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可展网状天线设计与在轨动力学行为预示 : 英文 / 付康佳著. -- 北京 : 清华大学出版社, 2026. 9. -- (清华大学优秀博士学位论文丛书). ISBN 978-7-302-70993-0
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Foreword
Large deployable mesh antennas are among the most valuable appendages of satel-lites; they are used for certain sophisticated tasks, such as high-resolution Earth observations and instant telecommunications. As a core technology in the fields of communication and national defense, it has been listed in the “Global Engineering Frontiers 2020” issued by the Chinese Academy of Engineering as one of the top ten engineering research frontiers in the fields of machinery and delivery. Structurally, this kind of mesh antennas is primarily composed of two components, a foldable supporting truss and a flexible cable network. Each of these two components fulfills a central function. The truss is in charge of unfolding the antenna from the launch-ready, stowed configuration to the working configuration in orbit. Once unfolded, the cable network is in charge of forming a geometric paraboloidal shape to reflect signals via the attached metallic mesh. If the truss fails to unfold the entire system, or the cable network fails to form the desired geometric shape within enough accu-racy, the antennas cannot work properly.
Research on guaranteeing the success of antennas has boosted the developments of several challenging interdisciplinary mechanical problems. I list a few of them:
1. Modeling, simulation, and design of complex flexible mechanisms. The deploy-able truss is a typical flexible mechanism suffering large spacial displacement and rotation; meanwhile, it is unfolded by a cable-pulley mechanism and active motors. Understanding and predicting deployment behaviors of the entire sys-tem, such as the loads on flexible components and driving forces of the motors, is of great concern. A flexible multibody system simulation is supposed to be a promising way to study the deployment dynamics. However, building a full-scale model to simulate the deployment dynamics is an extremely difficult task. It involves a lot of aspects on studying flexible multibody system and has attracted great research attention in the past years. For example, using Lie groups to formulate 3-dimensional movement of flexible beams and shells. Formulating flexible beam with warping effects. Modeling contacting problems between cable and pulleys with arbitrary Lagrangian-Euler formulations.
VIII Foreword
2.
Designing and evaluation of gravity compensation systems for flexible mecha-nisms to predict the in-obit behavior. Deploying mesh antennas in-orbit is a deli-cate and precise process. Thus, on-ground experiments have been conducted to attempt to predict the in-orbit deployment performance of mesh antennas. However, gravitational field on Earth presents a great obstacle in achieving this goal. It is necessary to understand and evaluate how effectively a gravity com-pensation system can counteract effects of gravity. Nowadays, research is actively focused on developing new gravity compensation mechanisms, either passive or active, to fulfill the on-ground verification purpose.
3.
Design and adjusting a flexible cable network structure to form a large parabo-loidal shape with enough accuracy. This is a tough job for several reasons. Firstly, the cables may lose tension if the tensile forces are not loaded property, then the network can generate rigid body motion and then result in large shape errors. Secondly, the cable network contains a large number of cable segments, and each of them has a manufacturing errors. These errors also contribute to shape errors. Thirdly, the flexible deformation of cable networks under tensile forces is another source of shape error. Therefore, to make the shape error con-trollable, it is demanded to understand the governing law adjusting shape of flexible cable network.
Designing a mesh antenna has to deal with the above-mentioned several problems. One of the core dynamic problems involved in its research is to ensure the success-ful unfolding and the formation of a high precision configuration of the antenna. Dr. Kangjia Fu has studied them when designing a type of mesh antenna, which is known as AstroMesh-II..He proposed a new modeling method to make the simula-tion a full-scale AstroMesh-II feasible. With the help of this model, he designed a synchronization mechanism to reduce the loads on flexible components and verified the result with an experiment. Then, he studied the design law of gravity compensa-tion system combining theoretically analysis, numerical simulations, and experi-mental verifications. Finally, he tried to understand how to keep all of the cable network taut under tensile forces. I believe Dr. Fu’s thesis is a great theoretical guide for designing mesh antennas. The research results can provide theoretical and design references for large deployable antennas and can also be extended to other space flexible structures such as solar sails and tensegrities. It is of great signifi-cance to enrich and develop the in-orbit dynamic prediction and the on-ground experimental design of space flexible structures.
Beijing, China Zhihua.Zhao November 2021
Related Publications in This Thesis
[1] Kangjia Fu, Zhihua Zhao*, Gexue Ren, Yong Xiao, Tao Feng, Jungang Yang, Paolo Gasbarri. From multiscale modeling to design of synchronization mecha-nisms in mesh antennas[J]. Acta Astronautica, 2019, 159: 156-165.
