Degrees of Freedom. Spherical (ball & socket) 3 (3 rotation) Two-Angle (universal) 2 (2 rotation)
|
|
- Briana McCarthy
- 5 years ago
- Views:
Transcription
1 ME 6590 Multbody Dynamcs Connectn Jonts Part I o Connectn jonts constran te relatve moton between adjonn bodes n a multbody system. Jonts rane rom allown no relatve moton (a rd jont) to allown all motons (a ree jont). o Wen derees o reedom are removed, constrant orces or torques result. Wen derees o reedom are not removed, orces, or torques at a jont may or may not be zero. For example, bodes may ave nterconnectve sprns dampers tat do not remove derees o reedom but do restrct te moton by applyn loads assocated wt te relatve motons. o A lstn o common connectn jonts s sown n te ollown table. Jont Derees o Freedom d 0 Hne (revolute) ( rotaton) Spercal (ball & socket) 3 (3 rotaton) wo-anle (unversal) ( rotaton) Prsmatc (slder) ( translaton) Cylndrcal ( translaton, rotaton) Free 6 (3 translaton, 3 rotaton) o For every deree o reedom tat a jont elmnates, a constrant equaton must be wrtten. e ollown pararaps outlne te orm o te constrant equatons or some o te common jonts. Spercal Jont: Absolute Coordnates o A spercal (or ball--socket) jont allows two bodes to sare a common pont, but stll rotate reely relatve to eac oter. Usn absolute coordnates, te constrant equaton can be wrtten as pg p G q r Kamman ME 6590 Multbody Dynamcs pae: /6
2 or pg p G q r 0 () o Eq. () represents a set o tree scalar constrant equatons tat elmnate te tree translatonal derees o reedom between te bodes. o For ncorporaton nto a set o equatons o moton, te constrant equatons may be derentated twce so tey are n te orm o second order derental equatons. Usn xed-rame anular velocty components ves 0 vg v G q r vg v G q r vg v G q r vg v G q r o Derentatn aan ves 0 ag ag q q r r () Calculaton o te elements o q r were dscussed n earler notes. o Usn body-rame anular velocty components ves 0 vg v G q r vg v G q r vg v G q r Derentatn aan ves or 0 ag a G q q r r Kamman ME 6590 Multbody Dynamcs pae: /6
3 0 ag a G q q r r (3) Spercal Jont: elatve Coordnates o Consder now te use o relatve coordnates to descrbe te postons o ponts wtn te multbody system as sown n te ure. o o dene a spercal jont tat attaces te ponts Q O, te constrant equatons are smply sss (4) 0 Hne (evolute) Jont: Absolute Coordnates o Lke te spercal jont, te ne jont connects two bodes at a snle pont, so te translatonal constrants are as ven n Eq. (). o In addton, te ne jont also restrcts te relatve rotatonal moton o te bodes by elmnatn two o te tree rotatonal derees o reedom. o Consder te two bodes sown n te daram. Let be a vector xed n parallel to te ne jont, let be vectors xed n tat are perpendcular to te ne axs (, ence, ). en, te rotatonal constrant can be expressed drectly n terms o te anular veloctes as ollows (5) 0 (6) Kamman ME 6590 Multbody Dynamcs pae: 3/6
4 o Intal condtons are used to ensure te alnment o wt. o Usn xed-rame anular velocty components, te rst o Eqs. (5) (6) may be wrtten 0 0 o ese equatons can be derentated to ve 0 0 So, nally, te two derentated constrant equatons are 0 0 o Usn body-rame anular velocty components ves 0 0 o ese equatons can be derentated to ve (7) (8) Kamman ME 6590 Multbody Dynamcs pae: 4/6
5 0 0 (9) smlarly, 0 (0) Hne (evolute) Jont: elatve Coordnates o e translaton constrants are te same as te spercal jont as ven n Eq. (4). o o derve te addtonal rotatonal constrant equatons, we start wt te same set-up used or absolute coordnates. at s, let be a vector xed n parallel to te ne jont, let be vectors xed n tat are perpendcular to te ne jont (, ence, ). en, te rotatonal constrants can be expressed drectly n terms o te anular veloctes as ollows ˆ 0 or ˆ 0 () Kamman ME 6590 Multbody Dynamcs pae: 5/6
6 ˆ 0 or o As beore, ntal condtons are used to ensure te alnment o ˆ 0 () o Derentatn te above equatons usn components o wt ˆ ves. ˆ 0 (3) ˆ 0 (4) o Derentatn te above equatons usn components o ves ˆ 0 (5) ˆ 0 (6) ˆ Kamman ME 6590 Multbody Dynamcs pae: 6/6
Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.
Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal
More informationLesson 5: Kinematics and Dynamics of Particles
Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and
More informationConservation of Energy
Lecture 3 Chapter 8 Physcs I 0.3.03 Conservaton o Energy Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcsall.html 95.4, Fall 03,
More informationORDINARY DIFFERENTIAL EQUATIONS EULER S METHOD
Numercal Analss or Engneers German Jordanan Unverst ORDINARY DIFFERENTIAL EQUATIONS We wll eplore several metods o solvng rst order ordnar derental equatons (ODEs and we wll sow ow tese metods can be appled
More informationLinear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
More informationSpring Force and Power
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
More informationChapter 5. Answers to Even Numbered Problems m kj. 6. (a) 900 J (b) (a) 31.9 J (b) 0 (c) 0 (d) 31.9 J. 10.
Answers to Even Numbered Problems Chapter 5. 3.6 m 4..6 J 6. (a) 9 J (b).383 8. (a) 3.9 J (b) (c) (d) 3.9 J. 6 m s. (a) 68 J (b) 84 J (c) 5 J (d) 48 J (e) 5.64 m s 4. 9. J 6. (a). J (b) 5. m s (c) 6.3
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationModule 11 Design of Joints for Special Loading. Version 2 ME, IIT Kharagpur
Module 11 Desgn o Jonts or Specal Loadng Verson ME, IIT Kharagpur Lesson 1 Desgn o Eccentrcally Loaded Bolted/Rveted Jonts Verson ME, IIT Kharagpur Instructonal Objectves: At the end o ths lesson, the
More information, rst we solve te PDE's L ad L ad n g g (x) = ; = ; ; ; n () (x) = () Ten, we nd te uncton (x), te lnearzng eedbac and coordnates transormaton are gve
Freedom n Coordnates Transormaton or Exact Lnearzaton and ts Applcaton to Transent Beavor Improvement Kenj Fujmoto and Tosaru Suge Dvson o Appled Systems Scence, Kyoto Unversty, Uj, Kyoto, Japan suge@robotuassyoto-uacjp
More informationDynamics 4600:203 Homework 08 Due: March 28, Solution: We identify the displacements of the blocks A and B with the coordinates x and y,
Dynamcs 46:23 Homework 8 Due: March 28, 28 Name: Please denote your answers clearly,.e., box n, star, etc., and wrte neatly. There are no ponts for small, messy, unreadable work... please use lots of paper.
More informationShuai Dong. Isaac Newton. Gottfried Leibniz
Computatonal pyscs Sua Dong Isaac Newton Gottred Lebnz Numercal calculus poston dervatve ntegral v velocty dervatve ntegral a acceleraton Numercal calculus Numercal derentaton Numercal ntegraton Roots
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationThe classical spin-rotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationPhysics 106 Lecture 6 Conservation of Angular Momentum SJ 7 th Ed.: Chap 11.4
Physcs 6 ecture 6 Conservaton o Angular Momentum SJ 7 th Ed.: Chap.4 Comparson: dentons o sngle partcle torque and angular momentum Angular momentum o a system o partcles o a rgd body rotatng about a xed
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationAMSC 660 Scientific Computing I. Term Project The Assignment
AS 66 Scentfc omputn I Prof. O Leary Term Project The Assnment odeln of a Seat Suspenson System Greory J. Hemen Dec., 6 ure depcts a lumped parameter system that may be used to model the response of a
More informationUIC University of Illinois at Chicago
DSL Dynamc Smulaton Laboratory UIC Unversty o Illnos at Chcago FINITE ELEMENT/ MULTIBODY SYSTEM ALGORITHMS FOR RAILROAD VEHICLE SYSTEM DYNAMICS Ahmed A. Shabana Department o Mechancal and Industral Engneerng
More informationPHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014
PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 Work-Knetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o
More informationPHYS 1441 Section 002 Lecture #16
PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Non-conservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!!
