压力载荷、变温和强迫振动作用下扁壳结构的突变和混沌运动的研究
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摘要
本文尝试利用非线性理论中的分岔、突变和混沌理论对一类扁壳结构进行非线性分析。在静载荷和变温共同作用的情况下,对扁壳结构进行了突变分析,并对Koiter过屈曲一般理论应用初等突变理论进行了研究。对在扰动载荷,静预加载和变温共同作用下的一类扁壳结构在理论上进行了混沌研究,并对若干种不同情况的实际例题进行了数值分析并研究了这些结构单元的动力学性质,给出了周期运动和混沌运动的时程曲线,相平面轨迹和Poincare映射。现将这些工作简述如下:
1.在本文第二章中,针对板壳结构屈曲临界载荷与实验结果存在较大差异这一历史话题,利用非线性理论中的初等突变理论及分岔概念研究了弹性结构过屈曲形态的Koiter一般理论的非线性性质,建立了该理论所具有的非线性动力系统。用突变形态解释结构后屈曲状态的非线性跳跃现象。本文提出,正是由于初始缺陷的存在使得那些对初始缺陷敏感的壳体的非线性动力系统的平衡态发生了全局性的改变,即由拆叠突变变成了类尖点突变,从而导至了另外一种平衡态。
2.在本文第三章中,研究了静载荷和变温作用下一类扁壳结构的后屈曲问题。在Donnel-Kàrmán非线性方程组的基础上推导了新的方程组,并对这个方程组进行了求解,求得了相应的应力函数和非线性动力系统。利用本文的结果可以直接对扁壳结构的超临界变形问题进行了定量计算,从而确定结构在后屈曲阶段的应力状态。在此基础上,本文对双曲扁壳,闭合圆柱壳进行了突变分析,同时考虑了侧压和初始缺陷的影响。双曲扁壳的结果可以推广到矩形板,开口圆柱壳和方底扁球壳等情况。
3.在固体力学领域内对混沌问题的研究大都局限于屈曲梁,并且大都归结为典型的Duffing方程。对于板壳问题中是否存在混沌运动则极少有人曾涉及,本文第四章以一类扁壳结构为对象进行了混沌运动方面的研究。在第三章的基础上,我们建立了这些壳体的非线性动力系统,对双曲扁壳、矩形板、开口圆柱壳和圆底扁球壳进行了理论分析。在分析中考虑了静力预加载,变温、初始缺陷和扰动载荷,并给出了新的非线性动力方程。
The bifurcation and the elementary catastrophe theory and chaos theory have been employed to investigate some shells for the purpose of nonlinear analysis in this paper.
The catastrophe analysis of shallow shells subjected to static pressure load and varying temperature is studied. The chaotic motion of shallow shells subjected to transverse periodic load and static load, varying temperature and initial defect of shells is investigated by Milnikov method and numerical analysis. We can introduce the beneficial study in this paper as follows:
1. The catastrophe analysis of Koiter general theoty of postbuckling problem has been completed in chapter2.
2. The postbuckling problem of shallow shells is studied in chapter3. This investigation is based on Donnel-Kàrmán eguations and a new nonlinear eguation is established, and the stress function and nonlinear dynamic system are carried out. In addition, the catastrophe analysis of shallow shells has been completed in chapter3.
3. The nonlinear dynamic system of double-curve shallow shells, plate, cylindrical shell and shallow spherical shell subjected to transverse periodic load and varying temperature are established in chapter4. The critical chaotic motion of cylindrical shell is given by Melnikov function, and the numerical analysis is carried out in chapter5, which contain rectangular plate, cylindrical shell and shallow spherical shell. The time-displacement history diagram and phase –plane diagram, chaos-bifurcation diagram and Poincare map are employed to determine the motion is chaotic or not.
Using these theoretical and numerical methods, the chaotic motion of the nonlinear dynamic systems are investigated in this paper.
