Investigation on large size deployable antenna truss mechanism

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INVESTIGATION ON LARGE SIZE DEPLOYABLE ANTENNA TRUSS MECHANISM

HEMANT ARORA

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2023

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© Indian Institute of Technology Delhi (IITD), New Delhi, 2023

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INVESTIGATION ON LARGE SIZE DEPLOYABLE ANTENNA TRUSS MECHANISM

by

HEMANT ARORA

Department of Mechanical Engineering

Submitted

in fulfillment of the requirements for the degree of Doctor of Philosophy to the

INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2023

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Dedicated to

my dear parents for their continuous love, support and blessings my loving wife & wonderful kids who supported me at each step to go

ahead to complete this research work And

The Gracious Almighty GOD

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Certificate

This is to certify that the thesis entitled Investigation on Large size Deployable Antenna Truss Mechanism being submitted by Mr. Hemant Arora to the Indian Institute of Technology Delhi for the award of the degree of Doctor of Philosophy in Mechanical Engineering is a bonafide record of original research work carried out by him under my supervision in conformity with rules and regulations of the institute. This work has not been submitted to any other University or Institute for the award of any Degree or Diploma.

Dr. Sudipto Mukherjee Dr. B. S. Munjal

Professor Ex. Group Director

Department of Mechanical Engineering Structure and Thermal Group Indian Institute of Technology Delhi Mechanical Engg. System Area

New Delhi-110016, India Space Applications Centre- ISRO

Ahmedabad-380015, India

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Acknowledgements

“There are no limitations to the mind except those we acknowledge”

—Napoleon Hill

I am feeling an immense pleasure for getting the opportunity to show my gratitude to the people without whom this thesis would not have been possible.

I thank my supervisor Prof. Sudipto Mukherjee, IIT Delhi for his continuous guidance and constant encouragement without which this thesis would not have completed. I am obliged to his patience with which he heard my queries, requests time to time and guided me with his full support and wisdom.

I thank my co-supervisor Dr. B. S. Munjal, SAC-ISRO, Ahmedabad for his continuous support, guidance and encouragement. His calmness and humble nature cannot be expressed in words. I am obliged to his wisdom on the subject and for his continusous direction to proceed further in completion of this thesis.

I am thankful to the members of Student Research Committee (SRC), Prof. S. K.

Saha, Prof. J. P. Khatait, Late Prof. Mashuq Un Nabi (Dept. of Electrical Engg.), for timely review and suggestions while evaluating the research plan and the synopsis of my work.

I express my gratitude towards Director, Space Applications Centre – ISRO, Ahmedabad, India for providing my opportunity to carry forward this research work. I also thank my senior colleagues Mr. P. J. Soni, Mr. H.R. Kansara, Mr. Arup Kumar Hait, Mr. D. K. Singh, Mr. Arup Rao Chodhuray and Mr. Anurag Verma at SAC-ISRO Ahmedabad for their continuous encouragement and support. I extend my thanks to my

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fellow colleagues Mr. Pradeep A. Iyer, Mr. Ankit Sharma, Mr. Shashank Pachauri, Mr.

Sunil Kumar Saw, Mr. Akhsay Khamkar at Space Applications Centre, SAC-ISRO, Ahmedabad for their support in technical as well as personal which cannot be expressed in words. I express my heartiest thanks to various students Mr. Vrushang Patel, Mr.

Dhwanil Sheth, Mr. Rutvik Dangarwala and Ms. Megha Thakkar who worked with me for their project work and supported me for completion of this work. I would like to thank Mr. Agraj, Mr. Sunil and Mr. Nikhil for their invaluable friendship and their nice gesture during my stay at IIT Delhi. I also thank Mr. Sachin Kansal and Mr. Rajesh Kumar and Ms. Shashikala Chauhan with their continuous support and help in timely submissions of semester report and regular information sharing. My thanks are also due to all the office staff of the department of mechanical engineering for their kind support and co-operation.

