数値計算:常微分方程式

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1 ( ) 1 / 82

2 ( ) 2 / 82

3 ( ) 3 / 82

4 C θ l y m O x mg λ ( ) 4 / 82

5 θ t C J = ml 2 C mgl sin θ θ C J θ = mgl sin θ = θ ( ) 5 / 82

6 ω = θ J ω = mgl sin θ ω J = ml 2 θ = ω, ω = g l sin θ = θ ω ( ) 6 / 82

7 θ = ω, ω = g l sin θ θ ω θ ω ( ) 7 / 82

8 x = [ f (x, t) = [ θ ω ] ω g sin θ l ] θ = ω, ω = g l sin θ ẋ = f (x, t) ( ) 8 / 82

9 Step 1 Step 2 ( ) ( ) 9 / 82

10 MATLAB pendulumconstants.m classdef pendulumconstants % properties (Constant) gravity = 9.8; % g length = 2.0; % l end end ( ) 10 / 82

11 MATLAB dotpendulum.m function dotq = dotpendulum(t,q) % % % theta = q(1); omega = q(2); dottheta = omega; dotomega = -(pendulumconstants.gravity/... pendulumconstants.length)*sin(theta); dotq = [dottheta; dotomega]; end ( ) 11 / 82

12 MATLAB pendulum.m pendulumconstants; % timeinterval=0:0.1:10; % ( 0.1s) q0=[pi/3;0]; % [time,q]=ode45( dotpendulum,timeinterval,q0); plot(time,q(:,1),time,q(:,2), -- ); % ( ) 12 / 82

13 MATLAB >> pendulum time ( ) 13 / 82

14 ( ) 14 / 82 MATLAB >> time time =

15 ( ) 15 / 82 MATLAB >> q q =

16 k m b f(t) mẍ = bẋ kx + f (t) ( ) 16 / 82

17 v = ẋ m v = bv kx + f (t) ẋ = v m v = bv kx + f (t) = x v ( ) 17 / 82

18 LCR L V(t) C R V (t) L i 1 C t 0 i(τ) dτ Ri = 0 ( ) 18 / 82

19 LCR q(t) = t 0 q = i i(τ) dτ V (t) L i 1 C q Ri = 0 q = i L i = Ri 1 C q + V (t) = q i ( ) 19 / 82

20 l2 y lc2 link 2 joint 1 lc1 l1 θ1 θ2 joint 2 link 1 x ( ) 20 / 82

21 H 11 θ1 + H 12 θ2 = h 12 θ h 12 θ1 θ2 G 1 G 12 + τ 1, H 22 θ2 + H 12 θ1 = h 12 θ2 1 G 12 + τ 2 H 11 = J 1 + m 1 l 2 c1 + J 2 + m 2 (l l 2 c2 + 2l 1 l c2 cos θ 2 ) H 12 = J 2 + m 2 (l 2 c2 + l 1 l c2 cos θ 2 ) H 22 = J 2 + m 2 l 2 c2 h 12 = m 2 l 1 l c2 sin θ 2 G 1 = (m 1 l c1 + m 2 l 1 ) g cos θ 1 G 12 = m 2 l c2 g cos(θ 1 + θ 2 ) ( ) 21 / 82

22 ω 1 = θ1 ω 2 = θ2 [ H11 H 12 H 12 H 22 [ ] θ1 θ 2 ] [ ] ω1 ω 2 = = [ ω1 ω 2 ], [ h12 ω h 12 ω 1 ω 2 G 1 G 12 + τ 1 h 12 ω 2 1 G 12 + τ 2 ] θ 1, θ 2, ω 1, ω 2 = = θ 1, θ 2, ω 1, ω 2 ( ) 22 / 82

23 x = θ 1 θ 2 ω 1 ω 2 ẋ = f (x, t) x θ1 θ2 ω1 ω2 solve linear equations θ1 θ2 ω1 ω2 x t ( ) 23 / 82

