# Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.

Size: px
Start display at page:

Download "Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f."

Transcription

1 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 to and n the same drecton o the orce actng on t Thrd Law: The orces o acton and reacton between nteractng artcles are equal n magntude, collnear, and ooste n drecton Dynamcs o A Partcle A artcle (ont) Mass m ( ) Force vector Acceleraton vector r or a Equaton o moton m ( ) r = Partcle Equaton o moton m r = In exanded orm m r = m 0 0 x 0 m 0 y = 0 0 m Dynamcs o A System o Partcles A system o artcles Center o mass (centrod) s ostoned wth vector r Equaton o moton or the centrod: mr = where m m and = Follow the dervaton o the equatons o moton or the mass center o a system o artcles n Sec Observe how the nternal reacton orces between the artcles cancel each other! Poston o the center o mass 1 Resultant equaton r 1 m m r m s = 0 x x ( x) ( y) ( ) r y y m ( ) r s r a j

2 Note that r locates artcle rom the orgn o the reerence rame where s locates the artcle rom the mass center! Translatonal Equatons o Moton or A Body Translatonal equatons o moton (centrodal) or a body are m r = In exanded orm m 0 0 x (x) 0 m 0 y = 0 0 m (y) () Mass o the body: m Sum o orces actng on the body: Acceleraton o the mass center: r r O mass center C Moment o A Force Moment o a orce actng on a body at ont P s comuted as n = s P where s P locates ont P rom the mass center o the body C s P P Rotatonal Equatons o Moton or A Body The dervaton shown here s much smler that the one gven n the textbook! Assume that the body s made o nnte () number o artcles For artcle the equaton o moton s m r = +, j where, s an external orce that may exst or some o the artcles, j ; j = 1,..., are the reacton orces exerted on ths artcle by other artcles Pre-multly ths equaton by s : m s r = s + s, j m s (r +s ) = s + s, j x 1 r 1 y j s,j j, j We wrte ths equaton or every artcle and then sum over all the artcles:

3 m s r + m s s = s + s, j (a) (b) We examne each o the our terms searately! (a) In ths term r can be moved outsde sgma. What remans wthn sgma, accordng to m s = 0, s equal to ero! (c) (m s r) = (m s ) r = 0 (b) From Lesson 10 we have s = s, s = s + s where s the angular velocty o the body. Substtutng n (b) and reerrng to Problem 2.15 yelds ( ) m s s m s (s + s ) m s s + m s s m s s + m s s m s s + m s s We now take and out o sgma n each term m s s + m s s (c) Ths term reresents the sum o all moments actng on the body n = (d) We can show that ths term s exactly equal to ero! I we exand the terms wthn the sgmas, we can ar every two terms as shown: s, j s + s, j + s j j, ++ s, j s j, j + + (s s j ), j + Accordng to the gure the vector s, j = s s j s arallel to the reacton orce, j. Thereore, the roduct (s s j ), j s ero! Now the equaton o moton has been smled to s j j, m s s + m s s = n j We relace each sgma by ntegral (nnte number o artcles) and then ntroduce the nerta matrx as J m s s = s s dm vol. The rotatonal equaton o moton becomes J + J = n The nerta matrx J s obtaned wth resect to a reerence rame attached to the mass center o the body and remanng arallel to the nonmovng x-y- rame. Thereore (d ) s s, j,j

4 the comonents o J vary wth changng orentaton o the body J ' The nerta matrx J ' s dened as J ' = A T JA The comonents o ths matrx are constants they are obtaned wth resect to a rame that s attached to and rotates wth the body The rotatonal equaton o moton can also be exressed as J ' '+ ' J ' ' = n' Equatons o Moton or A Rgd Body Newton-Euler equatons Both the translatonal and rotatonal equatons are descrbed n the x-y- comonents m r = J + J = n mi 0 r 0 + = 0 J J n mi 0 r = (a) 0 J n J The translatonal (Newton) equatons are descrbed n the x-y- comonents but the rotatonal (Euler) equatons are descrbed n the comonents m r = J ' '+ ' J ' ' = n' mi 0 r 0 + = 0 J' ' ' J ' ' n' mi 0 r = (b) 0 J' ' n' ' J ' ' Reer to the textbook or urther dscusson on these equatons! When Euler equatons are descrbed n terms o ther local comonents;.e., Eq. (b), the nerta matrx J ' remans a constant. Ths could be a bg advantage over Eq. (a) where the nerta matrx J needs to be re-evaluated every tme the rotatonal orentaton o the body changes! We can derve the multbody equatons o moton based on Eq. (a) or Eq. (b). I we derve the equatons n one orm, t takes a smle transormaton to obtan the equatons n the other orm! The rotatonal equatons o moton can also be derved n terms o the second tme dervatve o Euler arameters (ths s done n the textbook we wll not use ths orm n ths course!)

