Bewegungsgleichung für Loopingfahrten

contravariant · **Registriert:** Sa 14. Feb 2009, 20:16 **Beiträge:** 3029

Chief hat geschrieben:

Die Berechnung mit Fortran kann Contravariant übernehmen.

Kein Problem. Sagen wir 8 Stunden Arbeit, Stundensatz 150€, macht 1200€. Kannst dich melden, wenn ihr die Kohle beisammen habt.

contravariant · **Registriert:** Sa 14. Feb 2009, 20:16 **Beiträge:** 3029

Eigentlich wäre das hier doch auf eine super Gelegenheit für Highway. Hier kann er den ganzen Fortran Low-Tech Leuten mal zeigen, wie man sowas so richtig High-Tech IT-mäßig löst. Na los, keine falsche Scheu!

Trigemina · **Registriert:** So 11. Jan 2009, 14:30 **Beiträge:** 954

Ich habe meine Aufgabenstellung mit dem 20m hohen Looping als Funktion der Zeit numerisch über die Teilbogenlängen berechnet, was genauer ist als über ein n-Vieleck. Beginnend am untersten Punkt bei [x=0,y=-10] für eine volle Umrundung. In der ersten Spalte befindet sich die Zeit t, in der 2. Spalte die Position in rad, in der 3. Spalte die aktuelle Geschwindigkeit v, in der 4. Spalte die x-Koordinate und in der 5. Spalte die y-Koordinate.

Das Zeitintervall dt beträgt 0.01s, die dafür benötigte Umlaufzeit 4.07s.

Falls die Darstellung als Zeilenvektoren in einer anderen Form benötigt wird, kann ich's hoffentlich wunschgerecht formatieren.

Gruss

# Maple-Interpreter

restart;

Looping:=proc()

r:=10;g:=9.81;m:=1;

phi:=evalf(-1/2*Pi);
t:=0; x:=0; y:=-10;dt:=0.01; U:=evalf(2*r*Pi);
v_o:=sqrt(g*r); E:=m*g*2*r+m/2*v_o^2; v_u:=sqrt(2*E/m);

from 1 to 410 do
x:=r*cos(phi);
y:=r*sin(phi);
v:=sqrt(2/m*(E-m*g*(r+y)));
dphi:=evalf(v*dt*2*Pi/U);
phi:=phi+dphi;
print([t,phi,v,x,y]);
t:=t+dt;
od;
end;

Looping();

Code:

[t,phi,v,x,y]

