Bending deformation of four-bar tensegrity element

四杆张拉整体单元的弯曲变形 Bending deformation of four-bar tensegrity element first-author 罗阿妮（1978-），女，陕西人，副教授，2003年于哈尔滨工程大学获得硕士学位，2006年于哈尔滨工业大学获得博士学位，主要从事张拉整体结构和机构、空间可展结构、轻量化机构及结构设计理论的研究。E-mail： luoani@hrbeu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627040&type= 22.01332855 32.00408936 罗阿妮 （1978-），女，陕西人，副教授，2003年于哈尔滨工程大学获得硕士学位，2006年于哈尔滨工业大学获得博士学位，主要从事张拉整体结构和机构、空间可展结构、轻量化机构及结构设计理论的研究。E-mail： luoani@hrbeu.edu.cn luoani@hrbeu.edu.cn corresp E-mail： liuheping@hrbeu.edu.cn E-mail： liuheping@hrbeu.edu.cn 刘贺平（1975-），男，黑龙江佳木斯人，教授，2003年、2005年于哈尔滨工程大学分别获得硕士、博士学位，主要从事张拉整体机器人、张拉整体结构、机械运动学和动力学等方面的研究。E-mail： liuheping@hrbeu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627057&type= 22.01332855 32.00396347 刘贺平 （1975-），男，黑龙江佳木斯人，教授，2003年、2005年于哈尔滨工程大学分别获得硕士、博士学位，主要从事张拉整体机器人、张拉整体结构、机械运动学和动力学等方面的研究。E-mail： liuheping@hrbeu.edu.cn liuheping@hrbeu.edu.cn 哈尔滨工程大学 机电工程学院，黑龙江 哈尔滨 150001 哈尔滨工程大学 机电工程学院，黑龙江 哈尔滨 150001 College of Mechanical and Electrical Engineering， Harbin Engineering University， Harbin 150001， China College of Mechanical and Electrical Engineering， Harbin Engineering University， Harbin 150001， China 为了扩大张拉整体的内部空间，实现展开前空间占用小，展开后内部空间的有效利用率高，对张拉整体轴向折叠方式及折展特性进行了研究。根据张拉整体的理论基础，介绍了节点矩阵、连接矩阵和构件矩阵，分析了张拉整体轴向折叠的意义以及存在的问题，并介绍了两种张拉整体沿轴向折叠的方法。在这两种折叠方式的基础上提出弯杆张拉整体，并给出弯杆张拉整体节点坐标的求解方法。最后，分析弯杆张拉整体在完全展开状态直至上下水平面重合时的弹性变形量，利用有限元软件分析得到模型完全压缩时节点的弯曲角度临界值。高度为270 mm，半径为200 mm的四杆张拉整体在轴向完全压缩时，上、下底面的相对偏转角为57°，杆长压缩率为13%。利用竹节作为杆构件时弯曲节点连接两杆构件所成角度不得大于90°。张拉整体直杆变弯有效地解决了杆构件间的干涉问题，内部有效利用空间增大，实现了轴向折叠。 In order to expand the internal space in tensegrity elements， realize small space occupation before deployment， and effectively utilize the internal space after deployment， the axial folding mode and folding characteristics of tensegrity elements are studied. First， according to the tensegrity theory， the node， connection， and component matrices are introduced. Then， the axial-folding significance of tensegrity elements and the problems encountered during axial folding are proposed， and two axial folding methods of tensegrity elements are introduced. Then， on the basis of the two folding methods， the bending bar tensegrity is proposed， and the solution method of the node coordinates of the bending bar tensegrity is given. Finally， the elastic deformation of the bending bar tensegrity element in the fully expanded state until the upper and lower horizontal planes coincide is analyzed， and the limit-bending angle of the node is obtained using the finite element software. When the four-bar tensegrity element with a height of 270 mm and radius of 200 mm is fully compressed， the relative deflection angle of the upper and lower bottom surfaces is 57°， and the compression ratio of the bar length is 13%. When bamboo material is used for a bar member， the angle of the bending joint connecting the two bar members is at most 90°. The straight to bending bar tensegrity effectively avoids the interference between the bar components， which increases the effective utilization of internal space and realizes axial folding. 可展机构 张拉整体 空间利用率 弯曲变形 折叠 deployable mechanism tensegrity structure space utilization bending deformation folding 1 引　言 1 张拉整体是由连续受拉索和离散的受压杆构成的空间结构 ［ 1 1 ］ ，所有受拉索连续、受压杆分散，且只承受轴向力，因而材料利用率高。在稳定构型中，杆、索构件的自由度均为零，并且破坏掉任意一根杆、索构件，整个结构都会发生坍塌。结构既不是冗余的，也不是过约束的，具有一定的连续性 ［ 2 2 ］ ，可以通过调整个别构件来改变整体刚度。不论是自然状态下还是受到冲击振动，不利因素都可以自动分散至整个结构，找到稳定状态。 张拉整体作为一种空间可展结构，具有独特的构型方式，被广泛应用于伸展臂 ［ 3 3 - 4 4 3-4 ］ 、载体运输工具 ［ 5 5 ］ 、桥梁、建筑 ［ 6 6 - 7 7 6-7 ］ 、以及多种艺术品 ［ 8 8 ］ 中。Schorr ［ 9 9 ］ 等人利用曲杆代替直杆，通过控制模型的整体重心使张拉整体实现二维平面的滚动，张拉整体曲杆的建立方式为弯杆张拉整体弯角节点提供了帮助。