoshenko梁,本文首先尝试推导出了等截面与不等截面Timoshenko梁非线性振动的控制方程,较为全面地考虑了非线性轴力、曲率和剪切应变的影响,采用了传统微分求积法和样条微分求积法求解微分方程。计算结果表明非线性频率与线性频率之比随振动幅值的增大和梁长细比的增大而增大;三种非线性项对非线性频率的影响都是随着振动幅值的增大而增大,并且非线性剪切应变是仅次于非线性轴力的另一重要影响因素;双边简支梁的非线性特征最为明显,一端简支一端固支梁次之,双边固支梁最弱;在同等条件下,截面变高度对梁非线性频率的影响较截面变宽度的影响要大。此外,本文还就局部实施思想在传统微分求积法中的应用做了简要的讨论,利用一维和二维两个简单的例子说明了其具体操作步骤。数据结果显示,局部实施思想的引入,是对传统微分求积法的重要补充,它可以提高传统微分求积法的稳定性,从而扩大其应用范围。更多还原

【Abstract】 The Spline-based Differential Quadrature (SDQ) is a newly developed numerical method. The main distinction of the SDQ lies in the determination of weighting coefficients on the basis of cardinal B-spline interpolation functions. The application scope of the SDQ has been limited since the method is still in its infancy stage. One of the aims of the present research is to further extend the application of the method to more practical problems in structural analysis. The constructions of the quartic and the sextic SDQ are elaborated and the explicit expressions of weighting coefficients for approximation of derivatives are obtained. A brief summary is given for the development of differential quadrature method using ether odd-order or even-order B-splines. Excellent results are achieved in the provided examples. The SDQ method is shown to exhibit great flexibility and stability. The other main aim of the present study is the nonlinear (large amplitude) vibration analysis of beams. Two beam theories, the Bernoulli-Euler beam theory and the Timoshenko beam theory, are considered. The nonlinear vibrations of prismatic beams and tapered beams are investigated. The conventional differential quadrature method and the SDQ are used to resolve the nonlinear vibration problems. For the Bernoulli-Euler beams, the effects of the different boundary conditions on the formulation and the solution are discussed. For nonlinear vibrations of Timoshenko beams, the corresponding governing equations are established for the first time in present work. The nonlinear terms including the axial stretching, nonlinear bending curvature and shear strain are considered. It is shown that the nonlinear frequency ratio increases with the vibration amplitude and the slenderness ratio. The effects of nonlinear terms on the nonlinear frequency ratio are discussed at length and their contributions are found to be in the following descending sequence: axial stretching, shear strain and bending curvature. The change of nonlinear frequency against the effects of boundary conditions is always in the following descending order: <WP=4>beams with two simply supported ends, beams with two clamped ends and beams with one end simply supported and the other end clamed. For a specific vibration amplitude and a specific slenderness ratio, the effect of the cross-sectional height change on the nonlinear frequency ratio is larger than that of the cross-sectional width change.In addition, the local implementation strategy in the conventional differential quadrature is discussed briefly. It is shown that the philosophy of local implementation can efficiently improve the stability of the conventional differential quadrature and extend its application accordingly.更多还原