1.在本文第二章中,针对板壳结构屈曲临界载荷与实验结果存在较大差异这一历史话题,利用非线性理论中的初等突变理论及分岔概念研究了弹性结构过屈曲形态的Koiter一般理论的非线性性质,建立了该理论所具有的非线性动力系统。用突变形态解释结构后屈曲状态的非线性跳跃现象。本文提出,正是由于初始缺陷的存在使得那些对初始缺陷敏感的壳体的非线性动力系统的平衡态发生了全局性的改变,即由拆叠突变变成了类尖点突变,从而导至了另外一种平衡态。
2.在本文第三章中,研究了静载荷和变温作用下一类扁壳结构的后屈曲问题。在Donnel-Kàrmán非线性方程组的基础上推导了新的方程组,并对这个方程组进行了求解,求得了相应的应力函数和非线性动力系统。利用本文的结果可以直接对扁壳结构的超临界变形问题进行了定量计算,从而确定结构在后屈曲阶段的应力状态。在此基础上,本文对双曲扁壳,闭合圆柱壳进行了突变分析,同时考虑了侧压和初始缺陷的影响。双曲扁壳的结果可以推广到矩形板,开口圆柱壳和方底扁球壳等情况。
3.在固体力学领域内对混沌问题的研究大都局限于屈曲梁,并且大都归结为典型的Duffing方程。对于板壳问题中是否存在混沌运动则极少有人曾涉及,本文第四章以一类扁壳结构为对象进行了混沌运动方面的研究。在第三章的基础上,我们建立了这些壳体的非线性动力系统,对双曲扁壳、矩形板、开口圆柱壳和圆底扁球壳进行了理论分析。在分析中考虑了静力预加载,变温、初始缺陷和扰动载荷,并给出了新的非线性动力方程。
The bifurcation and the elementary catastrophe theory and chaos theory have been employed to investigate some shells for the purpose of nonlinear analysis in this paper.
The catastrophe analysis of shallow shells subjected to static pressure load and varying temperature is studied. The chaotic motion of shallow shells subjected to transverse periodic load and static load, varying temperature and initial defect of shells is investigated by Milnikov method and numerical analysis. We can introduce the beneficial study in this paper as follows:
1. The catastrophe analysis of Koiter general theoty of postbuckling problem has been completed in chapter2.
2. The postbuckling problem of shallow shells is studied in chapter3. This investigation is based on Donnel-Kàrmán eguations and a new nonlinear eguation is established, and the stress function and nonlinear dynamic system are carried out. In addition, the catastrophe analysis of shallow shells has been completed in chapter3.
3. The nonlinear dynamic system of double-curve shallow shells, plate, cylindrical shell and shallow spherical shell subjected to transverse periodic load and varying temperature are established in chapter4. The critical chaotic motion of cylindrical shell is given by Melnikov function, and the numerical analysis is carried out in chapter5, which contain rectangular plate, cylindrical shell and shallow spherical shell. The time-displacement history diagram and phase –plane diagram, chaos-bifurcation diagram and Poincare map are employed to determine the motion is chaotic or not.
Using these theoretical and numerical methods, the chaotic motion of the nonlinear dynamic systems are investigated in this paper.
引文
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(2)Donnell, L. H, A new theory for the buckling of thin cylinders under axial compression and bending. Trans. ASME 56(1934)
(3)N. J. Hoff, Buckling of thin cylindrical shells under hoop stresses varying in axial direction, Journal of App. Mech., Vo1. 24, No3.
(4)Koiter, W. T. On the stability of elastic equilibrium, (in Dutch, with English summary) Thesis, Delft, H. J. Paris, Amstardam (1945); English translation issued as NASA TTF-10 (1967)
(5)Koiter. W.T., Elastic and Postbuckling behavior. “Nonlinear problem” proc. of sygmp., conducted by Math. Res center, U. S Army, at the Uni. of Wisconsin, Apr (1962).
(6)瓦.扎·符拉索夫,壳体的一般理论及其在工程中的应用,中译本,人民教育出版社, 1960.