Li ［ 10 10 ］ 等人将张拉整体应用于交通吸振，提出一种混合张拉整体支撑柱，该支撑柱可有效减少因车辆碰撞或其他因素产生的不利振动，耗散因振动而产生的能量，将张拉整体优良的缓冲、吸振及自稳定性应用到实际工程中。Liu ［ 11 11 ］ 等人将球形张拉整体部分索构件用弹簧代替，增强了张拉整体的缓冲性能，为张拉整体压缩提供一种方式，但整体刚度变小、稳定性变差。González等人 ［ 12 12 ］ 通过驱动部分构件或全部构件实现模型从初始状态至完全压缩，找出模型在压缩过程中始终保持稳定的一种方式，为研究张拉整体在压缩过程中的节点坐标变化奠定基础。Kan ［ 13 13 ］ 等人分析了多层张拉整体节点的展开速度和展开过程中的振幅和振动频率。他们的分析方法同样可应用于研究张拉整体的压缩过程，如压缩速度对结构的动态响应、压缩过程系统的稳定性等。 张拉整体具有质量轻、折展比大、缓冲抗振能力强、材料利用率高及使用条件不严格等特点，因此成为了空间结构领域的研究热点。但是张拉整体自身几何尺寸的限制，在某些情况下载物与压杆间会产生干涉，如果偏转角选择不当，自身杆构件也会发生干涉。这就对如何保持原有的结构特性，增大内部有效利用空间，提高缓冲性能，实现一定的自身折展 ［ 14 14 - 15 15 14-15 ］ 提出了更高的要求。本文对张拉整体轴向折叠方式及折展特性进行了理论分析及实验研究，提出了两种张拉整体轴向折叠的方法，不仅避免了杆构件间的干涉，而且提高了有效利用空间。 2 张拉整体基本单元的构建 1 2.1　四杆张拉整体基本单元 2 图1 图1 为四杆张拉整体基本单元（彩图见期刊电子版）。构成单元的基本要素是：下底面圆上均匀分布4个节点 <math id="M1"><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627069&type= 3.04800010 3.72533321 ， <math id="M2"><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627083&type= 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627243&type= 98.63666534 30.14133072 . 2.3　连接矩阵 2 连接矩阵是将节点与节点间相互连接的介质，它能将节点坐标与矢量联系在一起。构件的两端点分别为起始点和终点，对于4杆张拉整体，每节点有4个构件与之相连，所组成的连接矩阵为 <math id="M17"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>i</mi><mi>j</mi><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mo>×</mo><mn mathvariant="normal">16</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627254&type= 11.26066685 3.97933316 。连接杆构件与杆构件间的矩阵成为杆连接矩阵 <math id="M18"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">B</mi><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mo>×</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627264&type= 10.41399956 3.97933316 ，连接索构件与索构件间的矩阵成为索构件矩阵 <math id="M19"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mo>×</mo><mn mathvariant="normal">12</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627275&type= 11.26066685 3.97933316 ，它们的关系如下： <math display="block" id="M20"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>i</mi><mi>j</mi><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mo>×</mo><mn mathvariant="normal">16</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">B</mi><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mo>×</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">8</mn><mo>×</mo><mn mathvariant="normal">12</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627287&type= 40.97866440 4.74133301 . 连接矩阵中元素仅由0， - 1，1构成， 表1 表1 为连接矩阵各元素的含义。 矩阵中代号 所表达的含义 -1 1 0 构件起始点与节点相连 构件终止点与节点相连 构件未连接位置 则上图中的杆连接矩阵为： <math display="block" id="M21"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627306&type= 44.61933517 38.43866730 . 索连接矩阵为： <math display="block" id="M22"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627316&type= 116.41667175 38.43866730 . 2.4　矢量矩阵 2 张拉整体构件矢量是将节点向向量进行转换的一种方式，通过终点坐标与起始点坐标做差得到构件矢量。构件矢量的向量长度即为杆构件长度，起始点向量已知，便可以表示出杆构件的空间位置。 图2 图2 为四杆张拉整体构件矢量示意图。 杆构件的矢量矩阵为： <math display="block" id="M23"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">B</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">]</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi>R</mi></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi>R</mi></mtd></mtr><mtr><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627341&type= 97.19733429 33.52799988 . 杆构件矢量矩阵与节点矩阵和连接矩阵的关系为： <math id="M24"><mi mathvariant="bold-italic">B</mi><mo>=</mo><mi mathvariant="bold-italic">N</mi><mo>⋅</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627352&type= 17.10266685 4.06400013 。 