    [0, -1.548649092, 22.14723459, 0, -10.]
[.01, -1.526502944, 22.14614829, .2214542430, -9.997547600]
[.02, -1.504360054, 22.14288980, .4427890099, -9.990192085]
[.03, -1.482222593, 22.13746073, .6638741108, -9.977939224]
[.04, -1.460092729, 22.12986395, .8845796454, -9.960799107]
[.05, -1.437972625, 22.12010361, 1.104776189, -9.938786122]
[.06, -1.415864440, 22.10818512, 1.324334972, -9.911918930]
[.07, -1.393770325, 22.09411517, 1.543128023, -9.880220438]
[.08, -1.371692423, 22.07790167, 1.761028366, -9.843717748]
[.09, -1.349632869, 22.05955381, 1.977910172, -9.802442112]
[.10, -1.327593787, 22.03908198, 2.193648906, -9.756428879]
[.11, -1.305577289, 22.01649781, 2.408121507, -9.705717429]
[.12, -1.283585475, 21.99181413, 2.621206536, -9.650351097]
[.13, -1.261620430, 21.96504493, 2.832784321, -9.590377104]
[.14, -1.239684225, 21.93620541, 3.042737118, -9.525846463]
[.15, -1.217778913, 21.90531188, 3.250949232, -9.456813898]
[.16, -1.195906531, 21.87238182, 3.457307183, -9.383337735]
[.17, -1.174069097, 21.83743377, 3.661699810, -9.305479810]
[.18, -1.152268610, 21.80048741, 3.864018412, -9.223305357]
[.19, -1.130507047, 21.76156342, 4.064156861, -9.136882893]
[.20, -1.108786363, 21.72068355, 4.262011736, -9.046284097]
[.21, -1.087108492, 21.67787056, 4.457482421, -8.951583685]
[.22, -1.065475344, 21.63314816, 4.650471187, -8.852859297]
[.23, -1.043888803, 21.58654105, 4.840883313, -8.750191355]
[.24, -1.022350728, 21.53807481, 5.028627178, -8.643662922]
[.25, -1.000862952, 21.48777594, 5.213614329, -8.533359575]
[.26, -.9794272802, 21.43567179, 5.395759557, -8.419369264]
[.27, -.9580454897, 21.38179052, 5.574980977, -8.301782165]
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[.44, -.6043700896, 20.22452904, 8.112002404, -5.847684755]
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[.49, -.5044741818, 19.81203632, 8.656814779, -5.005952245]
[.50, -.4847477346, 19.72644719, 8.754287482, -4.833471908]
[.51, -.4651078189, 19.63991575, 8.847925292, -4.659851717]
[.52, -.4455553392, 19.55247967, 8.937732120, -4.485191697]
[.53, -.4260911626, 19.46417657, 9.023714761, -4.309590690]
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[.75, -.2150949888e-1, 17.37605893, 9.992440520, -.3887575883]
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[.77, .1294758373e-1, 17.17935596, 9.999910461, -.4231759600e-1]
[.78, .3002856014e-1, 17.08097641, 9.999161812, .1294722198]
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[.81, .8068155626e-1, 16.78607263, 9.979593780, .6385201554]
[.82, .9736949942e-1, 16.68794316, 9.967470084, .8059405180]
[.83, .1139594445, 16.58994505, 9.952633344, .9721571523]
[.84, .1304515493, 16.49210479, 9.935136468, 1.137129441]
[.85, .1468459977, 16.39444837, 9.915032564, 1.300818685]
[.86, .1631429990, 16.29700125, 9.892374874, 1.463188081]
[.87, .1793427874, 16.19978837, 9.867216711, 1.624202690]
[.88, .1954456215, 16.10283414, 9.839611408, 1.783829403]
[.89, .2114517839, 16.00616243, 9.809612256, 1.942036916]
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[.93, .2745161740, 15.62276037, 9.666738682, 2.560110007]
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[.98, .3512147896, 15.15251441, 9.440605335, 3.297722078]
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[1.19, .6489210768, 13.33514306, 8.047239956, 5.936491311]
[1.20, .6621774343, 13.25635754, 7.967362847, 6.043271404]
[1.21, .6753557978, 13.17836350, 7.886553381, 6.148355533]
[1.22, .6884569678, 13.10116999, 7.804845646, 6.251750510]