24 ( ) 24 / 82

25 (a) ẋ = 3 x (b) ẋ 2 = 9x (c) ẋ 2 = 9x (ẋ 0) { ẋ + ẏ = x (d) ẋ ẏ = y { ẋ 2ẏ = x (e) 2ẋ + 4ẏ = y 2ẋ + p = x (f) 2ẏ + 3p = y ẋ + 3ẏ = x y ( ) 25 / 82

26 (1 ) ẋ = f (x, t) t n = nt (n = 0, 1, 2, ) x T ( ) 26 / 82

27 (Euler method) (Heun method) (Runge-Kutta method) t n x x n = x(t n ) t n+1 x x n+1 = x(t n+1 ) x 0 = x(0) x n = x(nt ) (n = 1, 2, ) ( ) 27 / 82

28 dx/dt = 2x x(0) = t x ( ) 28 / 82

29 CG ( ) ( ) 29 / 82

30 x n+1 = x n + Tf (x n, t n ) 1 x t (x n, t n ) f (x, t) x n+1 = x((n + 1)T ) = x(nt + T ) = x(t n + T ) x n+1 = x(t n ) + ẋ(t n )T = x(t n ) + f (x(t n ), t n )T = x n + Tf (x n, t n ) ( ) 30 / 82

31 x Euler method xn+tk1 xn k1=f(xn,tn) O tn tn+1 t ( ) 31 / 82

32 x n+1 = x n + T 2 (k 1 + k 2 ), k 1 = f (x n, t n ), k 2 = f (x n + Tk 1, t n + T ) 2 (x n, t n ) (x n + Tk 1, t n + T ) f (x, t) ( ) 32 / 82

33 x Heun method k2 xn+tk1 Euler method xn k1 O tn tn+1 t k 1 < k 2 ẍ > 0 ( ) 33 / 82

34 x xn+tk1 xn Euler method k2 k1 Heun method O tn tn+1 k 1 > k 2 ẍ < 0 t ( ) 34 / 82

35 x n+1 = x(t n + T ) ẋ = f (x, t) x n+1 = x(t n ) + ẋ(t n )T + 1 2ẍ(t n)t 2 ẍ = f dx x dt + f dt t dt = f x ẋ + f t = f xf + f t x n+1 = x n + f (x n, t n )T (f xf + f t ) T 2 ( ) 35 / 82

36 k 2 k 2 = f (x n + Tk 1, t n + T ) = f (x n, t n ) + f x Tk 1 + f t T = f + (f x f + f t )T x n + T 2 (k 1 + k 2 ) = x n + T 2 (f + f + (f xf + f t )T ) = x n + ft (f xf + f t ) T 2 ( ) 36 / 82

37 x n+1 = x n + T 6 (k 1 + 2k 2 + 2k 3 + k 4 ), k 1 = f (x n, t n ), k 2 = f (x n Tk 1, t n T ), k 3 = f (x n Tk 2, t n T ), k 4 = f (x n + Tk 3, t n + T ) 4 x t f (x, t) ( ) 37 / 82

38 xn+ xn+ T 2 T 2 x k1 k2 xn k1 k2 k3 xn+tk3 k4 O tn tn+ T 2 tn+1 t ( ) 38 / 82

39 ( ) 39 / 82

40 ( ) ẋ = f (x, t) x = x f = f k = k ( ) 40 / 82

41 MATLAB pendulum.m pendulumconstants; % timeinterval=0:0.1:10; % ( 0.1s) q0=[pi/3;0]; % [time,q]=ode45( dotpendulum,timeinterval,q0); plot(time,q(:,1),time,q(:,2), -- ); % ( ) 41 / 82