索构件的矢量矩阵分为上、下水平索以及分布在上下水平索中间的斜索3部分，其表达式为： <math display="block" id="M25"><mtable><mtr><mtd><mi mathvariant="bold-italic">S</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">]</mo><mo>=</mo><mo> </mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>R</mi></mtd><mtd><mo>-</mo><mi>R</mi></mtd><mtd><mi>R</mi></mtd><mtd><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot><mi>R</mi></mtd><mtd><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><mi>R</mi></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd></mtr><mtr><mtd><mi>R</mi></mtd><mtd><mo>-</mo><mi>R</mi></mtd><mtd><mo>-</mo><mi>R</mi></mtd><mtd><mi>R</mi></mtd><mtd><mo>-</mo><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mo stretchy="false">(</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><mi>R</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627360&type= 148.42066956 33.10466385 . 索构件矢量矩阵与节点矩阵和连接矩阵的关系为： <math id="M26"><mi mathvariant="bold-italic">S</mi><mo>=</mo><mi mathvariant="bold-italic">N</mi><mo>⋅</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627371&type= 16.84866714 4.06400013 。 3 四杆张拉整体轴向折叠 1 3.1　轴向折叠的意义 2 张拉整体是一种自平衡结构，任意构件受力时，该杆件的受力变形会发散至整体结构，使整体结构分散均匀受力，具有良好的稳定性。受拉构件分布在周围，受压构件分布在内部，有效地减小结构内部的损伤程度。由于只承受轴向力，并且构件材料利用率高，张拉整体的构件数目少、质量轻。 四杆张拉整体在构型不同时其力学性能是不同的，它共有3种稳定构型。其中，第二种稳定构型杆构件间是相互干涉的，并且杆数越多相互干渉越严重。为了使张拉整体结构使用前占用空间小，需对结构进行轴向压缩。对张拉整体轴向加力时，索构件进一步被拉伸，杆构件进一步被压缩，但杆构件的轴向变化率较小，超过某根杆构件可承受强度，还可能造成整体模型的破坏。 3.2　轴向压缩方式 2 张拉整体轴向压缩过程中，杆构件进一步被压缩，索构件进一步被拉伸，因此轴向压缩主要有两种方案： 方案一：将受拉索构件更换为具有较大弹性的构件，例如在索构件中添加弹簧，将张拉整体斜索用弹簧代替 ［ 16 16 ］ ，如 图3 图3 所示，也可以将水平索用弹簧代替。 方案二：将杆构件复杂化，将单根直杆变成带有套管的组合构件，套管中间放置弹簧 ［ 16 16 ］ 结构，两端分别与杆构件相连，实现杆构件的压缩，如 图4 图4 所示。 这两种方法均添加了额外的弹簧结构，当模型下降至全模型高度的60%时，杆构件间发生干渉，轴向无法继续压缩。 4 弯杆张拉整体设计 1 为了解决上述问题，本文将张拉整体直杆变成弯杆。张拉整体直杆变为弯杆不仅可以改变张拉整体承受轴向力时杆构件的受力方式，而且解决了直杆间干涉的问题。当施加轴向力时，整体构件会随着外力的增大发生明显的形变，这就使张拉整体在外力的作用下储存了一部分能量。很多领域将张拉整体作为缓冲装置，如NASA航天局外太空探测器的缓冲投放装置、深海领域探测器的缓冲投放装置、陆地补给箱的缓冲投放装置等。张拉整体直杆变弯可实现轴向压缩，减小占用空间，并且提升缓冲性能，为张拉整体的伸展性及折叠性提供了一种新的思路。 4.1　模型节点的构建方式 2 弯杆张拉整体是在传统直杆平行面张拉整体结构的基础上，添加弯曲角度 <math id="M27"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627409&type= 1.77800000 2.79399991 ，将内部的直杆变弯。张拉整体弯杆节点在整体高度 H 的一半处，这样建立起来的弯杆张拉整体仍然为完美的对称结构，便于力学研究。 弯角处节点坐标的求解方式有二种。首先，为保证结构对称性，弯角节点选在 α = H /2平面处，且过杆构件端点 <math id="M28"><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627421&type= 10.66800022 4.14866638 的中垂面 β 上。在这种条件约束下，弯角节点坐标被限制在一条直线上。 图5 图5 为弯角节点坐标与两平面间的关系，即直线 Am 为两平面相交所得。将直线 <math id="M29"><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627428&type= 9.14400005 3.72533321 的中点设为 A ，在交线上除 A 点外任取一点 m 与两端点相连会形成 <math id="M30"><mi mathvariant="normal">∠</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><mi>m</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627435&type= 15.49400043 4.06400013 。预设弯杆张拉整体弯曲角度，可以找到确定的两个点，令 m '为 m 点关于点 A 的对称点。为了增大内部空间， m 点与 m '点中有一点是向外弯曲的，另一点则是向内部弯曲（舍弃）。这样每给出一个确定的弯曲角度就会的到一个确定的弯点，这种思路适用于任意杆数 P 的弯杆张拉整体。 其次，为了方便实现弯杆张拉整体模型的绘制，通过向量合成方式得到图像。欲得到 m 点的准确坐标，需要得到 <math id="M31"><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>j</mi></mrow></msub><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627457&type= 6.18066692 3.72533321 ， <math id="M32"><mi mathvariant="normal">∠</mi><mi>m</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627467&type= 12.36133385 4.06400013 。通过水平面的法向量，即向量［0，0，1］与 <math id="M33"><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627475&type= 9.48266602 3.72533321 方向向量做外积得到 <math id="M34"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627505&type= 5.84200001 2.79399991 方向向量，方向向量取模后便得到单位方向向量。该向量长度为： <math display="block" id="M35"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub></mrow></mfenced><mo>/</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∠</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><mi>m</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub><mo>/</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627490&type= 59.09733200 4.91066647 . A 点坐标为 <math id="M36"><mo stretchy="false">(</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627498&type= 20.91266632 4.48733330 ，从 A 点出发做向量 <math id="M37"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627505&type= 5.84200001 2.79399991 ，向量终点 m 即为弯杆张拉整体弯角处坐标。