I would like to express my gratitutde towards my parents for their blessings, love, and care towards me which is inexpressible. I further extend my thanks to my dear wife Mrs. Deepika Arora and my kids, Prisha and Arjun for their continuous love and support towards completion of this thesis. I would like to express my affection towards my sisters Mrs. Neelam Khurana, Mrs Neerja Puri, my brother Mr. Dheeraj Arora and their family members for their continuous encouragement. I am grateful to all my family members especially my inlaws for their continuous wishes and support.

Last but not the least, I convey my thanks to one and all who have contributed to the realization of this thesis.

Hemant Arora

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Abstract

High-bandwidth satellite-based telecommunication demands large-diameter antenna reflectors (LDR). Launching of monolithic LDR antennas of sizes 6m or more is limited by the launch vehicle’s payload fairing space. A foldable antenna with space deployment mechanisms is hence used. The deployment process of the antenna is the process of transition from folded and strapped structure to a locked structure, which is complicated because of the presence of backlash, friction, and misalignment due to manufacturing accuracy. An attempt has been made to explore the various foldable configurations reported in the literature for smooth and effective deployment motion with a focus on configurations with space heritage. Mesh antennas are preferred for large-size deployable reflectors due to low manufacturing complexity, ease in deployment operation, and high deployed-to-stowed ratio. The AstroMesh configuration is seen to be the simplest configuration to investigate further through kinematics and dynamic analysis. The spatial configuration is converted to planar by introducing a transformation relationship between two adjacent bays to solve the kinematics. The 12m diameter configuration is analyzed for the given position, velocity, and acceleration parameters, which are used as an example. The deployment cable pull force, which estimates the amount of force required to achieve a fully deployed configuration, is predicted through dynamic analysis. Design parameters are identified and parametric optimization of configuration with the objective of achieving a compact stowed configuration is carried out. The parametric optimization study for a prototype configuration of 3m diameter leads to a volumetric efficiency of 99.7%. The optimized design configuration is then evaluated for structural requirements of space launch environmental specifications. A linearized formulation for support net

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analysis is carried out and an efficient numbering scheme was evolved to save memory and computation. Dynamic stiffness of stowed and deployed configurations is computed using the finite element method. The first fundamental mode of the stowed configuration of the 12m diameter truss structure is predicted to be 40 Hz. A prototype of a 3m diameter is realized to demonstrate the concept of the deployment mechanism. An SMA-based hold-down and release mechanism is also evaluated for functional aspects and hold-down capacity. Demonstration of deployment on 3m antenna reflector with support net and mesh configuration showed up issues with smooth deployment. To mitigate some of the problems, the use of tape flexure-based hinges is investigated. The buckling behavior of curved tapes is studied analytically with corresponding experimental strain measurement for equal sense and opposite sense bending cases under transverse loading. The absolute error between analytical formulation and experiments is found to be 9% and 19% for equal and opposite senses within the elastic limit. Characteristics of opposite and equal sense bending with combinations of a single and a double layer of tapes are analyzed in FEM for varying material options with CRFP and Ni36CrTiAl alloy seen to have benign stresses for 90^° rotation hinges. The feasibility of the concept is demonstrated with prototypes of joints incorporating tape flexures.

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सार

उच्च बैंडविड्थ उपग्रह आधारित दूिसंचाि बड़े व्यास िाल़े एंटीना रिफ्ल़ेक्टि

(एलडीआि) की मांग किता है। 6 मीटि या उसस़े अवधक आकाि क़े एलडीआि एंट़ेना का

प्रक्ष़ेपण, िाहन क़े प़ेलोड फ़ेयरिंग स्प़ेस द्वािा सीवमत है। इसवलए अंतरिक्ष परिवनयोजन तंत्र क़े

साथ एक फोल्ड़ेबल एंटीना का उपयोग वकया जाता है। एंट़ेना की खुलऩे की प्रविया, मुडी हुई औि बंधी हुई संिचना स़े एक पूिी तिह स़े विकवसत संिचना में संिमण की प्रविया है, जो वनमााण सटीकता क़े कािण बैकलैश, घर्ाण औि वमसवलग्न्मेंट की उपस्थथवत क़े कािण जवटल है। अंतरिक्ष वििासत क़े साथ विन्यास पि ध्यान द़ेऩे क़े साथ सहज औि प्रभािी परिवनयोजन गवत क़े वलए सावहत्य में रिपोटा वकए गए विवभन्न फोल्ड़ेबल कॉस्फ़िगि़ेशन का पता लगाऩे का प्रयास वकया गया

