Robotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.
|
|
- Vivian Spencer
- 5 years ago
- Views:
Transcription
1 Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L L L 3 3 able 1: able of DH parameers of a 3R spaial robo Given a posiion p R 3 of he origin of he end-effecor frame, provide he analyic expression of he soluion o he inverse kinemaics problem For L 1 = 1 [m] and L = L 3 = 15 [m], deermine all inverse kinemaics soluions in numerical form associaed o he end-effecor posiion p = [m] Exercise A robo join should move in minimum ime beween an iniial value a and a final value b, wih an iniial velociy a and a final velociy b, under he bounds and A Provide he analyic expression of he minimum feasible moion ime when = b a > 0 and he iniial and final velociies are arbirary in sign and magniude bu boh saisfy he velociy bound, ie, a and b Using he daa a = 90, b = 30, a = 45 /s, b = 45 /s, = 90 /s, A = 00 /s, deermine he numerical value of he minimum feasible moion ime and draw he velociy and acceleraion profiles of he join moion [180 minues, open books bu no compuer or smarphone] 1
2 Exercise 1 Soluion April 11, 017 From he direc kinemaics, using ab 1, we obain for he posiion of he origin of he end-effecor frame p 0 p H = = 0 A 1 1 A A L cos + L 3 cos + 3 cos 1 p = L cos + L 3 cos + 3 sin 1 L 1 + L sin + L 3 sin he analyic inversion of e 1 for p = p d = p dx p dy p dz proceeds as follows Afer moving L 1 o he lef-hand side of he hird euaion, suaring and adding he hree euaions yields he numeric value c 3 for cos 3 c 3 = p dx + p dy + p dz L 1 L L 3 L L 3 he desired end-effecor posiion will belong o he robo workspace if and only if c 3 [ 1, 1] Noe ha his condiion holds no maer if L and L 3 are eual or differen Under such premises, we compue s 3 = 1 c 3 3 and {+} 3 = AAN {s 3, c 3 }, { } 3 = AAN { s 3, c 3 }, 4 yielding by definiion wo opposie values { } 3 = {+} 3 If c 3 = ±1, he robo is in a kinemaic singulariy: he forearm is eiher sreched or folded, in boh cases on he boundary of he workspace In paricular, when c 3 = 1, {+} 3 and { } 3 are boh eual o 0; when c 3 = 1, he wo soluions will be aken 1 eual o π Insead, when c 3 [ 1, 1], he inverse kinemaics algorihm should oupu a warning message desired posiion is ou of workspace and exi When p dx + p dy > 0, from he firs wo euaions in 1 we can furher compue p dx + p dy = L cos + L 3 cos + 3 cos 1 = and hus p dx, sin 1 = ± p dx + p dy p dy ± p dx + p dy {+} 1 = AAN {p dy, p dx }, { } 1 = AAN { p dy, p dx } 5 hese wo values belong o π, π] and will always differ by π Insead, when p dx = p dy = 0, he firs join angle 1 remains undefined and he robo will be in a kinemaic singulariy wih he end-effecor placed along he axis of join 1 he soluion algorihm should oupu a warning message singular case: angle 1 is undefined, possibly se a flag sing 1 = ON, bu coninue 1 Remember ha we use as convenional range π, π], for all angles hus, if he oupu of a generic compuaion is π, we always replace i wih +π,
3 A his sage, we can rewrie a suiable combinaion of he firs wo euaions in 1 as well as he hird euaion in he following way: and cos 1 p dx + sin 1 p dy = L cos + L 3 cos + 3 = L + L 3 cos 3 cos L 3 sin 3 sin p dz L 1 = L sin + L 3 sin + 3 = L 3 sin 3 cos + L + L 3 cos 3 sin Plugging he muliple values found so far for 1 and 3, we obain four similar linear sysems in he rigonomeric unknowns c = cos and s = sin : L + L 3 c 3 L 3 s {+, } 3 c cos {+, } 1 p dx + sin {+, } 1 p dy = A {+, } x = b {+, } L 3 s {+, } 3 L + L 3 c 3 s p dz L 1 6 In 6, we should use and he values from 4 and 5 his gives rise o four possible combinaions for he marix/vecor pair A {+, }, b {+, }, which will evenually lead o four soluions for ha are in general disinc hese will be labeled as {f,u} {f,d} {b,u} {b,d} {f,u} {f,d} {b,u} {b,d} depending on wheher he robo is facing f of backing b he desired posiion uadran due o he choice of 1, and on wheher he elbow is up u or down d due o he combined choice of 1 and 3 If he common deerminan of he coefficien marix is differen from zero, ie, using