通过绘图命令将中点 m 分别与原直杆两端点相连，得到弯杆张拉整体结构示意图。该方法同样适用于任何构型，任何杆数的张拉整体。 弯杆角度为150°， H =270 mm， R u = R d =200 mm， P =4的3种结构示意图如 图6 图6 所示，从左到右单元内转角分别为135°，180°，225°。 4.2　基本参数之间的关系 2 张拉整体构型具有很强的规律性，已知一些参数，如上底面半径 R u ，下底面半径 R d ，高度 H ，偏转角 φ ，便可以得到 p- 1种确定的构型。当构型确定后，张拉整体结构也唯一确定。然后，进一步给出弯曲方向及弯曲角度 <math id="M38"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627585&type= 1.77800000 2.79399991 ，那么弯角处的节点坐标也唯一。这里从节点坐标出发，推导出弯杆长度、弯曲角度和斜索长度之间的关系。 在得到弯曲前杆构件长度后，根据斜杆弯曲角度得到每一根弯杆的长度，如 图7 图7 所示。利用相同的方法对任意 p 杆，任意 p- 1种构型进行推导，得到杆索长度的一般表达式。第 j 种构型节点1斜索与节点 p- （ j- 1）相连，节点1杆构件与节点 p +1相连，单元内转角 <math id="M39"><mi>φ</mi><mo>=</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627535&type= 24.21466637 3.97933316 ，弯杆角度为 <math id="M40"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627585&type= 1.77800000 2.79399991 。节点1、节点 p +1、节点 p- （ j- 1）的坐标如下： <math display="block" id="M41"><mtable><mtr><mtd><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">、</mi><mi>p</mi><mo>-</mo><mo stretchy="false">(</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>i</mi><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>p</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>i</mi><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>h</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627548&type= 68.83399963 32.85066605 其中： <math id="M42"><mi>i</mi><mo>=</mo><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>p</mi><mo>-</mo><mo stretchy="false">(</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627554&type= 33.78200150 3.97933316 。 杆构件长度为： <math display="block" id="M43"><mtable columnalign="left"><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><msup><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msup><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627570&type= 63.66933060 11.51466751 斜索长度为： <math display="block" id="M44"><mtable><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi><mo>-</mo><mo stretchy="false">(</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><msup><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">}</mo></mrow><mrow><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msup><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627578&type= 56.81133270 18.37266731 弯杆张拉整体弯曲角度为 <math id="M45"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627585&type= 1.77800000 2.79399991 ，由三角函数关系得每一根弯杆长为： <math display="block" id="M46"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627593&type= 11.43000031 4.82600021 = <math display="block" id="M47"><mfrac><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">π</mi><mo>-</mo><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627598&type= 18.71133423 19.81200027 = <math display="block" id="M48"><mfrac><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mfrac><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627604&type= 14.39333248 19.81200027 . 4.3　结构分析 2 在张拉整体模型中节点所连接杆构件的数目称为张拉整体阶数，一阶、二阶张拉整体如 图8 图8 所示。一阶张拉整体每个节点有且仅有一根杆构件相连，二阶张拉整体每个节点有且仅有两个杆构件相连。张拉整体的复杂程度随着阶数的增加而提高。 弯杆张拉整体在直杆基础上，添加角度利用弯杆代替直杆，其复杂程度介于一阶与二阶之间。在中间平面弯曲节点处，每个节点连接两根杆构件，而对于未发生变化的上、下水平面处的节点仍连接一根杆构件。在同一张拉整体模型中，既有连接一根杆构件的节点，也有连接两根杆构件的节点，故可将弯杆张拉整体理解为一阶向二阶过渡的一种结构。相比于直杆张拉整体而言，弯杆张拉整体在中间平面增加了弯角节点，因而原有的节点矩阵、连接矩阵和构件矩阵维度等均会发生变化。弯曲节点表达式稍复杂，这里仅用 <math id="M49"><msub><mrow><mi>m</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627622&type= 4.06400013 3.72533321 ， <math id="M50"><msub><mrow><mi>m</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627628&type= 3.97933316 3.72533321 表示。 