है। म़ेश एंट़ेना को बड़े आकाि क़े परिवनयोजन योग्य पिािताकों क़े वलए पसंद वकया जाता है

क्ोंवक कम विवनमााण जवटलता, तैनाती संचालन में आसानी औि उच्च अनुपात में तैनात होऩे क़े

कािण। वकऩेम़ेवटक्स औि गवतशील विश्ल़ेर्ण क़े माध्यम स़े आग़े की जांच किऩे क़े वलए एस्ट्रोम़ेश कॉस्फ़िगि़ेशन को सबस़े सिल कॉस्फ़िगि़ेशन क़े रूप में द़ेखा गया है। वकऩेम़ेवटक्स को

हल किऩे क़े वलए दो आसन्न खण्ों क़े बीच परिितान संबंध शुरू किक़े थथावनक विन्यास को

प्ल़ेनि में बदल वदया गया है। मीटि व्यास विन्यास का विश्ल़ेर्ण दी गई स्थथवत12, ि़ेग औि त्विण मापदंडों क़े वलए वकया गया है, जो एक उदाहिण क़े रूप में उपयोग वकए गए हैं। परिवनयोजन क़ेबल पुल बल, जो पूिी तिह स़े परिवनयोवजत कॉस्फ़िगि़ेशन को प्राप्त किऩे क़े वलए आिश्यक बल की मात्रा का अनुमान लगाता है, का अनुमान गवतशील विश्ल़ेर्ण क़े माध्यम स़े की गई है।

वडजाइन मापदंडों की पहचान की गई है औि एक कॉम्पैक्ट संग्रहीत कॉस्फ़िगि़ेशन प्राप्त किऩे

क़े उद्द़ेश्य स़े कॉस्फ़िगि़ेशन का पैिामीवटरक अनुकूलन वकया गया है। 3 मीटि व्यास क़े एक

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प्रोटोटाइप विन्यास क़े वलए पैिामीवटरक अनुकूलन अध्ययन स़े %99.7की िॉल्यूम़ेवटरक क्षमता

प्राप्त हुई है। वफि अंतरिक्ष प्रक्ष़ेपण पयााििण विवनदेशों की संिचनात्मक आिश्यकताओं क़े वलए अनुकूवलत वडजाइन विन्यास का मूल्यांकन वकया गया है। समथान शुद्ध विश्ल़ेर्ण क़े वलए एक

िैस्खक रूप स़े तैयाि वकया गया है औि स्मृवत औि गणना को बचाऩे क़े वलए एक कुशल नंबरिंग योजना विकवसत की गई है। परिवमत तत्व विवध का उपयोग किक़े संग्रहीत औि परिवनयोवजत कॉस्फ़िगि़ेशन की गवतशील कठोिता की गणना की गई है। 12 मीटि व्यास टरस संिचना क़े

संग्रहीत विन्यास का पहला मौवलक मोड हट्ाज होऩे 40का अनुमान लगाया गया है। तैनाती तंत्र की अिधािणा को प्रदवशात किऩे क़े वलए मीटि व्यास का एक प्रोटोटाइप 3वदखाया गया है।

कायाात्मक पहलुओं औि होल्डडाउन औि -आधारित होल्ड-डाउन क्षमता क़े वलए एक एसएमए- ल्यांकन वकया

रिलीज तंत्र का भी मू गया है। सपोटा ऩेट औि म़ेश कॉस्िगि़ेशन क़े साथ मीटि 3 एंट़ेना रिफ्ल़ेक्टि पि तैनाती का प्रदशान सुचारू तैनाती क़े साथ समस्याएँउत्पन हुई है। कुछ समस्याओं को कम किऩे क़े वलए, ट़ेप फ्ल़ेक्चिआधारित-जोड क़े उपयोग की जांच की गई है।

