Lecture Notes for PC2132 (Classical Mechanics)
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WUN2K for LECTURE 19 These Are Notes Summarizing the Main Concepts You Need to Understand and Be Able to Apply
Duke University Department of Physics Physics 361 Spring Term 2020 WUN2K FOR LECTURE 19 These are notes summarizing the main concepts you need to understand and be able to apply. • Newton's Laws are valid only in inertial reference frames. However, it is often possible to treat motion in noninertial reference frames, i.e., accelerating ones. • For a frame with rectilinear acceleration A~ with respect to an inertial one, we can write the equation of motion as m~r¨ = F~ − mA~, where F~ is the force in the inertial frame, and −mA~ is the inertial force. The inertial force is a “fictitious” force: it's a quantity that one adds to the equation of motion to make it look as if Newton's second law is valid in a noninertial reference frame. • For a rotating frame: Euler's theorem says that the most general mo- tion of any body with respect to an origin is rotation about some axis through the origin. We define an axis directionu ^, a rate of rotation dθ about that axis ! = dt , and an instantaneous angular velocity ~! = !u^, with direction given by the right hand rule (fingers curl in the direction of rotation, thumb in the direction of ~!.) We'll generally use ~! to refer to the angular velocity of a body, and Ω~ to refer to the angular velocity of a noninertial, rotating reference frame. • We have for the velocity of a point at ~r on the rotating body, ~v = ~! ×~r. ~ dQ~ dQ~ ~ • Generally, for vectors Q, one can write dt = dt + ~! × Q, for S0 S0 S an inertial reference frame and S a rotating reference frame. -
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. -
Classical Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 13
Classical mechanics - Wikipedia, the free encyclopedia Page 1 of 13 Classical mechanics From Wikipedia, the free encyclopedia (Redirected from Newtonian mechanics) In physics, classical mechanics is one of the two major Classical mechanics sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the Newton's Second Law action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of History of classical mechanics · the oldest and largest subjects in science, engineering and Timeline of classical mechanics technology. Branches Classical mechanics describes the motion of macroscopic Statics · Dynamics / Kinetics · Kinematics · objects, from projectiles to parts of machinery, as well as Applied mechanics · Celestial mechanics · astronomical objects, such as spacecraft, planets, stars, and Continuum mechanics · galaxies. Besides this, many specializations within the Statistical mechanics subject deal with gases, liquids, solids, and other specific sub-topics. Classical mechanics provides extremely Formulations accurate results as long as the domain of study is restricted Newtonian mechanics (Vectorial to large objects and the speeds involved do not approach mechanics) the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to Analytical mechanics: introduce the other major sub-field of mechanics, quantum Lagrangian mechanics mechanics, which reconciles the macroscopic laws of Hamiltonian mechanics physics with the atomic nature of matter and handles the Fundamental concepts wave-particle duality of atoms and molecules. In the case of high velocity objects approaching the speed of light, Space · Time · Velocity · Speed · Mass · classical mechanics is enhanced by special relativity. -
Rotating Reference Frame Control of Switched Reluctance
ROTATING REFERENCE FRAME CONTROL OF SWITCHED RELUCTANCE MACHINES A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Tausif Husain August, 2013 ROTATING REFERENCE FRAME CONTROL OF SWITCHED RELUCTANCE MACHINES Tausif Husain Thesis Approved: Accepted: _____________________________ _____________________________ Advisor Department Chair Dr. Yilmaz Sozer Dr. J. Alexis De Abreu Garcia _____________________________ _____________________________ Committee Member Dean of the College Dr. Malik E. Elbuluk Dr. George K. Haritos _____________________________ _____________________________ Committee Member Dean of the Graduate School Dr. Tom T. Hartley Dr. George R. Newkome _____________________________ Date ii ABSTRACT A method to control switched reluctance motors (SRM) in the dq rotating frame is proposed in this thesis. The torque per phase is represented as the product of a sinusoidal inductance related term and a sinusoidal current term in the SRM controller. The SRM controller works with variables similar to those of a synchronous machine (SM) controller in dq reference frame, which allows the torque to be shared smoothly among different phases. The proposed controller provides efficient operation over the entire speed range and eliminates the need for computationally intensive sequencing algorithms. The controller achieves low torque ripple at low speeds and can apply phase advancing using a mechanism similar to the flux weakening of SM to operate at high speeds. A method of adaptive flux weakening for ensured operation over a wide speed range is also proposed. This method is developed for use with dq control of SRM but can also work in other controllers where phase advancing is required. -
