GYRODYNAMICS Introduction to the Dynamics of Rigid Bodies
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Jean-Baptiste Charles Joseph Bélanger (1790-1874), the Backwater Equation and the Bélanger Equation
THE UNIVERSITY OF QUEENSLAND DIVISION OF CIVIL ENGINEERING REPORT CH69/08 JEAN-BAPTISTE CHARLES JOSEPH BÉLANGER (1790-1874), THE BACKWATER EQUATION AND THE BÉLANGER EQUATION AUTHOR: Hubert CHANSON HYDRAULIC MODEL REPORTS This report is published by the Division of Civil Engineering at the University of Queensland. Lists of recently-published titles of this series and of other publications are provided at the end of this report. Requests for copies of any of these documents should be addressed to the Civil Engineering Secretary. The interpretation and opinions expressed herein are solely those of the author(s). Considerable care has been taken to ensure accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user. Division of Civil Engineering The University of Queensland Brisbane QLD 4072 AUSTRALIA Telephone: (61 7) 3365 3619 Fax: (61 7) 3365 4599 URL: http://www.eng.uq.edu.au/civil/ First published in 2008 by Division of Civil Engineering The University of Queensland, Brisbane QLD 4072, Australia © Chanson This book is copyright ISBN No. 9781864999211 The University of Queensland, St Lucia QLD JEAN-BAPTISTE CHARLES JOSEPH BÉLANGER (1790-1874), THE BACKWATER EQUATION AND THE BÉLANGER EQUATION by Hubert CHANSON Professor, Division of Civil Engineering, School of Engineering, The University of Queensland, Brisbane QLD 4072, Australia Ph.: (61 7) 3365 3619, Fax: (61 7) 3365 4599, Email: [email protected] Url: http://www.uq.edu.au/~e2hchans/ REPORT No. CH69/08 ISBN 9781864999211 Division of Civil Engineering, The University of Queensland August 2008 Jean-Baptiste BÉLANGER (1790-1874) (Courtesy of the Bibliothèque de l'Ecole Nationale Supérieure des Ponts et Chaussées) Abstract In an open channel, the transition from a high-velocity open channel flow to a fluvial motion is a flow singularity called a hydraulic jump. -
Augustin-Louis Cauchy - Wikipedia, the Free Encyclopedia 1/6/14 3:35 PM Augustin-Louis Cauchy from Wikipedia, the Free Encyclopedia
Augustin-Louis Cauchy - Wikipedia, the free encyclopedia 1/6/14 3:35 PM Augustin-Louis Cauchy From Wikipedia, the free encyclopedia Baron Augustin-Louis Cauchy (French: [oɡystɛ̃ Augustin-Louis Cauchy lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic Cauchy around 1840. Lithography by Zéphirin principle of the Belliard after a painting by Jean Roller. generality of algebra exploited by earlier Born 21 August 1789 authors. He defined Paris, France continuity in terms of Died 23 May 1857 (aged 67) infinitesimals and gave Sceaux, France several important Nationality French theorems in complex Fields Mathematics analysis and initiated the Institutions École Centrale du Panthéon study of permutation École Nationale des Ponts et groups in abstract Chaussées algebra. A profound École polytechnique mathematician, Cauchy Alma mater École Nationale des Ponts et exercised a great Chaussées http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Page 1 of 24 Augustin-Louis Cauchy - Wikipedia, the free encyclopedia 1/6/14 3:35 PM influence over his Doctoral Francesco Faà di Bruno contemporaries and students Viktor Bunyakovsky successors. His writings Known for See list cover the entire range of mathematics and mathematical physics. "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)."[1] Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict Bourbon royalist, and a close associate of the Jesuit order. -
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 . -
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 -
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, -
An Appreciation and Discussion of Paul Germain’S “The Method of Virtual Power in the Mechanics of Continuous Media I: Second-Gradient Theory”
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 8 no. 2 2020 Mathematics and Mechanics of Complex Systems MARCELO EPSTEIN AND RONALD E. SMELSER AN APPRECIATION AND DISCUSSION OF PAUL GERMAIN’S “THE METHOD OF VIRTUAL POWER IN THE MECHANICS OF CONTINUOUS MEDIA I: SECOND-GRADIENT THEORY” msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 8, No. 2, 2020 dx.doi.org/10.2140/memocs.2020.8.191 ∩ MM AN APPRECIATION AND DISCUSSION OF PAUL GERMAIN’S “THE METHOD OF VIRTUAL POWER IN THE MECHANICS OF CONTINUOUS MEDIA I: SECOND-GRADIENT THEORY” MARCELO EPSTEIN AND RONALD E. SMELSER Paul Germain’s 1973 article on the method of virtual power in continuum me- chanics has had an enormous impact on the modern development of the disci- pline. In this article we examine the historical context of the ideas it contains and discuss their continuing importance. Our English translation of the French original appears elsewhere in this volume (MEMOCS 8:2 (2020), 153–190). Introduction Among the many contributions of Paul Germain (1920–2009) to mechanics, this classical 1973 article[1973a] on the method of virtual power in continuum mechan- ics stands out for its enormous impact on the modern development of the discipline, as evidenced by hundreds of citations and by its direct or indirect influence in establishing a paradigm of thought for succeeding generations. In this article we examine the historical antecedents of the ideas contained in the article and discuss their continuing relevance. The article was published in French in the Journal de Mécanique. -
La Notion De Couple En Mécanique : Réhabiliter Poinsot
