DOCSLIB.ORG

Dynamics of Open Chains: Newton Euler Approach

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dynamics of Open Chains: Newton Euler Approach ECE5463: Introduction to Robotics Lecture Note 13: Dynamics of Open Chains: Newton Euler Approach Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 22 Outline Introduction • Newton-Euler Inverse Dynamics of Open Chains • Forward Dynamics of Open Chains • Outline Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 22 Lagrangian vs. Newton-Euler Methods • We have introduced the Lagrangian approach in the previous lecture. This lecture will introduce the Newton-Euler method • Recall the differences between the two methods: Lagrangian Formulation Newton-Euler Formulation - Energy-based method - Balance of forces/torques - Dynamic equations in closed - Dynamic equations in form numeric/recursive form - Often used for study of dynamic - Often used for numerical properties and analysis of solution of forward/inverse control methods dynamics Introduction Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 22 Main Ideas for Inverse Dynamics • Goal: Given state (θ; θ_) and acelleration θ¨, compute the required τ Rn 2 τ = M(θ)θ¨ + h(θ; θ_) • We will not drive the analytical expressions for the overall M(θ) and h(θ; θ_). • Instead, we aim to write dynamic equations separately for each link/body. T total _ Chapter 8. Dynamicsi of= Openi Chainsi [ad i ] ( i i) 293 F G V − V G V joint axis i ˙ Vi Vi i { } Fi AdT ( ) − Ti+1,i Fi+1 Figure 8.6: Free-body diagram illustrating the moments and forces exerted on link i. Introduction Substituting this resultLecture into 13 Equation (ECE5463 Sp18)(8.46), we get Wei Zhang(OSU) 4 / 22 ˙ ¨ ˙ ˙ i = iθi + [AdTi,i 1 ] i 1 + [ad i ] iθi, (8.47) V A − V − V A i.e., the acceleration of link i is the sum of three components: a component due to the joint acceleration θ¨i, a component due to the acceleration of link i 1 expressed in i , and a velocity-product component. − Once we have{ } determined all the link twists and accelerations moving out- ward from the base, we can calculate the joint torques or forces by moving inward from the tip. The rigid-body dynamics (8.40) tells us the total wrench that acts on link i given and ˙ . Furthermore, the total wrench acting on Vi Vi link i is the sum of the wrench i transmitted through joint i and the wrench applied to the link through jointFi + 1 (or, for link n, the wrench applied to the link by the environment at the end-effector frame n + 1 ), expressed in the frame i. Therefore, we have the equality { } ˙ T T i i ad ( i i) = i AdT ( i+1); (8.48) G V − Vi G V F − i+1,i F see Figure 8.6. Solving from the tip toward the base, at each link i we solve for the only unknown in Equation (8.48): . Since joint i has only one degree Fi of freedom, five dimensions of the six-vector i are provided “for free” by the structure of the joint, and the actuator onlyF has to provide the scalar force or torque in the direction of the joint’s screw axis: τ = T . (8.49) i Fi Ai Equation (8.49) provides the torques required at each joint, solving the inverse dynamics problem. May 2017 preprint of Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org Main Ideas for Inverse Dynamics (Continued) • Note that i can be easily computed for each link. Thus, the main task is to find ; _ Gand the total wrench for each link in terms of the given (θ; θ;_ θ¨). Vi Vi • This can be done recursively: - The velocity Vi of link i is determined by Vi−1 of link i − 1 and the joint rate θ_i of joint i - The acceleration V_ i of link i is determined by V_ i−1 of link i − 1 and the acceleration θ¨i of joint i - The total wrench of link i can be determined by gravity and the wrench of adjacent links using the principle of action and reaction: Wrench applied by body i to body i+1 = − Wrench applied by body i+1 to body i Introduction Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 22 Outline Introduction • Newton-Euler Inverse Dynamics of Open Chains • Forward Dynamics of Open Chains • Newton-Euler Inverse Dynamics Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 6 / 22 Some Notations (1/3) • For each link i = 1; : : : ; n, Frame i is attached to its center of mass. f g • The base frame is denoted 0 f g • The end-effector frame is denoted n + 1 . Its configuration in n is fixed. f g f g • M : Configuration of j in i when the robot is at home position i;j f g f g • T (θ): Configuration of j in i when the robot