Dynic Physics for Siulion n Ge Progring F Discree Dynics. Force equls ss ies ccelerion (F=) v v v v Mike Biley This work is license uner Creive Coons Aribuion-NonCoercil- NoDerivives.0 Inernionl License jb@cs.oregonse.eu x v x v x x v Think Geek These cn be hough of s suing res uner curves physics-ynic.ppx Inegring he Physics Equions if Accelerion is Consn Wh if Accelerion is no Consn? v re v re
Wh is v(+δ)? 5 Wh is v(+δ)? 6 Accuule v so fr????? v Accelerion he sr of he ie sep The velociy is off by his uch! Accelerion he sr of he ie sep v v ( ) v () vv () Tie Sep Tie Sep v ( ) v ( ) vv ( ) This is close, bu is clerly no excly righ! The proble is h we re reing ll of he quniies s if hey lwys hve he vlue h hey h he sr of he Tie Sep, even hough hey on. This is known s Firs Orer soluion. Wh oes Firs Orer Soluion look like in Progr? 7 Wh oes Firs Orer Soluion look like in Progr? 8 You nee wy o hol he enire se of he syse. This will be he inpu o our nuericl inegror. You lso nee wy o reurn he erivives once you eerine he. sruc se flo ie; flo x; flo vx; ; x x v vx x sruc erivives flo vx; flo x; ; AvnceOneTieSep( ) GeDerivs( Se, Derivives ); // ge erivives Se.x = Se.x + Derivives.vx * Δ; // use erivives Se.vx = Se.vx + Derivives.x * Δ; // use erivives Se. = Se. + Δ ; sruc se Se; GeDerivs( Se, Derivives )... sruc erivives Derivives; The oupus re he erivives of he se vribles. The inpus re he se, which consiss of ll vribles necessry o copleely escribe he se of he physicl syse. The oupus re he erivives of ech se vrible. The pplicion, hen, consiss of: Iniilize( ); AvnceOneTieSep( ); Finish( );
Wh is v(+δ)? A Secon Orer soluion is obine by oing he Firs Orer soluion, eerining ll quniies ie hen verging he wih he quniies ie n hen reing he s consn hroughou he inervl. F() F( ) () ( ) v () () ( ) v vg. v ( ) v ( ) v 9 Wh oes Secon Orer Soluion look like in Progr? AvnceOneTieSep( ) GeDerivs( Se, Derivives); Se. = Se. + Δ; Se.x = Se.x + Derivives.vx * Δ; Se.vx = Se.vx + Derivives.x * Δ; GeDerivs( Se, Derivives ); flo vg = ( Derivives.x + Derivives.x) /.; flo vvg = ( Derivives.vx + Derivives.vx) /.; Se.x = Se.x + vvg * Δ; Se.vx = Se.vx + vg * Δ; Se. = Se. + Δ ; The pplicion, hen, consiss of: Iniilize( ); AvnceOneTieSep( ); Finish( ); 0 The Runge-Ku Fourh Orer Soluion v GeDerivs(, x, v) v GeDerivs(, x v, v ) v GeDerivs(, x v, v ) v GeDerivs(, x v, v ) x ( ) x v v v v ( ) v ( ) v 6 Ape fro: hp://en.wikipei.org/wiki/runge-ku The Runge-Ku Fourh Orer Soluion AvnceOneTieSep( ) GeDerivs( Se, Derivives ); Se. = Se. + Δ/.; Se.x = Se.x + Derivives.vx * (Δ/.); Se.vx = Se.vx + Derivives.x * (Δ/.); GeDerivs( Se, Derivives ); Se. = Se. + Δ/.; Se.x = Se.x + Derivives.vx * (Δ/); Se.vx = Se.vx + Derivives.x * (Δ/.); GeDerivs( Se, Derivives ); Se. = Se. + Δ; Se.x = Se.x + Derivives.vx * Δ; Se.vx = Se.vx + Derivives.x * Δ; GeDerivs( Se, Derivives ); Se.x = Se.x + (Δ/6.) * ( Derivives.vx +.*Derivives.vx +.*Derivives.vx + Derivives.vx ); Se.vx = Se.vx + (Δ/6.) * (Derivives.x +.*Derivives.x +.*Derivives.x + Derivives.x ); Ape fro: hp://en.wikipei.org/wiki/runge-ku
