球壳轴对称弯曲问题精确的挠度微分方程及其奇异摄动解
Refined Differential Equations of Deflections in Axial Symmetrical Bending Problems of Spherical Shell and Their Singular Perturbation Solutions
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摘要: 本文提出了球壳轴对称弯曲问题精确的挠度(ω)微分方程和精确的转角(dω/da)微分方程.本文重点研究了挠度微分方程的精度,基本思路是:首先假设边缘效应时经线中面位移u=0,从而建立挠度微分方程,然后再精确地证明挠度微分方程与原来微分方程内力解答完全相同.再精确地证明边缘效应时经线中面位移u=0是精确解.本文给出了挠度微分方程的奇异摄动解,最后验算了平衡条件,证明摄动解求出的内力和外荷载是完全平衡的.这一方面表明摄动解的计算是正确的;另一方面也再二次表明挠度微分方程是精确的微分方程.新微分方程的优点是:1.新微分方程和原来微分方程精度完全相同;2.新微分方程满足的边界条件非常简单;3.新微分方程便于使用摄动解;4.新微分方程可以得到挠度(ω)和转角(dω/da)的表达式.新微分方程使球壳的计算得到很大的简化.本文采用的符号与徐芝纶《弹性力学》第二版下册相同[1].Abstract: This paper deals with the research of accuracy of differential equations of deflections.The basic idea is as follows.Firstly,considering the boundary effect the meridian midsurface displacement u=0,thus we derive the deflection differential equations;secondly we accurately prove that by use of the deflection differential equations or the original differential equations the same inner forces solutions are obtained;finally,we accurately prove that considering the boundary effect the meridian surface displacement u=0 is an exact solution.In this paper we give the singular perturbation solution of the deflection differential equations.Finally we check the equilibrium condition and prove the inner forces solved by perturbation method and the outer load are fully equilibrated.It shows that perturbation solution is accurate.On the other hand,it shows again that the deflection differential equation is an exact equation. The features of the new differential equations are as follows: 1.The accuracies of the new differential equations and the original differential e-quations are the same. 2.The new differential equations can satisfy the boundary conditions simply. 3.It is advantageous to use perturbation method with the new differential equations. 4 We may obtain the deflection expression (ω) and slope expression (dw/da) by using the new differential equations. The new differential equations greatly simplify the calculation of spherical shell.The notation adopted in this paper is the same as that in Ref.[1].
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[1] 徐芝纶,稼弹性力学(下册),第二版,人民教育出版社(1982). [2] 钱伟长,《奇异摄动理沦及其在力学中的应用努,科学出版社(1981). [3] 钱伟长、叶开沉,弹性勺学九,科学出版社(1980). [4] 周焕文,圆板非线性理论的一种摄动解,应用数学和力学,2,5(1981,475-484. [5] 周焕文,合成展开法应用于球壳对称弯曲的边界层问题,应用数学和力学4,6(1983),763-770. [6] 王镇行,变壁厚轴对称球壳,应用数学和力学,9,2(1988),183-188. [7] 王新志,变厚度圆薄被在均布荷载下矢挠度问题,应用数学和力学,4,1(1983),103--112. [8] 杨耀乾,《薄壳理论》,中国铁道出版社(1981). [9] Ржаницын А.Р.,Cmpoumелъная,Высшая Щкола,Москва(1982). [10] Flüagge,W.,Sirecses in Shells,Springer-Verlag,Berlin(1960). [11] 范存旭,球壳轴对称弯曲问题的边缘效应,工程力学,5,4(1988). [12] 范存旭,任意旋转壳轴对称弯曲间题的研究,力学学报,21,6(1989). -
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