ハイブリッド微分方程式.nb
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1 Mathematica ã 15 ê 4 - Floor@y@tDD + DiracDelta@t - 2D Floor DiracDelta 2 ver.8 Mathematica == à ver.9 In[1]:= sol = NDSolve@8y'@tD ã 15 ê 4 - Floor@y@tDD + DiracDelta@t - 2D, y@0d ã.1<, y, 8t, 0, 6<D Out[1]= 88y Ø InterpolatingFunction@880., 6.<<, <>D<< (PlotLegends ver9 ) In[2]:= Out[2]= y@td ê. sol 8InterpolatingFunction@880., 6.<<, <>D@tD< In[3]:=?? *Legend* System` BarLegend ChartLegends LegendAppearance Legended LegendFunction LegendLabel LegendLayout LegendMargins LegendMarkers LegendMarkerSize LegendreP LegendreQ LegendreType LineLegend PlotLegends PointLegend SwatchLegend
2 2.nb In[237]:= ê. sold, 8t, 0, 6<, PlotLegends Ø 8"yHtL", "y'htl"<d 4 Out[237]= 3 2 yhtl y'htl ( ) Fillippov In[226]:= Out[226]= sxy = FirstüNDSolveB: ã IfAx@tD 2 + y@td 2 > 2, 1, x@tde, ã IfAx@tD 2 + y@td 2 > 2, -1, y@tde, x@0d ã , y@0d ã 1 >, 8x, y<, 8t, 0, 10<F 10 8x Ø InterpolatingFunction@880., 10.<<, <>D, y Ø InterpolatingFunction@880., 10.<<, <>D< x[t] y[t] ( )
3 .nb 3 In[227]:= pplot = ParametricPlot@ Evaluate@8x@tD, y@td< ê. sxyd, 8t, 0, 10<, PlotStyle Ø 8Thickness@0.01D, Red<, PlotRange Ø 88-2, 2<, 8-2, 2<<, Prolog Ø 8GrayLevel@.9D, Disk@80, 0<, Sqrt@2DD<, AxesLabel Ø 8"x", "y"<d y 2 1 Out[227]= x -1-2 In[228]:= Show@pplot, VectorPlot@If@x^2 + y^2 > 2, 81, -1<, 8x, y<d, 8x, -2, 2<, 8y, -2, 2<, VectorStyle Ø Gray, VectorPoints Ø 30, VectorScale Ø 8.02, 1, None<DD y 2 1 Out[228]= x c1 c2
4 4.nb In[231]:= 3 3 c1 c2 c1 = ContourPlot3DA 8 - y 2 - Hx + 3 ê 2L 2 - Hz - 1L 2, 8x, -2, 2<, 8y, -2, 2<, 8z, 0, 5<, Contours Ø 0, Mesh Ø FalseE; c2 = ContourPlot3D@ -y, 8x, -2, 2<, 8y, -2, 2<, 8z, 0, 5<, Contours Ø 0, Mesh Ø FalseD; Show@c1, c2d Out[233]= In[235]:= Out[235]= sol3d = FirstüNDSolve@8 x'@td ã Sign@8 - y@td^2 - Hx@tD + 3 ê 2L^2 - Hz@tD - 1L^2D, y'@td ã Sign@-y@tDD, z'@td ã 1, x@0d ã 2, y@0d ã -2, z@0d ã 0<, 8x, y, z<, 8t, 0, 5<D 8x Ø InterpolatingFunction@880., 5.<<, <>D, y Ø InterpolatingFunction@880., 5.<<, <>D, z Ø InterpolatingFunction@880., 5.<<, <>D< {2,-2,0} {-1.5,0,.5} In[236]:= Table@8x@tD, y@td, z@td< ê. sol3d, 8t, 0, 5<D Out[236]= 982., -2., 0.<, , -1., 1.<, , µ 10-8, 2.=, 80.5, 0., 3.<, 8-0.5, 0., 4.<, 8-1.5, 0., 5.<=
5 .nb 5 In[237]:= p1 = ParametricPlot3D@ Evaluate@8x@tD, y@td, z@td< ê. sol3dd, 8t, 0, 5<, PlotStyle Ø 8Thickness@.015D, Red<D Out[237]= 2 0 2
