基于角度坐标描述的三维柔性大变形梁动力学建模方法研究.pdf

摘要摘要随着计算机技术的飞速发展，依托于大规模数值仿真技术，多体系统动力 学更加广泛地应用于各种工程领域，如航天、机械、风力发电、机器人等。轻 质材料在工程中得到了广泛的应用，多体系统中的柔性体变形越来越大，柔性 体的动力学模型是否能够精确、高效地解决大位移、大变形、大转动以及刚柔 耦合问题，成为了柔性多体动力学理论研究的重点和难点。建立高效、精确的 梁、板、壳柔性多体模型，是柔性多体动力学理论研究的关键所在，也是开发 通用柔性多体仿真软件的理论基础。本文围绕柔性多体系统中的大变形柔性梁建模问题，建立了基于角度坐标 描述的三维柔性大变形梁动力学模型，不仅可以精确描述梁的大位移、大变形 和大转动，而且可以有效地减少所需单元数目和广义坐标数目。进一步将建立 的大变形柔性梁模型用于三维电梯随行缆绳建模，分析其动力学特性。具体研 究内容及结论如下：(1)针对三维不可伸长欧拉-伯努利梁，以欧拉四元数作为广义坐标，采 用超球面插值函数对其进行插值，梁中心线的位置向量将通过斜率的积分得到，从而实现了对梁的大位移、大变形和大转动的精确描述，通过第一类拉格朗日 方程，建立了欧拉-伯努利梁c连续单元。该梁单元使单元节点的广义坐标数目 达到了最小化，同时避免了空间转动的奇异性问题。与有限元和绝对节点坐标 法相比，该c单元可以使用较少的广义坐标数目以达到所需精度。(2)以欧拉四元数和曲率向量作为单元节点广义坐标，建立了欧拉四元数 的C1连续插值函数，可以在满足欧拉四元数归一化约束方程的同时保证单元节 点处曲率向量连续。以此插值函数为基础，通过第一类拉格朗日方程，建立了 欧拉-伯努利梁C】连续单元。与C单元相比，C单元可以进一步减少所需单元 数目。与有限元、绝对节点坐标法和几何精确梁模型相比，该C1单元可以极大 的减少建模所需的单元数目和广义坐标数目。(3)进一步考虑梁的伸长变形，在广义坐标中引入中心线的轴向应变，建 立了三维可伸长欧拉-伯努利梁单元。以此单元为基础，建立了三维电梯随行缆 绳的动力学模型。验证了电梯随行缆绳平衡位置标准U形回路的建立准则；计 算了随行缆绳的平衡位置、固有频率和模态，分析了电梯轿厢位置对其平衡位 置和固有频率的影响；分析了电梯轿厢竖直运动和建筑摇摆对电梯随行缆绳的哈尔滨工业大学工学博士学位论文动力学影响。其分析结果对电梯随行缆绳的的参数优化设计、安装和振动控制 提供了理论指导。(4)针对三维大变形剪切梁，建立了梁截面转动、轴向-剪切应变向量和 中心线斜率之间的关系。使用欧拉四元数作为转动坐标，采用。连续插值函数 对其进行插值；轴向-剪切应变向量采用多项式函数进行插值；通过第一类拉格 朗日方程，建立了基于欧拉四元数的三维大变形剪切梁的柔性多体模型。该模 型具有无奇异性、无剪切锁定和泊松锁定问题的优点，在满足所需精度的同时 可以大大减少所需单元数目和广义坐标数目。(5)采用转动向量取代欧拉四元数作为转动坐标，建立了基于转动向量的 三维大变形剪切梁的柔性多体模型。在该模型中，转动向量和轴向-剪切应变向 量均可采用多项式插值，因此在插值函数的选择上具有较大的灵活性。针对转 动向量的奇异性问题，提出了一种可以规避单个奇异点的缩放方法，可以用于 解决大多数工程问题。该模型没有剪切锁定和泊松锁定问题，在满足所需精度 的同时可以大大减少所需单元数目和广义坐标数目。关键词：柔性大变形梁模型；欧拉四元数；转动向量；无奇异性和剪切锁定问 题；电梯随行缆绳；减少单元数目和广义坐标数目-I I-Ab st ra ctAb str a ctWit h t h e ra pid devel o pment o f co mput er t ech no l o g y,mul t ib o dy sy st em dy na mics,w h ich is rel y ing o n l a rg e-sca l e numerica l simul a t io n t ech no l o g y,is mo re a nd mo re w idel y used in va rio us eng ineering f iel ds,such a s a ero spa ce,mech a nica l sy st ems,w ind t urb ines,ro b o t s a nd so o n.Wit h t h e w ide use o f l ig h t w eig h t ma t e ria l s,def o rma t io n o f f l ex ib l e b o dies b eco mes l a rg er a nd l a rg er.Wh et h e r dy na mic mo del s o f f l ex ib l e b o dies ca n ef f ect ive l y a nd ef f icient l y dea l w it h pro b l ems o f l a rg e displ a cement s,l a rg e def o rma t io ns,l a rg e ro t a t io ns a nd rig id-f l ex ib l e co upl ing,h a s b eco me k ey po int s a nd dif f icul t y o f f l ex ib l e mul t ib o dy dy na mic t h eo ry.Devel o ping ef f icient a nd a ccura t e f o rmul a t io ns o f a b ea m,pl a t e a nd sh el l f o r mul t ib o dy dy na mic a na l y sis is t h e k ey t o t h e st