Solving Moion where here is Spring Air Resisnce Force Fspring ky This is known s Hooke s lw F v AC rg y Drg Coefficien +y, +F F W ky v GeDerivs( Se, Derivives ) Derivives.vy = Se.vy; Derivives.y = ( W K*Se.y ) / MASS; Mechnicl Scheic Sybol: Boy Flling Flui ensiy +y Y Velociy Cross-secionl re Air Resisnce lwys cs in irecion opposie o he velociy of he objec Weigh W Soe Drg Coefficiens 5 Solving Moion where here is Air Resisnce 6 C Ie. sooh brick 0.9 ypicl bicycle plus cyclis 0. rough sphere 0. sooh sphere 0.00 linr fl ple 0.005 urbulen fl ple 0.95 bulle.0-. person (uprigh posiion).8 fl ple perpeniculr o flow.0-. skier.0-. wires n cbles.-.5 Epire Se Builing.8-.0 Eiffel Tower F W Sign( vy) vyac v kg ir.9 The Sign( ) funcion reurns +. or -., epening on he sign of rguen GeDerivs( Se, Derivives ) Derivives.vy = Se.vy; Derivives.y = ( W.5*Sign(Se.vy)*DENSITY* Se.vy * Se.vy *AREA*DRAG ) / MASS; hp://en.wikipei.org/wiki/drg_coefficien
Terinl Velociy 7 Hun Terinl Velociy 8 W vyac v The velociy becoes consn when Δv = 0 : When boy is in free fll, i is being ccelere by he force of grviy. However, s i cceleres, i is encounering ore n ore ir resisnce force. A soe velociy, hese wo forces blnce ech oher ou n he velociy becoes consn, h is, Δv=0. This is known s he erinl velociy. Assue: Weigh = 00 pouns = 890 Newons C =.8 A = 6 f = 0.558 kg ir.9 W vyac v W AC 0 W v.90 98ph AC sec How bou Cliff Juper on Bungee Cor? 9 Coulob Dping 0 Boy Flling Fspring ky Frg Sign( vy ) vyc A F W kysign( vy) vyac v This is very uch like rg force, bu i is he resisnce of flui being squeeze hrough sll opening. The resising force is proporionl o he velociy: F cv Velociy Dping Coefficien W GeDerivs( Se, Derivives ) Derivives.vy = Se.vy; Derivives.y = ( W K*Se.y -.5*Sign(Se.vy)*DENSITY*Se.vy*Se.vy*AREA*DRAG ) / MASS; Hole Oil-fille cyliner Mechnicl Scheic Sybol: 5
Lif Anoher Goo Force o Know Abou Coefficien of Lif vs. Angle of Ack F lif v ACL Coefficien of Lif, for given ngle of ck Air ensiy Airspee Plfor re Angle of Ack hp://en.wikipei.org/wiki/lif_%8force%9 hp://en.wikipei.org/wiki/lif_coefficien Lif n Drg Driclly Working Togeher Fligh of Frisbee Fricion Force Anoher Goo Force o Know Abou Ffricion N Norl force (i.e., oun of force h is perpeniculr o he surfce) Coefficien of Fricion N N W 6
Soe Coefficiens of Fricion Merils Dry & clen Lubrice Aluinu Seel 0.6 Copper Seel 0.5 Brss Seel 0.5 Cs iron Copper.05 Cs iron Zinc 0.85 Concree (we) Rubber 0.0 Concree (ry) Rubber.0 Concree Woo 0.6 Copper Glss 0.68 Glss Glss 0.9 Mel Woo 0. 0.6 0. (we) Polyhene Seel 0. 0. Seel Seel 0.80 0.6 Seel Teflon 0.0 0.0 Teflon Teflon 0.0 0.0 Woo Woo 0.5 0.5 0. (we) μ hp://en.wikipei.org/wiki/fricion 5 Buoyncy Anoher Goo Force o Know Abou Archiees Principle sys h he buoyncy force on n objec in flui is he weigh of he flui h is being isplce by he objec. ir.66x0 pouns / in heliu 0.65x0 pouns / in So, for heliu blloon h is one foo in ieer (i.e., rius=6 inches), i hs is weigh pulling i own n buoyncy force pushing i up. The ne force pushing i up becuse of he gs insie he blloon is: Fbuoyncy Vblloonheliu Vblloonir Vblloon( heliu ir ) Vblloon r 90.78in 5 5 Densiies 5 Fbuoyncy 90.78 in (.0x0 pouns / in ) 0.06 pouns Noe h his us sill counerblnce he weigh of he blloon eril, or he blloon will no fly. 6 T I T I T I Spinning Moion: Dynics Moen of Ineri n ngulr ss (newon-eers-sec =kg-eers ) Torque n ngulr force (newon-eers) 7 Spinning Moion: Wh oes his look like in Progr? sruc se flo ; flo x; flo vx; flo he; flo oeg; ; sruc erivives flo vx; flo x; flo oeg; flo lph; ; The se n erivive vecors now inclue ngulr coponens GeDerivs( Se, Derivives ) Derivives.vx = Se.vx; Derivives.x = SoeOfAllForces / MASS; Derivives.oeg = Se.oeg; Derivives.lph = SoeOfAllTorques / INERTIA 8 7