DOCSLIB.ORG

Effects of Patch Potentials on the GP-B Performance

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Effects of Patch Potentials on the GP-B Performance Effects of Patch Potentials on the GP-B Performance Results and Lessons Learned Sasha Buchman, John Turneaure HEPL June 24 2009 & GP-B March 24 2009 Page 1 GP-B Performance 6606 Geodetic effect Roll averaged 1,000 to 2,500 marcs/yr MISALIGNMENT 1000 Apparent linear drift LINEAR DRIFTS about 500 marcs/yr Repeat events RESONANCE to 200 marcs 100 39.0 Frame dragging effect 10 marcsec/yr 1 1.00 Original GP-B requirement Drift rate (deg/hr) rate Drift 0.50 Tighter req. 1995 DESIGN 0.21 Single gyro expectation 0.12 4 Gyro expectation 0.1 0.08 Readout alone (4 gyros) 0.01 1 marcsec/yr = 3.2 × 10-11 deg/hr Page 2 Goals and Outline ¾ Identify and understand “anomalous effects” ¾ Identify single cause of effects if possible ¾ Establish physical base for data analysis ¾ Experimental Observations ¾ Ground Measurements ¾ Classification of Effects on GP-B ¾ Discussion of Effects ¾ Remaining Work ¾ Lessons Learned Page 3 The Relativity Mission Concept 1 GM α GI ⎡3R ⎤ dγ 3 Ω d ⎛ ⎞ ⎛ 1 ⎞ ⎛ ⎞ Ω =⎜γ +⎟ 2× 3()R v +⎜γ +1 +⎟ 2 3⎢ ⋅ω 2 ()e ⋅R − e ω⎥ = ⎜ ⎟ ⎝ 2c⎠ R ⎝ 4⎠c 2 R ⎣ R ⎦ γ 2⎝ Ω ⎠geodet ic γ , α - 1 Frame Dragging 1 dγ ⎛ marcs⎞ Geodetic Effect 2= 3. × − 104⎜ ⎟ Space-time curvature Rotating matter drags space-time γPage 4 ⎝ yr ⎠ de Sitter (1916) Pugh and Schiff (1959, 1960) Experimental Observations Coupling of rotor-fixed frame to the Gyro Suspension System (GSS) 3 Modulation at low spin ωSL and polhode ωP frequency in the ~ 1.5 Hz GSS band -7 1 ¾ ωSL=1.3 Hz mod. control effort: 30% of ~1.5×10 N ¾ ωSL=1.3Hz mod. position at : 40 nmpp; ~0.1% gap -8 2 ¾ ωP mod. z (telescope axis) bias at 80 Hz spin : 10 N -8 3 ¾ ωP mod. control effort at 80 Hz spin: ~10 N “Triangle” = dipole + ‘all’ harmonics 2 1 Page 5 Possible Causes of Enhanced Coupling Rotor-fixed Mechanisms Explain “30% rotor bumps” (103 of expected) 1. Rotor geometry a. Mass unbalance: ~10nm measured (3×10-4 gap) @ Smaller than observed coupling by ~ 103 b. Surface waviness: ~10nm measured (3×10-4 gap) @ Smaller than observed coupling by ~ 103 2. Trapped flux interacting with magnetic fields Three independent calculations @ Smaller than observed coupling by > 103 3. Non uniform potential of rotor surface @ Coupling consistent with ~50 mV – 100 mV patch effect modulating Ve ≈ 200 mV suspension voltages ¾ Variation of electric potential over the surface ¾ It can arise due to the polycrystalline structure Gyroscope ¾ It can be affected by presence of contaminants d ¾ Modeled as dipole layer ¾ Patch fields present on rotor and housing walls Housing ¾ Cause forces and torques between surfaces PageData 6 explained by patch potentials of ~50-100 mV