6 6.nb In[238]:= c2, p1d Out[238]= WhenEvent 5 y''@td ã -9.81, y@0d ã 5, y'@0d ã 0 ( =0) y@td ã 0 95% y'@td -> y'@td In[34]:= sol = NDSolve@8y''@tD ã -9.81, y@0d ã 5, y'@0d ã 0, WhenEvent@y@tD ã 0, y'@td -> y'@tdd<, y, 8t, 0, 10<D;
7 .nb 7 In[35]:= Plot@y@tD ê. sol, 8t, 0, 10<D Out[35]= (WhenEvent) In[239]:= eqns = 8 x'@td ã -y@td - x@td^2, y'@td ã 2 x@td - y@td^3, x@0d ã y@0d ã 1<; x y In[241]:= sol = NDSolve@eqns, 8x, y<, 8t, 100<D Out[241]= 88x Ø InterpolatingFunction@880., 100.<<, <>D, y Ø InterpolatingFunction@880., 100.<<, <>D<<
8 8.nb In[244]:= ê. sol, 8t, 0, 100<, PlotRange Ø All, AxesLabel Ø 8x, y<d y Out[244]= x -0.5 X Y>0 Y<0 WhenEvent y[t]=0 t Sow In[245]:= Out[245]= 8sol1, points< = ReapüNDSolve@8eqns, WhenEvent@y@tD ã 0, Sow@tDD<, 8x, y<, 8t, 100<D 888x Ø InterpolatingFunction@880., 100.<<, <>D, y Ø InterpolatingFunction@880., 100.<<, <>D<<, , , 6.544, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , <<<
9 .nb 9 In[246]:= ParametricPlot@8x@tD, y@td< ê. sol1, 8t, 0, 100<, Mesh Ø points, MeshShading Ø 8Green, Orange<, PlotRange Ø All, AxesLabel Ø 8x, y<d y Out[246]= x -0.5 Sin[5 x[t]] In[249]:= 8sol2, points< = ReapüNDSolve@8eqns, WhenEvent@y@tD ã 1.1 Sin@5 x@tdd, Sow@tDD<, 8x, y<, 8t, 100<D;
10 10.nb In[250]:= ê. sol2, 8t, 0, 100<, Mesh Ø points, MeshShading Ø 8Green, Orange<, PlotRange Ø All, AxesLabel Ø 8x, y<d, 1.1 td<, 8t, -1, 1<, PlotStyle Ø DashedDD y Out[250]= x (WhenEvent) In[1]:= system = 8x''@tD ã u@td, x'@0d ã x@0d ã 0, u@0d ã 1<; (PD) In[16]:= kp = 1; td = 1; t = 1; xref = 1; control = WhenEvent@Mod@t, td == 0, u@td Ø kp Hxref - x@td - td x'@tdld; In[18]:= sol = NDSolve@8system, control<, 8x, u<, 8t, 0, 12<, DiscreteVariables Ø ud;
11 .nb 11 In[19]:= ê. sold, 8t, 0, 12<, PlotLegends Ø 9" 1.0", " " ImageSize Ø MediumE 1.0 Out[19]= u@td -0.5 DC-DC (WhenEvent) q[t] vi vd DC-DC Kirchoff In[261]:= Clear@"Global`*"D; In[262]:= system = 8 vo'@td ã -vo@td ê Hr cl + i@td ê c, i'@td ã -vo@td ê l + q@td vi ê l, vo@0d ã 0, i@0d ã 0<; q[t] t vd/vi In[263]:= control = 9 q@0d ã 1, H* *LWhenEvent@Mod@t, td ã Hvd ê vil t, q@td Ø 0D, H* *LWhenEvent@Mod@t, td ã 0, q@td Ø 1D=; vi=24 vd=16 In[264]:= In[265]:= Out[265]= pars = 8vi Ø 24, vd Ø 16, r Ø 22, l Ø 2 µ 10^-2, c Ø 4 µ 10^-6, t Ø 2 µ 10^-4<; sol = NDSolve@8system, control< ê. pars, 8vo, i, q<, 8t, 0,.01<, DiscreteVariables Ø q, MaxSteps Ø InfinityD 88vo Ø InterpolatingFunction@880., 0.01<<, <>D, i Ø InterpolatingFunction@880., 0.01<<, <>D, q Ø InterpolatingFunction@880., 0.01<<, <>D<<
