,., 1.2, (Mode), : 1. Draw Mode, Boundary Mode PDE Mode, 2. Mesh Mode Solve Mode 3. Plot Mode, PDETOOL
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- 列愁 翁
- 7 years ago
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1 MATLAB 1 (PDETOOL) PDETOOL MATLAB pdetool : PDETOOL,, Generic Scalar Generic system Structual Mechnics,Plane Stress Structual Mechnics,Plane Strain Electrostatics Magnetostatics Ac Power Electromagnetics Conductive Media DC Heat Transfer Diffusion
2 ,., 1.2, (Mode), : 1. Draw Mode, Boundary Mode PDE Mode, 2. Mesh Mode Solve Mode 3. Plot Mode, PDETOOL
3 2 2 SQ1, C1, (0, 0), 0.3. Set fomula SQ1 C1, h = 1, r = y, h = 1, r = 0. Eliptic c = 1, a = 0, f = 0.. Contour Arrrows 2. 0 < x < a, b/2 < y < b/2 u = x 2 y,u. 1 u = 0 u(x = 0) = 0; u(x = a) = 0 u(y = b/2) = 0; u(y = b/2) = 0 u = xy 12 (a3 x 3 )+ n=1 a 4 b[( 1) n n 2 π ( 1) n ] n 5 π 5 sinh(nπb/2a) u, a = 5, b = 5. ( nπy ) ( nπx ) sinh sin a a [X,Y]=meshgrid(0:0.1:5,-5/2:0.1:5/2); Z1=0; for n=1:1:10 Z2=5^4*5*((-1)^n*n^2*pi^2+2-2*(-1)^n)*... sinh(n*pi.*y/5).*sin(n*pi.*x/5)/... (n^5*pi^5*sinh(n*pi*5/10)); Z1=Z1+Z2; end Z=Z1+X.*Y.*(5^3-X.^3)/12; 1.. :,1998,,223
4 colormap(hot); mesh(x,y,z) view(119,7). Options/Axes Limits x 0 5, y , (0, 2.5), (5, 2.5), (5, 2.5), (0, 2.5).,,, h = 1, r = 0. Elliptic, c = 1, a = 0, f = 0.,. Color Hight(3D plot), colormap hot { u t = a 2 u xx u(x, t = 0) = ϕ(x) 1.. :,1998,,407
5 + [ ] 1 u(x, t) = ϕ(ξ) 2a (x ξ) 2 πt e 4a 2 t dξ ϕ(x) = { 1, (0 x 1) 0, (x 0, x 1) 1 1 u(x, t) = 0 2a (x ξ) 2 πt e 4a 2 t dξ. xx=-10:.5:10; tt=0.01:0.1:1; tau=0:0.01:1; a=2; [X,T,TAU]=meshgrid(xx, tt, tau); F=1/2/2./sqrt(pi*T).*exp(-(X-TAU).^2/4/2^2./T); js=trapz(f,3); waterfall(x(:,:,1), T(:,:,1), js)
6 , figure h=plot(xx,js(1,:)) set(h, erasemode, xor ); for j=2:10 end set(h, ydata,js(j,:)); drawnow; pause(0.1) u t = a 2 u xx u(0, t) = 0, u(l, t) = 0 u(x, t = 0) = ϕ(x) 0 x 20, l = 20, a = 10 u(x, t) = n=1 ϕ(x) = 2 nπ { 1, (10 x 11) [ cos 10nπ 20 0, (x < 10, x > 11) ] 11nπ cos e n2 π 2 a t sin nπ x 50,,. function jxj
7 N=50; t=1e-5: :0.005; x=0:0.21:20; w=rcdf(n,t(1)); h= plot(x,w, linewidth,5); axis([ 0, 20, 0, 1.5]) for n=2:length(t) w=rcdf(n,t(n)); set(h, ydata,w); drawnow; end function u=rcdf(n,t) x=0:0.21:20; u=0; for k=1:2*n cht=2/k/pi*(cos(k*pi*10/20)-... cos(k*pi*11/20))*sin(k*pi*x./20); u=u+cht*exp(-(k^2*pi^2*10^2/400*t)); end %a=10,l=20..