[2] Kangjia Fu, Jianbin Du, Jinyou Li, Zhihua Zhao*. Robust design of tension truss antennas against variation in tension forces[J]. AIAA Journal, 2018, 55: 1450-1459.
[3] Zhihua Zhao, Kangjia Fu*, Meng Li, Jinyou Li, Yong Xiao. Gravity compensa-tion system of mesh antennas for in-orbit prediction of deployment dynamics[J]. Acta Astronautica, 2020, 167: 1-13.
Preface
Deployable mesh antennas are central units of high-orbit satellites to gather electro-magnetic signals. They are composed of flexible cable networks attached with metal mesh to reflect signals and deployable trusses to unfold and support the cable net-works for forming a parabolic surface. The circumferential periodic trusses repre-sent an advanced technique for deployable antennas due to their large stiffness-to-mass ratio and high package efficiency. The main tasks in designing mesh antennas are to ensure a successful in-orbit deployment and to form a paraboloidal reflector with sufficiently high shape precision. However, it is difficult for on-ground experiments to predict the in-orbit deployment behavior and the surface accuracy after deploy-ment due to the gravitational difference between on-ground and in-orbit environ-ment. Therefore, it is necessary to use simulation to optimize the designs of the deployable truss, cable networks, as well as gravity compensation systems, to mini-mize the difference between the in-orbit and on-ground behavior of mesh antennas, and to achieve an accurate in-orbit prediction. This thesis addresses the following three aspects:
A multiscale dynamic method for modeling cable-pulley systems is proposed. In this method, the cable is separated into two segments from the contact border between the pulley and the cable. The non-contact segment is modeled with variable-length cable elements based on the Arbitrary Lagrangian-Eulerian formulation, and the contact segment is merely treated as a virtual segment to dynamically locate the contact border. This method avoids the problems of dense mesh and small step caused by traditional contact modeling methods, greatly improves the computa-tional efficiency, and makes it possible to simulate antenna deployment with multi-scale nature of time and length. Furthermore, two kinds of synchronization mechanisms with variant-wrap-angle and constant-wrap-angle were designed to synchronize each bay of the deployable trusses. The proposed multiscale method was used to evaluate the effects of synchronization mechanisms on the deployment dynamics of mesh antennas. Simulation results show that the variant-wrap-angle design leads to remarkable asynchronous deployment with a sharp increase of driv-ing forces by motors, while the constant-wrap-angle design leads to a nearly syn-chronous deployment accompanied by remarkably small driving forces.
XII Preface
A design methodology for achieving a robust cable network against variation in tension forces was proposed, which reduces the accuracy requirements of the ten-sion forces to guarantee the tautness of the cable network. The method consists of two parts: first, a force tolerance was defined to quantitatively characterize the robustness of the cable network to tension forces; second, an optimization algorithm was established to enlarge the force tolerance by optimizing the tension forces and the geometry of the cable network. The proposed methodology was applied to design the mesh for a 6-bay front-fed antenna and a 30-bay offset antenna. Results show that after optimization, the force tolerances were enlarged up to about four and two times without notably altering the tessellation error, which ensures the tautness of the cable network with or without gravity. The robust design provides a possible technical path to ensure the shape consistency of the cable network between on-ground and in-orbit environment.
The effects of on-ground gravity compensation systems for webs on deployment dynamics of mesh antennas were revealed. In terms of constant-length offloading and constant-force offloading, the position and number of suspension nodes were evaluated on the performance of gravity compensation. Simulation results show that without offloading for webs, the gravitational energy of the webs increases quickly during the early unfolding period since the center of mass of the webs goes up, resulting in larger driving forces for the on-ground test to unfold the mesh antenna. When the webs are offloaded, it proves that too few or too many suspension nodes will add more constraints to the deployment, and a moderate number of suspension nodes with nearly equal distribution on the webs is a better choice to reduce the dif-ference between on-ground and in-orbit deployment.
