大射电望远镜悬索式馈源支撑系统的非线性静力学、运动学和动力学理论及方法的研究
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摘要
本文研究了建立大射电望远镜FAST悬索式支撑馈源系统的非线性静力学，运动学和动力学的数学力学模型所涉及的相关理论和方法。悬索式支撑馈源系统属于并联机构，然而悬索具有的柔性特征使它与一般的刚性并联机构(如Stewart平台)有很大不同。由于悬索并联机构能够满足室外露天工程的要求，同时又能根据需要设计出巨大的工作空间，因此正逐步成为并联机构领域的研究热点；同时，作为建造新一代大射电望远镜的关键技术，已受到广泛的关注，然而关于这种新型机构的研究国内外仍处于探索阶段，尚没有形成统一的理论和方法。本文围绕FAST工程，对并联悬索结构的静力学、运动学及动力学分析的相关理论及方法进行了系统而深入的研究。全文分七章，各章所讨论的主要内容如下
第一章 绪论。本章介绍了新一代大射电望远镜的研究背景及机电光一体化的悬索馈源系统的技术特点与主要研究内容。回顾了并联悬索机构的研究现状。围绕悬索式馈源系统的非线性静力学、运动学和动力学分析，提出了本文付诸研究并取得进展的若干问题。最后介绍了本文的研究思路和内容。
第二章 悬索馈源系统的静力分析。本章建立了悬索式支撑馈源系统的非线性静力学模型。论述了柔性悬索的虚牵问题并提出了判定准则。定义并分析了系统的解空间。提出了悬索张力的均匀原则，实现了张力的优化配置。针对六悬索系统工作空间中存在的六个无解面，在综合考虑馈源舱运行规律及悬索受力特点的基础上，通过增加两根向下拉的冗余悬索，提出并设计了八悬索馈源系统。建立了八悬索系统的非线性静力学模型，提出了可行的求解方法——预估张力迭代法。彻底解决了六悬索馈源系统工作空间不连续的问题。
第三章 悬索馈源系统驱动力的研究。在已知馈源舱的位置，姿态，速度和加速度的情况下，本章对系统的驱动力进行了研究。建立了悬索馈源系统的动平衡方程。确定了在动平衡条件下各悬索的形状。通过将悬索离散为柔性索杆单元，建立了系统的逆运动学及逆动力学模型。分析了悬索上各离散点的位置、速度、加速度和惯性力。推导了在重力和惯性力共同作用下悬索的空间挠曲线微分方程，并研究了悬索的刚体位移和弹性变形。通过刚体-柔体的反复迭代算出悬索馈源系统的驱动力。
第四章 悬索馈源系统时变结构的顺风向风致响应研究。本章应用余弦级数模拟了符合Davenport功率谱的自然风。基于时不变结构的有限单元法，在综合考虑馈源舱运动规律的基础上，提出了时变系统的瞬时结构假定法，
通过将悬索离散为索杆单元，建立了系统的时变有限元模型，该模型充分考
虑了悬索的垂度和大变形等几何非线性因素，应用Ne。。ark－D数值积分方
法，在时域上分析了馈源舱停留在空间某一位置（典型点）和沿空间某一轨
迹运行（变结构）时，系统的顺风向风致响应。风振时程曲线表明在观测风
速内，无论是六悬索还是八悬索系统，均满足设计精度要求，但八悬索的两
根冗余下拉悬索对馈源舱Z方向的振动起到了很好的抑制作用。
第五章 悬索馈源系统的横风向共振和空气动力失稳的研究。大射电望
远镜悬索馈源系统是风敏感结构，在风力作用下可能导致横风向涡流脱落振
动和空气动力失稳。由于系统属于多自由度几何非线性结构，现有的线性结
构横风向共振分析方法和传统上用于节断模型分析的驰振临界风速判别式不
再适用。本章利用Newmark法和Newton－Raphson法的思想建立了系统的横风
向共振模型、平均风及脉动风空气动力模型，通过比较结构在不同风速作用
下的振动时程曲线，确定了横风向共振和驰振的临界风速，并应用瞬时结构
假定法研究了脉动风作用下系统的空气动力失稳情况。
第六章 大射电望远镜50米模型的研究。本章根据大射电望远镜50米
模型荷载、运动速度等要求，确定了悬索的型号。对模型的静力和驱动力进
行了研究。分析了模型的顺风向随机风振响应、横风向涡激共振及空气动力
失稳响应。完成了相关的实验。数值计算和实验数据的比较表明，本文建立
的大射电望远镜悬索馈源系统的数学力学模型是准确的，数值计算结果是可
靠的。
第七章 总结与展望。本章回顾和总结了新一代大射电望远镜机电光一体
化的悬索馈源系统所涉及的理论、技术和方法。归纳了本文的创新之处和特
色，并提出了设计和力学分析中仍需进一步研究的若干问题。
FAST (Five hundred meter Aperture Spherical Telescope) is a project of constructing the largest radio telescope of the world in Guizhou Province, China. Optomechatronics on Synthetic Design for Feed Support and Control System, with higher precision and low cost, is applied in the FAST project. Cable-suspension sustaining feed system characterized with the flexibility, as a parallel mechanics, is largely different from other rigid parallel manipulator commonly known as the Stewart platform. With a huge working space, low cost and less requirement of environments, this new type parallel mechanics has been a hot point in the field of parallel manipulator, simultaneously, as a key technology and the most challenging part of FAST, it has been focused in by the astronomer and antenna experts all over the world. However, at present time, it is still open, the existing studies have not established uniformed theories and methods yet. The purpose of this dissertation is to study the related theories and methods on establishing mathematics, mechanics and computational models of non-linear statics, kinematics and dynamics for cable-suspension feed system. There are seven chapters in this dissertation as follows.