8 PDETOOL. 10 x 10, 0.2 y 0.2,, q = 0, g = 0, Dirichlet, h = 1, r = 0. (Parabolic). time 0 : : 0.2, u(t0) 10 tanh(10000 x) 10 tanh(10000(x 0.05)),.. Color,Heigh(3-D plot) Animation u t a 2 u xx = bu x, (0 x l, t 0) u(x = 0) = 0, u(x = l) = 0 u(t = 0) = (x l/2) 2 u(x, t) = n=1 ( A n e n 2 π 2 a 2 ) l 2 + b2 4a 4 t bx e 2a 2 sin nπ l x A n = 2 l ( x l ) 2 e bx 2a l sin nπ l xdx.. 0 t ,. (a) t, x (b) (c) :,1998,,215
![(a) (b) (c), a 2 = 50, b = 5, l = 1,,. a2=50; b=5; [x, t] = meshgrid(0:0.01:1,0:0.000001:0.0005); Anfun=inline( 2*(x-0.5).^2.*exp(5*x./2./50).](/docs-images/50/26269620/images/9-0.png "*sin(n*pi*x), x, u=0 for n=1:30 An=quad(Anfun,0,1,[ ],[ ],n); un=an*exp(-(n*n*pi*pi*50+25/4/2500).*t).*... exp(5/2/50.*x).*sin(n*pi*x); u=u+un; size(u) end mesh(x,t,u) figure subplot(2,1,1) plot(u(1,:)) subplot(2,1,2) plot(u(end,:))")
9 (a) (b) (c), a 2 = 50, b = 5, l = 1,,. a2=50; b=5; [x, t] = meshgrid(0:0.01:1,0: :0.0005); Anfun=inline( 2*(x-0.5).^2.*exp(5*x./2./50).*sin(n*pi*x), x, u=0 for n=1:30 An=quad(Anfun,0,1,[ ],[ ],n); un=an*exp(-(n*n*pi*pi*50+25/4/2500).*t).*... exp(5/2/50.*x).*sin(n*pi*x); u=u+un; size(u) end mesh(x,t,u) figure subplot(2,1,1) plot(u(1,:)) subplot(2,1,2) plot(u(end,:))
10 (a),(b). (a), (b). N=500; dx=0.01; dt= ; c=50*dt/dx/dx; A=500; b=5; x=linspace(0,1,100) ; uu(1:100,1)=(x-0.5).^2; % figure h=plot(x,uu(:,1), linewidth,5); set(h, EraseMode, xor ) axis([0,1,0,0.25]); for k=2:200 uu(2:99,2)=(1-2*c)*uu(2:99,1)+c*(uu(3:100,1)+uu(1:98,1))-... b*dt/dx*(uu(3:100,1)-uu(2:99,1)); uu(1,2)=0; uu(100,2)=0; % uu(:,1)=uu(:,2); set(h, YData,uu(:,1)) ; drawnow; pause(0.01) end
11 3. 6. u tt = a 2 u xx, u(x, t = 0) = ϕ(x); u t (x, t = 0) = ψ(x) x+at u(x, t) = 1 1 [ϕ(x + at) + ϕ(x at)] + ψ(ξ)dξ 2 2a x at, 1. ϕ(x), ψ(x) sin 7π ( 3l ϕ(x) = l x, 7 x 4l ) 7 0,. u(x, t) = 1 [ϕ(x + at) + ϕ(x at)] 2, ϕ(x), x + at x at x, at, ϕ(x + at), ϕ(x at). l = 1, 140, 0 at 60.. u(1:140)=0; x=linspace(0,1,140); u(61:80)=0.05*sin(pi*x(61:80)*7); uu=u; h=plot(x,u, linewidth,3); axis([0,1,-0.05,0.05]);
12 set(h, EraseMode, xor ) for at=2:60 lu(1:140)=0; ru(1:140)=0; lx=[61:80]-at; rx=[61:80]+at; lu(lx)=0.5*uu(61:80); ru(rx)=0.5*uu(61:80); u=lu+ru; set(h, XData,x, YData,u) ; drawnow; pause(0.1) end ϕ(x), ψ(x) { 1, (0 x 1) ψ(x) = 0, 1, u(x, t) = 1 2a x+at x at ψ(ξ)dξ = 1 x+at ψ(ξ)dξ 1 x at ψ(ξ)dξ 2a 2a