42 MATLAB >> pendulum time ( ) 42 / 82

43 ( ) 43 / 82 MATLAB >> time time =

44 ( ) 44 / 82 MATLAB >> q q =

45 ( ) 45 / 82

46 x O t ( ) 46 / 82

47 6 k 1 = f (x n, t n ), k 2 = f (x n + T 4 k 1, t n T ), k 3 = f (x n + T 32 (3k 1 + 9k 2 ), t n T ), k 4 = f (x n + T 2179 (1932k k k 3 ), t n T ), k 5 = f (x n + T ( k 1 8k k k 4), t n + T ), k 6 = f (x n + T ( 8 27 k 1 + 2k k k k 5), t n T ) ( ) 47 / 82

48 5 x n+1 = x n + T ( k k k k k 6) 4 x n+1 = x n + T ( k k k k 5) x n+1 xn+1 x n+1 xn+1 ( ) 48 / 82

49 Step 1 Step 2 Step 3 5 x n+1 4 xn+1 ˆT { ˆT = αt ϵ x n+1 x n+1 } 1 5 Step 4 ϵ α T ˆT ( ) 49 / 82

50 MATLAB pendulumvar.m pendulumconstants; timeinterval=[0,10]; % q0=[pi/3;0]; [time,q]=ode45( dotpendulum,timeinterval,q0); plot(time,q(:,1),time,q(:,2), -- ); ( ) 50 / 82

51 MATLAB >> pendulumvar time ( ) 51 / 82

52 ( ) 52 / 82 MATLAB >> time time =

53 ( ) 53 / 82 MATLAB >> q q =

54 m t x(t) ( k) mẍ = f collision mg, { kx x < 0 f collision = 0 otherwise m = 1 k = 100 g = 9.8 x(0) = 100 ẋ(0) = 0 ( ) 54 / 82

55 MATLAB collideconstants.m classdef collideconstants % properties (Constant) gravity = 9.8; mass = 1.0; spring = 100.0; end end ( ) 55 / 82

56 MATLAB dotcollide.m function dotq = dotcollide(t,q) % x = q(1); v = q(2); dotx = v; if x < 0 dotv = -collideconstants.gravity... -collideconstants.spring*x/collideconstants.mass; else dotv = -collideconstants.gravity; end dotq = [dotx; dotv]; end ( ) 56 / 82

57 MATLAB collide.m collideconstants; timeinterval=[0,50]; % q0=[100.0;0.0]; % options = odeset( RelTol,1e-8, AbsTol,[1e-8 1e-10]); [time,q]=ode45( dotcollide,timeinterval,q0,options); plot(time,q(:,1),time,q(:,2), -- ); ( ) 57 / 82

58 MATLAB time x(t) ẋ(t) ( ) 58 / 82

59 MATLAB time interval time T ( ) 58 / 82

60 ( ) 59 / 82

61 ( ) C R<0 y l R=0 m R>0 O x mg λ ( ) 60 / 82

62 ( ) t (x, y) C l R(x, y) = { x 2 + (y l) 2} 1 2 l = 0 R(x, y) [R x, R y ] T R x (x, y) = R x = x { x 2 + (y l) 2} 1 R y (x, y) = R y = (y l) { x 2 + (y l) 2} 1 R(x, y) = 0 ( ) 61 / 82 2, 2.

63 ( ) [R x, R y ] T λ mẍ = λ R x (x, y), mÿ = λ R y (x, y) mg R(x, y) = 0 ( ) 62 / 82

64 ( ) v x = ẋ vy = ẏ ẋ = v x, ẏ = v y, m v x = λ R x (x, y), m v y = λ R y (x, y) mg, R(x, y) = 0 x y v x v y λ 4 1 ( ) 63 / 82

65 joint 2 θ2 tip point link 2 l2 joint 5 l4 link 4 joint 4 θ4 y O x l1 link 1 θ1 joint 1 (x1,y1) link 3 joint 3 (x3,y3) θ3 l3 ( ) 64 / 82

