McKenzie, David J. (2010) The dynamics of tethers and space-webs. PhD thesis, University of Glasgow. http://theses.gla.ac.uk/1483/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Ph.D. Thesis: The Dynamics of Tethers and Space-webs David J. McKenzie, BEng Submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy Departments of Mechanical and Aeronautical Engineering, Faculty of Engineering, University of Glasgow, G12 8QQ, Scotland, UK January 2010 ©David McKenzie, 2003-2010 Contents Abstract 15 Acknowledgements 16 1 Introduction 19 1.1 Overviewofthesis............................. 21 2 Literature review 23 2.1 Earlypioneers............................... 23 2.2 Tetherfundamentals ........................... 25 2.2.1 Fundamentaltetherconcepts . 26 2.2.2 Non-rotatingtethers . 26 2.2.3 Momentumexchange . 27 2.2.4 Motorizedmomentumexchangetethers . 29 2.2.5 Stagingwithmomentumexchange . 30 2.2.6 Deployingandrecoveryoftethers . 31 2.2.7 Captureandrendezvousoftethers . 33 2.2.8 WSBtrajectories ......................... 33 2.3 Tetherderivedstructures. 35 2 2.3.1 Deployedspacestructures . 35 2.3.2 Space-webs ............................ 36 3 Tether modelling 38 3.1 Constructingtheequationsofmotionofasystem . 38 3.1.1 Validating code modules . 39 3.2 Centreofmassmodelling......................... 40 3.2.1 Mass points & body selection . 40 3.2.2 Centre of mass calculations . 41 3.2.3 Positions of mass points . 43 3.3 ConstructingLagrange’sequations . 48 3.3.1 Energy modelling . 49 3.3.2 Choosing generalised coordinate systems . 50 3.4 Non-conservativeforces. 51 3.5 Rotationsystems ............................. 52 3.5.1 Rotationsummary ........................ 53 3.5.2 YZRotations ........................... 55 3.5.3 ZYRotations ........................... 56 3.5.4 Comparing the equations – position equations . 59 3.5.5 Comparing the equations – kinetic energies . 59 3.5.6 Comparing the equations – potential energies . 60 3.5.7 YZ rotation in inertial axes . 61 3.5.8 Singularities in the rotational systems . 61 3.5.9 Practicalusesofthedifferentequations . 62 3 3.6 Numericalintegrationtechniques . 62 4 Dynamics of tethers with inclination 64 4.1 Addition of inclination to tether model . 65 4.1.1 5degreeoffreedommodel . 66 4.2 Constructingtheequationsofmotion . 67 4.2.1 Rotations in the local coordinate system . 68 4.2.2 Lagrange’sequations . 71 4.2.3 Lagrange’sequations–localsystem . 71 4.2.4 Lagrange’sequations–globalsystem . 75 4.2.5 Non-conservativeforces. 77 4.3 Analysis of tether on an inclined orbit . 80 4.3.0.1 Behaviour of R, θ and ι with time . 81 4.3.0.2 Behaviour of α withtime ............... 81 4.3.0.3 Behaviour ofα ˙ withtime ............... 82 4.3.0.4 Behaviour of α with ι ................. 82 4.3.0.5 Behaviour of ψ, ψ,˙ ψ¨ ................. 83 4.3.0.6 Behaviour of α, α,˙ α¨ .................. 83 4.3.0.7 Behaviour of ∆V withtime .............. 84 4.3.0.8 Behaviour of tension with time . 84 4.3.0.9 Behaviour of stress with time . 84 4.3.1 Analysis of tension and stress in tether . 85 4.4 Tetherreleaseaiming........................... 86 4.4.1 Tethererroranalysis . .. .. 88 4 4.4.2 Lunarcapture........................... 90 4.5 A demonstration Lunar mission with a tether launch . 91 4.5.1 Spacecraftsizing ......................... 91 4.5.2 Analysis .............................. 93 4.6 Conclusions ................................ 94 4.7 Inclinationplots.............................. 98 5 Dynamics of tethers with length deployment 108 5.1 Deployment and recovery with the MMET . 108 5.2 Addition of length as a generalised coordinate . 109 5.3 Constructingtheequationsofmotion . .110 5.4 Non-conservativeforces. .115 5.4.1 Tetherbraking ..........................116 5.5 Analysis of length deployment of tether . 117 5.5.1 Discussion of results for deployment . 118 5.6 Refiningtheequationsofmotion . .120 5.7 Recoveryoftethers . .. .. .123 5.7.1 Recoveryprofile..........................124 5.7.2 Theequationsofmotionforrecovery . 125 5.7.3 Solving the equations of motion for recovery . 126 5.8 Deployment and recovery on an elliptical orbit . 128 5.9 Centreofmassmovement . .130 5.10Conclusions ................................131 5.11Figures...................................132 5 5.11.1 Linearkineticenergy . .143 6 Dynamics of space-webs 144 6.1 Introduction................................144 6.2 Modellingmethodology. .146 6.3 Webmeshing ...............................146 6.3.1 Webstructure...........................146 6.3.2 Dividing the web . 