13 0, (x + at 0) 1 x+at 1 ψ(ξ)dξ = (x + at), (0 (x + at) 1) 2a 2a 1 2a, (1 (x + at)) 0, (x at 0) 1 x at 1 ψ(ξ)dξ = (x at), (0 (x at) 1) 2a 2a 1 2a, (1 (x at)) x Ψ(x) = 1 ψ(ξ)dξ 2a a., 10 l 10, a = 1,. xpat = x + at, (0 (x + at) 1), xmat = x at, (0 (x at) 1). 4., if,.. t=0:0.005:8; x=-10:0.1:10; a=1; [X,T]=meshgrid(x,t); xpat=x+a*t; xpat(find(xpat<=0))=0; xpat(find(xpat>=1))=1; xmat=x-a*t; xmat(find(xmat<=0))=0; xmat(find(xmat>=1))=1; jf=1/2/a*(xpat-xmat);
14 h=plot(x,jf(1,:), linewidth,3) ; set(h, erasemode, xor ); axis([ ]) hold on for j=2:length(t) pause(0.01) set(h, ydata,jf(j,:)); drawnow; end %., jf=1/2/a*(xpat-xmat); jf=1/2/a*xpat); jf=-1/2/a*xmat; Ψ(x + at) Ψ(x + at). (a). Ψ(x + at) ; (b). Ψ(x + at) 7.,
15 u tt = a 2 u xx u(x = 0, t) = 0; u(x = l, t) = 0 u(x, t = 0) = ϕ(x); u t (x, t = 0) = ψ(x) u(x, t) = n=1 ( A n cos nπat l A n = 2 l 0 l 0 + B n sin nπat ) sin nπx l l ϕ(ξ) sin nπξ dξ l B n = 2 ψ(ξ) sin nπξ dξ nπα l l..,. 1., sin 7π ( 3l ϕ(x) = l x, 7 x 4l ) 7 0, l = 1, a = 1, [ 1 A n = sin(7 n)π 4 ] (7 n)π 7 sin(7 n)π3 [ 7 1 sin(7 + n)π 4 ] (7 + n)π 7 sin(7 + n)π3 (n 7) 7 A 7 = 1 7, B n = 0
16 ,. 3.1.,, n., n = 10, 50,. wfun.m k, jxj. function jxj N=50 t=0:0.005:2.0; x=0:0.001:1; ww=wfun(n,0); ymax=max(abs(ww)); h= plot(x,ww, linewidth,3); axis([ 0, 1, -ymax, ymax]) sy=[ ]; for n=2:length(t) ww=wfun(n,t(n)); set(h, ydata,ww); drawnow; sy=[sy,sum(ww)]; end function wtx=wfun(n,t) x=0:0.001:1; a=1; wtx=0; for I=1:N if I~=7 wtx=wtx+0.05*( (sin(pi*(7-i)*4/7)-sin(pi*(7-i)*3/7))... /(7-I)/pi-(sin(pi*(7+I)*4/7)-sin(pi*... (7+I)*3/7))/(7+I)/pi )*cos(i*pi*a*t).*sin(i*pi*x); else
17 end end wtx=wtx+0.05/7*cos(i*pi*a*t).*sin(i*pi*x); N, ,,,,,. 50
18 2., B.3. x = i x, t = j t,, u i,j+1 = c(u i+1,j + u i 1,j ) + 2(1 c)u i,j u i,j 1 c = a 2 ( t)2 ( x) 2 u i,1 = ϕ u i,2 = 1 2 [c(u i+1,1 + u i 1,1 ) + 2(1 c)u i,1 ] clear N=4010; dx=0.0024; dt=0.0005; c=dt*dt/dx/dx; x=linspace(0,1,420) ; u(1:420,1)=0; u(181:240,1)=0.05*sin(pi*x(181:240)*7); u(2:419,2)=u(2:419,1)+c/2*(u(3:420,1)-2*u(2:419,1)+u(1:418,1)); h=plot(x,u(:,1), linewidth,3); axis([0,1,-0.05,0.05]); set(h, EraseMode, xor, MarkerSize,18) for k=2:n set(h, XData,x, YData,u(:,2)) ; drawnow; u(2:419,3)=2*u(2:419,2)-u(2:419,1)+c*(u(3:420,2)... -2*u(2:419,2)+u(1:418,2)); u(2:419,1)=u(2:419,2); u(2:419,2)=u(2:419,3);
19 end.,. 3. PDETOOL,φ = exp( (x 1). 2. log10(2)/0.1 2 ), ψ = 0,. Options/Axes Limits x 0 2, y , (0, 0.15), (2, 0.15) (2, 0.15), (2, 0.15)., h = 1, r = 0,, g = 0, q = 0. hyperbolic, c = 1, a = 0, f = 0, d = 1.,. Solve Parameters, Time 0 : 0.05 : 3.6, u(to) exp(-(x-1).^2.*log10(2)/0.1^2) u (to) 0,. Plot Parameters,, Plot type, Color, Height(3-D Plot), Animation,.,,.