对于 P 杆直杆张拉整体，节点数为2 P ，节点矩阵维数为3 <math id="M51"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627658&type= 0.84666669 2.79399991 2 P ；弯杆张拉整体将直杆一分为二，因而节点数变为3 P ，节点矩阵维数为3 <math id="M52"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627687&type= 0.84666669 2.79399991 3 P 。当 P =4， j =1时，节点矩阵为： <math display="block" id="M53"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow></msub><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">]</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>x</mi></mrow></msub></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>R</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi>R</mi></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>y</mi></mrow></msub></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>R</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>h</mi></mtd></mtr></mtable></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627646&type= 130.55599976 34.88266373 直杆连接矩阵2 P 个节点， P 根杆，3 P 根索，直杆连接矩阵维数为：2 P <math id="M54"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627687&type= 0.84666669 2.79399991 4 P ；弯杆张拉整体索数量不变，杆数由 P 变为2 P ，因此弯杆连接矩阵维数为：3 P <math id="M55"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627658&type= 0.84666669 2.79399991 5 P 。当 P =4， j =1时，连接矩阵为： <math display="block" id="M56"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627662&type= 10.58333302 3.97933316 <math display="block" id="M57"><mfenced open="[" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn 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mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627666&type= 144.86466980 50.03799820 . 构件矩阵是节点矩阵与连接矩阵的乘积。直杆张拉整体共 P 根杆构件，3 P 根索构件，共4 P 个向量，直杆构件矩阵维数为：3 <math id="M58"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627687&type= 0.84666669 2.79399991 4 P ；弯杆张拉整体共2 P 根杆，索构件数量不变，弯杆构件矩阵维数为：3 <math id="M59"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627687&type= 0.84666669 2.79399991 5 P 。当 P =4， j =1时，构件矩阵为： <math id="M60"><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627678&type= 2.37066650 2.79399991 = <math id="M61"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627681&type= 6.94266701 3.80999994 <math id="M62"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627687&type= 0.84666669 2.79399991 <math id="M63"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627692&type= 6.51933336 3.97933316 ，仍然是弯杆节点矩阵与弯杆连接矩阵的乘积。 4.4　压缩过程中的弹性分析 2 当水平索、斜索长度不变时，对张拉整体结构进行轴向压缩，杆构件所连接的两节点间距离是不断变化的，压缩过程如 图9 图9 所示。最大距离为未开始形变时两节点的距离，最小距离则为上下水平面在外力作用下达到同一水平面时的距离。若索的形变量远远小于杆在压缩过程中的形变量，则得到初始位置和末态位置时的节点坐标，从而计算得到杆构件在压缩过程中的弹性形变。 对任意 p 杆，任意 <math id="M64"><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627702&type= 8.55133343 3.64066648 种构型进行推导，得到结构进行轴向压缩过程中杆、索长度及压缩量的表达式。任何构型节点1均与节点 <math id="M65"><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627707&type= 8.55133343 3.72533321 以杆构件相连，但斜索连接方式与构型有关，第 <math id="M66"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627713&type= 1.10066664 3.64066648 种构型节点1与 <math id="M67"><mn mathvariant="normal">2</mn><mi>p</mi><mo>-</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627717&type= 16.59466553 3.72533321 相连，同一根杆构件，下底面节点与上底面在下底面投影和下底面圆心间形成的角度称为单元内转角（下文均称单元内转角），其大小为： <math id="M68"><mi>φ</mi><mo>=</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi mathvariant="normal">p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627721&type= 24.55333328 3.97933316 。