考虑初始损伤双层扁球面网壳的非线性振动
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摘要
针对损伤网壳结构,基于Lemaitre损伤理论建立建筑用钢损伤本构关系,利用该本构方程建立损伤双层扁球面网壳的非线性动力学控制方程.利用中心最大振幅作为摄动变分法的摄动参数,将损伤扁球面网壳的能量变分方程和微分方程线性化处理,并对四边夹紧固定损伤扁球面网壳的非线性振动微分方程进行一次、二次边值问题求解,得到考虑损伤扁球面网壳频率与振幅之间的特征关系,并利用计算机模拟绘制出损伤双层扁球面网壳的非线性振动频率与振幅之间的关系曲线,研究了损伤对系统固有频率的影响.结果表明损伤对结构的非线性振动影响不容小觑.
Based on Lemaitre damage theory,the constitutive relation of steel damage for building was established,and the nonlinear dynamic governing equations of double-layer reticulated spherical shallow shells with damage were presented using the constitutive equation. The maximum amplitude of the center was taken as the perturbation parameter,and the vibrational equation and differential equation were linearized by the perturbation-variational method.Then the nonlinear vibration equation of the spherical reticulated shells with damage is solved under the fixed clamping boundary condition. The characteristic relation of the minimum natural frequency and the maximum amplitude in the center is obtained and it was simulated by computer. It is discovered that the effect of damage accumulation on the natural frequency of the system by computer simulation should be considered.
Based on Lemaitre damage theory,the constitutive relation of steel damage for building was established,and the nonlinear dynamic governing equations of double-layer reticulated spherical shallow shells with damage were presented using the constitutive equation. The maximum amplitude of the center was taken as the perturbation parameter,and the vibrational equation and differential equation were linearized by the perturbation-variational method.Then the nonlinear vibration equation of the spherical reticulated shells with damage is solved under the fixed clamping boundary condition. The characteristic relation of the minimum natural frequency and the maximum amplitude in the center is obtained and it was simulated by computer. It is discovered that the effect of damage accumulation on the natural frequency of the system by computer simulation should be considered.
引文
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[10]杜国君,胡宇达.阶梯圆形和环形薄板的大幅度基本振动[J].应用力学学报,1998(1):73-80,147.Du Guojun,Hu Yuda.Large amplitude fundamental vibration of step circular and annular thin plates[J].Chinese Journal of Applied Mechanics,1998(1):73-80,147(in Chinese).
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[2]刘人怀,朱高秋.夹层圆板大挠度问题底进一步研究[J].应用数学和力学,1989,10(12):1099-1106.Liu Renhuai,Zhu Gaoqiu.Further study on large deflection of circular sandwich plates[J].Applied Mathematics and Mechanics,1989,10(12):1099-1106(in Chinese).
[3]Vincent J J.The bending of a thin circular plate[J].Philosophial Magazine,1931,12:185-196.
[4]Chien W.Large deflection of a circular clamped plate under uniform pressure[J].Chinese Journal of Physics,1947,7(2):102-113.
[5]叶开沅,刘人怀,平庆元,等.圆底扁薄球壳的非线性稳定问题-Ⅰ.在对称线布载荷作用下的圆底扁薄球壳的非线性稳定问题[J].科学通报,1965(2):142-145.Ye Kaiyuan,Liu Renhuai,Ping Qingyun,et al.Nonlinear stability of flat spherical shells with circular bottom-Ⅰ.Nonlinear stability of flat spherical shells with circular bottom under symmetric linear loading[J].Chinese Science Bulletin,1965(2):142-145(in Chinese).
[6]叶开沅,刘人怀,张传智,等.圆底扁薄球壳的非线性稳定问题-Ⅱ.圆底扁薄球壳在边缘力矩作用下的非线性稳定问题[J].科学通报,1965(2):145-147.Ye Kaiyuan,Liu Renhuai,Zhang Chuanzhi,et al.Nonlinear stability of flat spherical shells with circular bottom-Ⅱ.Nonlinear stability of thin spherical shells with circular bottom subjected to edge moment force[J].Chinese Science Bulletin,1965(2):145-147(in Chinese).
[7]刘人怀.圆底扁薄球壳的非线性稳定问题-Ⅲ.在内边缘均布力矩作用下中心开孔圆底扁球壳的非线性稳定问题[J].科学通报,1965(3):253-255.Liu Renhuai.Nonlinear stability of flat spherical shells with circular bottom-Ⅲ.Nonlinear stability of a flat spherical shell with a aentral opening and a circular bottom under uniformly distributed moment force at the inner edge[J].Chinese Science Bulletin,1965(3):253-255(in Chinese).
[8]郑晓静,周又和.集中载荷作用下弹性地基圆薄板大挠度问题的精确解[J].力学学报,1988(2):161-172.Zheng Xiaojing,Zhou Youhe.On the exact plate on elastic foundation under a concentrated load[J].Chinese Journal of Theoretical and Applied Mechanics,1988(2):161-172(in Chinese).
[9]郑晓静,周又和.复合载荷下圆薄板大挠度问题的精确解[J].中国科学(A辑数学物理学天文学技术科学),1986(10):1045-1056.Zheng Xiaojing,Zhou Youhe.On convergence of interpolated iterative method of geometrically nonlinear equations of circular plates[J].Science in China,Ser.A,Mathematics,Physics,Astronomy,Science and Technology,1986(10):1045-1056(in Chinese).
[10]杜国君,胡宇达.阶梯圆形和环形薄板的大幅度基本振动[J].应用力学学报,1998(1):73-80,147.Du Guojun,Hu Yuda.Large amplitude fundamental vibration of step circular and annular thin plates[J].Chinese Journal of Applied Mechanics,1998(1):73-80,147(in Chinese).
[11]王林祥,王新志,邱平.圆薄板非对称大变形弯曲问题[J].应用数学和力学,1992(12):1103-1115.Wang Linxiang,Wang Xinzhi,Qiu Ping.Large deflection problem of circular plates with variable thickness[J].Applied Mathematics and Mechanics,1992(12):1103-1115(in Chinese).
[12]王新志,王林祥,徐鉴.圆薄板非对称的非线性问题[J].甘肃工业大学学报,1989(3):90-100.Wang Xinzhi,Wang Linxiang,Xu Jian.Nolinear problem of thin circular plates[J].Journal of Gansu University of Technology,1989(3):90-100(in Chinese).