Lecture : Kneatcs of Partcles Rectlnear oton Straght-Lne oton [.1] Analtcal solutons for poston/veloct [.1] Solvng equatons of oton Analtcal solutons (1 D revew) [.1] Nuercal solutons [.1] Nuercal ntegraton [Append A,.1] Poston vectors [.] Cartesan coordnate sste [.] Veloct/Acceleraton vectors [.] State vector, state space: Etensons to and 3 densons 1-densonal oton Poston, (t) Veloct, v(t) Acceleraton, a(t) Jer, (t) Snap, s(t) Two tpes of probles Gven forces, fnd oton Gven oton, fnd forces Eternal oton s nown, fnd force Eternal forces are nown, fnd oton Consder partcle wth oton gven b 3 = 6t t d v = = 1t 3t d a = = = 1 6t F = a Free bod dagra Inerta response dagra F Free bod dagra = Specal cases 1. Acceleraton s a gven functon of te, a(t) = f(t) a(t) = f(t) = constant. Acceleraton s a gven functon of poston, a() = f() 3. Acceleraton s a gven functon of veloct, a(v) = f(v) a Inerta response dagra F = a =1 6t
Specal case 3 a(v) = f(v) Eaple: Vscous Dapng [BJ, 11.3] (t) Brae echans used to reduce gun recol conssts of pston attached to barrel ovng n fed clnder flled wth ol. As barrel recols wth ntal veloct v, pston oves and ol s forced through orfces n pston, causng pston and clnder to decelerate at rate proportonal to ther veloct. Deterne v(t), (t), and v(). Integrate a = / = -v to fnd v(t). Soluton v( t ) t v( t) a = = v = ln = t v v v t v( t) v e = Integrate v(t) = d/ to fnd (t). d t v() t = = ve () t t t t 1 t d = v e () t = v e v t () t = ( 1 e ) Soluton (contnued) Integrate a = v /d = -v to fnd v(). v a = v = v = d = d d v v v = v = v Specal case : force =F(), a() = f() Man passve sstes Sple pendulu Sprng-ass sste Control sstes for postonng Gudance sstes for ssles Car Iagne a car beng accelerated (or decelerated) toward an ntersecton
F() =, Increasng What a(t) s an arbtrar functon of t? Eaple.6 [TS] 1.8 1.6 Kp=4 1.4 Kp= Apltude 1. 1.8.6.4. Kp=1 1 3 4 5 Te (sec.) Eaple: Suppose veloct s nown functon of te Gven v(t), ntal condton (t ) Nuercal Integraton Fnd (t) See Append A Need to solve d = v () t, ( t = ) o t t t Basc dea: Approate d t t
Algorth In steps of δt seconds, = + v( t ) δt Nuercal Integraton of ODEs d = f(, ) d Intal value proble: Gven the ntal state at =( ), to copute the whole traector () ' = 1 -, () = o t t o t t Eplct Euler or Bacward Euler (Append A) +1 = + h f (, ) Euler s ethod Truncaton errors Eplct: evaluate dervatve usng values at the begnnng of the te step Not ver accurate, requres sall te steps for stablt Global accurac O(h) +1 = + h f (, ) + O(h ) Local truncaton error Global truncaton error Iplct: Evaluate dervatve usng values at the end of the te step 1 = + h f (, + 1) O( h ) + + + 1 +1 +1 Ma requre teraton snce the answer depends upon what s calculated at the end. Stll not ver accurate (global accurac O(h)). Uncondtonall stable for all te step szes. o +1 o +1 +
Stablt A nuercal ethod s stable f errors occurrng at one stage of the process do not tend to be agnfed at later stages. A nuercal ethod s unstable f errors occurrng at one stage of the process tend to be agnfed at later stages. In general, the stablt of a nuercal schee depends on the step sze. Usuall, large step szes lead to unstable solutons. Iplct ethods are n general ore stable than eplct ethods. -Densonal oton Need ore powerful representaton Postons requre two coordnates (Cartesan coordnates) (t) (t) Need poston vectors r(t) = (t) + (t) Need a reference frae Fed reference fraes Frae fed to, orthogonal, unt vectors Classroo, Earth, Center of the Earth, Center of the Sun, Center of the Unverse, Movng reference fraes Fraes fed to ovng bodes Need to be able to dfferentate vectors v(t), a(t), (t), s(t), Dfferentaton of Vectors Transforatons between unt vectors B e 3 e e 1 u = u + u + u z Dfferentate wth respect to te: Understandng the relatonshp between sets of unt vectors s ver portant Vsualze Wrte down the dot products z e 1 e e 3 O u = u 1 e 1 + u e + u 3 e 3 Dfferentate wth respect to te:.e 1.e.e 3.e 1.e.e 3.e 1.e.e 3 Dot product of unt vectors = Cosne of angle between vectors