节点坐标如下： <math display="block" id="M69"><mtable><mtr><mtd><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mo>-</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>h</mi></mtd></mtr></mtable></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627725&type= 57.99666977 22.09799957 弯杆张拉整体轴向压缩过程中杆构件连接两点间距离相互靠近，而绳索伸长量远远小于杆构压缩量，在分析过程中将绳索受拉刚度看作无限大，即绳索不可伸长。在轴向压缩过程中，上下两端面有相对转动趋势。因此，在结构从高 h 压缩至上下水平面重合时，应综合分析高度和变化过程中偏转角的影响，如 图10 图10 所示。 图10 图10 中，上底面节点 C 在下底面投影为 <math id="M70"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627735&type= 3.72533321 3.72533321 ，下底面节点 A 、投影点 <math id="M71"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627738&type= 3.72533321 3.72533321 及下底面圆心 O 形成的夹角为 <math id="M72"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627742&type= 3.47133350 4.65666676 ；当高度被完全压缩 C 点落在下底面 <math id="M73"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627746&type= 3.72533321 3.72533321 ，投影点 <math id="M74"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627750&type= 3.72533321 3.72533321 ， <math id="M75"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627755&type= 3.72533321 3.72533321 及底面圆心角 O 形成的夹角为 <math id="M76"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627758&type= 3.47133350 4.65666676 。 压缩过程中，上底面相对于下底面做逆时针方向转动。上、下底面半径一定，高度越高，压缩时的相对转角越大，但高度过大时，转动至某一位置，继续压缩，斜索便会松弛。在压缩过程中， C 点运动至线段 <math id="M77"><mi>M</mi><mi>M</mi><mi>'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627762&type= 7.45066643 3.30200005 中间某一位置，假设运动至图中 C 3 点，此时斜索 <math id="M78"><mi>A</mi><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627767&type= 5.41866684 2.79399991 为直角△ <math id="M79"><mi>A</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627771&type= 10.49866676 3.72533321 斜边，两直角边分别为 <math id="M80"><mi>A</mi><mi>M</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627776&type= 6.94266701 3.30200005 和 <math id="M81"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi>M</mi><mi>'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627779&type= 8.38199997 4.23333359 ，此时为斜索最后绷紧时刻，继续压缩两直角边长度均会缩短，因而超过这一临界位置斜索会松弛，张拉整体结构松垮，失去轴向力而快速轴向压缩，结构不稳定。当发生上述不稳定情况时，杆长的压缩量超过整根原始杆长的50%，在实际模型搭建过程中是不会出现的，实际压缩过程中某根索发生松弛的主要原因是搭建模型精度不够高。只考虑稳定压缩过程，即 C 点不会转至 C 3 便可完成结构完全压缩。 完全压缩后斜索 <math id="M82"><mi>A</mi><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627783&type= 5.41866684 2.79399991 变化为 <math id="M83"><mi>A</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627787&type= 6.60400009 3.72533321 ， <math id="M84"><mo>△</mo><mi>A</mi><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627792&type= 13.20800018 3.97933316 的两边 <math id="M85"><mi>O</mi><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627798&type= 5.75733376 2.87866688 和 <math id="M86"><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627804&type= 6.60400009 3.72533321 长度为端面半径 R ，由此得到： <math display="block" id="M87"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">∠</mi><mi>A</mi><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfrac><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mo>-</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi><mo>⋅</mo><mi>R</mi></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mo>-</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627810&type= 68.41066742 24.80733490 则在压缩过程中上下端面的相对旋转角度为： <math display="block" id="M88"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627816&type= 29.63333130 4.74133301 . 同理可以得出 <math id="M89"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627822&type= 3.47133350 4.65666676 。 C 1 点为 C 点在下底面的投影， C 1 点坐标如下： <math display="block" id="M90"><msub><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627828&type= 65.53199768 17.44133186 . 