on rotor and housing Ground Patch Effect Investigations SEM images of gyro (Nb) & electrode (Ti) coatings ¾ Pre-launch investigation ¾Contact potential differences ~ 0.1V - 1V ¾Patches mitigated/eliminated by grain size 0.1 μm << 32 μm rotor-electrode gap ¾Kelvin probe measurements on flat samples ¾ Post-launch ground investigations dipole + harmonics1 39 40 2 3 38 0.65 4 ¾Work function profile by UV photoemission 37 0.6 5 36 0.55 6 35 0.5 7 ¾Detailed analytical modeling 34 0.45 8 0.4 33 0.35 9 32 0.3 10 ¾Kelvin probe measurements 0.25 31 0.2 11 30 12 Kelvin probe scans Kelvin probe 29 13 28 14 27 15 26 16 25 17 24 18 23 22 20 19 21 Work function polar plot, UV photoemission ST-7, LISA, LPF, LIGO find: Page 7 Patch potentials 30 -100 mV Causes of Patch Potential Variations Gyroscope ¾ The rotor coating process can lead to variations in the patch potential ¾ Coating is the result of many layers ¾ Each layer covers about 2/3 of the rotor surface ¾ Coating is axi-symmetric, but varies with angle to deposition source ¾ Thickness variations ¾ Impurity variations ¾ Crystal structure variations ¾ Contaminants Housing ¾ All suspension electrodes coated with same axi-symmetric process ¾ Small variation from center to edge ¾ 6 separate depositions ¾ Ground plane coating ¾ Substantial variation expected due to coating process / angle ¾ 2 separate depositions ¾ Thickness variations ¾ Impurity variations ¾ Crystal structure variations ¾ Contaminants Page 8 ‘Eight’ Patch Potential Effects on GP-B ¾ Coupling to GSS Î Vpp 50-100 mV ¾ z axis force ¾ At zero frequency ¾ At polhode harmonics (mentioned) ¾ Torques ¾ Misalignment Î following talks ¾ Resonance Î following talks ¾ Dissipation mechanisms ¾ Spin-down ¾ Polhode damping ¾ Charge measurement bias Page 9 Patch Potential in Spherical Harmonics Two Approximations: A. Dipole only B. Harmonic expansion ¾VR: rotor potential, VH: housing potential ¾ Potentials are real 1 ∞ l ′′ ′′ V(,)Rθ φ = ∑∑RVY l,,, m(,) lθ m φ m * (Y1 , 0 0 , 0l=−0 m ) = l VR,, l− = m ( −1) VR,, l m 1 ∞ l V = −1 m V * ′′′′ ′′′′ H,, l− m ()H,, l m VH(,)θ φ = ∑∑HVY l,,, m(,) lθ m φ (Y1 , 0 0 , 0l=−0 m ) = l Statistical Model ¾ Spherical harmonics terms are uncorrelated ¾ Assume a power spectrum characterized by the parameter r ¾ Assume that VR,l,m and VH,l,m are both Gaussian and have the same σ * * * VV, 1RR , 0 , 1 , 0 * VV, 1HH , 0 , 1 , 0 RVV l,,,, m R l= m r HVV l,,,, m H l= m r Page 10 l l z-axis Force Effects at f = 0 Axial thrust towards guide star (z axis) for the GP-B mission ¾ Average at f = 0 0.6 Day #310 z Bias Calibration Day #212 z Bias Calibration ¾ Varying with VR Gyros Discharged 0.4 0.2 N) μ 0 -0.2 Day #432 Day Day z Bias Calibration #157 #180 Gyro 1 Drag-Free -0.4 Gyro 1 Drag-Free to End of Mission GP-B orbit Semi-Major axis -0.6 Day #360 z Bias Calibration Gyros Discharged 1.3 Effective Z-axis thrust ( Effective -0.8 Day #151 - Sept 18, 2004, 140 190 240 290 340 390 440 490 Gyro 3 polhode resonates 1.2 with orbit period Time (days from Day #1; Apr. 20, 2004) 1.1 Day #318 - March 4, 2005, Drag-free off for 5 days 1.0 Semi-Major Axis ( km + 7,017 km ) km 7,017 + km ( Axis Semi-Major 0.9 0 100 200 300 400 500 600 Page 11 Time (days from Day #1; Apr. 20, 2004) Polhode Modulation of z-axis Force plitude of Control Effort at PomAAxis)Harmonics :#1 (a lhode Gyro 10-7 83 s,001=), T33-131 #1y04 (da-31 20g 29uA Mar 7- #321y9 2005 (da1 s= 3,12), T23-3 10-8 1 s), T= 3,1226-424 #4y05 (daJun 19-21 20 10-9 10-10 Control Effort (N) 10-11 100 101 102 Polhode Harmonic Number Polhode Modulation ¾ Patch potential model gives the force for the pth harmonic of polhode ¾ Using the observations and the statistical model 2 l lmax ⎡ ⎤ ⎛ a ⎞ 4ε 0 VR,,,0 l d 0()− 0 γ −H,, V0 l l −1 l+1 Fz, p = ⎜ ⎟ cos(pω P t× ) ⎢ V()R,,,1 l d 0p 0()−γ l +R V,,,1 l 0p 0−() dγ ⋅()1 + ⎥ l d 2 ∑ − + ⎝ ⎠ ()Y1,(,)0 0 l=20 ⎣⎢ +l 1 ⋅ 2() l − 1 + 1 ⎦⎥ Page 12 z-axis Forces Due to Rotor Charge z- xis force: zero frequency averagea Vd 0 = 72 mV ¾ Model: Vd0 rotor - oushing dipole component ¾ Force FZ = 9 nN; VR = 10 mV 8π ε r 2 F = 0 VV 3 d 2 R0 d z- axis force: variation with rotor charging Vd 0 = 81 mV ¾ Model: Vd0 rotor - oushing dipole component dV dV Δ =F 8 nN; ΔV =R t Δ10mV; = R 0= . 12mV/day Z R dt dt 8π ε r 2 ΔF =0 ⋅VV Δ' ⋅ Z 3 d 2 R d0 Data explained by patch potentials of ~50 - 100 mV on rotor and housing Page 13 Misalignment Torque ¾ Torque model for given l and m = 0 C d C N= −VVt [d l ()β =] lt l()1 + ) V V β l 3 R, , l 0 H , , ld 0β 0 , 0 6 R, , l 0 H , , l 0 ¾ Statistical model 2 Magnitudeateof Drift R v s. AngleMsaliig of nmetn ~1800 4 Ct ⎛l() l+1 ⎞ N = VV ⎜ ⎟ β Gy 1or RH1,,,, 0 1∑ 0 r 3.5 6 l=1 ⎝ l ⎠ Gy 2or 3 Gy 3or Gy 4or Equivalent dipole field as a function r 2.5 2 1.5 (as/day) Rate Drift )darcsec/yate (a RtfirD 1 0.5 0 0 20. 0.4 0.6 0.8 1 enmglignAlais Mfoe et (dnegre)s Data explained by patch potentials ~50-100 mV on rotor and housing Page 14 Roll – Polhode Resonance Torque ¾ Torque model C d C l( + l1) N= −Vt V()()l d γ []d l β = − t V Vl () d γ l 3R, , l 1 H , , l 1 0 m ,P dβ 0 , 1 β =0 6 R, , l 1 H , , l 1 0 m ,P ¾ Statistical Model 2 C ~1800 ⎛l() l+1 dl ()γ ⎞ N= −VVt ⎜ 0,m P ⎟ RH1,,,, 0 1∑ 0 ⎜ r ⎟ 6 l= m ⎝ l ⎠ Equivalent dipole field as a function r rcseca 343 393 443 ) 2004. 20,rp; A #1yDa romf syad(Time Data explained by patch potentials ~30-150 mV on rotor and housing Page 15 Spin-Down Measurements Gyro 1 dfs/dt vs. Time Gyro 2 dfs/dt vs. Time ‐0.15 ‐0.12 ‐0.13 ‐0.16 ‐0.14 /dt (nHz/sec) ‐0.17 /dt (nHz/sec) s s ‐0.15 df df ‐0.18 ‐0.16 163 254 345 436 163 254 345 436 Time (days from Day #1; Apr.