More informationAngular momentum. Instructor: Dr. Hoi Lam TAM ( 譚海嵐 )
Angular momentum Instructor: Dr. Ho Lam TAM ( 譚海嵐 ) Physcs Enhancement Programme or Gted Students The Hong Kong Academy or Gted Educaton and Department o Physcs, HKBU Department o Physcs Hong Kong Baptst
More information10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16
0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 9-3, 5-6. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall 03 -- Lecture 7 0/4/03
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationThe Karush-Kuhn-Tucker. Nuno Vasconcelos ECE Department, UCSD
e Karus-Kun-ucker condtons and dualt Nuno Vasconcelos ECE Department, UCSD Optmzaton goal: nd mamum or mnmum o a uncton Denton: gven unctons, g, 1,...,k and, 1,...m dened on some doman Ω R n mn w, w Ω
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 4. Moton Knematcs 4.2 Angular Velocty Knematcs Summary From the last lecture we concluded that: If the jonts
More informationwhere v means the change in velocity, and t is the
1 PHYS:100 LECTURE 4 MECHANICS (3) Ths lecture covers the eneral case of moton wth constant acceleraton and free fall (whch s one of the more mportant examples of moton wth constant acceleraton) n a more
More informationChapter 2. Pythagorean Theorem. Right Hand Rule. Position. Distance Formula
Chapter Moton n One Dmenson Cartesan Coordnate System The most common coordnate system or representng postons n space s one based on three perpendcular spatal axes generally desgnated x, y, and z. Any
More informationChapter 7: Conservation of Energy
Lecture 7: Conservaton o nergy Chapter 7: Conservaton o nergy Introucton I the quantty o a subject oes not change wth tme, t means that the quantty s conserve. The quantty o that subject remans constant
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationPHYS 1441 Section 002 Lecture #15
PHYS 1441 Secton 00 Lecture #15 Monday, March 18, 013 Work wth rcton Potental Energy Gravtatonal Potental Energy Elastc Potental Energy Mechancal Energy Conservaton Announcements Mdterm comprehensve exam
More informationChapter 07: Kinetic Energy and Work
Chapter 07: Knetc Energy and Work Conservaton o Energy s one o Nature s undamental laws that s not volated. Energy can take on derent orms n a gven system. Ths chapter we wll dscuss work and knetc energy.
More informationSolution Set #3
5-55-7 Soluton Set #. Te varaton of refractve ndex wt wavelengt for a transarent substance (suc as glass) may be aroxmately reresented by te emrcal equaton due to Caucy: n [] A + were A and are emrcally
More informationtotal If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.
Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationRotational and Translational Comparison. Conservation of Angular Momentum. Angular Momentum for a System of Particles
Conservaton o Angular Momentum 8.0 WD Rotatonal and Translatonal Comparson Quantty Momentum Ang Momentum Force Torque Knetc Energy Work Power Rotaton L cm = I cm ω = dl / cm cm K = (/ ) rot P rot θ W =
More informationPhysics 2A Chapters 6 - Work & Energy Fall 2017
Physcs A Chapters 6 - Work & Energy Fall 017 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on
More informationDynamics of a Spatial Multibody System using Equimomental System of Point-masses
Dynamcs of a Spatal Multbody System usng Equmomental System of Pont-masses Vnay Gupta Dept. of Mechancal Engg. Indan Insttute of echnology Delh New Delh, Inda vnayguptaec@gmal.com Hmanshu Chaudhary Dept.