Principle of Relativity and Inertial Dragging
ThePrincipleofRelativityandInertialDragging By ØyvindG.Grøn OsloUniversityCollege,Departmentofengineering,St.OlavsPl.4,0130Oslo, Norway Email: [email protected] Mach’s principle and the principle of relativity have been discussed by H. I. HartmanandC.Nissim-Sabatinthisjournal.Severalphenomenaweresaidtoviolate the principle of relativity as applied to rotating motion. These claims have recently been contested. However, in neither of these articles have the general relativistic phenomenonofinertialdraggingbeeninvoked,andnocalculationhavebeenoffered byeithersidetosubstantiatetheirclaims.HereIdiscussthepossiblevalidityofthe principleofrelativityforrotatingmotionwithinthecontextofthegeneraltheoryof relativity, and point out the significance of inertial dragging in this connection. Although my main points are of a qualitative nature, I also provide the necessary calculationstodemonstratehowthesepointscomeoutasconsequencesofthegeneral theoryofrelativity. 1 1.Introduction H. I. Hartman and C. Nissim-Sabat 1 have argued that “one cannot ascribe all pertinentobservationssolelytorelativemotionbetweenasystemandtheuniverse”. They consider an UR-scenario in which a bucket with water is atrest in a rotating universe,andaBR-scenariowherethebucketrotatesinanon-rotatinguniverseand givefiveexamplestoshowthatthesesituationsarenotphysicallyequivalent,i.e.that theprincipleofrelativityisnotvalidforrotationalmotion. When Einstein 2 presented the general theory of relativity he emphasized the importanceofthegeneralprincipleofrelativity.Inasectiontitled“TheNeedforan -
Conservation of Total Momentum TM1
L3:1 Conservation of total momentum TM1 We will here prove that conservation of total momentum is a consequence Taylor: 268-269 of translational invariance. Consider N particles with positions given by . If we translate the whole system with some distance we get The potential energy must be una ected by the translation. (We move the "whole world", no relative position change.) Also, clearly all velocities are independent of the translation Let's consider to be in the x-direction, then Using Lagrange's equation, we can rewrite each derivative as I.e., total momentum is conserved! (Clearly there is nothing special about the x-direction.) L3:2 Generalized coordinates, velocities, momenta and forces Gen:1 Taylor: 241-242 With we have i.e., di erentiating w.r.t. gives us the momentum in -direction. generalized velocity In analogy we de✁ ne the generalized momentum and we say that and are conjugated variables. Note: Def. of gen. force We also de✁ ne the generalized force from Taylor 7.15 di ers from def in HUB I.9.17. such that generalized rate of change of force generalized momentum Note: (generalized coordinate) need not have dimension length (generalized velocity) velocity momentum (generalized momentum) (generalized force) force Example: For a system which rotates around the z-axis we have for V=0 s.t. The angular momentum is thus conjugated variable to length dimensionless length/time 1/time mass*length/time mass*(length)^2/time (torque) energy Cyclic coordinates = ignorable coordinates and Noether's theorem L3:3 Cyclic:1 We have de ned the generalized momentum Taylor: From EL we have 266-267 i.e. -
Static Limit and Penrose Effect in Rotating Reference Frames
1 Static limit and Penrose effect in rotating reference frames A. A. Grib∗ and Yu. V. Pavlov† We show that effects similar to those for a rotating black hole arise for an observer using a uniformly rotating reference frame in a flat space-time: a surface appears such that no body can be stationary beyond this surface, while the particle energy can be either zero or negative. Beyond this surface, which is similar to the static limit for a rotating black hole, an effect similar to the Penrose effect is possible. We consider the example where one of the fragments of a particle that has decayed into two particles beyond the static limit flies into the rotating reference frame inside the static limit and has an energy greater than the original particle energy. We