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[2.17, 1.715519514, 9.993899009, -1.343220574, 9.909377301]
[2.18, 1.725527066, 10.00755221, -1.442185163, 9.895458653]
[2.19, 1.735549241, 10.02217485, -1.541140609, 9.880530635]
[2.20, 1.745587008, 10.03776677, -1.640085960, 9.864589097]
[2.21, 1.755641336, 10.05432777, -1.739020120, 9.847629614]
[2.22, 1.765713194, 10.07185763, -1.837941852, 9.829647488]
[2.23, 1.775803550, 10.09035616, -1.936849771, 9.810637745]
[2.24, 1.785913373, 10.10982311, -2.035742319, 9.790595141]
[2.25, 1.796043631, 10.13025825, -2.134617782, 9.769514160]
[2.26, 1.806195292, 10.15166131, -2.233474260, 9.747389021]
[2.27, 1.816369324, 10.17403202, -2.332309664, 9.724213677]
[2.28, 1.826566694, 10.19737008, -2.431121709, 9.699981816]
[2.29, 1.836788369, 10.22167519, -2.529907898, 9.674686870]
[2.30, 1.847035316, 10.24694697, -2.628665516, 9.648322010]
[2.31, 1.857308501, 10.27318507, -2.727391627, 9.620880153]
[2.32, 1.867608890, 10.30038908, -2.826083049, 9.592353966]
[2.33, 1.877937449, 10.32855857, -2.924736359, 9.562735865]
[2.34, 1.888295142, 10.35769310, -3.023347882, 9.532018023]
[2.35, 1.898682934, 10.38779215, -3.121913659, 9.500192372]
[2.36, 1.909101789, 10.41885517, -3.220429472, 9.467250605]
[2.37, 1.919552671, 10.45088161, -3.318890807, 9.433184182]
[2.38, 1.930036542, 10.48387082, -3.417292863, 9.397984331]
[2.39, 1.940554364, 10.51782215, -3.515630515, 9.361642061]
[2.40, 1.951107099, 10.55273487, -3.613898329, 9.324148158]
[2.41, 1.961695707, 10.58860820, -3.712090547, 9.285493190]
[2.42, 1.972321148, 10.62544132, -3.810201061, 9.245667519]
[2.43, 1.982984381, 10.66323334, -3.908223424, 9.204661301]
[2.44, 1.993686364, 10.70198331, -4.006150823, 9.162464493]
[2.45, 2.004428054, 10.74169019, -4.103976075, 9.119066859]
[2.46, 2.015210407, 10.78235292, -4.201691617, 9.074457976]
[2.47, 2.026034377, 10.82397033, -4.299289490, 9.028627242]
[2.48, 2.036900918, 10.86654114, -4.396761326, 8.981563886]
[2.49, 2.047810982, 10.91006408, -4.494098351, 8.933256965]
[2.50, 2.058765520, 10.95453772, -4.591291360, 8.883695382]
[2.51, 2.069765481, 10.99996055, -4.688330714, 8.832867888]
[2.52, 2.080811812, 11.04633098, -4.785206317, 8.780763093]
[2.53, 2.091905459, 11.09364732, -4.881907615, 8.727369480]
[2.54, 2.103047367, 11.14190776, -4.978423582, 8.672675402]
[2.55, 2.114238477, 11.19111041, -5.074742722, 8.616669096]
[2.56, 2.125479730, 11.24125326, -5.170853021, 8.559338703]
[2.57, 2.136772064, 11.29233413, -5.266741980, 8.500672262]
[2.58, 2.148116415, 11.34435081, -5.362396575, 8.440657733]
[2.59, 2.159513716, 11.39730089, -5.457803254, 8.379283003]
[2.60, 2.170964898, 11.45118184, -5.552947921, 8.316535901]
[2.61, 2.182470889, 11.50599102, -5.647815932, 8.252404207]
[2.62, 2.194032615, 11.56172562, -5.742392078, 8.186875669]
[2.63, 2.205650998, 11.61838268, -5.836660584, 8.119938006]
[2.64, 2.217326957, 11.67595911, -5.930605080, 8.051578937]
[2.65, 2.229061409, 11.73445163, -6.024208600, 7.981786187]
[2.66, 2.240855266, 11.79385680, -6.117453584, 7.910547494]
[2.67, 2.252709437, 11.85417101, -6.210321838, 7.837850641]
[2.68, 2.264624827, 11.91539049, -6.302794547, 7.763683462]
[2.69, 2.276602338, 11.97751124, -6.394852255, 7.688033860]
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[2.72, 2.312916546, 12.16923863, -6.668331059, 7.452070913]
[2.73, 2.325151467, 12.23492089, -6.758521101, 7.370372618]
[2.74, 2.337452949, 12.30148159, -6.848188932, 7.287133068]
[2.75, 2.349821864, 12.36891549, -6.937311058, 7.202340959]
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[2.78, 2.387341863, 12.57640075, -7.201157561, 6.938539455]