More informationChapter 21 - The Kinetic Theory of Gases
hapter 1 - he Knetc heory o Gases 1. Δv 8.sn 4. 8.sn 4. m s F Nm. 1 kg.94 N Δ t. s F A 1.7 N m 1.7 a N mv 1.6 Use the equaton descrbng the knetc-theory account or pressure:. hen mv Kav where N nna NA N
More informationStart with the equation of motion for a linear multi-degree of freedom system with base ground excitation:
SE 80 Earthquake Enneern November 3, 00 STEP-BY-STEP PROCEDURE FOR SETTING UP A SPREADSHEET FOR USING NEWMARK S METHOD AND MODAL ANALYSIS TO SOLVE FOR THE RESPONSE OF A MULTI-DEGREE OF FREEDOM (MDOF) SYSTEM
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationPhysics 131: Lecture 16. Today s Agenda
Physcs 131: Lecture 16 Today s Agenda Intro to Conseraton o Energy Intro to some derent knds o energy Knetc Potental Denton t o Mechancal Energy Conseraton o Mechancal Energy Conserate orces Examples Pendulum
More informationAdvanced Mechanical Elements
May 3, 08 Advanced Mechancal Elements (Lecture 7) Knematc analyss and moton control of underactuated mechansms wth elastc elements - Moton control of underactuated mechansms constraned by elastc elements
More informationSCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ
s SCHOOL OF COMPUTING, ENGINEERING ND MTHEMTICS SEMESTER EXMINTIONS 011/01 DYNMICS ME47 DR. N.D.D. MICHÉ Tme allowed: THREE hours nswer: ny FOUR from SIX questons Each queston carres 5 marks Ths s a CLOSED-BOOK
More informationTrajectory Optimization of Flexible Mobile Manipulators Using Open-Loop Optimal Control method
rajectory Optmzaton o Flexble Moble Manpulators Usng Open-Loop Optmal Control method M.H. Korayem and H. Rahm Nohooj Robotc Lab, Mechancal Department, Iran Unversty o Scence and echnology hkorayem@ust.ac.r,
More informationChapter 3 and Chapter 4
Chapter 3 and Chapter 4 Chapter 3 Energy 3. Introducton:Work Work W s energy transerred to or rom an object by means o a orce actng on the object. Energy transerred to the object s postve work, and energy
More informationOn a nonlinear compactness lemma in L p (0, T ; B).
On a nonlnear compactness lemma n L p (, T ; B). Emmanuel Matre Laboratore de Matématques et Applcatons Unversté de Haute-Alsace 4, rue des Frères Lumère 6893 Mulouse E.Matre@ua.fr 3t February 22 Abstract
More informationPHYSICS 203-NYA-05 MECHANICS
PHYSICS 03-NYA-05 MECHANICS PROF. S.D. MANOLI PHYSICS & CHEMISTRY CHAMPLAIN - ST. LAWRENCE 790 NÉRÉE-TREMBLAY QUÉBEC, QC GV 4K TELEPHONE: 48.656.69 EXT. 449 EMAIL: smanol@slc.qc.ca WEBPAGE: http:/web.slc.qc.ca/smanol/
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More information36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to
ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationExperiment 5 Elastic and Inelastic Collisions
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 1 Experment 5 Elastc and Inelastc Collsons Readng: Bauer&Westall: Chapter 7 (and 8, or center o mass deas) as needed Homework 5: turn n the
More informationCh. 7 Lagrangian and Hamiltonian dynamics Homework Problems 7-3, 7-7, 7-15, 7-16, 7-17, 7-18, 7-34, 7-37, where y'(x) dy dx Δ Δ Δ. f x.
Ch. 7 Laranan an Hamltonan namcs Homewor Problems 7-3 7-7 7-5 7-6 7-7 7-8 7-34 7-37 7-40 A. revew o calculus o varatons (Chapter 6. basc problem or J { ( '(; } where '( For e en ponts an ntereste n the
More informationCHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and non-conservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
More informationDynamics of Rotational Motion
Dynamcs of Rotatonal Moton Torque: the rotatonal analogue of force Torque = force x moment arm = Fl moment arm = perpendcular dstance through whch the force acts a.k.a. leer arm l F l F l F l F = Fl =
More informationPage 1. Clicker Question 9: Physics 131: Lecture 15. Today s Agenda. Clicker Question 9: Energy. Energy is Conserved.