घुमािदाि ट़ेपों क़े बकवलंग व्यिहाि का विश्ल़ेर्णात्मक रूप स़े अनुप्रथथ लोवडंग क़े तहत समान अथा औि विपिीत अथा झुकऩे िाल़े मामलों क़े वलए प्रयोगात्मक तनाि माप क़े साथ अध्ययन वकया गया है। लोचदाि सीमा क़े भीति समान औि विपिीत इंवियों क़े वलए विश्ल़ेर्णात्मक सूत्रीकिण औि प्रयोगों क़े बीच पूणा त्रुवट 9 % औि 19 % पाई गई है। वसंगल औि डबल ल़ेयि क़े

ट़ेप क़े संयोजन क़े साथ विपिीत औि समान अथा झुकऩे की विश़ेर्ताओं का विश्ल़ेर्ण ^FEMमें

CRFP औि Ni36CrTiAl वमश्र धातु क़े साथ अलगअलग सामग्री विकल्ों क़े वलए वकया-गया है, वजसमें 90 वडग्री िोट़ेशन जोडो क़े वलए सौम्य तनाि द़ेखा जा सकता है। अिधािणा की

व्यिहायाता को ट़ेप फ्ल़ेक्ससा को शावमल किऩे िाल़े जोडों क़े प्रोटोटाइप क़े साथ प्रदवशात वकया

गया है।

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(Courtesy: ISRO [40])... 2 Figure 1.3: Space Deployed Antenna (from ref. [3]) ... 4 Figure 1.4: Image captured of a table describing common metrics for evaluation of large space structures performance (from ref. [87]) ... 5 Figure 2.1: Deployment sequence of Classical Petal Reflector (from ref. [88]) ... 13 Figure 2.2: TRW Sunflower Solid Reflector Antenna (from ref. [3], [88]) ... 14 Figure 2.3: Schematic View of Deployment of Dornier Solid Reflector (from ref. [3]) ... 14 Figure 2.4: Deployment Sequence of Rigid panel Deployable Antenna Reflector (from ref. [14]) ... 15 Figure 2.5: Hinges of Isostatic Deployable Antenna Reflector (from ref. [88]) ... 15 Figure 2.6: Hinges of New petal hinge design for Isostatic Deployable Antenna Reflector (from ref. [88]) ... 16 Figure 2.7: Artistic Image of Inflatable Antenna deployed in Space (from ref. [5]) ... 17 Figure 2.8: Micro-gravity deployment tests of inflatable booms (a) a hexagonal Yoshimura pattern, and (b) a pentagonal Miura pattern (from ref. [89]) ... 17

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Figure 2.9: Mesh antenna (from ref. [11] and modified) ... 18

Figure 2.10: Wrap-Rib antenna for LMSC ATS-6 Flight (from ref. [5]) ... 19

Figure 2.11: Spring-back Antenna and Collapsible Rib-Tensioned Antenna (from ref. [8])... 20

Figure 2.12: HALCA Deployable Antenna (from ref. [8]) ... 20

Figure 2.13: Tensegrity Model and Tensegrity Antenna (from ref. [10]) ... 21

Figure 2.14: Modular Antenna (from ref. [8]) ... 22

Figure 2.15: AstroMesh (a) Full view of the antenna (b) Detailed view of the antenna (from ref. [6] and modified) ... 22

Figure 2.16: Deployment sequence of AstroMesh Deployable Configuration (from ref. [6]) ... 23

Figure 2.17: Deployment sequence of AM-2 configuration Reflector (from ref. [7] ) ... 24

Figure 2.18: (1) Double Pantograph (2) Pantograph with radial extending members . 26 Figure 2.19: (1) In-plane scissor mechanism (2) Shifted scissor mechanism (from ref. [8])... 27

Figure 2.20: (1) Sarrus Mechanism (2) Spatial mechanism (from ref. [17]) ... 27

Figure 2.21: (1,2) Conical ring structure (3) Conical ring structure with scissor mechanism (from ref. [4]) ... 28

Figure 3.1: AstroMesh configuration (1) deployed view (2) detailed view of a bay... 34

