dynamics of space tether systems

DYNAMICS OF SPACE TETHER SYSTEMS Volume 83 ADVANCES IN THE ASTRONAUTICAL SCIENCES by V. V. Beletsky E. M. Levin Published for the American Astronautical Society by Univelt, Incorporated, P.O. Box 281 30, San Diego, California 92198 Web Site: http://www.univelt.com Copyright 1993 (English Language Edition) by AMERICAN ASTRONAUTICAL SOCIETY AAS Publications Office P.O. Box 28130 San Diego, California 92198 Affiliated with the American Association for the Advancement of Science Member of the International Astronautical Federation First Printing 1993 Library of Congress Card No. 57-43769 ISSN 0065-3438 ISBN 0-87703-370-6 (Hard Cover) ISBN 0-87703-371-4 (Soft Cover) Published for the American Astronautical Society by Univelt, Incorporated, P.O. Box 28130, San Diego, California 92198 Web Site: http://www.univelt.com Printed and Bound in the U.S.A. (Russian Language Edition) © by V. V. Beletsky and E. M. Levin 1990 vi CONTENTS Foreword . . . . . . . . . . . . . iii Historical . . . . . . . . . . . . . 7 The Concept of the Book . . . . . . . . . . 9 Computer Aid . . . . . . . . . . . . 13 Nomenclature . . . . . . . . . . . . 15 1. Tethers in Space . . . . . . . . . . . 19 1.1. Space Tethers Prospects . . . . . . . . . 20 1.2. Science Fiction and Reality . . . . . . . . 34 1.3. Mechanical Models . . . . . . . . . . 42 1.3.1. General Model . . . . . . . . . . 42 1.3.2. Equations of Tether Motion . . . . . . . 43 1.3.3. Motion of the End-Bodies . . . . . . . 45 1.3.4. Simpli¯ed Models . . . . . . . . . 47 1.3.5. Tether Systems (TS) . . . . . . . . 48 1.4. Perturbations . . . . . . . . . . . 48 1.5. The Main Points . . . . . . . . . . 59 2. Dynamic Problems . . . . . . . . . . 61 2.1. Model with a Massless Tether . . . . . . . . 62 2.2. Massive Tether: Motion of the Mass Center . . . . . 66 2.3. Pendular Motions . . . . . . . . . . 71 2.4. Oscillations Against a Pendular Motion . . . . . 78 2.4.1. Full Equations . . . . . . . . . . 78 2.4.2. Linearization . . . . . . . . . . 80 2.4.3. Decomposition . . . . . . . . . . 83 2.5. Stationary Motions . . . . . . . . . . 83 1 2 CONTENTS 2.6. Investigation of Stability . . . . . . . . . 87 2.6.1. Variation Equations . . . . . . . . 87 2.6.2. Eigenvalue Problem . . . . . . . . 88 2.6.3. Eigenvalue Calculation Technique . . . . . 91 2.7. Forced Oscillations . . . . . . . . . . 95 2.8. Fast and Slow Components of Tether Motion . . . . 98 2.8.1. Di±culties of Numerical Simulation . . . . . 98 2.8.2. Modi¯ed Minakov's Equations . . . . . . 100 2.8.3. Tether Compression . . . . . . . . 103 2.8.4. Gradient of External Forces . . . . . . . 106 2.8.5. Boundary Conditions . . . . . . . . 107 2.8.6. Concluding Relations . . . . . . . . 109 2.9. The Main Points . . . . . . . . . . 111 3. Tether in the Newtonian Field . . . . . . . . 115 3.1. Classi¯cation of Motions with a Massless Tether . . . . 116 3.2. Equilibria of a Massive Tether . . . . . . . . 120 3.2.1. Equilibrium Equations . . . . . . . . 120 3.2.2. Integrating Tether Equilibrium Equations . . . . 122 3.2.3. Tether Wave Shapes . . . . . . . . 125 3.2.4. Out-of-Plane Con¯gurations . . . . . . . 128 3.2.5. Tapered Constant-Stress Tether . . . . . . 131 3.3. Conditions at the Tether Ends . . . . . . . 132 3.4. Bifurcations of Equilibria and Stability . . . . . . 137 3.4.1. Out-of-Plane Oscillations . . . . . . . 138 3.4.2. Instability of Wave-Like Con¯gurations . . . . 139 3.4.3. Stability of Monotonous Con¯gurations . . . . 142 3.4.4. Analogy to Euler's Problem . . . . . . . 147 3.5. Radial Con¯gurations . . . . . . . . . 148 3.6. Stability of Radial Con¯gurations . . . . . . . 153 3.6.1. Inextensible Tether . . . . . . . . . 153 3.6.2. Extensible Tether . . . . . . . . . 157 3.6.3. Super-Long Tether . . . . . . . . . 