is at θ position i;j f g f g Newton-Euler Inverse Dynamics Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 7 / 22 Some Notations (2/3) • All the following quantities are expressed in frame i f g - Vi: Twist of link fig - ¯mi: mass; Ii: rotational inertia matrix; Ii 0 - Gi = : Spatial inertia matrix 0 ¯miI Chapter 8. Dynamics of Open Chains 293 - Ai: Screw axis of joint i, expressed in fig joint axis i ˙ Vi i - Fi = (mi; fi): wrench transmitted through V joint i to link fig i { } Fi AdT ( ) − Ti+1,i Fi+1 Figure 8.6: Free-body diagram illustrating the moments and forces exerted on link i. Newton-Euler Inverse Dynamics Substituting this resultLecture into 13 Equation (ECE5463 Sp18)(8.46), we get Wei Zhang(OSU) 8 / 22 ˙ ¨ ˙ ˙ i = iθi + [AdTi,i 1 ] i 1 + [ad i ] iθi, (8.47) V A − V − V A i.e., the acceleration of link i is the sum of three components: a component due to the joint acceleration θ¨i, a component due to the acceleration of link i 1 expressed in i , and a velocity-product component. − Once we have{ } determined all the link twists and accelerations moving out- ward from the base, we can calculate the joint torques or forces by moving inward from the tip. The rigid-body dynamics (8.40) tells us the total wrench that acts on link i given and ˙ . Furthermore, the total wrench acting on Vi Vi link i is the sum of the wrench i transmitted through joint i and the wrench applied to the link through jointFi + 1 (or, for link n, the wrench applied to the link by the environment at the end-effector frame n + 1 ), expressed in the frame i. Therefore, we have the equality { } ˙ T T i i ad ( i i) = i AdT ( i+1); (8.48) G V − Vi G V F − i+1,i F see Figure 8.6. Solving from the tip toward the base, at each link i we solve for the only unknown in Equation (8.48): . Since joint i has only one degree Fi of freedom, five dimensions of the six-vector i are provided “for free” by the structure of the joint, and the actuator onlyF has to provide the scalar force or torque in the direction of the joint’s screw axis: τ = T . (8.49) i Fi Ai Equation (8.49) provides the torques required at each joint, solving the inverse dynamics problem. May 2017 preprint of Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org Some Notations (3/3) • Some simple facts: −1 −1 - Mi−1;i = Mi−1Mi and Mi;i−1 = Mi Mi−1 h i - Ai = AdM 1 Si, where Si is the screw axis of joint i (relative to f0g frame) i− when θ = 0 [A ]θ −[A ]θ - Ti−1;i(θ) = Ti−1;i(θi) = Mi−1;ie i i and Ti;i−1(θi) = e i i Mi;i−1 Newton-Euler Inverse Dynamics Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 9 / 22 Forward Propagation of Twist (1/3) • Goal: Given i 1 and (θi; θ_i), we want to find i V − V • Generally speaking, there are two key elements in defining a rigid body velocity: - The inertia frame relative to which the motion is considered - The representation frame in which the coordinate of the velocity vector is determined. • So far, we have mainly played with the representation frames, while the motion is always considered to be relative to the fixed s frame. f g • Sometimes, it is convenient consider a relative velocity with respect to a moving frame. Newton-Euler Inverse Dynamics Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 10 / 22 Forward Propagation of Twist (2/3) b=c • Define a as the velocity (twist) of b relative to c , expressed in a . V f g f g f g • Connecting to our previous shorthand notations: = b=s; = b=s; = i=s Vs Vs Vb Vb Vi Vi • With Newtonian approximation: we have c=s = b=s + c=b V∗ V∗ V∗ - The "*" can be replaced with fsg or fbg or any other stationary frames. - Note the representation frame must be the same for all the three twists. Newton-Euler Inverse Dynamics Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 11 / 22 Forward Propagation of Twist (3/3) • Now consider velocity of link i: we will have i=s i=i 1 i 1=s = − + − Vi Vi Vi i=i 1 • Note that i − can be viewed as the body twist of a 1-joint robot where V i=i 1 i-1 is viewed as the “fixed” frame − = θ_ f g )V i Ai i • Moreover, with our shorthand notation: i−1=s i−1=s Vi = AdTi;i 1 Vi−1 = AdTi;i 1 Vi−1 − − • Therefore, velocity (twist) can be propagated by: _ i = iθi + AdTi;i 1 i 1 (1) V A − V − Newton-Euler Inverse Dynamics Lecture 13 (ECE5463 Sp18) Wei Zhang(OSU) 12 / 22 Forward Propagation of Acceleration (1/2) • Goal: Given _ i 1, i 1 and (θi; θ_i; θ¨i), we want to find _ i, V − V − V • We first mention a result about the derivative of the adjoint operator.