66 joint 2 θ2 tip point link 2 l2 xleft [ yleft ] xright [ yright] link 4 tip point l4 joint 4 θ4 link 1 joint 1 l1 θ1 link 3 joint 3 θ3 l3 (x1,y1) (x3,y3) ( ) 65 / 82

67 [ ] [ ] x1 C1 + l y 1 1 S 1 [ ] [ ] x3 C3 + l y 3 3 C 3 + l 2 [ C1+2 S l 4 [ C3+4 S 3+4 ] ] P(x, y) = l 1 C 1 + l 2 C 1+2 l 3 C 3 l 4 C x 1 x 3 = 0 Q(x, y) = l 1 S 1 + l 2 S 1+2 l 3 S 3 l 4 S y 1 y 3 = 0 ( ) 66 / 82

68 H 11 ω 1 + H 12 ω 2 = f 1 + λ x ( l 1 S 1 l 2 S 1+2 ) + λ y (l 1 C 1 + l 2 C 1+2 ), H 22 ω 2 + H 12 ω 1 = f 2 + λ x ( l 2 S 1+2 ) + λ y l 2 C 1+2, H 33 ω 3 + H 34 ω 4 = f 3 + λ x (l 3 S 3 + l 2 S 3+4 ) + λ y ( l 3 C 3 l 4 C 3+4 ), H 44 ω 4 + H 34 ω 3 = f 4 + λ x l 4 S λ y ( l 4 C 3+4 ) f 1 = h 12 ω h 12 ω 1 ω 2 G 1 G 12 + τ 1, f 2 = h 12 ω1 2 G 12, f 3 = h 34 ω h 34 ω 3 ω 4 G 3 G 34 + τ 3, f 4 = h 34 ω3 2 G 34 ( ) 67 / 82

69 θ 1 = ω 1, θ2 = ω 2, θ3 = ω 3, θ4 = ω 4, H 11 ω 1 + H 12 ω 2 = f 1 + λ x ( l 1 S 1 l 2 S 1+2 ) + λ y (l 1 C 1 + l 2 C 1+2 ), H 22 ω 2 + H 12 ω 1 = f 2 + λ x ( l 2 S 1+2 ) + λ y l 2 C 1+2, H 33 ω 3 + H 34 ω 4 = f 3 + λ x (l 3 S 3 + l 2 S 3+4 ) + λ y ( l 3 C 3 l 4 C 3+4 ), H 44 ω 4 + H 34 ω 3 = f 4 + λ x l 4 S λ y ( l 4 C 3+4 ), P(x, y) = 0, Q(x, y) = 0 θ 1, θ 2, θ 3, θ 4 ω 1, ω 2, ω 3, ω 4 λ x λ y 8 2 ( ) 68 / 82

70 R = 0 R + 2νṘ + ν2 R = 0 ν R = ( ) R 0 = R = 0 ( ) 69 / 82

71 R(x, y) = 0 R(x, y) Ṙ R x (x, y) R y (x, y) Ṙ = R dx x dt + R y = R x ẋ + R y ẏ Ṙ x = R x dx x dt + R x y = R xx ẋ + R xy ẏ Ṙ y = R y dx x dt + R y y = R yx ẋ + R yy ẏ ( ) 70 / 82 dy dt dy dt dy dt

72 R(x, y) R R = Ṙxẋ + R x ẍ + Ṙyẏ + R y ÿ = R x ẍ + R y ÿ + (R xx ẋ + R xy ẏ)ẋ + (R yx ẋ + R yy ẏ)ẏ = R x ẍ + R y ÿ + [ ẋ ẏ ] [ ] [ ] R xx R xy ẋ R yx R yy ẏ R + 2νṘ + ν 2 R = 0 R x ẍ R y ÿ = C(x, y, ẋ, ẏ) C(x, y, ẋ, ẏ) = [ ẋ ẏ ] [ R xx R xy R yx R yy ] [ ẋ ẏ ] + 2ν(R x ẋ + R y ẏ) + ν 2 R ( ) 71 / 82