Beijing, China Kangjia.Fu
Acknowledgment
I would like to extend my sincere gratitude to my supervisor, Associate Professor Zhihua Zhao. At the beginning of my PhD journey, when I often felt uncertain, Professor Zhao provided patient and meticulous guidance. He helped me master fundamental research skills, fostered my enthusiasm for scientific exploration, and supported my transition from an undergraduate to a doctoral researcher. During challenging research periods, his profound knowledge, insightful perspective, and creative thinking greatly broadened my horizons, inspired me to pursue innovation, and allowed me to experience the joy of scientific inquiry. I vividly remember the night before my first academic conference when Professor Zhao stayed up with me until 4.a.m. refining my presentation. For my first academic paper, he reviewed every sentence with utmost care. While studying abroad, whenever I encountered challenges in research or daily life, I often asked myself how Professor Zhao would approach the situation.A lab mate once remarked that I appeared to carry some of Professor Zhao’s influence, a comment I regard as a significant compliment. The lessons I have learned from Professor Zhao will benefit me throughout my lifetime.
I am also deeply thankful to Professor Gexue Ren. Whenever I felt inclined to relax, I recalled his consistent habit of working in the lab on weekends and holidays. In the early stages of my doctoral studies, when I struggled to immerse myself in research, my conversations with Professor Ren always provided valuable inspira-tion. Whenever we met, he kindly inquired about my progress and offered warm encouragement. His diligence and dedication will continue to serve as a model in my future career.
I wish to express my sincere appreciation to Professor Lihua Jin for providing me the opportunity to conduct research on mechanical metamaterials at UCLA..My study abroad experience stands out as one of the most rewarding and memorable chapters of my doctoral journey. I am also deeply grateful to Professor Jianbin Du at Tsinghua University for his invaluable guidance in structural optimization. My heartfelt thanks extend to Yongpeng Gu and Yun Peng for their assistance with mul-tibody dynamics and coding, as well as to Huan Zhang and Jiawei He for their sup-port in various aspects of daily life. Additionally, I would like to thank Meng Li, Jinyou Li, and Jiang Cui for their constructive suggestions regarding my
XIV Acknowledgment
dissertation. Finally, I wish to express my gratitude to every member of our lab for their unwavering support and camaraderie throughout my PhD studies.
I would like to express my deepest gratitude to my parents for their guidance and support throughout my journey from childhood to the completion of my doctorate. I am also profoundly thankful to my brother, with whom I shared a joyful childhood and countless cherished memories as we grew up together. Finally, I extend my heartfelt thanks and love to my wife, whose unwavering support and dedication to our family have enabled me to fully devote myself to my research and studies. I am equally grateful to my parents-in-law for their steadfast support to us. I am also thankful to my son, whose arrival has brought our family boundless joy and happiness.
I extend my sincere gratitude to the professors who reviewed this dissertation for their invaluable suggestions. I also wish to express my appreciation to the Tsinghua University Outstanding Doctoral Dissertation Publication Fund for its support. I am grateful to my editor, Yang Cheng, for his dedicated work, and to all the editors and staff at Tsinghua University Press and Springer for their diligent efforts that made the publication of this thesis possible.
Abbreviation
A: Area of cross section ALE: Arbitrary Lagrange-Euler β: Dummy variable C, C: Constraint equation(s)
d: Wrapping direction of the cable on a pulley ehp: Half-path-length RMS error ehpbf: Half-path-length best-fit RMS error ethreshold: Threshold of tessellation error of the web Etruss: Potential of the ring truss Eweb: Potential of cable networks ftrans: Friction force of sliding joints
g: Acceleration of gravity
G: Antenna gain
M: Mass matrix Mbend: Peak bending moment of the ring truss
N: Shape function
P: Center of the pulley
q: Generalized coordinate
Q: Generalized force RMS: Root mean square sign(x): Sign function
t: Tangent direction of the pulley tf: Tension of the cable networks Tdriving: Peak driving force by motor vmotor: Reel-in velocity by motor η: Coefficient of tension decay of the cable passing through pulleys μ: Coefficient of friction of a sliding joint along the batten δ: Force tolerance δf: Absolute force tolerance
XVI Abbreviation
δrel:
Relative force tolerance
ρ: Density
ψ: Cartesian rotation vector
λ: Vector of the Lagrange multipliers
φ, φ: Wrapping angle of the pulley
Λ: Coordinate transformation matrix
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Current Research and Unresolved Issues. . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Design of Deployable Structures. . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Modeling and Simulation of Deployable Antennas. . . . . . . 6