(7)Chien Wei-Zang, The intrinsic theory of thin shells and plates; Part ⅠGeneral theory Quart, Appl. Mech., l (1944), 297-327; PartⅡ, Application to thin plates, ibid, 2 (1944). 43-59; Part Ⅲ, Application to thin shells, ibid, 2 (1944), 120-135
(8)鹫津久一郎,弹性和塑性力学中的变分法 中译本 科学出版社, 1984
(9)杨耀乾,薄壳理论 中国铁道出版社 1981
(10)竹内洋一郎 热应力 中译本 科学出版社 1977
(11)A. Tylikowski Dynamic Stability of a Nonlinear Cylindrical Shells Vol. 51 December 1984. Trans. ASME (Mechanics)
(12)Berger, M. S.; and Fife, P. C.; On Von Karman Equations and the Buckling of a Thin Elastic plate, Bull, Amer. Math. Soc., Vol.72, 1966 PP.1006-1011.
(12)钱伟长、叶开沅、圆薄板大挠度问题,物理学报,10(1954)
(13)Chien Wei-zang, Large deflection of a circular clamped plate under 154 uniform pressure, Chines J. of physics, 7 (1947)
(14)黄黔,摄动初参数法解轴对称壳几何非线性问题, 应用数学和力学 6(1986)。
(15)黄黔,轴对称壳任意大挠度问题的一阶微分方程组,最小位能原理,广义变分原理及奇异无量纲化方法。
(16)黄黔,用数值积分的初参数法解波纹管,应用数学和力学 3,1(1982)
(17)叶志明,均布载荷作用下变厚度圆球扁薄壳的非线性轴对称弯曲和稳定性,应用数学和力学 2(1988)
(18)J.Q.Ye. Postbuckling Analysis of Plates Under Combined Loads by Mixed Finite Element and Boundary Element Method, Pressure Vessel Tech. Vol.115 (1993), transactions of the ASME ( 19 ) W. Wojewodzki, R. Bukowski Influence of the Yield Function Nonlinearity and Temperature in Dynamic Buckling of Viscoplastic Cylindrical Shells, Vo1.51,(1984) Mechanics Trans of the ASME
(20)沈惠申,陈铁云 圆柱薄壳在外压作用下屈曲的边界层理论,应用数学和力学 6 (1988)
(21)Budiansky, B.and J.C. Amazigo, Initial post-buckling behavior of cylindrical shells under external pressure, J. Math. Phys, 47(1968)
(22)周承倜 薄壳弹塑性稳定性理论 国际工业出版社(1979)
(23)周承倜 复合材料迭层板的非线性动力稳定性理论 应用数学和力学 2 (1991)
(24)徐加初,王乘,刘人怀 变厚度夹层截顶扁锥壳的非线性稳定性分析 应用数学和力学 9(2000)
(25)G.F.塞蒙斯,微分方程 人民教育出版社(1981)
(26)沈惠川 弹性大挠度问题 Von Karman 方程与量子本征值问题Schr?dinser 方程的关系,应用数学和力学 5,8 (1985)
(27)沈惠川,薄壳理论中的 Schr?dinger 方程,应用数学和力学,6,10(1985)
(28)黄克智等,板壳理论,清华大学出版社(1987)
(29)郭柏灵 非线性演化方程 上海科技教育出版社 1995
(30)E.H. Dowell, Aeroelasticity of plates and shells, Noordhoff, 1975
(31)凌复华,魏焕民,初等突变理论在 Hamilton 系统中应用的一个例子力学学报 Vol. 17, No.1 Jan, 1985
(32)刘式达,刘式适,大气中对流的分岔和突变模型,力学学报,16, 1 (1984)10-18
(33)Poston, T. and Stewart, I., Catastrophe Theory and Its Applications, pitman, London(1978)
(34)凌复华, 突变理论及其应用,上海科技教育出版社, 1995
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