Chapter 1 Preface
The research background and target of new generation large radio telescope are presented. The technical features and research contents of cable-suspension feed system are described. The current study on parallel cable mechanics is reviewed. Several issues that have been studied and the progresses have been made in this dissertation are proposed.
Chapter 2 Non-linear static analysis of the cable-suspension system
Non-linear static mathematics and computational model of cable-suspension feed system is established. Pseudo-drag problem of flexible cable is described and the critical criterion is deduced. The problems of solution space and unique solution of system are defined and discussed. Two optimization principles are proposed to distribute the load ideally among cables, thereby avoid pseudo-drag. Based on the movement laws of cabin, and considering the force condition in cables, a 8-cable suspension feed system, which can completely remove the no solution surfaces in 6-cable system, is presented and designed by means of adding 2 down-ward cables. A solving method, whose feasibility is proved, is developed
for static equilibrium equations of the cabin.
Chapter 3 Research on driving forces in the cable-suspension feed system
Dynamics equilibrium equations of the feed cabin are deduced. Inverse kinematic and dynamic mathematics and computational model are established by discretizing cable into flexible cable-rob body. When cable interacted under the load both of gravity and inertia force, derivative equation of cable's defection curve in space is deduced to study cable's rigidity movement, and flexible deformation is also determined by finite element method. An iterative algorithm of rigid-flexible body is developed to compute driving force of system if the position, posture, velocity and acceleration of feed cabin are known.
Chapter 4 Time-dependent computational simulation of the response of cable-suspension feed system in gusty wind
Based on the time-independent structure finite element method, and with consideration of the movement laws of cabin, a 3-D time-dependent structure finite element model formulated by a new method named ISS method (Instantaneous Structure Supposition method) to deal with the vibration response of cable-suspension feed system in gusty wind is presented. In the model, all sources of geometric non-linearity, cable sag and changes of cable geometry due to large displacement, are fully considered. In time domain, turbulent wind load is simulated by applying cosine series. The Newmark- P step-by-step numerical integration algorithm is used to calculate the response behavior of system when feed cabin locates at a certain position (typical location) and when feed cabin moves along a certain trace (time-dependent structure). Compared with the data
第一章 绪论。本章介绍了新一代大射电望远镜的研究背景及机电光一体化的悬索馈源系统的技术特点与主要研究内容。回顾了并联悬索机构的研究现状。围绕悬索式馈源系统的非线性静力学、运动学和动力学分析，提出了本文付诸研究并取得进展的若干问题。最后介绍了本文的研究思路和内容。
第二章 悬索馈源系统的静力分析。本章建立了悬索式支撑馈源系统的非线性静力学模型。论述了柔性悬索的虚牵问题并提出了判定准则。定义并分析了系统的解空间。提出了悬索张力的均匀原则，实现了张力的优化配置。针对六悬索系统工作空间中存在的六个无解面，在综合考虑馈源舱运行规律及悬索受力特点的基础上，通过增加两根向下拉的冗余悬索，提出并设计了八悬索馈源系统。建立了八悬索系统的非线性静力学模型，提出了可行的求解方法——预估张力迭代法。彻底解决了六悬索馈源系统工作空间不连续的问题。
第三章 悬索馈源系统驱动力的研究。在已知馈源舱的位置，姿态，速度和加速度的情况下，本章对系统的驱动力进行了研究。建立了悬索馈源系统的动平衡方程。确定了在动平衡条件下各悬索的形状。通过将悬索离散为柔性索杆单元，建立了系统的逆运动学及逆动力学模型。分析了悬索上各离散点的位置、速度、加速度和惯性力。推导了在重力和惯性力共同作用下悬索的空间挠曲线微分方程，并研究了悬索的刚体位移和弹性变形。通过刚体-柔体的反复迭代算出悬索馈源系统的驱动力。
第四章 悬索馈源系统时变结构的顺风向风致响应研究。本章应用余弦级数模拟了符合Davenport功率谱的自然风。基于时不变结构的有限单元法，在综合考虑馈源舱运动规律的基础上，提出了时变系统的瞬时结构假定法，
通过将悬索离散为索杆单元，建立了系统的时变有限元模型，该模型充分考
虑了悬索的垂度和大变形等几何非线性因素，应用Ne。。ark－D数值积分方
法，在时域上分析了馈源舱停留在空间某一位置（典型点）和沿空间某一轨
迹运行（变结构）时，系统的顺风向风致响应。风振时程曲线表明在观测风
速内，无论是六悬索还是八悬索系统，均满足设计精度要求，但八悬索的两
根冗余下拉悬索对馈源舱Z方向的振动起到了很好的抑制作用。
第五章 悬索馈源系统的横风向共振和空气动力失稳的研究。大射电望
远镜悬索馈源系统是风敏感结构，在风力作用下可能导致横风向涡流脱落振
动和空气动力失稳。由于系统属于多自由度几何非线性结构，现有的线性结
构横风向共振分析方法和传统上用于节断模型分析的驰振临界风速判别式不
再适用。本章利用Newmark法和Newton－Raphson法的思想建立了系统的横风
向共振模型、平均风及脉动风空气动力模型，通过比较结构在不同风速作用
下的振动时程曲线，确定了横风向共振和驰振的临界风速，并应用瞬时结构
假定法研究了脉动风作用下系统的空气动力失稳情况。
第六章 大射电望远镜50米模型的研究。本章根据大射电望远镜50米
模型荷载、运动速度等要求，确定了悬索的型号。对模型的静力和驱动力进
行了研究。分析了模型的顺风向随机风振响应、横风向涡激共振及空气动力
失稳响应。完成了相关的实验。数值计算和实验数据的比较表明，本文建立
的大射电望远镜悬索馈源系统的数学力学模型是准确的，数值计算结果是可
靠的。
第七章 总结与展望。本章回顾和总结了新一代大射电望远镜机电光一体