Load more

Recommended publications
  • Lecture 10: Impulse and Momentum
    ME 230 Kinematics and Dynamics Wei-Chih Wang Department of Mechanical Engineering University of Washington Kinetics of a particle: Impulse and Momentum Chapter 15 Chapter objectives • Develop the principle of linear impulse and momentum for a particle • Study the conservation of linear momentum for particles • Analyze the mechanics of impact • Introduce the concept of angular impulse and momentum • Solve problems involving steady fluid streams and propulsion with variable mass W. Wang Lecture 10 • Kinetics of a particle: Impulse and Momentum (Chapter 15) - 15.1-15.3 W. Wang Material covered • Kinetics of a particle: Impulse and Momentum - Principle of linear impulse and momentum - Principle of linear impulse and momentum for a system of particles - Conservation of linear momentum for a system of particles …Next lecture…Impact W. Wang Today’s Objectives Students should be able to: • Calculate the linear momentum of a particle and linear impulse of a force • Apply the principle of linear impulse and momentum • Apply the principle of linear impulse and momentum to a system of particles • Understand the conditions for conservation of momentum W. Wang Applications 1 A dent in an automotive fender can be removed using an impulse tool, which delivers a force over a very short time interval. How can we determine the magnitude of the linear impulse applied to the fender? Could you analyze a carpenter’s hammer striking a nail in the same fashion? W. Wang Applications 2 Sure! When a stake is struck by a sledgehammer, a large impulsive force is delivered to the stake and drives it into the ground.
    [Show full text]
  • Chapter 3 Dynamics of Rigid Body Systems
    Rigid Body Dynamics Algorithms Roy Featherstone Rigid Body Dynamics Algorithms Roy Featherstone The Austrailian National University Canberra, ACT Austrailia Library of Congress Control Number: 2007936980 ISBN 978-0-387-74314-1 ISBN 978-1-4899-7560-7 (eBook) Printed on acid-free paper. @ 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com Preface The purpose of this book is to present a substantial collection of the most efficient algorithms for calculating rigid-body dynamics, and to explain them in enough detail that the reader can understand how they work, and how to adapt them (or create new algorithms) to suit the reader’s needs. The collection includes the following well-known algorithms: the recursive Newton-Euler algo- rithm, the composite-rigid-body algorithm and the articulated-body algorithm. It also includes algorithms for kinematic loops and floating bases.
    [Show full text]
  • Impact Dynamics of Newtonian and Non-Newtonian Fluid Droplets on Super Hydrophobic Substrate
    IMPACT DYNAMICS OF NEWTONIAN AND NON-NEWTONIAN FLUID DROPLETS ON SUPER HYDROPHOBIC SUBSTRATE A Thesis Presented By Yingjie Li to The Department of Mechanical and Industrial Engineering in partial fulfillment of the requirements for the degree of Master of Science in the field of Mechanical Engineering Northeastern University Boston, Massachusetts December 2016 Copyright (©) 2016 by Yingjie Li All rights reserved. Reproduction in whole or in part in any form requires the prior written permission of Yingjie Li or designated representatives. ACKNOWLEDGEMENTS I hereby would like to appreciate my advisors Professors Kai-tak Wan and Mohammad E. Taslim for their support, guidance and encouragement throughout the process of the research. In addition, I want to thank Mr. Xiao Huang for his generous help and continued advices for my thesis and experiments. Thanks also go to Mr. Scott Julien and Mr, Kaizhen Zhang for their invaluable discussions and suggestions for this work. Last but not least, I want to thank my parents for supporting my life from China. Without their love, I am not able to complete my thesis. TABLE OF CONTENTS DROPLETS OF NEWTONIAN AND NON-NEWTONIAN FLUIDS IMPACTING SUPER HYDROPHBIC SURFACE .......................................................................... i ACKNOWLEDGEMENTS ...................................................................................... iii 1. INTRODUCTION .................................................................................................. 9 1.1 Motivation ........................................................................................................