e, de A {+, } = L + L 3 c 3 + L 3 s {{f,b},{u,d}} s {+, } 3 = L + L 3 + L L 3 c 3 = p dx + p dy + p dz L 1 > 0, he soluion for of each of he above four cases is uniuely deermined from {{f,b},{u,d}} c = L + L 3 c 3 cos {+, } 1 p dx + sin {+, } 1 p dy L + L 3 c 3 p dz L 1 L 3 s {+, } 3 and henceforh + L 3 s {+, } 3 p dz L 1 cos {+, } 1 p dx + sin {+, } 1 p dy { } {{f,b},{u,d}} = AAN s {{f,b},{u,d}}, c {{f,b},{u,d}} 7 Insead, when p dx = p dy = 0 and p dy = L 1, he robo will be in a double kinemaic singulariy, wih he arm folded and he end-effecor placed along he axis of join 1 Noe ha his siuaion can only occur in case he robo has L = L 3 oherwise he singular Caresian poin would be ou of he robo workspace he soluion algorihm should oupu a warning message singular case: angle is undefined, possibly se a second flag sing = ON, and hen exi In his case, only a single value 3 = π for he hird join angle will be defined Moving nex o he reuesed numerical case wih L 1 = 1, L = 15, and L 3 = 15 [m], and for he desired posiion 1 p d = 1 [m], 15 A special case arises when he join angle 1 remains undefined a singulariy wih flag sing 1 = ON he firs componen of he known vecor b in 6 will vanish p dx = p dy = 0 and only wo soluions would be lef for he case in which hese wo well-defined soluions collapse ino a single value is lef o he reader s analysis, 3
4 we can see ha p d belongs o he robo workspace and ha his is no a singular case since c 3 = 05 [ 1, 1], p dx + p dy = > 0 We noe ha he desired posiion is in he second uadran x < 0, y > 0 hus, he four inverse kinemaics soluions obained from 4, 5 and 7 are: 356 3π/ {f,u} = = [rad] = π/ {f,d} = {b,u} = {b,d} = = = = 3π/4 356 π/3 π/ [rad] = π/3 π/4 434 π/3 [rad] = [rad] = As a double-check of correcness, i is always highly recommended o evaluae he direc kinemaics wih he obained soluions 8 In reurn, one should ge every ime he desired posiion p d Exercise his exercise is a generalizaion of he minimum-ime rajecory planning problem for a single join under velociy and acceleraion bounds, wih zero iniial and final velociy res-o-res as boundary condiions I is useful o recap firs he soluion o he res-o-res problem he minimum-ime moion is given by a rapezoidal velociy profile or a bang-coas-bang profile in acceleraion, wih minimum moion ime and symmeric iniial and final acceleraion/deceleraion phases of duraion s given by = + A > s, s = > 0 9 A his soluion is only valid when he disance o ravel in absolue value and he limi velociy and acceleraion values > 0 and A > 0 saisfy he ineualiy 8 A, 10 namely, when he disance is sufficienly long wih respec o he raio of he suared velociy limi o he acceleraion limi When he eualiy holds in 10, he maximum velociy is reached only a he single insan / = s, when half of he moion has been compleed Insead, when 10 is violaed, he minimum-ime moion is given by a bang-bang acceleraion profile ie, wih a riangular velociy profile having only he acceleraion/deceleraion phases, each of duraion s = A = s 11 4
5 he crusing phase wih maximum velociy is no reached in his case For all he above cases, when < 0 he opimal velociy and acceleraion profiles are simply changed of sign flipped over he ime axis a b a a > 0, b > 0 a b b a > 0, b < 0 b a a b c a < 0, b > 0 d a < 0, b < 0 Figure 1: Qualiaive asymmeric velociy profiles of he rapezoidal ype for he four combinaions of signs of he iniial and final velociy a and b I is assumed ha > 0, and ha his disance is sufficienly long so as o have a non-vanishing cruising inerval a maximum velociy = Consider now he problem of moving in minimum ime he join by a disance = b a > 0, bu wih generic non-zero boundary condiions 0 = a and = b on he iniial and final velociy he reuiremen ha a and b is obviously mandaory in order o have a feasible soluion Wih reference o he ualiaive rapezoidal velociy profiles skeched in Fig 1, we see ha non-zero iniial and final velociies may help in reducing he moion ime or work agains i In paricular, when boh a and b are posiive case a i is clear ha less ime will be needed o ramp