Physcs 3: Lecture 5 Today s Agenda Intro to Conseraton o Energy Intro to some derent knds o energy Knetc Potental Denton o Mechancal Energy Conseraton o Mechancal Energy Conserate orces Examples Pendulum
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationChapter 3: Kinematics and Dynamics of Multibody Systems
Chapter 3: Knematcs and Dynamcs of Multbody Systems 3. Fundamentals of Knematcs 3.2 Settng up the euatons of moton for multbody-systems II 3. Fundamentals of Knematcs 2 damper strut sprng chasss x z y
More informationLinear Momentum. Equation 1
Lnear Momentum OBJECTIVE Obsere collsons between two carts, testng or the conseraton o momentum. Measure energy changes durng derent types o collsons. Classy collsons as elastc, nelastc, or completely
More informationPHYSICS 231 Review problems for midterm 2
PHYSICS 31 Revew problems for mdterm Topc 5: Energy and Work and Power Topc 6: Momentum and Collsons Topc 7: Oscllatons (sprng and pendulum) Topc 8: Rotatonal Moton The nd exam wll be Wednesday October
More informationClassical Mechanics ( Particles and Biparticles )
Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles
More informationTechnical Report TR05
Techncal Report TR05 An Introducton to the Floatng Frame of Reference Formulaton for Small Deformaton n Flexble Multbody Dynamcs Antono Recuero and Dan Negrut May 11, 2016 Abstract Ths techncal report
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationChapter 7. Potential Energy and Conservation of Energy
Chapter 7 Potental Energy and Conservaton o Energy 1 Forms o Energy There are many orms o energy, but they can all be put nto two categores Knetc Knetc energy s energy o moton Potental Potental energy
More informationWork is the change in energy of a system (neglecting heat transfer). To examine what could
Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationPhysics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall
Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons
More informationConservation Laws (Collisions) Phys101 Lab - 04
Conservaton Laws (Collsons) Phys101 Lab - 04 1.Objectves The objectves o ths experment are to expermentally test the valdty o the laws o conservaton o momentum and knetc energy n elastc collsons. 2. Theory
More informationOne can coose te bass n te 'bg' space V n te form of symmetrzed products of sngle partcle wavefunctons ' p(x) drawn from an ortonormal complete set of
8.54: Many-body penomena n condensed matter and atomc pyscs Last moded: September 4, 3 Lecture 3. Second Quantzaton, Bosons In ts lecture we dscuss second quantzaton, a formalsm tat s commonly used to
More informationChapter 11: Angular Momentum
Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationEstimation of Natural Frequency of the Bearing System under Periodic Force Based on Principal of Hydrodynamic Mass of Fluid
Internatonal Journal o Mechancal and Mechatroncs Engneerng Vol:, No:7, 009 Estmaton o Natural Frequency o the Bearng System under Perodc Force Based on Prncpal o Hydrodynamc Mass o Flud M. H. Pol, A. Bd,
More informationGoal Programming Approach to Solve Multi- Objective Intuitionistic Fuzzy Non- Linear Programming Models
Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 Goal Prorammn Approach to Solve Mult- Objectve Intutonstc Fuzzy Non- Lnear Prorammn Models S.Rukman #, R.Sopha Porchelv
More informationPhysics 201 Lecture 4
Phscs 1 Lectue 4 ltoda: hapte 3 Lectue 4 v Intoduce scalas and vectos v Peom basc vecto aleba (addton and subtacton) v Inteconvet between atesan & Pola coodnates Stat n nteestn 1D moton poblem: ace 9.8
More informationA Tale of Friction Student Notes
Nae: Date: Cla:.0 Bac Concept. Rotatonal Moeent Kneatc Anular Velocty Denton A Tale o Frcton Student Note t Aerae anular elocty: Intantaneou anular elocty: anle : radan t d Tanental Velocty T t Aerae tanental
More informationAPPENDIX F A DISPLACEMENT-BASED BEAM ELEMENT WITH SHEAR DEFORMATIONS. Never use a Cubic Function Approximation for a Non-Prismatic Beam
APPENDIX F A DISPACEMENT-BASED BEAM EEMENT WITH SHEAR DEFORMATIONS Never use a Cubc Functon Approxmaton for a Non-Prsmatc Beam F. INTRODUCTION { XE "Shearng Deformatons" }In ths appendx a unque development
More informationMEEM 3700 Mechanical Vibrations
MEEM 700 Mechancal Vbratons Mohan D. Rao Chuck Van Karsen Mechancal Engneerng-Engneerng Mechancs Mchgan echnologcal Unversty Copyrght 00 Lecture & MEEM 700 Multple Degree of Freedom Systems (ext: S.S.