    [Show full text]
  • Post-Newtonian Approximation
    Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism Post-Newtonian Approximation Piotr Jaranowski Faculty of Physcis, University of Bia lystok,Poland 01.07.2013 P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Relativistic binary systems exist in nature, they comprise compact objects: neutron stars or black holes. These systems emit gravitational waves, which experimenters try to detect within the LIGO/VIRGO/GEO600 projects.
    [Show full text]
  • Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism
    sensors Article Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism Linkang Wang 1 , Jingjing You 1,* , Xiaolong Yang 2, Huaxin Chen 1, Chenggang Li 3 and Hongtao Wu 3 1 College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China; [email protected] (L.W.); [email protected] (H.C.) 2 School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210037, China; [email protected] 3 School of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; [email protected] (C.L.); [email protected] (H.W.) * Correspondence: [email protected] Abstract: The solution of the dynamic equations of the six-axis accelerometer is a prerequisite for sensor calibration, structural optimization, and practical application. However, the forward dynamic equations (FDEs) and inverse dynamic equations (IDEs) of this type of system have not been completely solved due to the strongly nonlinear coupling relationship between the inputs and outputs. This article presents a comprehensive study of the FDEs and IDEs of the six-axis accelerometer based on a parallel mechanism. Firstly, two sets of dynamic equations of the sensor are constructed based on the Newton–Euler method in the configuration space. Secondly, based on the analytical solution of the sensor branch chain length, the coordination equation between the output signals of the branch chain is constructed. The FDEs of the sensor are established by combining the coordination equations and two sets of dynamic equations. Furthermore, by introducing generalized momentum and Hamiltonian function and using Legendre transformation, the vibration differential equations (VDEs) of the sensor are derived.
    [Show full text]
  • Apollonian Circle Packings: Dynamics and Number Theory
    APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND NUMBER THEORY HEE OH Abstract. We give an overview of various counting problems for Apol- lonian circle packings, which turn out to be related to problems in dy- namics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lec- tures given at RIMS, Kyoto in the fall of 2013. Contents 1. Counting problems for Apollonian circle packings 1 2. Hidden symmetries and Orbital counting problem 7 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 15 5. Expanders and Sieve 19 References 25 1. Counting problems for Apollonian circle packings An Apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of Apollonius of Perga: Theorem 1.1 (Apollonius of Perga, 262-190 BC). Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Figure 1. Pictorial proof of the Apollonius theorem 1 2 HEE OH Figure 2. Possible configurations of four mutually tangent circles Proof. We give a modern proof, using the linear fractional transformations ^ of PSL2(C) on the extended complex plane C = C [ f1g, known as M¨obius transformations: a b az + b (z) = ; c d cz + d where a; b; c; d 2 C with ad − bc = 1 and z 2 C [ f1g. As is well known, a M¨obiustransformation maps circles in C^ to circles in C^, preserving angles between them.
    [Show full text]
  • New Rotational Dynamics- Inertia-Torque Principle and the Force Moment the Character of Statics Guagsan Yu* Harbin Macro, Dynamics Institute, P
    Computa & tio d n ie a l l p M Yu, J Appl Computat Math 2015, 4:3 p a Journal of A t h f e o m l DOI: 10.4172/2168-9679.1000222 a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access New Rotational Dynamics- Inertia-Torque Principle and the Force Moment the Character of Statics GuagSan Yu* Harbin Macro, Dynamics Institute, P. R. China Abstract Textual point of view, generate in a series of rotational dynamics experiment. Initial research, is wish find a method overcome the momentum conservation. But further study, again detected inside the classical mechanics, the error of the principle of force moment. Then a series of, momentous the error of inside classical theory, all discover come out. The momentum conservation law is wrong; the newton third law is wrong; the energy conservation law is also can surpass. After redress these error, the new theory namely perforce bring. This will involve the classical physics and mechanics the foundation fraction, textbooks of physics foundation part should proceed the grand modification. Keywords: Rigid body; Inertia torque; Centroid moment; Centroid occurrence change? The Inertia-torque, namely, such as Figure 3 the arm; Statics; Static force; Dynamics; Conservation law show, the particle m (the m is also its mass) is a r 1 to O the slewing radius, so it the Inertia-torque is: Introduction I = m.r1 (1.1) Textual argumentation is to bases on the several simple physics experiment, these experiments pass two videos the document to The Inertia-torque is similar with moment of force, to the same of proceed to demonstrate.