则向量 <math id="M91"><mi mathvariant="bold-italic">A</mi><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627834&type= 6.51933336 3.72533321 的长度为： <math display="block" id="M92"><mtable><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><msup><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><msup><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mi>p</mi><mo>+</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>j</mi><mi mathvariant="normal">π</mi><mo>/</mo><mi>p</mi></mrow></mfenced><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">}</mo></mrow><mrow><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msup><mo>.</mo><mtext>         </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">20</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627840&type= 76.87733459 18.79599953 <math id="M93"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627846&type= 3.47133350 4.65666676 为： <math display="block" id="M94"><mtable><mtr><mtd><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfrac><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi><mo>⋅</mo><mi>R</mi></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627850&type= 52.40866470 26.92399979 在旋转过程中上下端的面相对转角为： <math display="block" id="M95"><mtable><mtr><mtd><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mo>-</mo><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627856&type= 52.74732971 35.05199814 当张拉整体结构完全压缩， C 点落在 C 2 点，杆构件最终位置为线段 <math id="M96"><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627862&type= 6.18066692 3.72533321 ， C 2 点坐标，压缩后杆长 <math id="M97"><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627868&type= 6.18066692 3.72533321 为： <math display="block" id="M98"><mtable columnalign="left"><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>R</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">∠</mi><mi>A</mi><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>R</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">∠</mi><mi>A</mi><mi>O</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mo> </mo><mfenced open="|" close="|" 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mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627886&type= 93.98000336 25.82333374 . 将 <math id="M101"><mi>P</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627892&type= 9.14400005 2.79399991 ，高度 h =270 mm，上下底面半径 <math id="M102"><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">200</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627897&type= 20.57399940 3.72533321 mm的第一种构型代入上式，通过理论计算分别求得偏转角 <math id="M103"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627903&type= 3.47133350 4.65666676 =56.79°，压缩率 <math id="M104"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627909&type= 1.43933344 2.79399991 =13.07%。 动力学软件柔性建模如 图11 图11 所示，测量杆构件从初始状态至完全压缩长度的变化量如 图12 图12 所示。通过动力学软件柔性建模，得到偏转角和压缩率分别为56.6°和13.46%。 搭建相同参数的张拉整体模型，并对它进行轴向压缩，测量偏转角及杆长变化率，如 图13 图13 所示。压缩前杆长430 mm，压缩后杆长370 mm，得到杆长压缩率为13.95%，端面相对偏转角为50°。 通过理论计算和动力学柔性建模对相同参数模型进行分析，误差控制在0.5%以内，以验证推导偏转角和杆长压缩率表达式的正确性。微小误差来源是仿真软件本身算法或刚体间的接触力等不确定因素。而实物模型上、下端面的偏转角误差在7°，偏大，可能是由于压缩过程中受拉索被拉长或微小误差的累计造成的。 4.5　压缩过程受力分析 2 仅得到弯杆张拉整体杆构件两节点间的相对位移长度，还不足得到弯杆的形变情况，形变情况还与弯曲角度，材料基本属性，如弹性模量、泊松比等有关。假设两点间距离为400 mm，形变后的长度为280 mm，分别对90°，120°，150°的弯杆张拉整体进行分析。 实际分析中，弯角通过三维建模后3D打印得到，其打印材料为光敏树脂。通过施加力得到破坏时可施加的最大外力。 图14 图14 采用一端固定一段加力的方式。考虑到测量关节强度，将实际测量过程中的竹杆变为金属铝管。 对模型加力至28.5 N，外力关节并未发生变形，此时金属管发生明显弯曲，说明竹结构先发生断裂，弯曲关节强度大于杆构件强度。 在有限元软件中关节材料密度为3×10 2 <math id="M105"><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi><mo>/</mo><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23627946&type= 9.56733322 4.14866638 ，杨氏模量为2×10 5 MPa，泊松比为0.3，屈服极限为50 MPa，初始应变为零；竹节类材料的参数则与多种因素有关，例如：竹节生长环境、竹节受光照时间、以及生长海拔等，另外对于同一根竹节不同部位的材料参数也是不同的，这里取竹节平均值近似研究。