5 In ths course, we derve and use the equatons o moton n the orm o (a). Ths wll be consstent wth the knematc constrants rom Lesson 11. These equatons or body are exressed n comact orm as M v = g where, M = m I 0 0 J, v = r, g = n J Note that M and g n your textbook (Eqs. 8.36, 8.38, and 8.39) are dened derently! A System o Unconstraned Bodes In an unconstraned multbody system there are no knematc jonts; hence there are no knematc constrants. The srngs, damers, actuators are orce elements t s assumed that a orce element does not mose any constrants on a system. In the next lesson we wll learn how to construct the array o orces or several commonly used orce elements Equatons o moton are derved by constructng the Newton-Euler equatons or every body n the system. Assume that there are b bodes n the system: M v = g where M 1 M M = 2 M b v 1 g 1 v, v = 2 g, g = 2 v b g b A System o Constraned Bodes In a constraned multbody system one or more knematc jonts are resent; hence knematc constrants must be ncororated nto the equatons o moton. For a system o b bodes the equatons o moton are exressed as M v = g + g (c) where g (c) reresents the array o reacton orces, and g contans the srng, damer,... orces and moments. Other elements n the above equaton are the same as n the unconstraned equatons o moton How do we determne the reacton orces? Assume that the constrants and ther rst and second tme dervatves are reresented as

6 (q) = 0 Dv= 0 D v + Dv= 0 The Jacoban matrx, D, s used n determnng the array o reacton orces as g (c) = D T where contans as many coecents as the number o constrants. These coecents are called Lagrange multlers At ths ont these multlers are unknowns! In the ucomng lessons we wll learn how to determne these multlers. The equatons o moton can be wrtten as M v D T = g In Secton o the textbook you nd urther dscusson on the array o reacton orces In the ucomng lessons we wll look at the reacton orces and moments or some secc knematc jonts.

### Lesson 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

### Rigid 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

### A 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

### Iterative 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

### So 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

### Week 9 Chapter 10 Section 1-5

Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,

### Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa

Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.

### PHYS 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)]

### Physics 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

### Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis

Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta

### Linear 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

### Conservation 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

### Degrees of Freedom. Spherical (ball & socket) 3 (3 rotation) Two-Angle (universal) 2 (2 rotation)

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

### Chapter 11 Torque and Angular Momentum

Chapter Torque and Angular Momentum I. Torque II. Angular momentum - Defnton III. Newton s second law n angular form IV. Angular momentum - System of partcles - Rgd body - Conservaton I. Torque - Vector

### First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.

Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act

### Angular 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

### Physics 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

### NMT 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

### Spin-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

### Moments 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

### Rotational 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 =

### A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

### Linear 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

### Module 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

### Classical Mechanics Virtual Work & d Alembert s Principle

Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often

### Chapter 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

### From Newton s 2 nd Law: v v. The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle.

From Newton s 2 nd Law: F ma d dm ( ) m dt dt F d dt The tme rate of change of the lnear momentum of a artcle s equal to the net force actng on the artcle. Conseraton of Momentum +x The toy rocket n dee

### Physics 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

### PHYSICS 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/

### Indeterminate pin-jointed frames (trusses)

Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all

### One Dimensional Axial Deformations

One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

### Spring 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

### Physics 40 HW #4 Chapter 4 Key NEATNESS COUNTS! Solve but do not turn in the following problems from Chapter 4 Knight

Physcs 40 HW #4 Chapter 4 Key NEATNESS COUNTS! Solve but do not turn n the ollowng problems rom Chapter 4 Knght Conceptual Questons: 8, 0, ; 4.8. Anta s approachng ball and movng away rom where ball was

### ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15

NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound

### EN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics

N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.

### NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy

### Classical 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

### 6. Hamilton s Equations

6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant

### Chapter 3 Differentiation and Integration

MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton

### Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the

### Physics 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

### Lesson 16: Basic Control Modes

0/8/05 Lesson 6: Basc Control Modes ET 438a Automatc Control Systems Technology lesson6et438a.tx Learnng Objectves Ater ths resentaton you wll be able to: Descrbe the common control modes used n analog

### PHYS 705: Classical Mechanics. Calculus of Variations II

1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

### ELASTIC 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

### Physics 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

### Physics for Scientists and Engineers. Chapter 9 Impulse and Momentum

Physcs or Scentsts and Engneers Chapter 9 Impulse and Momentum Sprng, 008 Ho Jung Pak Lnear Momentum Lnear momentum o an object o mass m movng wth a velocty v s dened to be p mv Momentum and lnear momentum

### Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or

### THERMODYNAMICS. Temperature

HERMODYNMICS hermodynamcs s the henomenologcal scence whch descrbes the behavor of macroscoc objects n terms of a small number of macroscoc arameters. s an examle, to descrbe a gas n terms of volume ressure

### Lie 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

### Physics 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

### Please review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem n the space

### LAB # 4 - Torque. d (1)

LAB # 4 - Torque. Introducton Through the use of Newton's three laws of moton, t s possble (n prncple, f not n fact) to predct the moton of any set of partcles. That s, n order to descrbe the moton of

### An Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors

An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,

### Chapter 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

### Chapter 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

### CHAPTER 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

### OPTIMISATION. 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

### The 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

### Week3, 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

### Technical 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

### Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004

Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a

### Study Guide For Exam Two

Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force

### MEV442 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

### BAR & TRUSS FINITE ELEMENT. Direct Stiffness Method

BAR & TRUSS FINITE ELEMENT Drect Stness Method FINITE ELEMENT ANALYSIS AND APPLICATIONS INTRODUCTION TO FINITE ELEMENT METHOD What s the nte element method (FEM)? A technqe or obtanng approxmate soltons

### 36.1 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

### coordinates. 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,

### 11. 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

### Notes 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

### total 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

### Center of Mass and Linear Momentum

PH 221-2A Fall 2014 Center of Mass and Lnear Momentum Lectures 14-15 Chapter 9 (Hallday/Resnck/Walker, Fundamentals of Physcs 9 th edton) 1 Chapter 9 Center of Mass and Lnear Momentum In ths chapter we

### NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles

### ONE-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

### Spring 2002 Lecture #13

44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term

### Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)

Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng

### Physics 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

### PHYS 1443 Section 002

PHYS 443 Secton 00 Lecture #6 Wednesday, Nov. 5, 008 Dr. Jae Yu Collsons Elastc and Inelastc Collsons Two Dmensonal Collsons Center o ass Fundamentals o Rotatonal otons Wednesday, Nov. 5, 008 PHYS PHYS

### Lagrange 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

### PHYS 1101 Practice problem set 12, Chapter 32: 21, 22, 24, 57, 61, 83 Chapter 33: 7, 12, 32, 38, 44, 49, 76

PHYS 1101 Practce problem set 1, Chapter 3: 1,, 4, 57, 61, 83 Chapter 33: 7, 1, 3, 38, 44, 49, 76 3.1. Vsualze: Please reer to Fgure Ex3.1. Solve: Because B s n the same drecton as the ntegraton path s

### SCALARS 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.

### χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown

### Mathematical 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

### Chapter 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

### 10/24/2013. 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

### Digital PI Controller Equations

Ver. 4, 9 th March 7 Dgtal PI Controller Equatons Probably the most common tye of controller n ndustral ower electroncs s the PI (Proortonal - Integral) controller. In feld orented motor control, PI controllers

### Physics 4C. Chapter 19: Conceptual Questions: 6, 8, 10 Problems: 3, 13, 24, 31, 35, 48, 53, 63, 65, 78, 87

Physcs 4C Solutons to Chater 9 HW Chater 9: Concetual Questons: 6, 8, 0 Problems:,, 4,,, 48,, 6, 6, 78, 87 Queston 9-6 (a) 0 (b) 0 (c) negate (d) oste Queston 9-8 (a) 0 (b) 0 (c) negate (d) oste Queston

### Mechanics Physics 151

Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and

### PHYS 1443 Section 003 Lecture #17

PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!

### University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014

Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure

### Conservation 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,

### Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen

omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense. ontents ontour ntegrals auchy-goursat theorem

### Poisson brackets and canonical transformations

rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order

### Physics 207 Lecture 27

hyscs 07 Lecture 7 hyscs 07, Lecture 7, Dec. 6 Agenda: h. 0, st Law o Thermodynamcs, h. st Law o thermodynamcs ( U Q + W du dq + dw ) Work done by an deal gas n a ston Introducton to thermodynamc cycles

### Physics 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

### 8. THE CONTACT DYNAMICS METHOD

8 THE CONTACT DYNAMICS METHOD 8 Introducton The Contact Dynamcs ( CD ) method was ntroduced at the begnnng of the 990es by M Jean and JJ Moreau (Jean és Moreau, 992; Jean, 999) The method turned out to

### Modeling of Dynamic Systems

Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how

### ERRATA. COMPUTER-AIDED ANALYSIS OF MECHANICAL SYSTEMS Parviz E. Nikravesh Prentice-Hall, (Corrections as of November 2014)

ERRATA COMPUTER-AIDED ANALYSIS OF MECHANICAL SYSTEMS Parvz E. Nkravesh Prentce-Hall, 1988 (Correctons as of November 2014 Address to an error s gven n the frst column by the page number and n the second

### = 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]

Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:

### MEEM 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.