160 3.6.4. Tether Twisting . . . . . . . . . 162 3.7. Natural Frequencies and Modes of Oscillations . . . . 164 3.7.1. Eigenvalue Problem . . . . . . . . 164 3.7.2. Longitudinal Oscillations . . . . . . . 166 3.7.3. Transverse Oscillations . . . . . . . . 167 3.7.4. Coupled Longitudinal-Transverse Oscillations . . . 173 3.8. Nonlinear Oscillations . . . . . . . . . 176 3.8.1. Galerkin's Formulation . . . . . . . . 176 CONTENTS 3 3.8.2. Integrating Equations of Oscillations . . . . . 179 3.8.3. Amplitude-Frequency Relation . . . . . . 182 3.9. Rotation of the End-Bodies . . . . . . . . 183 3.9.1. Equations of Rotational Motion . . . . . . 183 3.9.2. Uniaxial Orientation along the Tether Line . . . 186 3.9.3. Rotation about the Tether Line . . . . . . 189 3.9.4. Torsional Oscillations of a Tape . . . . . . 194 3.10. The Main Points . . . . . . . . . . 195 4. Tethered Atmospheric Probe . . . . . . . . 199 4.1. Aerodynamic Forces . . . . . . . . . . 200 4.2. Classi¯cation of Motions with a Massless Tether . . . . 202 4.3. Equilibria of a Massive Tether . . . . . . . . 208 4.3.1. Equilibrium Equations . . . . . . . . 208 4.3.2. Wave-Like Con¯gurations . . . . . . . 209 4.3.3. Monotonous Con¯gurations . . . . . . . 212 4.3.4. The E®ect of the Atmospheric Rotation . . . . 214 4.4. Stability of Stationary Motions . . . . . . . 216 4.4.1. Increments to the Aerodynamic Forces . . . . 216 4.4.2. Damping Out-of-Plane Oscillations . . . . . 216 4.4.3. In-Plane Oscillations . . . . . . . . 218 4.4.4. Orbital Instability . . . . . . . . . 220 4.4.5. Evolution of Eigenvalues . . . . . . . 222 4.5. Aerogradient Instability . . . . . . . . . 226 4.5.1. Massless Tether . . . . . . . . . 226 4.5.2. Massive Tether . . . . . . . . . 229 4.5.3. Stability Regions . . . . . . . . . 233 4.5.4. E®ective Longitudinal Sti®ness of the Tether . . . 235 4.6. Probing Feasibility Assessment . . . . . . . 241 4.6.1. Tether Strength Condition . . . . . . . 241 4.6.2. Stability Condition . . . . . . . . . 243 4.6.3. Thrusting . . . . . . . . . . . 248 4.6.4. Heating . . . . . . . . . . . 248 4.6.5. Conclusions . . . . . . . . . . 253 4.7. Periodic Perturbations . . . . . . . . . 253 4.7.1. Rotation of the Atmosphere . . . . . . . 253 4.7.2. Variations of the Air Density . . . . . . 256 4.7.3. Orbit Ellipticity . . . . . . . . . 258 4.7.4. Resonace in Out-of-Plane Oscillations . . . . 260 4.7.5. Pendular Approximation . . . . . . . 262 4.8. The Main Points . . . . . . . . . . 264 4 CONTENTS 5. Electrodynamic Tether . . . . . . . . . 267 5.1. Physical Description . . . . . . . . . . 268 5.2. Analysis of Motion with a Massless Tether . . . . . 272 5.2.1. Equations of Motion . . . . . . . . 272 5.2.2. Equilibrium Positions . . . . . . . . 274 5.2.3. Instability under a Constant Current . . . . . 276 5.3. Equilibria of a Massive Tether . . . . . . . . 281 5.3.1. Equilibrium Equations . . . . . . . . 281 5.3.2. Integrating Tether Equilibrium Equations . . . . 283 5.3.3. Conditions at the Tether Ends . . . . . . 286 5.3.4. Current Limitation . . . . . . . . . 287 5.3.5. Tether Shape . . . . . . . . . . 289 5.4. Stability of Stationary Motions . . . . . . . 291 5.4.1. Out-of-Plane and In-Plane Oscillations . . . . 291 5.4.2. Instability under a Constant Current . . . . . 292 5.4.3. Current Control . . . . . . . . . 294 5.4.4. Calculation of Eigenvalues . . . . . . . 295 5.4.5. Asymptotic Stability Conditions . . . . . . 297 5.4.6. E®ective Longitudinal Sti®ness of the Tether . . . 299 5.4.7. Stability Regions . . . . . . . . . 303 5.5. Generation and Propulsion Feasibility Assessment . . . 304 5.5.1. Stability Limitations . . . . . . . . 304 5.5.2. Ohmic Losses . . . . . . . . . . 308 5.5.3. Conclusions . . . . . . . . . . 309 5.6. Forced Oscillations and Orbit Evolution . . . . . 