Load more

Recommended publications
  • GYRODYNAMICS Introduction to the Dynamics of Rigid Bodies
    2 GYRODYNAMICS Introduction to the dynamics of rigid bodies Introduction. Though Newton wrote on many topics—and may well have given thought to the odd behavior of tops—I am not aware that he committed any of that thought to writing. But by Euler was active in the field, and it has continued to bedevil the thought of mathematical physicists. “Extended rigid bodies” are classical abstractions—alien both to relativity and to quantum mechanics—which are revealed to our dynamical imaginations not so much by commonplace Nature as by, in Maxwell’s phrase, the “toys of Youth.” That such toys behave “strangely” is evident to the most casual observer, but the detailed theory of their behavior has become notorious for being elusive, surprising and difficult at every turn. Its formulation has required and inspired work of wonderful genius: it has taught us much of great worth, and clearly has much to teach us still. Early in my own education as a physicist I discovered that I could not understand—or, when I could understand, remained unpersuaded by—the “elementary explanations” of the behavior of tops & gyros which are abundant in the literature. So I fell into the habit of avoiding the field, waiting for the day when I could give to it the time and attention it obviously required and deserved. I became aware that my experience was far from atypical: according to Goldstein it was in fact a similar experience that motivated Klein & Sommerfeld to write their 4-volume classic, Theorie des Kreisels (–). In November I had occasion to consult my Classical Mechanics II students concerning what topic we should take up to finish out the term.
    [Show full text]
  • Accurate Simulation of Rigid Body Rotation and Extensions to Lie Groups
    Accurate Simulation of Rigid Body Rotation and Extensions to Lie Groups Sam Buss Dept. of Mathematics U.C. San Diego Visible Luncheon Gwyddor Cyfrifiador University of Swansea March 3, 2011 Topics: ◮ Algorithms for simulating rotating rigid bodies. ◮ All algorithms preserve angular momentum. ◮ Algorithms can be made energy preserving. ◮ Generalization to Lie group setting. Talk outline: 1. Rigid body rotations. 1st thru 4th order algorithms. Unexpected terms. 2. Generalization to Taylor series methods over Lie groups/Lie algebras. 3. Energy preservation based on Poinsot ellipsoid. 4. Numerical simulations and efficiency. 1 Part I: The simple rotating, rigid body I = Inertia matrix (tensor). ω L = Angular momentum. ω = Rotation axis & rate, L L = Iω (Euler’s equation) ω = I−1L ω˙ = I−1(L˙ ω Iω) − × ω¨ = ω ω˙ + I−1(L¨ ω˙ L 2ω L˙ + ω (ω L)) × − × − × × × ... ... ω = 2ω ω¨ ω (ω ω˙ ) + I−1[L 3ω L¨ 3ω˙ L˙ ω¨ L × − × × − × − × − × + ω˙ (ω L) + 2ω (ω˙ L) + 3ω (ω L˙ ) ω (ω (ω L))] × × × × × × − × × × Wobble: ω˙ = 0 even when no applied torque (L˙ = 0). 6 vL˙ = Rate of change of momentum = Applied Torque. 2 Framework for simulating rigid body motion We assume the rigid body has a known angular momentum, and the external torques are completely known. The orientation (and hence the angular velocity) is updated in discrete time steps, at times t0, t1, t2,.... Update Step: At a given time t , let h = ∆t = t t , and assume i i+1 − i orientation Ωi at time ti is known, and that momentum is known (at all times). Update step calculates a net rotation rate vector, ω¯, and sets Ωi+1 = Rhω¯ Ωi, where Rν performs a rotation around axis ν of angle ν .
