dx dt = f x,t ( ) t
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- ああす いまいだ
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1 MATLAB Runge-Kutta
2 dx dt = f x,t ( ) t
3 dx( t) dt = lim Δt x( t + Δt) x( t) Δt t = nδt n = 0,1,2,3,4,5, x = x( nδt) n Δt dx ( ) dt = f x,t x n +1 x n Δt = f ( x,nδt) n
4 1 x n = x 0 n = 0 2 x = x + Δtf ( x,nδt) n +1 n n n = n +1 t = nδt Δt
5 Runge-Kutta x = x + Δtf x,t + Δt n +1 n 2 t = nδt x = x n + Δt ( ) 2 f x n,t
6 Euler.m Runge-Kutta RK2.m
7 Euler.m x0 x
8 Runge-Kutta RK2.m ( ) x0 x Runge-Kutta
9 MATLAB Runge-Kutta
10 τ dx ( t ) dt = x( t) + u( t) x L{ u( t) } = U( s) L{ x( t) } = X( s) ( ) = X ( s ) U( s) = 1 G s 1+ τs
11 U( s) = 1 s X( s) = 1 1+ τs 1 s = 1 s 1 s +1 τ x( t) =1 exp( t τ)
12
13 MATLAB Runge-Kutta
14 Euler.m Runge-Kutta RK2.m FOL.m FOL_main.m
15 FOL.m dx( t) dt = x ( t ) + u( t) τ
16 FOL_main.m function main tau = 2; % In = 1; % dt = 0.3; % T = 10; % Runge- Kutta xinit =[0]; % para = [tau,in]; % %@FOL [t1,x] = Euler(@FOL,[0,T],xinit,dt,para); % xe = x(1,:); xmin = min(xe)-0.1; xmax = max(xe)+0.1; %@FOL [t2,x] = RK2(@FOL,[0,T],xinit,dt,para); %2 Runge-Kutta xr2 = x(1,:); t3 = 0:dt:T; % xas = In.*(1-exp(-t3./tau)); % plot(t3,xas, -,t1,xe, -.,t2,xr2, : ); % axis([0 T xmin xmax]); end
17 MATLAB Runge-Kutta
18 Runge-Kutta RK2.m FOL_FR.m FOL_FR_main.m
19 FOL_FR.m function x = FOL_FR(t,y,para) x = (-y(1) + para(2).*sin(para(3).*t))./para(1); end dx( t) dt = x ( t ) + Asin( ωt) τ
20 FOL_FR_main. m function main tau = 1; % In = 1; % omega = 10; dt = 0.02; % T = 20; % xinit =[0]; % para = [tau,in,omega]; % %@FOL_FR [t,x] = RK2(@FOL_FR,[0,T],xinit,dt,para); %2 Runge-Kutta xr = x(1,:); xmin = min(xr)-0.1; xmax = max(xr)+0.1; plot(t,xr); % axis([0 T xmin xmax]);
21 dx( t) dt = x ( t ) + sin( ωt) τ ( ) x( t) = G( jω) sin ωt + tan 1 G( jω) G( jω) = 1 1+ jτω
22
23 TF_FOL.m FOL_boad_main.m
24 TF_FOL.m s function [ y ] = TF_FOL(s,para) y = 1./(ones(size(s))+para(1).*s); s G( s) = 1 1+ τs Matlab
25 FOL_boad_ma in.m function main tau = 1; % j=sqrt(-1);% para = [tau];% omega = 10.^(-2:0.1:2); % G = TF_FOL(j.*omega,para);% gain = 20.*log10(abs(G)); % kaku = angle(g); % subplot(2,1,1); %2 semilogx(omega,gain); %x xlabel( omega ); ylabel( db );% subplot(2,1,2); %2 semilogx(omega,kaku); %x xlabel( omega ); ylabel( rad ); % end
26 MATLAB Runge-Kutta
27 d 2 x( t) dt 2 dx( t) + 2ζω 2 n dt + ω 2 x( t) = ω 2 u( t) n n xx (0)=0 L{ u( t) } = U( s) L{ x( t) } = X( s) ( ) = X ( s ) U( s) = ω 2 n G s s 2 + 2ζω n s + ω n 2
28 U( s) = 1 s X( s) = ω n 2 s 2 + 2ζω n s + ω n 2 1 s 3 ζ <1 ζ =1 ζ >1
29 ζ <1 X s ( ) = 1 s ω 2 n 2 1 β β + jα ( ) α = ω n ζ 1 s + α jβ ω 2 n 2 1 β β jα ( ) β = ω n 1 ζ 2 1 s + α + jβ x( t) =1 1 1 ζ exp ( ω nζt)sin 1 ζ 2 ω 2 n t + φ ( ) φ = tan 1 1 ζ 2 ζ ζ =1 X( s) = 1 s 1 s + ω n ω n ( ) 2 s + ω n x( t) =1 ( 1+ ω t)exp( ω t) n n
30 ζ >1 X( s) = 1 s + s s 1 s 2 s s 1 s 1 1 s 2 s 1 s s 2 s 1 = ζω n + ω n ζ 2 1 s 2 = ζω n ω n ζ 2 1 x( t) =1+ s 2 exp( s 1 t) + s 1 s 2 s 1 exp( s 2 t) s 2 s 1