[2.79, 2.399989134, 12.64727058, -7.287847635, 6.847428485]
[2.80, 2.412708118, 12.71898392, -7.373863759, 6.754711930]
[2.81, 2.425499652, 12.79153405, -7.459178079, 6.660380048]
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[2.84, 2.464317817, 13.01413399, -7.710620423, 6.367600231]
[2.85, 2.477407776, 13.08995862, -7.792833930, 6.266716791]
[2.86, 2.490574358, 13.16658223, -7.874195024, 6.164174943]
[2.87, 2.503818354, 13.24399635, -7.954671273, 6.059967405]
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[2.90, 2.544022599, 13.48089234, -8.190455122, 5.737285499]
[2.91, 2.557583976, 13.56137738, -8.267052273, 5.626352879]
[2.92, 2.571226582, 13.64260566, -8.342590838, 5.513726337]
[2.93, 2.584951149, 13.72456670, -8.417033748, 5.399402086]
[2.94, 2.598758399, 13.80724962, -8.490343158, 5.283377051]
[2.95, 2.612649042, 13.89064322, -8.562480460, 5.165648881]
[2.96, 2.626623778, 13.97473585, -8.633406286, 5.046215999]
[2.97, 2.640683294, 14.05951554, -8.703080517, 4.925077614]
[2.98, 2.654828264, 14.14496989, -8.771462289, 4.802233784]
[2.99, 2.669059350, 14.23108610, -8.838509993, 4.677685442]
[3.00, 2.683377201, 14.31785097, -8.904181301, 4.551434429]
[3.01, 2.697782452, 14.40525089, -8.968433168, 4.423483527]
[3.02, 2.712275724, 14.49327180, -9.031221845, 4.293836511]
[3.03, 2.726857623, 14.58189925, -9.092502897, 4.162498176]
[3.04, 2.741528741, 14.67111831, -9.152231214, 4.029474384]
[3.05, 2.756289655, 14.76091364, -9.210361034, 3.894772089]
[3.06, 2.771140924, 14.85126945, -9.266845962, 3.758399381]
[3.07, 2.786083093, 14.94216945, -9.321638981, 3.620365549]
[3.08, 2.801116690, 15.03359695, -9.374692487, 3.480681078]
[3.09, 2.816242225, 15.12553476, -9.425958310, 3.339357712]
[3.10, 2.831460190, 15.21796521, -9.475387733, 3.196408502]
[3.11, 2.846771060, 15.31087018, -9.522931524, 3.051847833]
[3.12, 2.862175291, 15.40423103, -9.568539969, 2.905691459]
[3.13, 2.877673320, 15.49802866, -9.612162903, 2.757956548]
[3.14, 2.893265563, 15.59224349, -9.653749738, 2.608661724]
[3.15, 2.908952418, 15.68685540, -9.693249501, 2.457827111]
[3.16, 2.924734262, 15.78184378, -9.730610877, 2.305474346]
[3.17, 2.940611450, 15.87718758, -9.765782243, 2.151626636]
[3.18, 2.956584315, 15.97286516, -9.798711710, 1.996308798]
[3.19, 2.972653169, 16.06885441, -9.829347167, 1.839547298]
[3.20, 2.988818302, 16.16513271, -9.857636331, 1.681370263]
[3.21, 3.005079979, 16.26167692, -9.883526793, 1.521807525]
[3.22, 3.021438442, 16.35846340, -9.906966063, 1.360890677]
[3.23, 3.037893910, 16.45546798, -9.927901630, 1.198653085]
[3.24, 3.054446576, 16.55266598, -9.946281017, 1.035129906]
[3.25, 3.071096608, 16.65003222, -9.962051831, .8703581521]
[3.26, 3.087844149, 16.74754100, -9.975161827, .7043766949]
[3.27, 3.104689315, 16.84516608, -9.985558968, .5372262930]
[3.28, 3.121632196, 16.94288075, -9.993191491, .3689496299]
[3.29, 3.138672854, 17.04065780, -9.998007967, .1995913218]
[3.30, 3.155811323, 17.13846948, -9.999957374, .2919795441e-1]
[3.31, 3.173047611, 17.23628756, -9.998989164, -.1421819032]
[3.32, 3.190381694, 17.33408333, -9.995053336, -.3144977067]
[3.33, 3.207813522, 17.43182757, -9.988100508, -.4876968672]
[3.34, 3.225343013, 17.52949060, -9.978081994, -.6617248036]
[3.35, 3.242970055, 17.62704224, -9.964949881, -.8365248786]
[3.36, 3.260694507, 17.72445186, -9.948657108, -1.012038417]
[3.37, 3.278516195, 17.82168837, -9.929157545, -1.188204714]
[3.38, 3.296434915, 17.91872024, -9.906406082, -1.364961006]
[3.39, 3.314450430, 18.01551547, -9.880358702, -1.542242498]