La notion de couple en mécanique : réhabiliter Poinsot par Ivor Grattan-Guinness Historien des mathématiques et philosophe des sciences Professeur émérite à l’université du Middlesex (UK) Figure 1: Louis Poinsot, gravure de Boilly. Jules Boilly (1796-1874) est un peintre et lithographe spécialisé dans la gravure de personnalités, dont les membres de l’Institut (P.S. Girard, J.D. Cassini, Alexis Bouvard). Il était le fils d’un peintre plus renommé, Léopold Boilly (1761-1845), auteur de peintures de genre et de portraits, notamment de Sadi Carnot. I - PRÉLIMINAIRES En 1803, Louis Poinsot publia un traité de statique, à caractère révolutionnaire puisqu’il posait clairement le sujet non seulement en termes de forces mais aussi en terme de « couples » (c’est son expression), c’est-à-dire des paires de forces non colinéaires égales en amplitude et en direction mais en sens opposés. Plus tard, il adapta cette notion pour induire en dynamique une relation 1 nouvelle entre mouvement linéaire et mouvement de rotation. Le présent article résume ces développements et examine leur réception, qui fut lente parmi ses contemporains mathématiciens et quasi inexistante parmi les « historiens » de la mécanique plus tard. 1. Les organisations Un beau jour, lors des années révolutionnaires II ou III, période à présent plus connue sous le nom d’année 1794, un adolescent orphelin, étudiant au collège Louis le Grand à Paris, tomba sur un prospectus annonçant la création d’une nouvelle institution d’enseignement supérieur. Intrigué, il se porta candidat et fut accepté, ce qui détermina la suite de sa longue carrière. Cette institution constituait un des deux projets du gouvernement français pour résoudre l’une des crises sociales causées par cinq années de révolution et de ruptures. -
La Notion De Couple En Mécanique : Réhabiliter Poinsot
Bibnum Textes fondateurs de la science Physique La notion de couple en mécanique : réhabiliter Poinsot Ivor Grattan-Guiness Traducteur : Alexandre Moatti Édition électronique URL : http://journals.openedition.org/bibnum/725 ISSN : 2554-4470 Éditeur FMSH - Fondation Maison des sciences de l'homme Référence électronique Ivor Grattan-Guiness, « La notion de couple en mécanique : réhabiliter Poinsot », Bibnum [En ligne], Physique, mis en ligne le 01 janvier 2013, consulté le 19 avril 2019. URL : http:// journals.openedition.org/bibnum/725 © BibNum La notion de couple en mécanique : réhabiliter Poinsot par Ivor Grattan-Guinness Historien des mathématiques et philosophe des sciences Professeur émérite à l’université du Middlesex (UK) Figure 1: Louis Poinsot, gravure de Boilly. Jules Boilly (1796-1874) est un peintre et lithographe spécialisé dans la gravure de personnalités, dont les membres de l’Institut (P.S. Girard, J.D. Cassini, Alexis Bouvard). Il était le fils d’un peintre plus renommé, Léopold Boilly (1761-1845), auteur de peintures de genre et de portraits, notamment de Sadi Carnot. I - PRÉLIMINAIRES En 1803, Louis Poinsot publia un traité de statique, à caractère révolutionnaire puisqu’il posait clairement le sujet non seulement en termes de forces mais aussi en terme de « couples » (c’est son expression), c’est-à-dire des paires de forces non colinéaires égales en amplitude et en direction mais en sens opposés. Plus tard, il adapta cette notion pour induire en dynamique une relation 1 nouvelle entre mouvement linéaire et mouvement de rotation. Le présent article résume ces développements et examine leur réception, qui fut lente parmi ses contemporains mathématiciens et quasi inexistante parmi les « historiens » de la mécanique plus tard. -
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 . -
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. -
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. -
Lessons from the Avalanche of Numbers: Big Data in Historical Perspective
I/S: A JOURNAL OF LAW AND POLICY FOR THE INFORMATION SOCIETY Lessons from the Avalanche of Numbers: Big Data in Historical Perspective MEG LETA AMBROSE, JD, PHD* Abstract: The big data revolution, like many changes associated with technological advancement, is often compared to the industrial revolution to create a frame of reference for its transformative power, or portrayed as altogether new. This article argues that between the industrial revolution and the digital revolution is a more valuable, yet overlooked period: the probabilistic revolution that began with the avalanche of printed numbers between 1820 and 1840. By comparing the many similarities between big data today and the avalanche of numbers in the 1800s, the article situates big data in the early stages of a prolonged transition to a potentially transformative epistemic revolution, like the probabilistic revolution. The widespread changes in and characteristics of a society flooded by data results in a transitional state that creates unique challenges for policy efforts by disrupting foundational principles relied upon for data protection. The potential of a widespread, lengthy transition also places the law in a pivotal position to shape and guide big data-based inquiry through to whatever epistemic shift may lie ahead. * Assistant Professor, Communication, Culture & Technology, Georgetown University. Many thanks to my Governing Algorithms course and Samantha Fried, whose thesis Quantify This: Statistics, The State, and Governmentality (on file at CCT, available upon request by emailing [email protected]) directed me to many of the wonderful historical secondary sources in this article. Additional thanks to John Grant at Palantir, Professor Paul Ohm, Solon Barocas and all the wonderful participants at the 2014 Privacy Law Scholars Conference for their insightful comments and suggestions.