udy o f f l ex ib l e mul t ib o dy dy na mic t h eo ry,a nd it is a l so t h e t h eo ret ica l b a sis o f t h e devel o pment o f g enera l f l ex ib l e mul t ib o dy dy na mic so f t w a re.In t h e t h esis,t h ree-dime nsio na l b ea m f o rmul a t io ns b a sed o n ro t a t io na l co o rdina t e descript io ns a re devel o ped f o r b ea ms w it h l a rg e displ a cement s,l a rg e def o rma t io ns a nd l a rg e ro t a t io ns.Th e f o rmul a t io ns ca n g rea t l y reduce numb ers o f el ement s a nd g enera l iz ed co o rdina t es.Th e current f o rmul a t io n is t h en used t o ca l cul a t e t h e equil ib rium a nd dy na mic respo nses o f a n el eva t o r t ra ve l ing ca b l e w it h a rb it ra ril y mo ving ends.Specif ic co nt ent s a nd so me co ncl usio ns a re sh o w n a s f o l l o w s.(1)A sing ul a rit y-f re e C-co nt inuo us f o rmul a t io n o f a t h ree-dimensio na l inex t ensib l e Eul er-Berno ul l i b ea m w it h l a rg e def o rma t io ns a nd l a rg e ro t a t io ns is devel o ped.Th e po sit io n o f t h e cent ro id l ine o f t h e b ea m is int eg ra t ed f ro m it s sl o pe,a nd po sit io n vect o rs o f no des o f b ea m el ement s a re no l o ng er used a s g enera l iz ed co o rdina t es.Eul er pa ra met ers a re used t o ch a ra ct eriz e o rient a t io ns o f cro ss-sect io ns o f t h e b ea m,w h ich ca n reso l ve t h e sing ul a rit y pro b l em ca used b y Eul er a ng l es.Th e h y per-sph erica l int erpo l a t io n f unct io n is used t o g ua ra nt ee t h a t t h e no rma l iz a t io n co nst ra int equa t io n o f Eul er pa ra met ers is a l w a y s sa t isf ied.It ca n o nl y g ua ra nt ee t h a t Eul er pa ra met ers a re co nt inuo us a t no des o f b ea m el ement s a nd is C co nt inuo us.Severa l numerica l ex a mpl es a re present ed t o demo nst ra t e t h e perf o rma nce o f t h e current f o rmul a t io n.It is sh o w n t h a t t h e current f o rmul a t io n ca n a ch ieve t h e sa me a ccura cy a s t h e f init e el ement met h o d(FEM)a nd a b so l ut e no de co o rdina t e f o rmul a t io n(ANCF)w it h a f ew er numb er o f co o rdina t es.