1.2.3 Design of Cable Networks. . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 On-Ground Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Research Content of this Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 From Modeling to Design of Synchronization Mechanisms in Mesh Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Multiscale Modeling Problems of Mesh Antennas . . . . . . . . . . . . . 18
2.3 Multiscale Modeling Methods of Cable-Pulley Systems. . . . . . . . . 19
2.3.1 Modeling of the Non-contact Segment . . . . . . . . . . . . . . . . 20
2.3.2 Border Location of the Non-sliding Case . . . . . . . . . . . . . . 22
2.3.3 Border Locations and Friction of the Sliding Case . . . . . . . 23
2.3.4 Governing Equations of a Cable-Pulley System . . . . . . . . . 25
2.3.5 Multiscale Modeling Methods of Cable-Cam Systems . . . . 26
2.3.6 Multiscale Modeling Validations. . . . . . . . . . . . . . . . . . . . . 29
2.4 Multibody Dynamic Model of a Mesh Antenna . . . . . . . . . . . . . . . 34
2.4.1 Tension Decay of Driving Cables Through Pulleys. . . . . . . 36
2.4.2 Modeling of Flexible Sliding Joints. . . . . . . . . . . . . . . . . . . 37
2.4.3 Stick-Slip Friction of the Sliding Joint . . . . . . . . . . . . . . . . 42
2.4.4 Assembly and Calculation of the Governing Equations . . . 46
2.5 Design and Verification of Synchronization Mechanism. . . . . . . . . 46
2.5.1 Variant-Wrap-Angle Design . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.2 Constant-Wrap-Angle Design . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.3 Experimental Validation of the Constant-Wrap-Angle Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.4 Material and Pretension of the Synchronization Cable . . . . 51
XVIII Contents
2.6 Evaluation of Synchronization Mechanisms on Deployment Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.1 Kinematic Latch Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.6.2 Bending Moments of the Ring Truss. . . . . . . . . . . . . . . . . . 53
2.6.3 Potential Energy and Butterfly Effect on the Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6.4 Overall Comparison of the Two Synchronization Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.7 Friction Sensitivity Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Robust Design of Cable Networks Against Variation in Tension Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 Error Source and Classification of the Cable Networks . . . 63
3.1.2 Robust Cable Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Definition and Calculation of Force Tolerance . . . . . . . . . . . . . . . . 65
3.2.1 Taut Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 Definition of Force Tolerance . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Formulation of the Force Tolerance of a Tension Truss Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.4 Simple Analytical Example . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Optimization of the Relative Force Tolerance. . . . . . . . . . . . . . . . . 71
3.3.1 Optimization of the Tie Forces . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Simultaneous Optimization of the Tie Forces and the Geometry of the Truss. . . . . . . . . . . . . . . . . . . . . . . 72
3.3.3 Constraint of Tessellation Error. . . . . . . . . . . . . . . . . . . . . . 73
3.3.4 Optimization Procedure of Force Tolerance . . . . . . . . . . . . 74
3.4 Examples of Robust Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Robust Design of a Six-Bay Two-Meter Front-Fed Truss Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.2 Robust Design of a Thirty-Bay Offset-Fed Truss Antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Effect and Design of Gravity Compensation System of Mesh Antennas on Deployment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.1 Gravity Compensation of Ring Truss . . . . . . . . . . . . . . . . . 84
4.1.2 Gravity Compensation of Webs. . . . . . . . . . . . . . . . . . . . . . 85
4.1.3 Contents of this Chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Gravity Compensation Design of the Ring Truss . . . . . . . . . . . . . . 86
4.2.1 Problems of the Constant-Length Offloading . . . . . . . . . . . 86
4.2.2 Constant-Force Offloading Design of a Single Bay . . . . . . 88
4.2.3 Constant-Force Offloading Design of the Ring Truss . . . . . 90
Contents XIX
4.3 Constant-Force Offloading Experiment. . . . . . . . . . . . . . . . . . . . . . 92
4.3.1 Constant-Force Offloading Experiment of a Single Bay . . . 92
4.3.2 Constant-Force Offloading Experiment of the Ring Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4 Effect of Gravity of Webs on Deployment Dynamics . . . . . . . . . . . 95
4.4.1 Modeling a Mesh Antenna Utilizing a Flexible Multibody Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.2 In-Orbit Deployment Dynamics . . . . . . . . . . . . . . . . . . . . . 99
4.4.3 On-Ground Deployment Dynamics Without Web Suspension Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 On-Ground Deployment Dynamics with Web Suspension Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5.1 Two Designs of Web Suspension Systems. . . . . . . . . . . . . . 102
4.5.2 Evaluation of Gravity Compensation. . . . . . . . . . . . . . . . . . 103
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.7 Conclusions and Prospects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.7.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Conclusions and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