Figure 3.2: Description of 5J and 3J joint configuration ... 34

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Figure 3.3: Deployment sequence of each bay configuration ... 35

Figure 3.4: Description of a four-bar mechanism ... 36

Figure 3.5: Single Bay configuration ... 37

Figure 3.6: Double bay configuration ... 37

Figure 3.7: Three bay configuration ... 39

Figure 3.8: Six bay Configuration ... 39

Figure 3.9: Kinematic Configuration of Linkage ... 40

Figure 3.10: Dimensional synthesis of Vertical member ... 43

Figure 3.11: AstroMesh deployment Sequence 1-6 ... 44

Figure 3.12: Deployment positions of the deployable ring structure ... 45

Figure 3.13: Position analysis of ring structure mechanism ... 46

Figure 3.14: Top-view of position analysis of the mechanism ... 46

Figure 3.15: A screenshot of Matlab symbolic formulation ... 51

Figure 3.16: Plotted profiles of Angular velocity and position ... 51

Figure 3.17: Angular acceleration and cable pull force ... 52

Figure 4.1: Fixed Reference Frame for Optimization Problem Formulation ... 56

Figure 4.2: Description of various critical points on the diagonal member ... 56

Figure 4.3: Description of various critical points on 5J joint ... 57

Figure 4.4: Description of remaining critical points on 5J joint ... 57

Figure 4.5: Design Parameters- Top View ... 58

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Figure 4.6: Design Parameters- Front View ... 58

Figure 4.7: Description of stowed radius ... 60

Figure 4.8: Interference of 3J joint with diagonal member and 5J joint with adjacent 5J joint body ... 62

Figure 4.9: Interference of 5J joint bodies ... 65

Figure 4.10: Interference of pulley with the inner wall of 5J joint ... 68

Figure 4.11: Interference of rope with slider inner surface ... 70

Figure 4.12: Convention for intersection of rope and slider ... 71

Figure 4.13: Genetic algorithm output ... 74

Figure 4.14: Iterations and step size of the optimization algorithm ... 75

Figure 4.15: MATLAB results for Optimization of 1m diameter 14 bays ... 76

Figure 4.16: Validation of optimum radius in CAD software ... 76

Figure 4.17: Visualization of optimized design parameters ... 77

Figure 4.18: Plot of variation of (a) stowed radius and (b) Volumetric efficiency v/s number of bays for 12m deployed diameter configuration ... 78

Figure 5.1: CAD simulation of launch configuration of ISRO’s typical spacecraft .... 82

Figure 5.2: Deployed and stowed configuration of 12m AstroMesh configuration (views shown are not to scale) ... 86

Figure 5.3: Screenshots of a few modes of 12m Antenna ring truss in the stowed configuration ... 89

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Figure 5.4: Displacements (mm) and Stress profile (MPa) under the quasi-static load

of 30g along the x-axis (Parallel to the mounting plane) ... 90

Figure 5.5: Displacement (mm) and Stress (MPa) profile under the quasi-static load of 30g along the z-axis (Parallel to the mounting plane) ... 90

Figure 5.6: Displacement (mm) and Stress (MPa) profile under the quasi-static load of 20g along the y-axis (Normal to the mounting plane) ... 91

Figure 5.7: Stowed configuration with node locations for random response measurement ... 92

Figure 5.8: Random response of specified nodes along the X-axis ... 92

Figure 5.9: Random response of specified nodes along the Y-axis ... 93

Figure 5.10: Random response of specified nodes along the Z-axis ... 93

Figure 5.11: Schematic view of the working mechanism of HDRM ... 95

Figure 5.12: CAD model of SMA based HDRM with the description of parts... 96

Figure 5.13: Free body diagram of ejector nut, ball and latch cup ... 96

Figure 5.14: HDRM assembly under testing ... 100

Figure 5.15: Tensile test setup and force-elongation response ... 102

Figure 5.16: Release time plots at various current values ... 102

Figure 5.17: FE models of various concepts of the ring structure ... 103

Figure 5.18: Fundamental modes of ring structure variant 1 (AstroMesh Configuration) ... 105