(2)A sing ul a rit y-f re e C1-co nt inuo us f o rmul a t io n o f a t h ree-dimensio na l inex t ensib l e Eul er-Berno ul l i b ea m w it h l a rg e def o rma t io ns a nd l a rg e ro t a t io ns is-ill-哈尔滨工业大学工学博士学位论文devel o ped.Th e b ea m ca n b e eit h e r st ra ig h t o r init ia l l y curved.A new Cco nt inuo us int erpo l a t io n f unct io n f o r Eul er pa ra met ers is devel o ped in t h e curre nt f o rmul a t io n,w h ich resul t s in co nt inuo us f irst deriva t ives o f Eul er pa ra met ers a nd ma t eria l curva t ures a t no des o f b ea m el ement s.Severa l numerica l ex a mpl es a re present ed t o demo nst ra t e t h e perf o rma nce o f t h e current f o rmul a t io n.Th e current f o rmul a t io n ca n b e a ppl ied t o st a t ic a nd dy na mic pro b l ems a nd ca n a ch ieve t h e sa me a ccura cy a s t h e FEM,ANCF,a nd g eo met rica l l y ex a ct b ea m f o rmul a t io n w it h much f ew er el ement s a nd g enera l iz ed co o rdina t es.(3)A sing ul a rit y-f re e f o rmul a t io n o f a t h ree-dime nsio na l ex t ensib l e Eul er-Berno ul l i b ea m w it h l a rg e def o rma t io ns a nd l a rg e ro t a t io ns is devel o ped.Th e po sit io n vect o r o f t h e cent ro id l ine o f t h e b ea m is int eg ra t ed f ro m it s sl o pe,w h ich is ex pressed b y Eul er pa ra met ers a nd t h e no rma l st ra in.Go verning equa t io ns o f t h e b ea m a nd co nst ra int equa t io ns a re derived using La g ra ng es equa t io ns f o r sy st ems w it h co nst ra int s.Af t er verif ica t io n o f t h e current f o rmul a t io n w it h a b ench ma rk pro b l em,it is used t o ca l cul a t e t h e equil ib rium a nd dy na mic respo nses o f a ro und el eva t o r t ra ve l ing ca b l e w it h a rb it ra ril y mo ving ends.Equil ib ria o f t h e t ra ve l ing ca b l e w it h dif f erent ca r po sit io ns a re ca l cul a t ed;o nl y t rivia l equil ib ria o f t h e ca b l e w it h no cl o sed l o o ps ca n b e f o und f ro m it s t h ree-dimensio na l mo del.Na t ura l f requencies o f t h e t ra vel ing ca b l e a nd co rrespo nding mo de sh a pes a re ca l cul a t ed.In-pl a ne na t ura l f requencies do no t ch a ng e much w it h t h e ca r po sit io n co mpa red w it h o ut-o f-pl a ne o nes.Free respo nses o f t h e t ra vel ing ca b l e due t o vert ica l mo t io n o f t h e ca r a nd f o rced respo nses due t o in-pl a ne a nd o ut-o f-pl a ne b uil ding sw a y s a re st udied.Th e vert ica l mo t io n o f t h e ca r ca n a f f ect t h e l a t era l respo nse o f t h e t ra vel ing ca b l e,b ut it h a s a l mo st no ef f ect o n it s o ut-o f-pl a ne respo nse.Wh il e b uil ding sw a y s ca n a f f ect b o t h l a t era l a nd o ut-o f-pl a ne respo nses o f t h e t ra vel ing ca b l e,t h ey h a ve l it t l e ef f ect o n it s vert ica l respo nses.