12 12.nb In[268]:= ê. sol, vd ê. pars<, 8t, 0,.01<, PlotRange Ø All, PlotLegends Ø 8"voHtL", "vd"<, ImageSize Ø MediumD, ê. sol, 8t, 0,.001<, AxesLabel Ø 8t, ImageSize Ø MediumD<D vohtl vd Out[268]= 1.0 qhtl t ( )
13 .nb 13 m x '' HtL = F x m y (t)= F y In[272]:= deqns = : m 1 ã l1@td x1@td l2@td Hx2@tD - x1@tdl -, m 1 ã l 1 l1@td y1@td l2@td Hy2@tD - y1@tdl - - m 1 g, m 2 ã l 1 m 2 ã l 2 l 2 l2@td Hy2@tD - y1@tdl - m 2 g>; l 2 In[269]:= aeqns = 9 x1@td 2 + y1@td 2 ã l 1 2, Hx2@tD - x1@tdl 2 + Hy2@tD - y1@tdl 2 ã l 2 2 =; In[270]:= ics = 8 x1@0d ã 1, y1@0d ã 0, x1'@0d ã 0, y1'@0d ã 0, x2@0d ã 1, y2@0d ã -1, x2'@0d ã 0, y2'@0d ã 0<; In[271]:= params = 8g Ø 9.81, m 1 Ø 1, m 2 Ø 1, l 1 Ø 1, l 2 Ø 1<; l2@td Hx2@tD - x1@tdl l 2, In[275]:= soldp = FirstüNDSolve@ 8deqns, aeqns, ics< ê. params, 8x1, y1, x2, y2, l1, l2<, 8t, 0, 15<, Method Ø 8"IndexReduction" Ø 8Automatic, "IndexGoal" Ø 0<<D; ParametricPlotA Evaluate@88x1@tD, y1@td<, 8x2@tD, y2@td<< ê. soldpd, 8t, 0, 15<, PlotStyle Ø 8Red, Blue<, ImageSize Ø Medium, PlotLegends Ø 9" 1 ", " 2 "=E Out[276]=
14 14.nb In[278]:= AnimateAGraphicsA9 H* *L 8Red, 8Blue, 0<, ê. soldp, H* *L8Gray, ê. soldpdd, t, 0.025DDD<=, PlotRange Ø 88-2, 2<, 8-2, 0<<, Axes Ø True, Ticks Ø False, ImageSize Ø 500E, 8t, 0, 10,.025<, SaveDefinitions Ø True, AnimationRunning Ø FalseE Out[278]= FLB ZHU 2
15 .nb 15 In[1]:= In[2]:= Fin = kla HpCO2 ê H - CO2@tDL; r1 = k1 FLB@tD^2 CO2@tD^0.5; r2 = k2 FLBT@tD ZHU@tD; r3 = Hk2 ê KKL FLB@tD ZLA@tD; r4 = k3 FLB@tD ZHU@tD^2 CO2@tD; r5 = k4 FLBZHU@tD CO2@tD^0.5; In[7]:= eqns = 8 FLB'@tD ã -2 r1 + r2 - r3 - r4, CO2'@tD ã -0.5 r1 - r4-0.5 r5 + Fin, FLBT'@tD ã r1 - r2 + r3, ZHU'@tD ã -r2 + r3-2 r4, ZLA'@tD ã r2 - r3 + r5<; In[8]:= eqeqn = Ks FLB@tD ZHU@tD ã FLBZHU@tD; In[9]:= params = 8k1 Ø 18.7, k2 Ø 0.58, k3 Ø 0.09, k4 Ø 0.42, KK Ø 34.4, kla Ø 3.3, Ks Ø , pco2 Ø 0.9, H Ø 737<; In[10]:= ics = 8FLB@0D ã 0.444, CO2@0D ã , FLBT@0D ã 0, ZHU@0D ã 0.007, ZLA@0D ã 0<;
16 16.nb In[13]:= sol = NDSolve@8eqns, eqeqn, ics< ê. params, 8FLB, ZHU, CO2, ZLA<, 8t, 0, 200<D; Column@ Plot@Evaluate@Ò@tD ê. sold, 8t, 0, 200<, PlotRange Ø All, PlotLabel Ø Ò@tDD & êü 8FLB, ZHU, CO2, ZLA<D FLBHtL Out[14]= ZHUHtL CO2HtL ZLAHtL
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