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Figure 5.19: Plot of comparison of the first fundamental modal frequency of various

options of 6m diameter ring structures ... 105

Figure 5.20: CAD model of 3J joint ... 107

Figure 5.21: CAD model of 5J joint ... 107

Figure 5.22: Antenna in deployed and stowed 2-bay configuration ... 108

Figure 5.23: Deployed configuration of 12m antenna ring truss structure ... 109

Figure 5.24: Schematic of a cable network with fixed and free nodes (adopted from [81])... 111

Figure 5.25: Numbering scheme ... 114

Figure 5.26: Mesh configuration of 3m deployable ring truss... 115

Figure 5.27: Force distribution in triangular faceted cables due to variation in tension ties computed using force density method ... 116

Figure 5.28: Prototype 3J joint... 118

Figure 5.29: Prototype of 5J joint ... 118

Figure 5.30: Prototype of 3m diameter AstroMesh configuration ... 119

Figure 5.31: Prototype with support net and simulated RF mesh ... 120

Figure 6.1: Sensitivity analysis of transmission efficiency and friction coefficient on deployment (adopted from [13] ) ... 122

Figure 6.2: Flexure-based support for the optical mirror fixing device ... 124

Figure 6.3: (a) Flexure-based support for optical mirror fixing device (b) Setup for alignment correction ... 125

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Figure 6.4: CAD simulation of prototype of Delta manipulator with flexure hinges

[49] ... 126

Figure 6.5: Concept of STEM configuration (from ref. [50])... 126

Figure 6.6: Conceptual design of Flex-16 Flexible hinge (from ref. [51]) ... 128

Figure 6.7: CAD model of tape flexure ... 129

Figure 6.8: Description of Equal sense and opposite sense bending ... 131

Figure 6.9: Bending behaviour of Tape Flexure ... 132

Figure 6.10: Representation of Wüst’s general theory (from ref. [56]) ... 134

Figure 6.11: Experimental method for finding residual moment (from ref. [67]) ... 136

Figure 6.12: Deformation zones of tape spring ... 139

Figure 6.13: Fold behaviour when (a) force and moment applied on one end and other end is fixed (b) equal forces and moments applied at both ends ... 140

Figure 6.14: loading and deformation under consideration ... 140

Figure 6.15: Cross-sectional view of tape spring ... 146

Figure 6.16: Experimental setup for loading unloading of tape spring ... 146

Figure 6.17: Representative sketch of cross-section at the fold ... 147

Figure 6.18: Projection of rotation angle about fold of equal and opposite sense bending ... 149

Figure 6.19: Representation of fold parameters ... 150

Figure 6.20: Strain measurement test set-up ... 151

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Figure 6.21: Strain for equal and opposite sense bending ... 152

Figure 6.22: Strain comparison plot – Opposite sense bending... 154

Figure 6.23: Strain comparison plot – Equal sense bending ... 154

Figure 6.24: Configuration of tape flexures explored for flexure hinge ... 156

Figure 6.25: Finite element model of tape flexure configuration with boundary condition ... 157

Figure 6.26: The boundary conditions applied on tape flexure configuration ... 158

Figure 6.27: Displacement profile of combined double layered tape flexures of single case ... 159

Figure 6.28: Von Mises stress profile of combined double-layered tape flexures of single case ... 159

Figure 6.29: Deployment of a folded boom structure ... 161

Figure 6.30: Deployment stages of a reflector with compliant joint ... 161

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List of Tables

Table 5.1: Environmental Test level specifications [24] ... 87

Table 5.2: Material properties of Link elements ... 88

Table 5.3: Performance parameters of SMA NITINOL wire ... 98

Table 5.4: SMA wire force calculation ... 98

Table 5.5: Response release time at no-load condition... 100

Table 5.6: Response Release time at various loads ... 101

Table 5.7: First Fundamental modes of Linkage with mass detail of various 12m diameter ring truss structure... 106

Table 5.8: Parameters of Triangular Faceted Mesh configuration obtained by force density method ... 117