    [Show full text]
  • Polhode Motion, Trapped Flux, and the GP-B Science Data Analysis
    July 8, 2009 • Stanford University HEPL Seminar Polhode Motion, Trapped Flux, and the GP-B Science Data Analysis Alex Silbergleit, John Conklin and the Polhode/Trapped Flux Mapping Task Team July 8, 2009 • Stanford University HEPL Seminar Outline 1. Gyro Polhode Motion, Trapped Flux, and GP-B Readout (4 charts) 2. Changing Polhode Period and Path: Energy Dissipation (4 charts) 3. Trapped Flux Mapping (TFM): Concept, Products, Importance (7 charts) 4. TFM: How It Is Done - 3 Levels of Analysis (11 charts) A. Polhode phase & angle B. Spin phase C. Magnetic potential 5. TFM: Results ( 9 charts) 6. Conclusion. Future Work (1 chart) 2 July 8, 2009 • Stanford University HEPL Seminar Outline 1. Gyro Polhode Motion, Trapped Flux, and GP-B Readout 2. Changing Polhode Period and Path: Energy Dissipation 3. Trapped Flux Mapping (TFM): Concept, Products, Importance 4. TFM: How It Is Done - 3 Levels of Analysis A. Polhode phase & angle B. Spin phase C. Magnetic potential 5. TFM: Results 6. Conclusion. Future Work 3 July 8, 2009 • Stanford University HEPL Seminar 1.1 Free Gyro Motion: Polhoding • Euler motion equations – In body-fixed frame: – With moments of inertia: – Asymmetry parameter: (Q=0 – symmetric rotor) • Euler solution: instant rotation axis precesses about rotor principal axis along the polhode path (angular velocity Ωp) • For GP-B gyros 4 July 8, 2009 • tanford UniversityS HEPL Seminar 1.2 Symmetric vs. Asymmetric Gyro Precession • Symmetric (II 1 = 2 , Q=0): γp= const (ω3=const, polhode path=circular cone), • =φp Ω p const = , motion
    [Show full text]
  • The Gravity Probe B Test of General Relativity
    Home Search Collections Journals About Contact us My IOPscience The Gravity Probe B test of general relativity This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Class. Quantum Grav. 32 224001 (http://iopscience.iop.org/0264-9381/32/22/224001) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 162.247.72.27 This content was downloaded on 31/05/2016 at 09:49 Please note that terms and conditions apply. Classical and Quantum Gravity Class. Quantum Grav. 32 (2015) 224001 (29pp) doi:10.1088/0264-9381/32/22/224001 The Gravity Probe B test of general relativity C W F Everitt1, B Muhlfelder1, D B DeBra1, B W Parkinson1, J P Turneaure1, A S Silbergleit1, E B Acworth1, M Adams1, R Adler1, W J Bencze1, J E Berberian1, R J Bernier1, K A Bower1, R W Brumley1, S Buchman1, K Burns1, B Clarke1, J W Conklin1, M L Eglington1, G Green1, G Gutt1, D H Gwo1, G Hanuschak1,XHe1, M I Heifetz1, D N Hipkins1, T J Holmes1, R A Kahn1, G M Keiser1, J A Kozaczuk1, T Langenstein1,JLi1, J A Lipa1, J M Lockhart1, M Luo1, I Mandel1, F Marcelja1, J C Mester1, A Ndili1, Y Ohshima1, J Overduin1, M Salomon1, D I Santiago1, P Shestople1, V G Solomonik1, K Stahl1, M Taber1, R A Van Patten1, S Wang1, J R Wade1, P W Worden Jr1, N Bartel6, L Herman6, D E Lebach6, M Ratner6, R R Ransom6, I I Shapiro6, H Small6, B Stroozas6, R Geveden2, J H Goebel3, J Horack2, J Kolodziejczak2, A J Lyons2, J Olivier2, P Peters2, M Smith3, W Till2, L Wooten2, W Reeve4, M Anderson4, N R Bennett4,
    [Show full text]
  • Effects of Energy Dissipation on the Free Body Motions of Spacecraft
    . * , NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 4 Technical Report No. 32-860 Effects of Energy Dissipation on the Free Body Motions of Spacecraft Peter W. Likins N66 30123 (ACCESSION NUMBER1 (THRUI .I ' IPAdESI E/// INASh CR OR TMX ORADNUMBER1 \ . GPO PRICE $ I I 1 'I ' CFSTI PRICE(S) $ , < Hard c2py (HC) ?/ fs-i') l f e 7c Microfiche (MF) I / \ ff 653 July 65 JETI PROPULSION LABORATORY CALIFORNIAINSTITUTE OF TECHNOLOGY PASAD ENA. CALIFOR N I A July 1, 1966 _- NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Report No. 32-860 Effects of Energy Dissipation on the Free Body Motions of Spacecraft Peter W. Likins Spacecraff Control Section JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA,CALIFORNIA July 1, 1966 i Copyright @ 1966 Jet Propulsion Laboratory California Institute of Technology Prepared Under Contract No. NAS 7-100 National Aeronautics & Space Administration JPL TECHNICAL REPORT NO. 32-860 CONTENTS 1. Introduction to the Problem ............... 