obtain constraints on the relative velocity of the decay products during the Penrose process in the rotating reference frame. We consider the problem of defining energy in a noninertial reference frame. For a uniformly rotating reference frame, we consider the states of particles with minimum energy and show the relation of this quantity to the radiation frequency shift of the rotating body due to the transverse Doppler effect. Keywords: rotating reference frame, negative-energy particle, Penrose effect 1. Introduction The study of relativistic effects related to rotation began immediately after the theory of relativity appeared [1], and such effects are still actively discussed in the literature [2]. The importance of studying such effects is obvious in relation to the Earth’s rotation. In [3], we compared the properties of particles in rotating reference frames in a flat space with the properties of particles in the metric of a rotating black hole. -
Chapter 6 the Equations of Fluid Motion
Chapter 6 The equations of fluid motion In order to proceed further with our discussion of the circulation of the at- mosphere, and later the ocean, we must develop some of the underlying theory governing the motion of a fluid on the spinning Earth. A differen- tially heated, stratified fluid on a rotating planet cannot move in arbitrary paths. Indeed, there are strong constraints on its motion imparted by the angular momentum of the spinning Earth. These constraints are profoundly important in shaping the pattern of atmosphere and ocean circulation and their ability to transport properties around the globe. The laws governing the evolution of both fluids are the same and so our theoretical discussion willnotbespecifictoeitheratmosphereorocean,butcanandwillbeapplied to both. Because the properties of rotating fluids are often counter-intuitive and sometimes difficult to grasp, alongside our theoretical development we will describe and carry out laboratory experiments with a tank of water on a rotating table (Fig.6.1). Many of the laboratory experiments we make use of are simplified versions of ‘classics’ that have become cornerstones of geo- physical fluid dynamics. They are listed in Appendix 13.4. Furthermore we have chosen relatively simple experiments that, in the main, do nor require sophisticated apparatus. We encourage you to ‘have a go’ or view the atten- dant movie loops that record the experiments carried out in preparation of our text. We now begin a more formal development of the equations that govern the evolution of a fluid. A brief summary of the associated mathematical concepts, definitions and notation we employ can be found in an Appendix 13.2. -
ME451 Kinematics and Dynamics of Machine Systems
ME451 Kinematics and Dynamics of Machine Systems Elements of 2D Kinematics September 23, 2014 Dan Negrut Quote of the day: "Success is stumbling from failure to failure with ME451, Fall 2014 no loss of enthusiasm." –Winston Churchill University of Wisconsin-Madison 2 Before we get started… Last time Wrapped up MATLAB overview Understanding how to compute the velocity and acceleration of a point P Relative vs. Absolute Generalized Coordinates Today What it means to carry out Kinematics analysis of 2D mechanisms Position, Velocity, and Acceleration Analysis HW: Haug’s book: 2.5.11, 2.5.12, 2.6.1, 3.1.1, 3.1.2 MATLAB (emailed to you on Th) Due on Th, 9/25, at 9:30 am Post questions on the forum Drop MATLAB assignment in learn@uw dropbox 3 What comes next… Planar Cartesian Kinematics (Chapter 3) Kinematics modeling: deriving the equations that describe motion of a mechanism, independent of the forces that produce the motion. We will be using an Absolute (Cartesian) Coordinates formulation Goals: Develop a general library of constraints (the mathematical equations that model a certain physical constraint or joint) Pose the Position, Velocity and Acceleration analysis problems Numerical Methods in Kinematics (Chapter 4) Kinematics simulation: solving the equations that govern position, velocity and acceleration analysis 4 Nomenclature Modeling Starting with a physical systems and abstracting the physics of interest to a set of equations Simulation Given a set of equations, solve them to understand how the physical system moves in time NOTE: People sometimes simply use “simulation” and include the modeling part under the same term 5 Example 2.4.3 Slider Crank Purpose: revisit several concepts – Cartesian coordinates, modeling, simulation, etc. -
Classical Mechanics