310 5.7. Petal-Like Con¯gurations . . . . . . . . . 314 5.7.1. Equilibrium Shapes . . . . . . . . . 314 5.7.2. Stability Proof . . . . . . . . . . 316 5.7.3. Forced Oscillations . . . . . . . . . 320 5.8. The Main Points . . . . . . . . . . 321 6. Traveling Tether . . . . . . . . . . . 323 6.1. Equations of Motion with a Traveling Tether . . . . 324 6.1.1. General Equations . . . . . . . . . 324 6.1.2. Integrating Tether Equilibrium Equations . . . . 325 6.1.3. Out-of-Plane Con¯gurations . . . . . . . 328 6.2. The Loops of a Traveling Tether . . . . . . . 332 6.2.1. \Right" and \Left" Loops . . . . . . . 332 6.2.2. Stability of the \Right" Loops . . . . . . 336 6.3. The Main Points . . . . . . . . . . 337 CONTENTS 5 7. Damping Oscillations . . . . . . . . . . 339 7.1. Possible Ways of Damping . . . . . . . . 340 7.1.1. Internal Friction in the Tether . . . . . . 340 7.1.2. Active Longitudinal Damper . . . . . . . 342 7.1.3. Nonlinear Damping of Out-of-Plane Oscillations . . 344 7.1.4. Other Strategies . . . . . . . . . 346 7.2. Decrements for a Massive Tether . . . . . . . 348 7.3. Damping with a Movable Boom . . . . . . . 353 7.4. The Main Points . . . . . . . . . . 358 8. Rotation and Libration . . . . . . . . . 361 8.1. Classi¯cation of Motions with a Massless Tether . . . . 362 8.1.1. Equations of Motion . . . . . . . . 362 8.1.2. Circular Orbit . . . . . . . . . . 362 8.1.3. Elliptic Orbit . . . . . . . . . . 366 8.2. Motions with a Slack Tether . . . . . . . . 369 8.2.1. Tether Bounces . . . . . . . . . 369 8.2.2. Evolution of Motion . . . . . . . . 371 8.2.3. Final Motions . . . . . . . . . . 378 8.3. Pendular Motions with a Massive Tether . . . . . 378 8.4. Modal Decomposition . . . . . . . . . 383 8.4.1. Transverse Oscillations . . . . . . . . 383 8.4.2. Parametric Resonances . . . . . . . . 386 8.4.3. Oscillations of Tension . . . . . . . . 391 8.5. The Main Points . . . . . . . . . . 395 9. Deployment and Retrieval . . . . . . . . . 397 9.1. Exponential Deployment and Retrieval . . . . . . 398 9.2. Uniform Deployment and Retrieval . . . . . . 401 9.2.1. Nearly Uniform Tether Length Variation . . . . 402 9.2.2. Natural Tether Tension Control . . . . . . 407 9.2.3. Adjusted Tension Control . . . . . . . 412 9.3. Some General Limitations . . . . . . . . 415 9.4. Tether Con¯guration During Deployment and Retrieval . . 420 9.4.1. Quasi-Stationary Tether Shapes . . . . . . 420 9.4.2. Deployment from the Main Satellite . . . . . 423 9.4.3. Deployment from the Subsatellite . . . . . 424 9.4.4. Deployment from Both Ends . . . . . . 426 9.5. Modal Expansion and Analysis of Transient Processes . . 427 9.5.1. Transverse Oscillations . . . . . . . . 427 9.5.2. Longitudinal Oscillations . . . . . . . 432 9.5.3. Numerical Simulation . . . . . . . . 433 6 CONTENTS 9.6. Motion after Tether Release or Failure . . . . . 435 9.7. The Main Points . . . . . . . . . . 440 10. Anchored Tether . . . . . . . . . . . 443 10.1. Motions of a Lunar Tethered Satellite . . . . . 444 10.1.1 Equations of Motion . . . . . . . . 444 10.1.2. Approximation of Circular Orbit . . . . . 446 10.1.3. Elliptic Orbit . . . . . . . . . 450 10.2. Stationary Con¯gurations . . . . . . . . 451 10.3. Stability of Radial Con¯gurations . . . . . . 457 10.4. Motion of a Phobos' Anchored String . . . . . 461 10.5. The Main Points . . . . . . . . . . 463 11. Satellite Ring . . . . . . . . . . . 465 11.1. Discrete and Continuum Models . . . . . . . 466 11.2. Stability of a Continuum Ring . . . . . . . 468 11.3. Stability of a Discrete Ring . . . . . . . . 476 11.4. Analogy to Meteor Rings . . . . . . . . 479 11.5. The Main Points . . . . . . . . . . 482 References . . . . . . . . . . . . . 485 Subject Index . . . . . . . . . . . . 495 List of Tables . . . . . . . . . . . . 499