1 II. Preliminary Considerations ............... 2 A . Definitions .................... 2 B . Rigid Body Rotational Motion and Stability ......... 3 C . Attitude Stability of Nonrigid Bodies ........... 3 D . The Space Environment ................ 4 111. Alternative Methods of Solution ............. 5 A . The Energy Sink Model Method ............. 5 B . The Discrete Parameter Model Method .......... 5 C . The Modal Model Method ............... 5 IV. The Energy Sink Model ................ 8 A . Poinsot Motion Modified for Energy Dissipation ....... 8 B . The Acceleration of a Particle Attached to a Free Rigid Body ... 13 C . Application to SYNCOM ...............15 D . Application to Explorer Z ............... 16 E . Applications with Structural Damping ........... 18 F . Applications with Annular Dampers ............20 G . Applications with Pendulum Dampers ...........20 H .
    [Show full text]
  • 1. Do All Torque-Free Systems Necessarily Exhibit Polhode Motion Or Is It a Special Case for Specific Geometries (E.G
    Ph610 Fall 2012 Students’ questions #6 1. 1. Do all torque-free systems necessarily exhibit Polhode motion or is it a special case for specific geometries (e.g. Poinsot's Ellipsoid)? 2. How good of an approximation is the classical precession without accounting for a variable precession and variable obliquity, the Earth being non-rigid and non-relativistic precession effects? 3. Is there a straight-forward way of identifying which convention is being used for Euler's angles? The convention seems to be switched between quantum and classical books. Do either fields use the x-convention that mathematicians use? 2. Questions 9/30 The notation for the tutorial of problem 6.1 confuses me. Your arrows above the I indicate a tensor but this is a scalar. It is the "moment of inertia about the axis of rotation" Goldstein p. 192. The moment of inertia tensor is the between the dot product. So in part a) when you ask for the rotational inertia tensor, it's confusing as to which I you want. I've tried to compare the terminology with Goldstein, Marion, and Taylor mechanics, as well as various websites. Rotational inertia tensor is not in common use, but I think you mean the moment of inertia tensor. For part c), or in general, I don't understand how include time. My tensor in part a) is supposed to be I(t) at t = 0, but seems identical to what I'd get for the I independent of time. I don't understand how to make it time dependent.
    [Show full text]
  • Inertial Properties Estimation of a Passive On-Orbit Object Using Polhode Analysis
    Inertial Properties Estimation of a Passive On-orbit Object Using Polhode Analysis The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Setterfield, Timothy P., David W. Miller, Alvar Saenz-Otero, Emilio Frazzoli, and John J. Leonard. “Inertial Properties Estimation of a Passive On-Orbit Object Using Polhode Analysis.” Journal of Guidance, Control, and Dynamics (June 29, 2018): 1–18. As Published https://doi.org/10.2514/1.G003394 Publisher American Institute of Aeronautics and Astronautics Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/117602 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Inertial Properties Estimation of a Passive On-Orbit Object Using Polhode Analysis Timothy P. Setterfielda, David W. Millerb, and Alvar Saenz-Oteroc MIT Department of Aeronautics and Astronautics, Cambridge, MA Emilio Frazzolid ETH Zürich Department of Mechanical and Process Engineering, Zürich, Switzerland John J. Leonarde MIT Department of Mechanical and Ocean Engineering, Cambridge, MA Many objects in space are passive, with unknown inertial properties. If attempting to dock autonomously to an uncooperative object (one not equipped with working sensors or actuators), a motion model is required to predict the location of the desired docking location into the future. Additionally, for cooperative satellites that failed to deploy hardware, accurate knowledge of the object’s principal axes and inertia ratios may aid in diagnosing the problem. This paper develops algorithms for estimation of the analytical motion model, principal axes, and inertia ratios of a passive on-orbit object.