20 4. ( ) ( 3l 1, ψ(x) = 7 x 4l ) 7 0, S u = n=1 B n sin nπa t sin nπ l l x B n = 2l (cos 37 n 2 π 2 a na cos 47 ) na a = 1, l = 1,.,. function psi N=50; t=0:0.005:2.0; x=0:0.001:1; ww=psi1fun1(n,0); h= plot(x,ww, linewidth,3); axis([ 0, 1, -0.08, 0.08]) sy=[ ]; for n=2:length(t) ww=psi1fun1(n,t(n)); set(h, ydata,ww); drawnow;
21 end sy=[sy,sum(ww)]; function wtx=psi1fun1(n,t) x=0:0.001:1; a=1; wtx=0; for k=1:n Bk=2/(k*k*pi*pi)*(cos(3*k*pi/7)-cos(4*k*pi/7)); wtx=wtx+bk*sin(k*pi*t)*sin(k*pi*x); end 50,,,,, u i,2 = ψ t,,. clear N=4025; dx=0.0024;
22 dt=0.0005; c=dt*dt/dx/dx; u(1:420,1)=0; x=linspace(0,1,420) ; u(1:420,1)=0; u(180:240,2)=dt*0.5; h=plot(x,u(:,1), linewidth,5); axis([0,1,-0.05,0.05]); set(h, EraseMode, xor, MarkerSize,18) for k=2:n set(h, XData,x, YData,u(:,2)) ; drawnow; %pause u(2:419,3)=2*u(2:419,2)-u(2:419,1)+c*(u(3:420,2)... -2*u(2:419,2)+u(1:418,2)); u(2:419,1)=u(2:419,2); u(2:419,2)=u(2:419,3); end 8. r 0 v = v 0 cos θ cos ωt, u, r 0 λ. 3,, { u tt a 2 u = 0 u r (r = r 0 ) = v 0 P 1 (cos ϑ)e iωt cos ωt Re(e iωt ), e iωt,. u = v 0r0 3 ( 1 2 r + i ω ) p 2 1 (cos θ)e iω a (r at) ar.,, 3.. :,1998,,369
23 u = v 0ωr0 3 2ar p 1(cos θ) sin ω (r at) a, r0 = 0.2; v0 = 2; k = 60; a = 2; clear r0=0.2; v0=2; k=60; a=2; theta=linspace(0,2*pi,50); rho=0.2:0.1:4; [Th,Rh]=meshgrid(theta,rho); [X,Y]=pol2cart(Th,Rh); rh=sqrt(x.^2+y.^2); th=atan(y./x); for t=0:0.001:0.03 u=real(v0/2*r0^3*(-1./rh.^2+i*k./rh).*... cos(th).*exp(k*(rh+2*t)*i)); surf(x,y,u) view(-32,28) pause(0.5) end;,.,,,,.