Table 6.1: Input parameters for moment calculation ... 142

Table 6.2: Cases description of longitudinal curvature ... 143

Table 6.3: Computation of Dimensionless parameters for Case-I ... 143

Table 6.4: Computation of Dimensionless parameters for Case-II ... 143

Table 6.5: Comparison of bending moment computed analytically and experimentally ... 144

Table 6.6: Material Properties of Carpenter’s Steel Tape spring... 145

Table 6.7: Geometric Properties of tape spring ... 145

Table 6.8: Calculated results of bending moment, stress and strain ... 150

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Table 6.9: Material properties considered for tape flexure configurations ... 157 Table 6.10: Maximum Von mises stresses (MPa) for various tape flexure

configurations ... 160

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List of Symbols

The important symbols and abbreviations are given below in alphabetical order, where each one is explained. Notational rules followed in this thesis are as follows:

 Italic Roman/Greek letters (lower case) refer to scalars.

 BOLDFACE Roman/Greek upper-case letters denote matrices and vectors.

Latin Letters Description

A Surface Area of Reflector

𝑪_𝒖 Connectivity Relationship matrix for free node elements 𝑪_𝒇 Connectivity Relationship matrix for fixed node elements

D Dissipation Function of joints

𝛿_𝑟𝑚𝑠 Surface faceting root mean square error

E Elastic Potential Energy of Torsional Spring and pre-tension of net

ea Efficiency of Antenna

F/D Ratio of Focal length to Diameter of Antenna Reflector g Acceleration due to gravity

G Gain of Antenna

H Height of Vertical Member of Truss Configuration

i Degree of freedom

j Number of Joints

k1 Stiffness coefficient of torsional spring

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k2 Stiffness coefficient of tension support net structure 𝑘_𝑙 Longitudinal curvature of tape spring

𝑘_𝑡 Transverse curvature of tape spring λ Wavelength of Electromagnet Radiation

m Mass of linkage

mr Mass of release nut

M Bending moment of tape flexure

N Total number of Links

n Total number of bays

Pi Co-ordinates of point P in coordinate frame XiOiYi

𝑷̇_𝟏 Velocity Matrix of Point P 𝑷̈_𝟏 Acceleration Matrix of Point P

Pi+1 Co-ordinates of point P in coordinate frame Xi+1Oi+1Yi+1

∅ ̇ Angular velocity of link

QØ Generalized moment corresponding to the generalized coordinates

q Force density vector

R Reaction force generated by steel ball 𝑻_𝐢+𝟏^𝐢 Transformation Matrix

𝑻_𝐣^𝟏 Global Transformation Matrix T Total Kinetic Energy of Links

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t Total time period of deployment

θ In-plane Angle of horizontal member of truss V Gravitational Potential Energy of System

𝜀 Damping coefficient

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Acronyms

DC Direct Current

DOF Degrees of Freedom

EIRP Effective Isotropic Radiation Power GSLV Geo Synchronous Satellite Launch Vehicle HDRM Hold down and Release Mechanism ISRO Indian Space Research Organization LDR Large Diameter Antenna Reflector NGST Next Generation Space Telescopes OPI Opening Per Inch

PSD Power Spectral Density PSLV Polar Satellite Launch Vehicle RMS Root mean square

SAC Space Applications Centre, Ahmedabad, India SAR Synthetic Aperture Radar

SRS Shock Response Spectrum SMA Shape Memory Alloy SME Shape Memory Effect

STEM Storable Tubular Extendible Members TRW Thompson Ramo Wooldridge

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Investigation on large size deployable antenna truss mechanism

INVESTIGATION ON LARGE SIZE DEPLOYABLE ANTENNA TRUSS MECHANISM

HEMANT ARORA

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2023

© Indian Institute of Technology Delhi (IITD), New Delhi, 2023

INVESTIGATION ON LARGE SIZE DEPLOYABLE ANTENNA TRUSS MECHANISM

HEMANT ARORA

INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2023

Dedicated to

my dear parents for their continuous love, support and blessings my loving wife & wonderful kids who supported me at each step to go

ahead to complete this research work And

The Gracious Almighty GOD

Certificate

Acknowledgements

Abstract

सार

Contents

List of Figures

List of Tables

List of Symbols

Acronyms