[3.40, 3.332562472, 18.11204168, -9.850972577, -1.719982353]
[3.41, 3.350770738, 18.20826603, -9.818206145, -1.898111718]
[3.42, 3.369074893, 18.30415530, -9.782019208, -2.076559707]
[3.43, 3.387474569, 18.39967588, -9.742373011, -2.255253447]
[3.44, 3.405969363, 18.49479378, -9.699230336, -2.434118094]
[3.45, 3.424558838, 18.58947465, -9.652555590, -2.613076842]
[3.46, 3.443242522, 18.68368379, -9.602314899, -2.792050963]
[3.47, 3.462019908, 18.77738619, -9.548476196, -2.970959832]
[3.48, 3.480890455, 18.87054651, -9.491009315, -3.149720969]
[3.49, 3.499853584, 18.96312914, -9.429886072, -3.328250090]
[3.50, 3.518908682, 19.05509819, -9.365080375, -3.506461117]
[3.51, 3.538055100, 19.14641753, -9.296568295, -3.684266269]
[3.52, 3.557292151, 19.23705079, -9.224328162, -3.861576098]
[3.53, 3.576619112, 19.32696140, -9.148340663, -4.038299532]
[3.54, 3.596035225, 19.41611260, -9.068588921, -4.214343956]
[3.55, 3.615539692, 19.50446749, -8.985058575, -4.389615292]
[3.56, 3.635131681, 19.59198902, -8.897737884, -4.564018027]
[3.57, 3.654810321, 19.67864004, -8.806617787, -4.737455346]
[3.58, 3.674574704, 19.76438332, -8.711692004, -4.909829165]
[3.59, 3.694423886, 19.84918158, -8.612957105, -5.081040238]
[3.60, 3.714356884, 19.93299751, -8.510412582, -5.250988259]
[3.61, 3.734372678, 20.01579379, -8.404060933, -5.419571924]
[3.62, 3.754470211, 20.09753316, -8.293907729, -5.586689054]
[3.63, 3.774648389, 20.17817841, -8.179961676, -5.752236693]
[3.64, 3.794906081, 20.25769242, -8.062234681, -5.916111218]
[3.65, 3.815242119, 20.33603820, -7.940741910, -6.078208447]
[3.66, 3.835655298, 20.41317893, -7.815501843, -6.238423754]
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[3.68, 3.876708075, 20.56369889, -7.553870639, -6.552788595]
[3.69, 3.897345081, 20.63700556, -7.417533483, -6.706727743]
[3.70, 3.918054043, 20.70896209, -7.277557071, -6.858364461]
[3.71, 3.938833576, 20.77953294, -7.133977139, -7.007593751]
[3.72, 3.959682259, 20.84868295, -6.986832961, -7.154310950]
[3.73, 3.980598636, 20.91637732, -6.836167386, -7.298411846]
[3.74, 4.001581218, 20.98258171, -6.682026843, -7.439792824]
[3.75, 4.022628480, 21.04726222, -6.524461345, -7.578351018]
[3.76, 4.043738865, 21.11038547, -6.363524508, -7.713984433]
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[3.84, 4.214696955, 21.55549657, -4.962269827, -8.681928251]
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[3.90, 4.344965913, 21.81339174, -3.794820987, -9.251990795]
[3.91, 4.366815509, 21.84959607, -3.592116892, -9.332561076]
[3.92, 4.388699314, 21.88380500, -3.387363014, -9.408813518]
[3.93, 4.410615313, 21.91599880, -3.180667735, -9.480683138]
[3.94, 4.432561472, 21.94615878, -2.972141901, -9.548108322]
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[3.97, 4.498560313, 22.02426569, -2.336723907, -9.723153881]
[3.98, 4.520606439, 22.04612591, -2.122029179, -9.772256247]
[3.99, 4.542672315, 22.06587579, -1.906090572, -9.816660264]
[4.00, 4.564755819, 22.08350354, -1.689030920, -9.856326626]
[4.01, 4.586854818, 22.09899856, -1.470974545, -9.891220040]
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[4.03, 4.631090723, 22.12355420, -1.032375421, -9.946567297]
[4.04, 4.653223323, 22.13259981, -.8120873143, -9.966971164]
[4.05, 4.675362806, 22.13948269, -.5913114446, -9.982502230]
[4.06, 4.697507004, 22.14419848, -.3701771487, -9.993146095]
[4.07, 4.719653748, 22.14674409, -.1488142706, -9.998892654]
[4.08, 4.741800866, 22.14711770, .7264703714e-1, -9.999736117]
[4.09, 4.763946185, 22.14531878, .2940764530, -9.995675017]