Stony Brook University Academic Commons Essential Graduate Physics Department of Physics and Astronomy 2013 Part CM: Classical Mechanics Konstantin Likharev SUNY Stony Brook, [email protected] Follow this and additional works at: https://commons.library.stonybrook.edu/egp Part of the Physics Commons Recommended Citation Likharev, Konstantin, "Part CM: Classical Mechanics" (2013). Essential Graduate Physics. 2. https://commons.library.stonybrook.edu/egp/2 This Book is brought to you for free and open access by the Department of Physics and Astronomy at Academic Commons. It has been accepted for inclusion in Essential Graduate Physics by an authorized administrator of Academic Commons. For more information, please contact [email protected], [email protected]. Konstantin K. Likharev Essential Graduate Physics Lecture Notes and Problems Beta version Open online access at http://commons.library.stonybrook.edu/egp/ and https://sites.google.com/site/likharevegp/ Part CM: Classical Mechanics Last corrections: September 10, 2021 A version of this material was published in 2017 under the title Classical Mechanics: Lecture notes IOPP, Essential Advanced Physics – Volume 1, ISBN 978-0-7503-1398-8, with model solutions of the exercise problems published in 2018 under the title Classical Mechanics: Problems with solutions IOPP, Essential Advanced Physics – Volume 2, ISBN 978-0-7503-1401-5 However, by now this online version of the lecture notes and the problem solutions available from the author, have been better corrected Note also: Konstantin K. Likharev (ed.) Essential Quotes for Scientists and Engineers Springer, 2021, ISBN 978-3-030-63331-8 (see https://essentialquotes.wordpress.com/) © K. -
Generalized Coordinates
GENERALIZED COORDINATES RAVITEJ UPPU Abstract. This primarily deals with the conception of phase space and the uses of it in classical dynamics. Then, we look into the other fundamentals that are required in dynamics before we start with perturbation theory i.e. topics like generalised coordi- nates and genral analysis of dynamical systems. Date: 12 October, 2006. cmi. 1 2 RAVITEJ UPPU The general analysis of dynamics using the conception of phase space is a very deep physical aspect explained in mathematical structure. Let’s intially look what is a phase space and the requirement of a phase space. Well before that let’s see the use of generalized coordinates. 1. Generalized coordinates 1.1. Why generalized coordinates? I have not particularily used the vector symbol over x. It is implicity asssumed to be so and when I speak of components, then I use it with an index (either a subscript or supeerscript) We always try to select coordinates such that they have a visual- izable geometric significance. They are mostly chosen on the basis (being independent usually helps us). They are system dependent un- like the usual coordiantes. Such coordinates are helpful principally in Lagrangian Dynamics, where the forms of the principal equations describing the motion of the system are unchanged by a shift to gen- eralized coordinates from any other coordinate system. 1.2. An approach mentioned. Say we have a N particle system with K holonomic constraint(i.e. constraints of the form fi(x1, x2, ...xN , t) = 0 and i = 1, 2, ...K < 3N) on the system, where, xj are position vectors of N particles. -
11. Dynamics in Rotating Frames of Reference Gerhard Müller University of Rhode Island, [email protected] Creative Commons License
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 11. Dynamics in Rotating Frames of Reference Gerhard Müller University of Rhode Island, [email protected] Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: http://digitalcommons.uri.edu/classical_dynamics Abstract Part eleven of course materials for Classical Dynamics (Physics 520), taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "11. Dynamics in Rotating Frames of Reference" (2015). Classical Dynamics. Paper 11. http://digitalcommons.uri.edu/classical_dynamics/11 This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Classical Dynamics by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Contents of this Document [mtc11] 11. Dynamics in Rotating Frames of Reference • Motion in rotating frame of reference [mln22] • Effect of Coriolis force on falling object [mex61] • Effects of Coriolis force on an object projected vertically up [mex62] • Foucault pendulum [mex64] • Effects of Coriolis force and centrifugal force on falling object [mex63] • Lateral deflection of projectile due to Coriolis force [mex65] • Effect of Coriolis force on range of projectile [mex66] • What is vertical? [mex170] • Lagrange equations in rotating frame [mex171] • Holonomic constraints in rotating frame [mln23] • Parabolic slide on rotating Earth [mex172] Motion in rotating frame of reference [mln22] Consider two frames of reference with identical origins.