147 6.3.3 Web divisions . 148 6.4 Equationsofmotion ...........................149 6.4.1 Centre of mass modelling . 149 6.4.2 Rotations .............................150 6.4.3 Position equations . 152 6.5 Energymodelling .............................153 6.5.1 Virtualworkterms . .. .. .154 6.5.2 Simplifying Assumptions . 155 6.5.3 Robot Position Modelling . 156 6.6 Space-webdynamics ...........................157 6.6.1 Dynamical simulation . 157 6.7 Investigating the stability of the space-web . 158 6.7.1 Massoftheweb..........................158 6.7.2 Robotmass ............................159 6.7.3 Webasymmetry..........................160 6.7.4 Robotcrawlvelocity . .162 6 6.8 Statistical investigation into stability . 162 6.8.1 Massesofthewebandsatellite . .163 6.8.2 Massesofthewebandrobot. .164 6.8.3 Massofthewebandangularvelocity . .165 6.8.4 Massoftherobotandangularvelocity . 167 6.9 Conclusionsandrecommendations. .168 6.9.1 Conclusions ............................168 6.9.2 Recommendations. .168 7 Conclusions 169 7.1 Tetherrotations..............................169 7.2 Tethers with inclination . 169 7.3 Deploymentandrecoveryoftethers . .170 7.4 Space-webs ................................170 7.5 Furtherstudy ...............................171 8 Bibliography 172 Glossary 183 A Equations of motion with inclination 184 B Space-webs – additional graphics 188 B.1 Case 1 – CoM plots of stability while increasing web mass . 188 B.2 Case 1 – CoM plots of stability while increasing robot mass . 190 B.3 CoM plots of stability while changing robot paths . 191 7 B.4 Case 1 – CoM plots of stability while decreasing robot velocity . 192 C Space-webs – additional data 195 D Material strengths 200 E Motor properties 202 8 List of Figures 1.1 DiagramoftheMMET. ......................... 20 2.1 Tether length increment ∆L, from [Cosmo and Lorenzini, 1997, p175] 27 3.1 Symmetricaldumbbelltether . 41 3.2 Symmetrical dumbbell tether on a circular orbit (solid). After pay- load release, the facility system path at apogee (dash) and payload atperigee(dots) ............................. 42 3.3 Asymmetrical dumbbell in inertial axes . 44 3.4 Localcoordinateconstruction . 45 3.5 Local coordinate construction after rotation through angle ψ about thefacilitymass.............................. 45 3.6 Local coordinate construction after translation from the facility to theCoM.................................. 46 3.7 Local coordinate construction after translation from the CoM to the inertialcoordinatesystem . 46 3.8 Local coordinate construction after rotation through θ about the in- ertialcoordinatesystem . 47 9 3.9 Rotation sequence for YZ rotation: Rotating α about Y-axis then ψ about Z-axis. α is shown as a negative rotation here for ease of visualisation. ............................... 57 3.10 Rotation sequence for ZY rotation.. Rotating ψ about Z-axis then α about Y-axis. α is shown as a negative rotation here for ease of visualisation. ............................... 58 4.1 Ephemeris of Lunar orbital inclination [JPL, 2008] . 65 4.2 Rotation sequence for YZ rotation with inclination. α is shown as a negative rotation here for ease of visualisation. 69 4.3 Rotation sequence for YZ rotation with inclination. α and ι are shown as negative rotations here for ease of visualisation. 70 4.4 Eclipse checking function (not to scale) . 79 4.5 Sample plot of the Eclipse function in Mathematica . 79 4.6 Diagram showing an example of the WSB trajectory, from ESA Bul- letin103: [BiesbroekandJanin,2000]. 87 4.7 Change in apogee of the payload after release from the tether sys- tem after one half of an orbit compared to small changes of system parameters................................. 95 4.8 Change in apogee and inclination of the payload after release from the tether system after one half of an orbit compared to small changes ofsystemparameters . .. .. 96 4.9 The orbital debris environment from LEO to GEO, from [Wikipedia, 2008].................................... 97 4.10 Plots of R, θ and ι vs. timeforthefivecases. 99 4.11 Plots of α vs.timeforthefivecases. .100 10 4.12 Plots ofα ˙ vs.timeforthefivecases. .101 4.13 Plots of α vs. ι forthefivecases. .102 4.14 Phase plots of ψ forthefivecases.. .103 4.15 Phase plots of α forthefivecases.. .104 4.16 Plots of ∆V vs. timeforthefivecases. .105 4.17 PlotsofTensionvs. timeforthefivecases. 106 4.18 PlotsofStressvs. timeforthefivecases. 107 5.1 Discretised tether diagram with n =5masspoints.