Trigemina · **Registriert:** So 11. Jan 2009, 14:30 **Beiträge:** 954

OK, dann formatiere ich das jetzt um. x und y alternierend in Hundertstelsekunden Schritten.

Gruss

Code:

Trigemina · **Registriert:** So 11. Jan 2009, 14:30 **Beiträge:** 954

Danke Chief für Deine animierte Darstellung.

Hallo Solkar
Das Gravitationspotential kann man getrost weglassen und mit einem konstanten g rechnen. Die maximale Abweichung beträgt gerade mal einen Zehntelsmilimeter. Dennoch habe ich umgestellt und das Gravitationspotential mit dem Nullpunkt am Boden (bei g=9.81m/s^2) gemäss Deiner Zusatzbedingung berücksichtigt. Zudem, was viel stärker ins Gewicht fällt (bis zu 4cm) sind die Schrittweiten von dt. In der Ursprungsprozedur verwendete ich dt=0.01s bei 10 Dezimalstellen. In der Nachfolgeversion rechnete ich mit dt=1e-6s, also einer Millionstelsekunde bei 15 Dezimalstellen. Ausgabe alle 0.02s.

Die vollständige Umlaufszeit beträgt 4.076738s. (Letzte Kommastelle +-1)

Gruss

Code:

# Looping mit Gravitationspotential:

restart; Digits:=15;

Looping:=proc()

global r,g,m,phi,dphi,t,v,x,y,U,v_o,E,v_u,dt,ml;

r:=10;g:=9.81;m:=1; ml:=6371000^2*9.81; # ml=m_Erde*Lambda

phi:=evalf(-1/2*Pi);
t:=0; x:=0; y:=-10;dt:=0.00001; U:=evalf(2*r*Pi);

v_o:=sqrt(r*ml/6371020^2); E:=-m*ml*(1/6371020-1/6371000)+m/2*v_o^2; v_u:=sqrt(2*E/m);
x:=r*cos(phi);
y:=r*sin(phi);
# print([t,phi,v_u,x,y]);
printf("%f\n%f\n",x,y);

  while phi <= 4.712388981 do
  v:=sqrt(2/m*(E+ml*(1/(6371000+r+y)-1/6371000))); 
  dphi:=evalf(v*dt*2*Pi/U);
  phi:=phi+dphi;
  x:=r*cos(phi);
  y:=r*sin(phi);
  t:=t+dt;

    if round(t/0.02)-t/0.02=0 then       # Ausgabe alle 0.02s
    # print([t,phi,v,x,y]);
    printf("%f\n%f\n",x,y);
    fi;  
  od;
# print([t,phi,v,x,y]);          # vollständige Umdrehung
printf("%f\n%f\n",x,y);
end;

Looping();

Ausgabe alle 0.02s. Erstes Tupel [x,y] mit Zeilenumbruch bei t=0. Letztes Tupel bei t= 4.076738s.