化的悬索馈源系统所涉及的理论、技术和方法。归纳了本文的创新之处和特
色，并提出了设计和力学分析中仍需进一步研究的若干问题。
FAST (Five hundred meter Aperture Spherical Telescope) is a project of constructing the largest radio telescope of the world in Guizhou Province, China. Optomechatronics on Synthetic Design for Feed Support and Control System, with higher precision and low cost, is applied in the FAST project. Cable-suspension sustaining feed system characterized with the flexibility, as a parallel mechanics, is largely different from other rigid parallel manipulator commonly known as the Stewart platform. With a huge working space, low cost and less requirement of environments, this new type parallel mechanics has been a hot point in the field of parallel manipulator, simultaneously, as a key technology and the most challenging part of FAST, it has been focused in by the astronomer and antenna experts all over the world. However, at present time, it is still open, the existing studies have not established uniformed theories and methods yet. The purpose of this dissertation is to study the related theories and methods on establishing mathematics, mechanics and computational models of non-linear statics, kinematics and dynamics for cable-suspension feed system. There are seven chapters in this dissertation as follows.
Chapter 1 Preface
The research background and target of new generation large radio telescope are presented. The technical features and research contents of cable-suspension feed system are described. The current study on parallel cable mechanics is reviewed. Several issues that have been studied and the progresses have been made in this dissertation are proposed.
Chapter 2 Non-linear static analysis of the cable-suspension system
Non-linear static mathematics and computational model of cable-suspension feed system is established. Pseudo-drag problem of flexible cable is described and the critical criterion is deduced. The problems of solution space and unique solution of system are defined and discussed. Two optimization principles are proposed to distribute the load ideally among cables, thereby avoid pseudo-drag. Based on the movement laws of cabin, and considering the force condition in cables, a 8-cable suspension feed system, which can completely remove the no solution surfaces in 6-cable system, is presented and designed by means of adding 2 down-ward cables. A solving method, whose feasibility is proved, is developed
for static equilibrium equations of the cabin.
Chapter 3 Research on driving forces in the cable-suspension feed system
Dynamics equilibrium equations of the feed cabin are deduced. Inverse kinematic and dynamic mathematics and computational model are established by discretizing cable into flexible cable-rob body. When cable interacted under the load both of gravity and inertia force, derivative equation of cable's defection curve in space is deduced to study cable's rigidity movement, and flexible deformation is also determined by finite element method. An iterative algorithm of rigid-flexible body is developed to compute driving force of system if the position, posture, velocity and acceleration of feed cabin are known.
Chapter 4 Time-dependent computational simulation of the response of cable-suspension feed system in gusty wind
Based on the time-independent structure finite element method, and with consideration of the movement laws of cabin, a 3-D time-dependent structure finite element model formulated by a new method named ISS method (Instantaneous Structure Supposition method) to deal with the vibration response of cable-suspension feed system in gusty wind is presented. In the model, all sources of geometric non-linearity, cable sag and changes of cable geometry due to large displacement, are fully considered. In time domain, turbulent wind load is simulated by applying cosine series. The Newmark- P step-by-step numerical integration algorithm is used to calculate the response behavior of system when feed cabin locates at a certain position (typical location) and when feed cabin moves along a certain trace (time-dependent structure). Compared with the data
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