    [Show full text]
  • Leonhard Euler : Life, Work and Legacy
    Prelims-N52728.fm Page i Wednesday, November 15, 2006 1:35 PM LEONHARD EULER: LIFE, WORK AND LEGACY Prelims-N52728.fm Page ii Wednesday, November 15, 2006 1:35 PM STUDIES IN THE HISTORY AND PHILOSOPHY OF MATHEMATICS Volume 5 Cover art: “Leonhard Euler”, by Susan Petry, 2002, 26 by 33cm, cherry wood. After the 1753 portrait by Handmann. Photo by C. Edward Sandifer AMSTERDAM - BOSTON - HEIDELBERG - LONDON - NEW YORK - OXFORD - PARIS SAN DIEGO - SAN FRANCISCO - SINGAPORE - SYDNEY - TOKYO Prelims-N52728.fm Page iii Wednesday, November 15, 2006 1:35 PM LEONHARD EULER: LIFE, WORK AND LEGACY Edited by Robert E. Bradley Department of Mathematics and Computer Science Adelphi University 1 South Avenue Garden City, NY 11530, USA C. Edward Sandifer Department of Mathematics Western Connecticut State University 181 White Street Danbury, CT 06810, USA Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris San Diego - San Francisco - Singapore - Sydney - Tokyo Prelims-N52728.fm Page iv Wednesday, November 15, 2006 1:35 PM Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department
    [Show full text]
  • Chapter 4 Rigid Body Motion
    Chapter 4 Rigid Body Motion In this chapter we develop the dynamics of a rigid body, one in which all interparticle distances are fixed by internal forces of constraint. This is, of course, an idealization which ignores elastic and plastic deformations to which any real body is susceptible, but it is an excellent approximation for many situations, and vastly simplifies the dynamics of the very large number of constituent particles of which any macroscopic body is made. In fact, it reduces the problem to one with six degrees of freedom. While the ensuing motion can still be quite complex, it is tractible. In the process we will be dealing with a configuration space which is a group, and is not a Euclidean space. Degrees of freedom which lie on a group manifold rather than Eu- clidean space arise often in applications in quantum mechanics and quantum field theory, in addition to the classical problems we will consider such as gyroscopes and tops. 4.1 Configuration space for a rigid body A macroscopic body is made up of a very large number of atoms. Describing the motion of such a system without some simplifications is clearly impos- sible. Many objects of interest, however, are very well approximated by the assumption that the distances between the atoms in the body are fixed1, ~r ~r = c = constant. (4.1) | α − β| αβ 1In this chapter we will use Greek letters as subscripts to represent the different particles within the body, reserving Latin subscripts to represent the three spatial directions. 85 86 CHAPTER 4.
    [Show full text]
  • Aas 09-123 Rigid Body in Er Tia Es Ti Ma Tion with Ap Pli Ca Tions to the Cap
    AAS 09-123 RIGID BODY IN ER TIA ES TI MA TION WITH AP PLI CA TIONS TO THE CAP TURE OF A TUM BLING SAT EL LITE Dan iel Sheinfeld* and Ste phen Rock† A frame work for rigid-body in er tia es ti ma tion is pre sented which is gen eral and may be used for any rigid body un der go ing ei ther torque-free or non- torque-free motion. It is ap plied here to the case of a tum bling satel lite. In - cluded are a geo met ric in ter pre ta tion of the es ti ma tion prob lem which pro vides an in tu itive un der stand ing of when to ex pect good es ti ma tion re sults and sim u- la tion re sults to dem on strate the vi a bil ity of the method. INTRODUCTION A critical issue that must be addressed to enable on-orbit servicing of satellites is how to cap- ture and de-spin a satellite that may be tumbling. A satellite can lose attitude control and begin to tumble for any number of reasons: thruster failure, avionics failure, failed docking attempts, colli- sion with space debris, etc. In general, it cannot be assumed that the tumbling motion is benign. De-spinning is accomplished by applying forces to the tumbling satellite, and knowledge of the inertia properties of the satellite is necessary to enable the calculation of efficient and/or op- timal control forces and moments to accomplish this task (the satellite is assumed to be rigid).