Code:

M.S · **Registriert:** Di 12. Apr 2011, 17:07 **Beiträge:** 918

solkar hat geschrieben:

Ich möchte vorschlagen, dass wir die Diskussion über Programmierung und Programmiersprachen im "Looping"-Thread
weiterführen; dort sind wir näher an der konkreten Fallstudie.

Gut, dann schreib ich's halt hier rein.

Was ist eure Meinung zu einer Sprache wie haskell.
Da müsste sich doch ein Mathematiker wie im Paradies fühlen.
Lambda Kalkül, keine Seiteneffekte; Eigentlich ideal - oder?
Die Ausführungsgeschwindigkeit kann man an die von C hintunen (allerdings muss man dann teilweise kryptischen Code schreiben).

M.S · **Registriert:** Di 12. Apr 2011, 17:07 **Beiträge:** 918

Solkar hat geschrieben:

Kann man da eigentlich Unicode-Bezeichner verwenden, also singemäss z.B.

Code:

φ = π

statt

Code:

phi = PI

schreiben?

Offengestanden, ich weiss es nicht (vermutlich nein).

Zitat:

Aber Deine letzte Bemerkung zum Tuning gibt mir zu denken - da würd ich dann doch lieber glattes C als eckiges Haskell schreiben wollen...

Dieser Meinung bin ich (leider) auch. Diese Art von Sprachen wäre so interessant. Kurze prägnante Formulierungen.
Aber leider gehen die dann nicht, wenn man wirklich auf Geschwindigkeit optimieren will. Da muss man dann wieder aufpassen, welchen Buffertyp man verwendet,
Bescheid wissen, wie intern optimiert wird usw.
Wirklich schade. Naja, vielleicht schau ich mir haskell in ein paar Jahren wieder mal an.

contravariant · **Registriert:** Sa 14. Feb 2009, 20:16 **Beiträge:** 3029

Wie versprochen meine C-ische Implementation. Ich benutzte Operator Overloading um den Code übersichtlicher zu machen. Ist alles in allem nicht besonders optimiert. Um die Standardintegratoren zu verwenden, braucht man DGLs erster Ordnung. Dazu formt man
d^2 phi/dt^2 = -(g/r)*sin(phi)
um,
d phi/dt = omega
d omega/dt = -(g/r)*sin(phi).
Darauf kann man dann den klassischen Runge-Kutta los lassen.

Code:

#include <cmath>
#include <iostream>
#include <cstdlib>

using namespace std;

#define DIM 2
#define g 9.81
#define r 10.0

typedef struct {
    double vektor[DIM];
} Vektor;

/* Vektor + Vektor */
Vektor operator+(Vektor a, Vektor b);
/* Skalar * Vektor */
Vektor operator*(double ska, Vektor vek);
/* "Rechte Seite" der DGL
 * d^2 phi/dt^2 = - (g/r) sin(phi)
 * Zerlegung in zwei gekoppelte DGL erster Ordnung
 * d phi/dt = omega
 * d omega/dt = - (g/r) sin(phi)
 */
Vektor f(Vektor vek);
/* Ein Zeitschritt mit Runge-Kutta vierter Ordnung*/
Vektor rk4(double dt, Vektor phi);

int main(int argc, char **argv) {
    double dt, t, tMax;
    Vektor phi;

    if (argc != 5) {
        cerr << "looping dt tMax phi0 omega0" << endl;
        exit(EXIT_FAILURE);
    }

    dt = atof(argv[1]);
    tMax = atof(argv[2]);
    phi.vektor[0] = atof(argv[3]);
    phi.vektor[1] = atof(argv[4]);

    t = 0.0;
    while (t<tMax) {
        cout << t << "\t" << phi.vektor[0] << "\t" << phi.vektor[1] << endl;
        phi = rk4(dt, phi);
        t += dt;
    }