up from a > 0 o, raher han from 0 o he same is rue for slowing down from o b > 0, raher han down o 0 On he conrary, when boh a and b are negaive case d, an exra ime will be spen for reversing moion from a < 0 o 0 in his ime inerval, he join will coninue o move in he opposie direcion o he desired one, unil i sops, when finally a posiive velociy can be achieved, and, similarly, anoher exra ime will be spen oward he end of he rajecory for bringing he velociy from 0 o b < 0 also in his second inerval, he join will move in he opposie direcion o he desired one Cases b and c in Fig 1 are inermediae siuaions beween a and d, and can be analyzed in a similar way 5
6 As a resul: in general, he acceleraion/deceleraion phases will have differen duraions 0 raher han he single s 0 of he res-o-res case; 0 and he original reuired disance o ravel > 0 will become in pracice longer, since we need o counerbalance he negaive displacemens inroduced during hose inervals where he velociy is negaive; since we need o minimize he oal moion ime, inervals wih negaive velociy should be raversed in he leas possible ime, hus wih maximum posiive or negaive acceleraion = ±A Wih he above general consideraions in mind, we perform now uaniaive calculaions In he posiive acceleraion and negaive deceleraion phases, we have = a A, = b A 1 We noe ha boh hese ime inervals will be shorer han s = /A for a posiive boundary velociy and longer han s for a negaive one he area wih sign underlying he velociy profile should provide, over he oal moion ime > 0, he reuired disance > 0 We compue his area as he sum of hree conribuions, using he rapezoidal rule for he wo inervals where he velociy is changing linearly over ime: a b = 13 Subsiuing 1 in 13 and rearranging erms gives + a a + A A + a + b + + b b = 14 A A Solving for he moion ime, we obain finally he opimal value = + a + b 15 A his is he generalizaion for > 0 of he minimum moion ime formula 9 of he res-o-res case which we recover by seing a = b = 0 his soluion is only valid when he disance o ravel > 0, he velociy and acceleraion limi values > 0 and A > 0, and he boundary velociies a and b saisfy he ineualiy a + b A 0, 16 which is again he generalizaion of condiion 10 his ineualiy is obained by imposing ha he sum of he firs and hird erm in he lef-hand side of 14, ie, he space raveled during he acceleraion and deceleraion phases, does no exceed a cruising phase a maximum speed > 0 would no longer be necessary I is ineresing o noe ha, for a given riple,, and A, he ineualiy 16 would be easier o enforce as soon as a 0 and/or b 0, independenly from heir signs he physical reason, however, is slighly differen for a posiive or negaive boundary velociy, say of a When a > 0, less ime is needed in order o reach he maximum velociy > 0; hus, i is more likely ha 6
7 he same problem daa will imply a cruising velociy phase Insead, when a < 0, a negaive displacemen will occur in he iniial phase, which needs o be recovered; hus, i is more likely ha a cruising phase a maximum velociy will be needed laer Finally, we poin ou ha: when ineualiy 16 is violaed, or for special values of a or b eg, a =, a number of sub-cases arise; heir complee analysis is ou of he presen scope and is lef as an exercise for he reader; for < 0, i is easy o show ha he formulas corresponding o 1, 15, and 16 are = + a A, = + b A, = + + a + + b, A a + b A Indeed, he velociy profiles in Fig 1 will use he value as cruising velociy [ /s]! = 90! a = 45 b = - 45 = 05 = 0675 [s]! [ /s ]! A = 00! * = = 05 cruise = 0996 = 0675 [s]! - 00! Figure : ime-opimal velociy and acceleraion profiles for he numerical problem in Exercise Moving o he given numerical problem, from = b a = = 10 > 0, a = 45 /s, b = 45 /s, = 90 /s, and A = 00 /s, we evaluae firs he ineualiy 16 and verify ha 10 > = 30375, 00 so ha he general formula 15 applies his yields while from 1 we obain = 05 [s], = [s], = 0675 [s], wih an inerval of duraion cruise = = [s] in which he join is cruising a = 90 /s he associaed ime-opimal velociy and acceleraion profiles are repored in Fig 7