    [Show full text]
  • 494 Karin Reich Meusnier Never Came Back to Mathematics, but He
    494 Karin Reich Fig. 4. Meusnier’s figure 8, the catenoid Meusnier never came back to mathematics, but he contributed to var- ious other fields. For example, he was an aeronautical theorist. After the first flights of Montgolfier’s balloon he designed an elliptical shaped airship instead of the spherical balloon, but his balloon was never built. Meusnier also collaborated with Antoine de Lavoisier (1743-1794) to separate water into its constituents. Like Tinseau, Meusnier had an accomplished military career, but Meus- nier supported the revolution. He became even a g´en´eral de division. He was severely wounded during a battle between the French and the Prussians near Mainz. Goethe observed that battle and gave a detailed description [Goethe 1793]. 2.3.2. Graduate students from the Ecole´ Polytechnique In 1795, almost immediately after the foundation of the Ecole´ Poly- technique in Paris, Monge published his Feuilles d’Analyse appliqu´ee`ala G´eom´etrie `al’usage de l’Ecole´ Polytechnique [Monge 1805-1850]. It reap- peared in later editions, entitled Application de l’Analyse `ala G´eom´etrie [Monge 1795-1807]. The last edition from 1850 was prepared by Joseph Li- ouville (1809-1882), who remarkably also published Monge’s Application and Gauß’ Disquisitiones generales circa superficiem curvas. The lectures that Monge gave at the Ecole´ polytechnique in Paris had much greater influence than the lectures at M´ezi`eres. Taton mentioned the following students of Monge: Lacroix, 21 Hachette, Fourier, Lancret, Dupin, Livet, Brianchon, Amp`ere, Malus, Binet, Gaultier, Sophie Germain, Gergonne, O. Rodrigues, Poncelet, Berthot, Roche, Lam´e,Fresnel, Chasles, Olivier, Val´ee, Coriolis, Bobillier, Barr´ede Saint-Venant and many others [Taton 1951, p.235f].
    [Show full text]
  • 18Th Australian International Aerospace Congress 27Th International Symposium on Space Flight Dynamics
    PEER REVIEW 18th Australian International Aerospace Congress 27th International Symposium on Space Flight Dynamics Please select category below: Normal Paper Student Paper Young Engineer Paper Enhancement of the Spacecraft Attitude Dynamics Capabilities via Combination of the Inertial Morphing and Reaction Wheels (Keynote Paper and Presentation) Pavel M. Trivailo 1 and Hirohisa Kojima 3 1 RMIT University, School of Engineering, GPO Box 2476V, Melbourne, Victoria, 3001, Australia, [email protected] 2 Tokyo Metropolitan University, Department of Aeronautics and Astronautics, Asahigaoka 6-6, Hino, Tokyo, Japan, 191-0065, [email protected] Abstract In our previous works [1-4] we discovered an efficient method of control of the inversion of the spinning spacecraft. This method was prompted by the “Dzhanibekov’s Effect” or “Tennis Racket Theorem” [4], which are often seen by many as odd or even mysterious. The proposed spacecraft inversion method does not employ conventional RWs, but, instead, is using proposed morphing of the inertial properties of the spinning spacecraft. It allows to completely stop flipping motion or activate flipping motion on the spacecraft, being in its stable spin. In this paper, we are considering a spacecraft, which has both, inertial morphing and RW, capabilities. It is believed that performance of the RW systems could be enhanced with the inertial morphing. For general formulation, we first assume ability of the spacecraft to change its inertial properties, associated with all three principal axes of inertia. Secondly, we also assume that the spacecraft is equipped with three control wheels, located along the x, y and z- axes of the body-fixed frame.
    [Show full text]