    exit(EXIT_SUCCESS);
}

Vektor rk4(double dt, Vektor phi) {
    Vektor k1, k2, k3, k4;

    k1 = dt*f(phi);
    k2 = dt*f(phi + 0.5*k1);
    k3 = dt*f(phi + 0.5*k2);
    k4 = dt*f(phi + k3);

    return phi + 1.0/6.0*(k1 + 2.0*k2 + 2.0*k3 + k4);
}

Vektor operator+(Vektor a, Vektor b) {
    Vektor erg;

    erg.vektor[0] = a.vektor[0] + b.vektor[0];
    erg.vektor[1] = a.vektor[1] + b.vektor[1];

    return erg;
}

Vektor operator*(double ska, Vektor vek){
    Vektor erg;

    erg.vektor[0] = ska*vek.vektor[0];
    erg.vektor[1] = ska*vek.vektor[1];

    return erg;
}

Vektor f(Vektor vek) {
    Vektor erg;

    erg.vektor[0] = vek.vektor[1];
    erg.vektor[1] = -(g/r)*sin(vek.vektor[0]);

    return erg;
}

Da gewisse Leute die Weisheit mit Suppenkellen gefressen haben, aber selten liefern können, habe ich das ganze nochmal in Python/Numpy geschrieben.

Code:

#!/usr/bin/python

import sys
from math import *
from numpy import *

g = 9.81
r = 10.0

def f(phi):
   return array([phi[1], -g/r*sin(phi[0])])

def rk4(dt, phi):
   k1 = dt*f(phi)
   k2 = dt*f(phi + 0.5*k1)
   k3 = dt*f(phi + 0.5*k2)
   k4 = dt*f(phi + k3)

   return phi + 1.0/6.0*(k1 + 2.0*k2 + 2.0*k3 + k4)

if (len(sys.argv) != 5):
   sys.stderr.write("./looping.py dt tMax phi0 omega0\n")
   sys.exit(2)

dt = float64(sys.argv[1])
tMax = float64(sys.argv[2])
phi0 = float64(sys.argv[3])
omega0 = float64(sys.argv[4])

t = 0.0
phi = array([phi0, omega0])
while t<tMax:
   print t, phi[0], phi[1]
   phi = rk4(dt, phi)
   t += dt

Ich habe die Laufzeit für eine Millionen Integrationsschritte der beiden Implementation (inklusive I/O) auf meiner Maschine (Intel Core i7 Q720, Ubuntu 11.10) mit time gemessen.
Intel icc 12.1.2 mit -fast:

Python 2.7.2:

Ich bin dennoch hoch gespannt auf etwaige hochgeschwindigkeits Implementationen in C# und Java.

contravariant · **Registriert:** Sa 14. Feb 2009, 20:16 **Beiträge:** 3029

Der Aufruf ist
./looping[.py] dt tMax phi0 omega
dt ist der Zeitschritt für die Integration, tMax die maximale Integrationzeit, phi0 Startwinkel und omega0 Startwinkelgeschwindigkeit. Beide schreiben den Output als "t phi omega" nach stdout.

contravariant · **Registriert:** Sa 14. Feb 2009, 20:16 **Beiträge:** 3029

Solkar hat geschrieben:

contravariant hat geschrieben:

Der Aufruf ist
./looping[.py] dt tMax phi0 omega
dt ist der Zeitschritt für die Integration, tMax die maximale Integrationzeit, phi0 Startwinkel und omega0 Startwinkelgeschwindigkeit. Beide schreiben den Output als "t phi omega" nach stdout.

Ich brauche die aktuellen Parameter, die Du beim Profiling auf der Kommandozeile übergeben hast; die formalen Parameter sieht man ja im Code.

Grüsse, Solkar

Ich hab dt=0.001, tMax=1000.0, phi0=0.0 und omega0=2.215 genommen.

Das Prinzip des Seins

Bewegungsgleichung für Loopingfahrten

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