    [Show full text]
  • Existence and Uniqueness of Rigid-Body Dynamics With
    EXISTENCE AND UNIQUENESS OF RIGIDBODY DYNAMICS WITH COULOMB FRICTION Pierre E Dup ont Aerospace and Mechanical Engineering Boston University Boston MA USA ABSTRACT For motion planning and verication as well as for mo delbased control it is imp ortanttohavean accurate dynamic system mo del In many cases friction is a signicant comp onent of the mo del When considering the solution of the rigidb o dy forward dynamics problem in systems such as rob ots wecan often app eal to the fundamental theorem of dierential equations to prove the existence and uniqueness of the solution It is known however that when Coulomb friction is added to the dynamic equations the forward dynamic solution may not exist and if it exists it is not necessarily unique In these situations the inverse dynamic problem is illp osed as well Thus there is the need to understand the nature of the existence and uniqueness problems and to know under what conditions these problems arise In this pap er we show that even single degree of freedom systems can exhibit existence and uniqueness problems Next weintro duce compliance in the otherwise rigidb o dy mo del This has two eects First the forward and inverse problems b ecome wellp osed Thus we are guaranteed unique solutions Secondlyweshow that the extra dynamic solutions asso ciated with the rigid mo del are unstable INTRODUCTION To plan motions one needs to know what dynamic b ehavior will o ccur as the result of applying given forces or torques to the system In addition the simulation of motions is useful to verify motion
    [Show full text]
  • Newton Euler Equations of Motion Examples
    Newton Euler Equations Of Motion Examples Alto and onymous Antonino often interloping some obligations excursively or outstrikes sunward. Pasteboard and Sarmatia Kincaid never flits his redwood! Potatory and larboard Leighton never roller-skating otherwhile when Trip notarizes his counterproofs. Velocity thus resulting in the tumbling motion of rigid bodies. Equations of motion Euler-Lagrange Newton-Euler Equations of motion. Motion of examples of experiments that a random walker uses cookies. Forces by each other two examples of example are second kind, we will refer to specify any parameter in. 213 Translational and Rotational Equations of Motion. Robotics Lecture Dynamics. Independence from a thorough description and angular velocity as expected or tofollowa userdefined behaviour does it only be loaded geometry in an appropriate cuts in. An interface to derive a particular instance: divide and author provides a positive moment is to express to output side can be run at all previous step. The analysis of rotational motions which make necessary to decide whether rotations are. For xddot and whatnot in which a very much easier in which together or arena where to use them in two backwards operation complies with respect to rotations. Which influence of examples are true, is due to independent coordinates. On sameor adjacent joints at each moment equation is also be more specific white ellipses represent rotations are unconditionally stable, for motion break down direction. Unit quaternions or Euler parameters are known to be well suited for the. The angular momentum and time and runnable python code. The example will be run physics examples are models can be symbolic generator runs faster rotation kinetic energy.
