空间框架结构弹性动力学非传统Hamilton型变分原理
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摘要
根据古典阴阳互补和现代对偶互补的基本思想,通过罗恩提出的一条简单而统一的新途径,系统地建立了空间框架结构弹性动力学的各类非传统Hamilton型变分原理.文中首先给出空间框架结构弹性动力学的广义虚功原理的表式,然后从该式出发,不仅能得到空间框架结构弹性动力学的虚功原理,而且通过所给出的广义Legendre变换,还能系统地成对导出空间框架结构弹性动力学的5类变量、3类变量、2类变量变分原理的互补泛函,以及1类变量和相空间非传统Hamilton型变分原理的泛函.同时,通过这条新途径还能清楚地阐明这些原理的内在联系.
According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified new way proposed by Luo,the unconventional Hamilton-type variational principles for elasto-dynamics of space frame structures were established systematically. The unconventional Hamilton-type variational principle can fully characterize the initial-boundary-value problem of space frame structures’ elasto-dynamics. In this paper,an important integral relation was given,which can be considered as the expression of the generalized principle of virtual work for elasto-dynamics of space frame structures. Based on this relation,it is possible not only to obtain the principle of virtual work for elasto-dynamics of space frame structures,but also to derive systematically the complementary functionals for five-field,three-field and two-field unconventional Hamilton-type variational principles,and the functional for one-field and the unconventional Hamilton-type variational principle in phase space by the generalized Legendre transformations given in this paper. Furthermore,with this new approach,the intrinsic relationship among various principles can be explained clearly.
According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified new way proposed by Luo,the unconventional Hamilton-type variational principles for elasto-dynamics of space frame structures were established systematically. The unconventional Hamilton-type variational principle can fully characterize the initial-boundary-value problem of space frame structures’ elasto-dynamics. In this paper,an important integral relation was given,which can be considered as the expression of the generalized principle of virtual work for elasto-dynamics of space frame structures. Based on this relation,it is possible not only to obtain the principle of virtual work for elasto-dynamics of space frame structures,but also to derive systematically the complementary functionals for five-field,three-field and two-field unconventional Hamilton-type variational principles,and the functional for one-field and the unconventional Hamilton-type variational principle in phase space by the generalized Legendre transformations given in this paper. Furthermore,with this new approach,the intrinsic relationship among various principles can be explained clearly.
引文
1熊丹安.钢筋混凝土框架结构设计与实例.武汉:武汉理工大学出版社,2005(Xiong Dan’an.Design and examples for RC frame structures.Wuhan:Press of WUHANUniversi-ty of Technology,2005(in Chinese))
2方鄂华.多层及高层建筑结构设计.北京:地震工程出版社,2000(Fang Ehua.Multi and high-rise building structur-al design.Beijing:Earthquake engineering press,2000(inChinese))
3S Vasudevan,H Okada and S N.Atluri Development of a new frame finite element for crash analysis,using a mixed variational principle and rotations as independent variables.Finite Elements in Analysis and Design,1996,23(2-4):155~171
4朱杰江,徐道远,王颍.框架侧移的连续化模型分析.河海大学学报,2001,29(1):12~16(Zhu Jiejiang,Xu Daoyuan,Wang Yin.Analysis of frame lateral displacement using continuum model.Journal ofHehai University,2001,29(1):12~16(in Chinese))
5罗恩.关于线弹性动力学中各种Gurtin型变分原理.中国科学(A辑),1987,(9):936~948(Luo En.On the Gurtin-type variational principles in linear elastodynamics.Science in China(Series A),1987,(9):936~948(in Chinese))
6Luo E,Cheung Y K.On the variational principles in linear elastodynamics.Acta Mechanica Sinica,1988,4(4):337~349
7Finlayson B A,Scriven L.On the search for variational prin-ciples.Int JHeat Mass Transfer,1967,10:799
8谷迎松,杨智春,李斌.一种充分局部化的子结构并行有限元方法的变分原理.机械科学与技术2007,26(7):836~840(Gu Yinsong,Yang Zhichun,Li Bin.Variation principle for the localized FETI method.Mechanical Science and Technology for Aerospace Engineering,2007,26(7):836~840(in Chinese))
9李纬华,罗恩.弹性膜结构动力学的各类非传统Hamil-ton型变分原理.动力学与控制学报,2007,5(3):209~215(Li Weihua,Luo En.Unconventional Hamilton-type variational principles for nonlinear elastodynamics of mem-brane structures.Journal ofDynamics and Control,2007,5(3):209~215(in Chinese))
2方鄂华.多层及高层建筑结构设计.北京:地震工程出版社,2000(Fang Ehua.Multi and high-rise building structur-al design.Beijing:Earthquake engineering press,2000(inChinese))
3S Vasudevan,H Okada and S N.Atluri Development of a new frame finite element for crash analysis,using a mixed variational principle and rotations as independent variables.Finite Elements in Analysis and Design,1996,23(2-4):155~171
4朱杰江,徐道远,王颍.框架侧移的连续化模型分析.河海大学学报,2001,29(1):12~16(Zhu Jiejiang,Xu Daoyuan,Wang Yin.Analysis of frame lateral displacement using continuum model.Journal ofHehai University,2001,29(1):12~16(in Chinese))
5罗恩.关于线弹性动力学中各种Gurtin型变分原理.中国科学(A辑),1987,(9):936~948(Luo En.On the Gurtin-type variational principles in linear elastodynamics.Science in China(Series A),1987,(9):936~948(in Chinese))
6Luo E,Cheung Y K.On the variational principles in linear elastodynamics.Acta Mechanica Sinica,1988,4(4):337~349
7Finlayson B A,Scriven L.On the search for variational prin-ciples.Int JHeat Mass Transfer,1967,10:799
8谷迎松,杨智春,李斌.一种充分局部化的子结构并行有限元方法的变分原理.机械科学与技术2007,26(7):836~840(Gu Yinsong,Yang Zhichun,Li Bin.Variation principle for the localized FETI method.Mechanical Science and Technology for Aerospace Engineering,2007,26(7):836~840(in Chinese))
9李纬华,罗恩.弹性膜结构动力学的各类非传统Hamil-ton型变分原理.动力学与控制学报,2007,5(3):209~215(Li Weihua,Luo En.Unconventional Hamilton-type variational principles for nonlinear elastodynamics of mem-brane structures.Journal ofDynamics and Control,2007,5(3):209~215(in Chinese))
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