(4)An a ccura t e sing ul a rit y-f ree a nd l o ck ing-f ree f o rmul a t io n o f a t h ree-dimensio na l sh ea r-def o rma b l e b ea m w it h l a rg e def o rma t io ns a nd l a rg e ro t a t io ns is devel o ped.Th e b ea m is ex t ensib l e a nd sh ea r-def o rma b l e,a nd it ca n b e init ia l l y st ra ig h t o r curved.Th e po sit io n o f t h e cent ro id l ine o f t h e b ea m is int eg ra t ed f ro m it s sl o pe t h a t is rel a t ed t o t h e ro t a t io n o f a co rrespo nding cro ss-sect io n a nd st ret ch a nd sh ea r st ra ins.Wh il e st ret ch a nd sh ea r st ra ins a re int erpo l a t ed using po l y no mia l f unct io ns,Eul er pa ra met ers a re int erpo l a t ed using a C co nt ino us int erpo l a t io n f unct io n t o g ua ra nt ee t h a t curva t ures o f t h e cent ro id l ine o f t h e b ea m a re co nt inuo us a t no des o f it s el ement s.Numerica l resul t s sh o w t h a t t h e current f o rmul a t io n do es no t suf f er f ro m sh ea r l o ck ing a nd P o isso n l o ck ing pro b l ems a nd it ca n g rea t l y reduce numb ers o f el e ment s a nd g enera l iz ed co o rdina t es.-I V-Ab st ra ct(5)A new l o ck ing-f ree f o rmul a t io n o f a t h ree-dimensio na l sh ea r-de f o rma b l e b ea m w it h l a rg e def o rma t io ns a nd l a rg e ro t a t io ns is devel o ped.Th e ro t a t io n is pa ra met riz ed b y a ro t a t io n vect o r,w h ich h a s a cl ea r a nd int uit ive ph y sica l mea ning.Since t h e ro t a t io n vect o r a nd st ret ch-sh e a r st ra in vect o r a re int erpo l a t ed using po l y no mia l f unct io ns,int erpo l a t io n f unct io n is no t unique a nd h a s much f l ex ib il it y.It no t o nl y a vo ids t h e co mpl ex int e rpo l a t io n o f Eul er pa ra me t ers,b ut a l so a vo ids no rma l iz a t io n co nst ra int equa t io ns,w h ich g rea t l y reduces t h e dif f icul t y o f numerica l simul a t io n.A resca l ing st ra t eg y is a do pt ed t o reso l ve t h e sing ul a rit y pro b l em w h en t h ere is o nl y o ne sing ul a r po int a t a t ime inst a nt,w h ich is t h e ca se f o r mo st eng ineering a ppl ica t io ns.Resul t s sh o w t h a t t h e current f o rmul a t io n do no t suf f er f ro m sh ea r a nd P o isso n l o ck ing pro b l ems a nd it uses much f ew er numb ers o f el ement s a nd g enera l iz ed co o rdia nt es t o y iel d co nverg e d resul t s.Key w o rds:f l ex ib l e l a rg e-def o rma t io n b ea m f o rmul a t io ns;Eul er pa ra met ers;ro t a t io n vect o r;sing ul a rit y-f ree a nd sh ea r-l