73 ( ) mẍ = λ R x (x, y), mÿ = λ R y (x, y) mg, R(x, y) = 0 ẋ = v x, ẏ = v y, m v x λ R x (x, y) = 0, x y v x v y λ 5 m v y λ R y (x, y) = mg, R x v x R y v y = C(x, y, v x, v y ) ( ) 72 / 82

74 P(x, y) = x 2 + (y l) 2 R(x, y) = P 1 2 l, R x (x, y) = R x = 1 xp R y (x, y) R xx (x, y) R yy (x, y) R xy (x, y) = R yx (x, y) = R y = 2 R x 2 2, 1 = (y l)p 2, = P 1 2 x 2 P 3 2, = 2 R y = P (y l) 2 P 3 2, = 2 R x y 3 = x(y l)p 2. ( ) 73 / 82

75 m v x λ R x (x, y) = 0, m v y λ R y (x, y) = mg, R x v x R y v y = C(x, y, v x, v y ) m 0 R x 0 m R y R x R y 0 v x v y λ = 0 mg C(x, y, v x, v y ) x, y, v x, v y = = v x, v y ( ) 74 / 82

76 q = x y v x v y q = f (q, t) x x y q y vx vy solve linear equations vx vy q t ( ) 75 / 82

77 MATLAB m = 0.01 l = 2 g = 9.8 θ(0) = (π/3) ω(0) = 0 x(0) = l sin θ(0) y(0) = l(1 cos θ(0)) v x (0) = 0 v y (0) = 0 ( ) 76 / 82

78 MATLAB dotpendulumcartesian.m function dotq = dotpendulumcartesian (t,q) % x = q(1); y = q(2); vx = q(3); vy = q(4); nu = pendulumcartesianconstants.nu; mass = pendulumcartesianconstants.mass; gravity = pendulumcartesianconstants.gravity; [R,Rx,Ry,Rxx,Ryy,Rxy] = calculate_r_derivatives(x,y); C = Rxx*vx*vx + Ryy*vy*vy + 2*Rxy*vx*vy *nu*(Rx*vx + Ry*vy) + nu*nu*r; ( ) 77 / 82

79 MATLAB A = [ mass, 0, -Rx;... 0, mass, -Ry;... -Rx, -Ry, 0 ]; b = [ 0; -mass*gravity; C ]; d = A\b; dotvx = d(1); dotvy = d(2); dotq = [vx; vy; dotvx; dotvy]; end ( ) 78 / 82

80 MATLAB calculate R derivatives.m function [ R, Rx, Ry, Rxx, Ryy, Rxy ] =... calculate_r_derivatives( x, y ) % R, R_x, R_y, R_xx, R_yy, R_xy length = pendulumcartesianconstants.length; P = x*x + (y-length)*(y-length); Psqrt = sqrt(p); R = Psqrt - length; Rx = x/psqrt; Ry = (y-length)/psqrt; Rxx = 1/Psqrt - x*x/p/psqrt; Ryy = 1/Psqrt - (y-length)*(y-length)/p/psqrt; Rxy = -x*(y-length)/p/psqrt; end ( ) 79 / 82

81 MATLAB pendulumcartesian.m pendulumconstants; %timeinterval=0:0.1:10; % timeinterval=[0,10]; % length = pendulumcartesianconstants.length; theta0 = pi/3; q0=[length*sin(theta0); length*(1-cos(theta0)); 0; 0]; [time,q]=ode45( dotpendulumcartesian,timeinterval,q0); plot(time,q(:,1),time,q(:,2), -- ); ( ) 80 / 82

82 MATLAB position time x, y ( ) 81 / 82

83 MATLAB velocity time v x, v y ( ) 81 / 82

84 MATLAB angle time θ ( ) 81 / 82

85 MATLAB constraint time R ( ) 81 / 82

86 ẋ = f (x, t) ( ) 82 / 82

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

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