密度为1.1×10 2 kg/m 3 ，杨氏模量为9.6×10 3 ，泊松比为0.3，屈服极限为30 MPa，初始应变为零。建立长度相同弯曲角度不同的弯角模型，竹杆最大受力点产生的应力如 图15 图15 所示。 1;2;3; 由 图15 图15 可知，最大应力值发生在弯角关节与杆构件连接处外侧，最大应力值分别为46，44，30 MPa。竹杆材料的屈服极限近似为30 MPa，由仿真结果可知，弯杆所成角度越小，材料越不易发生破坏，轴向折展性越好。其他尺寸的弯杆变形张拉整体可根据相同方法得到弯曲角度极限值，弯角小于极限值时变形四杆张拉整体均可完全压缩。对于上述材料，弯杆角度不大于90°时能够实现轴向完全压缩。 4.6　实物搭建 2 综合上述分析论述，结合张拉整体现有的理论基础，分别搭建单层和多层四杆弯曲张拉整体模型，实现了它们的轴向压缩与自动展开，如 图16 图16 所示。此方法可以用于搭建任意层数的弯杆张拉整体结构。 5 结　论 1 本文针对张拉整体基本单元的要求，即张拉整体展开前占用空间小，展开后空间利用率高，提出了张拉整体沿轴向进行折叠压缩的必要性。给出了索伸长和杆压缩两种实现轴向折叠的方式，并将两种折叠方式进行对比，进而提出了弯杆张拉整体，得出了弯杆张拉整体节点坐标的构建方式和节点矩阵与各参数间的关系。对弯杆张拉整体压缩过程的杆长压缩率和上、下端面相对偏转角度进行理论推导和动力学柔性建模分析，将两种结果进行对比，验证了结果的正确性。对材料强度进行校核，利用有限元找出特定材料下弯曲角度的极限应力值和受力形变。结果表示：将直杆张拉整体变为弯杆张拉整体，在没有添加其他构件的基础上实现了轴向完全压缩，扩大内部空间的利用率，有效解决了杆构件间的干涉问题。高度为270 mm，半径为200 mm的四杆张拉整体轴向完全压缩时，上、下底面的相对偏转角为57°，杆长压缩率为13%。利用竹节作为杆构件时，弯曲节点连接两杆构件所成角度不得大于90°。 参考文献 ZHANG J Y ， OHSAKI M . Regular Truncated Tetrahedral Structures ［M］. Tokyo ： Springer Japan ， 2015 ： 233 - 248 . doi: 10.1007/978-4-431-54813-3_8 http://dx.doi.org/10.1007/978-4-431-54813-3_8 Regular Truncated Tetrahedral Structures 蒋淑慧 . 结构冗余度理论及其在张拉整体穹顶中的应用 ［D］. 杭州 ： 浙江大学 ， 2018 . doi: 10.5152/tjg.2019.18418 http://dx.doi.org/10.5152/tjg.2019.18418 结构冗余度理论及其在张拉整体穹顶中的应用 JIANG SH H . Redundancy Theory and it's Application in Tensegrity Dome Comprised of 3 -bar Prisms ［D］. Hangzhou ： Zhejiang University ， 2018 . （in Chinese） . doi: 10.5152/tjg.2019.18418 http://dx.doi.org/10.5152/tjg.2019.18418 Redundancy Theory and it's Application in Tensegrity Dome Comprised of 3-bar Prisms FADEYEV D ， ZHAKATAYEV A ， KUZDEUOV A ， et al . Generalized dynamics of stacked tensegrity manipulators ［J］. IEEE Access ， 2019 ， 7 ： 63472 - 63484 . doi: 10.1109/access.2019.2916681 http://dx.doi.org/10.1109/access.2019.2916681 Generalized dynamics of stacked tensegrity manipulators MIRLETZ B T ， PARK I W ， QUINN R D ， et al . Towards bridging the reality gap between tensegrity simulation and robotic hardware ［C］. 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems （IROS）. 282，2015 ， Hamburg， Germany. IEEE ， 2015 ： 5357 - 5363 . doi: 10.1109/iros.2015.7354134 http://dx.doi.org/10.1109/iros.2015.7354134 Towards bridging the reality gap between tensegrity simulation and robotic hardware GEBARA C A ， CARPENTER K C ， WOODMANSEE A . Tensegrity ocean world landers ［C］. AIAA Scitech 2019 Forum. San Diego， California. Reston， Virginia ： AIAA ， 2019 . doi: 10.2514/6.2019-0868 http://dx.doi.org/10.2514/6.2019-0868 Tensegrity ocean world landers BECCARELLI P . The Design ， Analysis and Construction of Tensile Fabric Structures ［M］. Springer International Publishing ， 2015 . doi: 10.1007/978-3-319-02228-4_2 http://dx.doi.org/10.1007/978-3-319-02228-4_2 The Design， Analysis and Construction of Tensile Fabric Structures 葛家琪 ， 张爱林 ， 刘鑫刚 ， 等 . 索穹顶结构张拉找形与承载全过程仿真分析 ［J］. 建筑结构学报 ， 2012 ， 33 （ 4 ）： 1 - 11 . 索穹顶结构张拉找形与承载全过程仿真分析 GE J Q ， ZHANG A L ， LIU X G ， et al . Analysis of tension form-finding and whole loading process simulation of cable dome structure ［J］. Journal of Building Structures ， 2012 ， 33 （ 4 ）： 1 - 11 . （in Chinese） Analysis of tension form-finding and whole loading process simulation of cable dome structure ALOUI O ， FLORES J ， ORDEN D ， et al . Cellular morphogenesis of three-dimensional tensegrity structures ［J］. Computer Methods in Applied Mechanics and Engineering ， 2019 ， 346 ： 85 - 108 . doi: 10.1016/j.cma.2018.10.048 http://dx.doi.org/10.1016/j.cma.2018.10.048 Cellular morphogenesis of three-dimensional tensegrity structures SCHORR P ， LI E R C ， KAUFHOLD T ， et al . 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