Trajectory planning in Cartesian space
Roboics 1 Trajecory planning in Caresian space Prof. Alessandro De Luca Roboics 1 1 Trajecories in Caresian space in general, he rajecory planning mehods proposed in he join space can be applied also in
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationMEI Mechanics 1 General motion. Section 1: Using calculus
Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationSPH3U1 Lesson 03 Kinematics
SPH3U1 Lesson 03 Kinemaics GRAPHICAL ANALYSIS LEARNING GOALS Sudens will Learn how o read values, find slopes and calculae areas on graphs. Learn wha hese values mean on boh posiion-ime and velociy-ime
More informationPhysics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008
Physics 221 Fall 28 Homework #2 Soluions Ch. 2 Due Tues, Sep 9, 28 2.1 A paricle moving along he x-axis moves direcly from posiion x =. m a ime =. s o posiion x = 1. m by ime = 1. s, and hen moves direcly
More informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationModule 3: The Damped Oscillator-II Lecture 3: The Damped Oscillator-II
Module 3: The Damped Oscillaor-II Lecure 3: The Damped Oscillaor-II 3. Over-damped Oscillaions. This refers o he siuaion where β > ω (3.) The wo roos are and α = β + α 2 = β β 2 ω 2 = (3.2) β 2 ω 2 = 2
More informationDisplacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationBasilio Bona ROBOTICA 03CFIOR 1
Indusrial Robos Kinemaics 1 Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
More informationPhys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole
Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationWelcome Back to Physics 215!
Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More informationSolution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration
PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc
More information72 Calculus and Structures
72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s
More informationAnswers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
More informationand v y . The changes occur, respectively, because of the acceleration components a x and a y
Week 3 Reciaion: Chaper3 : Problems: 1, 16, 9, 37, 41, 71. 1. A spacecraf is raveling wih a veloci of v0 = 5480 m/s along he + direcion. Two engines are urned on for a ime of 84 s. One engine gives he
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationPhysics 20 Lesson 5 Graphical Analysis Acceleration
Physics 2 Lesson 5 Graphical Analysis Acceleraion I. Insananeous Velociy From our previous work wih consan speed and consan velociy, we know ha he slope of a posiion-ime graph is equal o he velociy of
More informationa 10.0 (m/s 2 ) 5.0 Name: Date: 1. The graph below describes the motion of a fly that starts out going right V(m/s)
Name: Dae: Kinemaics Review (Honors. Physics) Complee he following on a separae shee of paper o be urned in on he day of he es. ALL WORK MUST BE SHOWN TO RECEIVE CREDIT. 1. The graph below describes he
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationMotion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationBrock University Physics 1P21/1P91 Fall 2013 Dr. D Agostino. Solutions for Tutorial 3: Chapter 2, Motion in One Dimension
Brock Uniersiy Physics 1P21/1P91 Fall 2013 Dr. D Agosino Soluions for Tuorial 3: Chaper 2, Moion in One Dimension The goals of his uorial are: undersand posiion-ime graphs, elociy-ime graphs, and heir
More informationKinematics and kinematic functions
Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationTheory of! Partial Differential Equations-I!
hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationStructural Dynamics and Earthquake Engineering
Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp://www.c.up.ro/users/aurelsraan/
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationPhysics for Scientists and Engineers I
Physics for Scieniss and Engineers I PHY 48, Secion 4 Dr. Beariz Roldán Cuenya Universiy of Cenral Florida, Physics Deparmen, Orlando, FL Chaper - Inroducion I. General II. Inernaional Sysem of Unis III.