31
32 MATLAB Runge-Kutta
33 2 d 2 x t ( ) dt 2 ( ) 2 dx t + 2ζω n dt + ω n 2 x t ( ) = ω n 2 u t ( ) dx t ( ) dt = y t ( ) d 2 x t ( ) dt 2 = dy ( t ) dt = 2ζω n 2 y t ( ) ω n 2 x t ( ) + ω n 2 u t ( ) d dt x = y ω n 2ζω n x y + 0 ω n 2 u t ( ) x = [ 1 0] x y
34 Runge-Kutta RK2.m SOE.m SOE_main.m
35 (SOE.m) d dt x = y y ω 2 n x 2ζω 2 n y + ω 2 n u t ( )
36 SOE_main.m function main xi = 0.2; % wn = 2; % In = 1;% dt = 0.1;% T = 15;% xinit =[0;0];% para = [xi,wn,in];% %@SOE [t1,xy] = RK2(@SOE,[0,T],xinit,dt,para); %2 Runge-Kutta x = xy(1,:); xmin = min(x)-0.1; xmax = max(x)+0.1; y = xy(2,:); ymin = min(y)-0.1; ymax = max(y)+0.1; subplot(2,1,1); %2 plot(t1,x);% axis([0 T xmin xmax]); xlabel('t'); ylabel('x'); subplot(2,1,2); %2 plot(x,y);% axis([xmin xmax ymin ymax]); xlabel('x'); ylabel('dx/dt'); end
37 MATLAB Runge-Kutta
38 Runge-Kutta RK2.m SOE_FR.m SOE_FR_main.m
39 (SOE.m) function xy = SOE(t,y,para) xy = [y(2); % -para(2).^2.*y(1)-2.*para(1).*para(2).*y(2)+para(2).^2.*para(3).*sin(para(4).*t)]; end d dt x y = y ω 2 n x 2ζω 2 n y + ω 2 n Asin ωt ( )
40 SOE_FR_mai n.m xi = 0.5; % wn = 1; % In = 1; % omega = 1.5; % dt = 0.01; % T = 60; % xinit =[0;0]; % para = [xi,wn,in,omega]; % %@SOE_FR [t1,xy] = RK2(@SOE_FR,[0,T],xinit,dt,para); %2 Runge- Kutta x = xy(1,:); xmin = min(x)-0.1; xmax = max(x)+0.1; y = xy(2,:); ymin = min(y)-0.1; ymax = max(y)+0.1; subplot(2,1,1); %2 plot(t1,x); % axis([0 T xmin xmax]); xlabel('t'); ylabel('x'); subplot(2,1,2); %2 plot(x,y); % axis([xmin xmax ymin ymax]); xlabel('x'); ylabel('dx/dt');
41 d 2 x t ( ) dt 2 ( ) 2 dx t + 2ζω n dt + ω n 2 x t ( ) = ω n 2 sin ωt ( ) ( ) x( t) = G( jω) sin ωt + tan 1 G( jω) G( jω) = = ω n 2 ω jζω ω + ω 2 n n 1 ω 2 ω jζ ω ω +1 n n
42 ω ω n
43 TF_SOE.m FOL_boad_main.m
44 TF_SOE.m s function [ y ] = TF_SOE(s,para) y = para(2).^2./(s.^2 + 2.*para(1).*para(2).*s + para(2).^2.*ones(size(s))); end s G( s) = ω n 2 s 2 + 2ζω n s + ω n 2 Matlab
45 SOE_boad_ma in.m function main xi = 0.05;% wn = 1;% j=sqrt(-1);% para = [xi,wn]; % omega = 10.^(-1:0.05:1); % G = TF_SOE(j.*omega,para); % gain = 20.*log10(abs(G)); % kaku = angle(g); % subplot(2,1,1); %2 semilogx(omega,gain); %x xlabel('omega'); ylabel('db'); % subplot(2,1,2); %2 semilogx(omega,kaku); %x xlabel('omega'); ylabel('rad'); % end
46 RK2.m 4 Runge-Kutta 4 Runge-Kutta 2
47 4 Runge-Kutta x n +1 = x n + Δt t = nδt [ ] 6 F 1 + 2F 2 + 2F 3 + F 4 F = f ( x,t) 1 n F = f x + Δt 2 n 2 F Δt,t F = f x + Δt 3 n 2 F Δt,t F = f ( x + ΔtF,t + Δt) 4 n 3
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More information(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
More informationgrad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
More information18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
More information1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
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