o ck ing-f ree;el eva t o r t ra vel ing ca b l e;reduce numb ers o f el e ment s a nd g enera l iz ed co o rdina t es-v-哈尔滨工业大学工学博士学位论文目录摘要.IABSTRACT.Il l目录.VICONTENTS.X第1章绪论.11.1 课题背景及研究意义.11.1.1 课题背景.11.1.2 研究目的及意义.41.2 柔性梁多体建模研究现状.41.2.1 运动-弹性动力学方法.41.2.2 混合坐标法.41.2.3 几何精确梁模型.71.2.4 绝对节点坐标法.111.2.5 基于角度坐标描述的平面梁单元研究进展.121.3 电梯绳索系统研究现状.141.4 本文主要内容.15第2章基于欧拉四元数的三维不可伸长欧拉-伯努利梁C0单元建模研究.172.1 引言.172.2 三维欧拉伯努利梁运动学描述.172.2.1 基本运动学描述.182.2.2 欧拉四元数基础理论.192.3 空间离散.212.4 控制方程.232.4.1 动能.232.4.2 广义弹性力.262.4.3 广义外力.272.4.4 动力学控制方程.28-V I-目录2.5 平衡位置、固有频率和模态.282.5.1 平衡位置.282.5.2 固有频率和模态.292.6 数值求解算法.302.6.1 降阶为指数1的微分代数方程进行求解.302.6.2 降阶为指数2的微分代数方程进行求解.312.6.3 直接求解指数3的微分代数方程.312.7 仿真算例.332.7.1 静力学问题.332.7.2 固有频率和模态.362.7.3 动力学响应.392.8 本章小结.43第3章基于欧拉四元数的三维不可伸长欧拉-伯努利梁Ci单元建模研究.443.1 引言.443.2 运动学描述和空间离散.443.3 控制方程.463.3.1 动能.463.3.2 广义弹性力.483.3.3 广义外力.503.3.4 动力学控制方程.513.4 仿真算例.513.4.1 悬臂曲梁的大变形分析.513.4.2 螺旋弹簧刚度.523.4.3 悬臂梁动力学响应.553.4.4 旋转梁.563.4.5 简支-自由梁自由下落.603.4.6 旋转轴.613.5 本章小结.63第4章三维可伸长欧拉-伯努利梁建模及其在电梯随行缆绳建模中的应用.644.1 引言.644.2 二维可伸长欧拉梁的动力学描述.64-V I I-哈尔滨工业大学工学博士学位论文4.2.1 运动学描述.644.2.2 空间离散.654.3 控制方程.664.3.1 动能.664.3.2 广义弹性力和广义外力.684.3.3 动力学控制方程.694.3.4 平衡位置，固有频率和模态.694.3.5 标准问题验证.714.4 电梯随行缆绳建模与分析.724.4.1 平衡位置.734.4.2 固有频率和模态.754.4.3 电梯轿厢竖直运动引起的随行缆绳自由振动.794.4.4 建筑摇摆引起的随行缆绳强迫振动.814.5 本章小结.85第5章基于欧拉四元数的二维剪切梁建模方法研究.865.1 引言.865.2 二维剪切梁动力学描述.865.2.1 运动学描述.865.2.2 空间离散.885.3 控制方程.885.3.1 动能.885.3.2 广义弹性力和广义外力.915.3.3 动力学控制方程.925.4 数值仿真.925.4.1 悬臂梁自由端作用集中力.925.4.2 固定-较接圆拱作用集中力.945.4.3 轴向压力作用下悬臂梁的固有频率.955.4.4 旋转曲梁.985.5 本章小结.99第6章基于转动向量的三维剪切梁建模方法研究.1006.1 引言.100-V I I I-目录6.2 三维剪切梁动力学描述.1006.2.1 转动向量.1006.2.2 空间离散.1016.3 控制方程.1026.3.1 动能.102632广义弹性力，阻尼力和外力.1046.3.3 动力学控制方程.1066.4 转动向量的奇异性.1066.5 数值仿真.1086.5.1 悬臂梁弯曲成圆环.1086.5.2 悬臂梁自由端作用集中力.1096.5.3 简支-自由梁自由下落.1146.5.4 旋转梁.1156.6 本章小结.119结论.120参考文献.123附录A皿，包%,包如,表达式.135加欠 河 沟上风附录B答，畀，k k畀表达式.138附录C可伸长平面剪切悬臂梁精确解.141攻读博士学位期间发表的学术论文.144哈尔滨工业大学学位论文原创性声明和使用权限.146致谢.147个人简历.148-I X-哈尔滨工业大学工学博士学位论文ContentsAb str a ct(In Ch inese).IAb str a ct(In Eng l ish).Il lContents(In Ch ine se).VContents(In Eng l ish).IXCha pter 1 Int ro duct io n.11.1 Ba ck g ro und,o b ject ive a nd sig nif ica nce o f t h e sub ject.11.1.1 Ba ck g ro und.11.1.2 Ob ject ive a nd sig nif ica nce.41.2 Current resea rch o n f l ex ib l e mul t ib o dy dy na mics.41.2.1 Kinet o-el a st o dy na mics met h o d.41.2.2 Hy b rid co o rdina t e met h o d.41.2.3 Geo met rica l l y ex a ct b ea m f o rmul a t io n.61.2.4 Ab so l ut e no da l co o rdina t e f o rmul a t io