More informationINTO CHAINED FORM USING DYANMIC FEEDBACK D. TILBURY, O. SRDALEN, L. BUSHNELL; S. SASTRY
A MULTI-STEERING TRAILER SYSTEM: CONVERSION INTO CHAINED FORM USING DYANMIC FEEDBACK D. TILBURY, O. SRDALEN, L. BUSHNELL; S. SASTRY Electroncs Research Laboratory, Department of Electrcal Engneerng and
More informationMEV442 Introduction to Robotics Module 2. Dr. Santhakumar Mohan Assistant Professor Mechanical Engineering National Institute of Technology Calicut
MEV442 Introducton to Robotcs Module 2 Dr. Santhakumar Mohan Assstant Professor Mechancal Engneerng Natonal Insttute of Technology Calcut Jacobans: Veloctes and statc forces Introducton Notaton for tme-varyng
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationONE-DIMENSIONAL COLLISIONS
Purpose Theory ONE-DIMENSIONAL COLLISIONS a. To very the law o conservaton o lnear momentum n one-dmensonal collsons. b. To study conservaton o energy and lnear momentum n both elastc and nelastc onedmensonal
More informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More informationLinear Momentum. Center of Mass.
Lecture 16 Chapter 9 Physcs I 11.06.2013 Lnear oentu. Center of ass. Course webste: http://faculty.ul.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.ul.edu/danylov2013/physcs1fall.htl
More informationPhysics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall
Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons
More informationTHE NEW TRANSFORMATION EQUATION AND TRANSVERSE PHYSICAL EFFECT K.VAITHIYANATHAN
THE NEW TRANSFORMATION EQUATION AND TRANSVERSE PHYSICAL EFFECT BY K.VAITHIYANATHAN E.Mal : vathyanathan_k@yahoo.co.n Andkkadu 61473. Taml Nadu. South Inda. 1 Introducton PART 1 Any one can see the new
More informationApplication to Plane (rigid) frame structure
Advanced Computatonal echancs 18 Chapter 4 Applcaton to Plane rgd frame structure 1. Dscusson on degrees of freedom In case of truss structures, t was enough that the element force equaton provdes onl
More informationLecture 16. Chapter 11. Energy Dissipation Linear Momentum. Physics I. Department of Physics and Applied Physics
Lecture 16 Chapter 11 Physcs I Energy Dsspaton Lnear Momentum Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Department o Physcs and Appled Physcs IN IN THIS CHAPTER, you wll learn
More informationChapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation
Chapter 20 Rgd Body: Translaton and Rotatonal Moton Knematcs for Fxed Axs Rotaton 20.1 Introducton... 1 20.2 Constraned Moton: Translaton and Rotaton... 1 20.2.1 Rollng wthout slppng... 5 Example 20.1
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. Mult-Partcle
More informationLie Group Formulation of Articulated Rigid Body Dynamics
Le Group Formulaton of Artculated Rgd Body Dynamcs Junggon Km 11/9/2012, Ver 1.01 Abstract It has been usual n most old-style text books for dynamcs to treat the formulas descrbng lnearor translatonal
More information