More informationMath 23 Spring Differential Equations. Final Exam Due Date: Tuesday, June 6, 5pm
Mah Spring 6 Differenial Equaions Final Exam Due Dae: Tuesday, June 6, 5pm Your name (please prin): Insrucions: This is an open book, open noes exam. You are free o use a calculaor or compuer o check your
More informationSPH3U: Projectiles. Recorder: Manager: Speaker:
SPH3U: Projeciles Now i s ime o use our new skills o analyze he moion of a golf ball ha was ossed hrough he air. Le s find ou wha is special abou he moion of a projecile. Recorder: Manager: Speaker: 0
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More informationAP CALCULUS AB 2017 SCORING GUIDELINES
AP CALCULUS AB 17 SCORING GUIDELINES 16 SCORING GUIDELINES Quesion For, a paricle moves along he x-axis. The velociy of he paricle a ime is given by v ( ) = 1 + sin. The paricle is a posiion x = a ime.
More informationThe motions of the celt on a horizontal plane with viscous friction
The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationPHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections
PHYSICS 220 Lecure 02 Moion, Forces, and Newon s Laws Texbook Secions 2.2-2.4 Lecure 2 Purdue Universiy, Physics 220 1 Overview Las Lecure Unis Scienific Noaion Significan Figures Moion Displacemen: Δx
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationMOMENTUM CONSERVATION LAW
1 AAST/AEDT AP PHYSICS B: Impulse and Momenum Le us run an experimen: The ball is moving wih a velociy of V o and a force of F is applied on i for he ime inerval of. As he resul he ball s velociy changes
More informationWall. x(t) f(t) x(t = 0) = x 0, t=0. which describes the motion of the mass in absence of any external forcing.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES 7 Mechanics applicaions of second-order ODEs Second-order linear ODEs wih consan coefficiens arise in many physical applicaions. One physical sysems whose behaviour
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationThe Contradiction within Equations of Motion with Constant Acceleration
The Conradicion wihin Equaions of Moion wih Consan Acceleraion Louai Hassan Elzein Basheir (Daed: July 7, 0 This paper is prepared o demonsrae he violaion of rules of mahemaics in he algebraic derivaion
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationInterpretation of special relativity as applied to earth-centered locally inertial
Inerpreaion of special relaiviy as applied o earh-cenered locally inerial coordinae sysems in lobal osiioning Sysem saellie experimens Masanori Sao Honda Elecronics Co., Ld., Oyamazuka, Oiwa-cho, Toyohashi,
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationPHYSICS 149: Lecture 9
PHYSICS 149: Lecure 9 Chaper 3 3.2 Velociy and Acceleraion 3.3 Newon s Second Law of Moion 3.4 Applying Newon s Second Law 3.5 Relaive Velociy Lecure 9 Purdue Universiy, Physics 149 1 Velociy (m/s) The
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationCh.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationGiambattista, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76
Giambaisa, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76 9. Sraeg Le be direced along he +x-axis and le be 60.0 CCW from Find he magniude of 6.0 B 60.0 4.0 A x 15. (a) Sraeg Since he angle
More informationE β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.
Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke
More informationPhysics Notes - Ch. 2 Motion in One Dimension
Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationRoller-Coaster Coordinate System
Winer 200 MECH 220: Mechanics 2 Roller-Coaser Coordinae Sysem Imagine you are riding on a roller-coaer in which he rack goes up and down, wiss and urns. Your velociy and acceleraion will change (quie abruply),
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationk 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series
Final Review A Puzzle... Consider wo massless springs wih spring consans k 1 and k and he same equilibrium lengh. 1. If hese springs ac on a mass m in parallel, hey would be equivalen o a single spring
More information02. MOTION. Questions and Answers
CLASS-09 02. MOTION Quesions and Answers PHYSICAL SCIENCE 1. Se moves a a consan speed in a consan direcion.. Reprase e same senence in fewer words using conceps relaed o moion. Se moves wi uniform velociy.
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationDynamics. Option topic: Dynamics
Dynamics 11 syllabusref Opion opic: Dynamics eferenceence In his cha chaper 11A Differeniaion and displacemen, velociy and acceleraion 11B Inerpreing graphs 11C Algebraic links beween displacemen, velociy
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationCircuit Variables. AP 1.1 Use a product of ratios to convert two-thirds the speed of light from meters per second to miles per second: 1 ft 12 in
Circui Variables 1 Assessmen Problems AP 1.1 Use a produc of raios o conver wo-hirds he speed of ligh from meers per second o miles per second: ( ) 2 3 1 8 m 3 1 s 1 cm 1 m 1 in 2.54 cm 1 f 12 in 1 mile
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More information