n.111.2.5 Ro t a t io na l-co o rdina t e-b a sed pl a na r b ea m f o rmul a t io n.121.3 Current resea rch o n el eva t o r ca b l e sy st ems.141.4 Ma in co nt ent s o f t h is sub je ct.15Cha pter 2 A C-co nt inuo us f o rmul a t io n o f a t h ree-dime nsio na l inex t ensib l e Eul er-Berno ul l i b ea m using Eul er pa ra met e rs.172.1 Int ro duct io n.172.2 Kinema t ics o f a t h ree-dimensio na l Eul er-Berno ul l i b ea m.172.2.1 Ba sic k inema t ics.182.2.2 Ba sic t h eo ry o f Eul er pa ra me t ers.192.3 Spa t ia l discret iz a t io n.212.4 Go verning equa t io ns.232.4.1 Kinet ic energ y.232.4.2 Gene ra l iz ed el a st ic f o rce.262.4.3 Genera l iz ed ex t erna l f o rce.272.4.4 Dy na mic g o verning equa t io ns.282.5 St a t ic equil ib rium,na t ura l f requencies a nd mo de sh a pe s.282.5.1 St a t ic equil ib rium.282.5.2 Na t ura l f requencies a nd mo de sh a pes.292.6 Numerica l met h o d f o r DAEs.302.6.1 So l ving reduced index-1 DAEs.302.6.2 So l ving reduced index-2 DAEs.312.6.3 So l ving index-3 DAEs direct l y.312.7 Numerica l simul a t io ns.332.7.1 St a t ic pro b l ems.332.7.2 Na t ura l f requencies a nd mo de sh a pes.362.7.3 Dy na mic pro b l e ms.38-x-Co nt ent s2.8 Summa ry.43Cha pter 3 A C-co nt inuo us f o rmul a t io n o f a t h ree-dimensio na l inex t ensib l e Eul er-Berno ul l i b ea m using Eul er pa ra met e rs.443.1 Int ro duct io n.443.2 Kinema t ic descript io n a nd spa t ia l discret iz a t io n.443.3 Dy na mic equa t io ns.463.3.1 Kinet ic energ y.463.3.2 Genera l iz ed el a st ic f o rce.483.3.3 Gene ra l iz ed ex t erna l f o rce.503.3.4 Dy na mic g o verning equa t io ns.513.4 Numerica l simul a t io ns.513.4.1 La rg e displ a cement a na l y sis o f a curved ca nt il ever b ea m.513.4.2 St if f ness o f a h el ix spring.523.4.3 Dy na mic respo nses o f a ca nt il ever b ea m.553.4.4 Ro t a t ing b ea m.563.4.5 Free f a l l ing pinned-f ree b ea m under g ra vit y.603.4.6 Ro t a t ing sh a f t.613.5 Summa ry.63Cha pter 4 A new f o rmul a t io n o f a t h ree-dime nsio na l ex t ensib l e Eul er-Berno ul l ib ea m w it h a ppl ica t io n t o a n el eva t o r t ra ve l ing ca b l e.644.1 Int ro duct io n.644.2 Kinema t ics o f a t h ree-dime nsio na l ex t ensib l e Eul er-Berno ul l i b ea m.644.2.1 Ba sic k inema t ics.644.2.2 Spa t ia l discret iz a t io n.654.3 Go verning equa t io ns.664.3.1 Kinet ic energ y.664.3.2 Gene ra l iz ed el a st ic a nd ex t erna l f o rces.684.3.3 Dy na mic g o verning equa t io n