    [Show full text]
  • Post-Newtonian Approximations and Applications
    Monash University MTH3000 Research Project Coming out of the woodwork: Post-Newtonian approximations and applications Author: Supervisor: Justin Forlano Dr. Todd Oliynyk March 25, 2015 Contents 1 Introduction 2 2 The post-Newtonian Approximation 5 2.1 The Relaxed Einstein Field Equations . 5 2.2 Solution Method . 7 2.3 Zones of Integration . 13 2.4 Multi-pole Expansions . 15 2.5 The first post-Newtonian potentials . 17 2.6 Alternate Integration Methods . 24 3 Equations of Motion and the Precession of Mercury 28 3.1 Deriving equations of motion . 28 3.2 Application to precession of Mercury . 33 4 Gravitational Waves and the Hulse-Taylor Binary 38 4.1 Transverse-traceless potentials and polarisations . 38 4.2 Particular gravitational wave fields . 42 4.3 Effect of gravitational waves on space-time . 46 4.4 Quadrupole formula . 48 4.5 Application to Hulse-Taylor binary . 52 4.6 Beyond the Quadrupole formula . 56 5 Concluding Remarks 58 A Appendix 63 A.1 Solving the Wave Equation . 63 A.2 Angular STF Tensors and Spherical Averages . 64 A.3 Evaluation of a 1PN surface integral . 65 A.4 Details of Quadrupole formula derivation . 66 1 Chapter 1 Introduction Einstein's General theory of relativity [1] was a bold departure from the widely successful Newtonian theory. Unlike the Newtonian theory written in terms of fields, gravitation is a geometric phenomena, with space and time forming a space-time manifold that is deformed by the presence of matter and energy. The deformation of this differentiable manifold is characterised by a symmetric metric, and freely falling (not acted on by exter- nal forces) particles will move along geodesics of this manifold as determined by the metric.
    [Show full text]
  • Equation of Motion for Viscous Fluids
    1 2.25 Equation of Motion for Viscous Fluids Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 2001 (8th edition) Contents 1. Surface Stress …………………………………………………………. 2 2. The Stress Tensor ……………………………………………………… 3 3. Symmetry of the Stress Tensor …………………………………………8 4. Equation of Motion in terms of the Stress Tensor ………………………11 5. Stress Tensor for Newtonian Fluids …………………………………… 13 The shear stresses and ordinary viscosity …………………………. 14 The normal stresses ……………………………………………….. 15 General form of the stress tensor; the second viscosity …………… 20 6. The Navier-Stokes Equation …………………………………………… 25 7. Boundary Conditions ………………………………………………….. 26 Appendix A: Viscous Flow Equations in Cylindrical Coordinates ………… 28 ã Ain A. Sonin 2001 2 1 Surface Stress So far we have been dealing with quantities like density and velocity, which at a given instant have specific values at every point in the fluid or other continuously distributed material. The density (rv ,t) is a scalar field in the sense that it has a scalar value at every point, while the velocity v (rv ,t) is a vector field, since it has a direction as well as a magnitude at every point. Fig. 1: A surface element at a point in a continuum. The surface stress is a more complicated type of quantity. The reason for this is that one cannot talk of the stress at a point without first defining the particular surface through v that point on which the stress acts. A small fluid surface element centered at the point r is defined by its area A (the prefix indicates an infinitesimal quantity) and by its outward v v unit normal vector n .
    [Show full text]
  • Investigation Into the Utility of the MSC ADAMS Dynamic Software for Simulating
    Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating Robots and Mechanisms A thesis presented to The faculty of The Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Xiao Xue May 2013 © 2013 Xiao Xue. All Rights Reserved 2 This thesis titled Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating Robots and Mechanisms by XIAO XUE has been approved for the Department of Mechanical Engineering and the Russ College of Engineering and Technology by Robert L. Williams II Professor of Mechanical Engineering Dennis Irwin Dean, Russ College of Engineering and Technology 3 ABSTRACT XUE, XIAO, M.S., May 2013, Mechanical Engineering Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating Robots and Mechanisms (pp.143) Director of Thesis: Robert L. Williams II A Slider-Crank mechanism, a 4-bar mechanism, a 2R planar serial robot and a Stewart-Gough parallel manipulator are modeled and simulated in MSC ADAMS/View. Forward dynamics simulation is done on the Stewart-Gough parallel manipulator; Inverse dynamics simulation is done on Slider-Crank mechanism, 2R planar serial robot and 4- bar mechanism. In forward dynamics simulation, the forces are applied in prismatic joints and the Stewart-Gough parallel manipulator is actuated to perform 4 different expected motions. The motion of the platform is measured and compared with the results in MATLAB. An example of inverse dynamics simulation is also performed on it. In the inverse dynamics simulation, the Slider-Crank mechanism and four-bar mechanism both run with constant input angular velocity and the actuating forces are measured and compared with the results in MATLAB.
    [Show full text]