24 ,. surf(x,y,u); view(-32,28) contour(x,y,u); axis([ ]); axis square ,
25 u + k 2 u = 0 C1 C2, 0.2, 1.C1,C2. C2-C1., C1, h=1, r=cos(atan(y./x)), C2, g=0,q= 60i. A8.1. Elliptic, c = 1, a = 3600, f = 0.,. Plot Parameters, Plot type, Color Height (3-D Plot),.,,. OK., : Mesh/Export Mesh OK p,e,t. Solve/Export Solution
26 OK u,. sswt.m,.,. % program sswt.m h=newplot; hf=get(h, Parent ); set(hf, Renderer, zbuffer ) axis tight set(gca, DataAspectRatio,[1 1 1]); axis off M=moviein(10,hf); maxu=max(abs(u)); colormap(cool) for j=1:10, ur=real(exp(-j*2*pi/10*sqrt(-1)) u); pdeplot(p,e,t, xydata,ur, colorbar, off, mesh, off ); caxis([-maxu maxu]); axis tight, set(gca, DataAspectRatio,[1 1 1]); axis off M(:,j)=getframe; end movie(hf,m,10);
27 legendre(n,x) N X, >>legendre(2,0:0.1:0.2) x = 0 x = 0.1 x = 0.2 m = 0 P2 0 (0)= P2 0 (0.1)= P2 0 (0.2)= m = 1 P2 1 (0)=0 P2 1 (0.1)= P2 1 (0.2)= m = 2 P2 2 (0)= P2 2 (0.1)= P2 2 (0.2)=
28 >>x=0:0.01:1; >>y1=legendre(1,x); >>y2=legendre(2,x); >>y3=legendre(3,x); >>y4=legendre(4,x); >>y5=legendre(5,x); >>y6=legendre(6,x); >>plot(x,y1(1,:),x,y2(1,:),x,y3(1,:),... x,y4(1,:),x,y5(1,:),x,y6(1,:)) 1.3 P 0,1,2, x >>x=0:0.01:1; >>y=legendre(3,x); >>plot(x, y(1,:),,x, y(2,:), -.,... x, y(3,:), :,x, y(4,:), -- ) >>legend( P 3 0, P 3 1,... P 3 2, P 3 3 ); θ P (0,1) 1, P (0,1,2) 2, P (0,1,2,3) 3
29 rho=legendre(1,cos(0:0.1:2*pi)); t=0:0.1:2*pi; rho1=legendre(2,cos(0:0.1:2*pi)); rho2=legendre(3,cos(0:0.1:2*pi)); subplot(3,4,1),polar(t,rho(1,:)) subplot(3,4,2), polar(t,rho(2,:)) subplot(3,4,5) polar(t,rho(2,:)) subplot(3,4,6) polar(t,rho(2,:)) subplot(3,4,7) polar(t,rho(2,:)) subplot(3,4,9) polar(t,rho(2,:)) subplot(3,4,10) polar(t,rho(2,:)) subplot(3,4,11) polar(t,rho(2,:)) subplot(3,4,12) polar(t,rho(2,:)) P 1, P 2, P MATLAB 5, besselj bessely besselh besseli (Bessel) (Neumann) (Hankel) ( ) besselk ( ) 2.1 (Bessel)
30 >> y=besselj(0:3,(0:.2:10) ); >> figure(1) >> plot((0:.2:10),y) >> legend( J 0, J 1, J 2, J 3 ) J 0,1,2, interp1, interp1q interp2, 0 x 50 x=0:0.05:50; y=besselj(0,x); LD=[]; for k=1:1000, if y(k)*y(k+1)<0 end end LD h=interp1(y(k:k+1), x(k:k+1),0); LD=[LD,h] 2.2 (Neumann) N 0,1 >> y=bessely(0:1,(0:.2:10) ); >> plot((0:.2:10),y) >> grid on
31 2.3 I 0,1 >>I=besseli(0:1,(0.1:0.1:3) ); >>plot((0.1:0.1:3),i) 2.4 K 0,1 >> K=besselk(0:1,(0.1:0.1:3) ); >> plot((0.1:0.1:3),k) 2.5 j 0,1,2,3 x = 0 0/0. x=eps:0.2:15; y1=sqrt(pi/2./x).*besselj(1/2,x); y2=sqrt(pi/2./x).*besselj(3/2,x); y3=sqrt(pi/2./x).*besselj(5/2,x); y4=sqrt(pi/2./x).*besselj(7/2,x); plot(x,y1,x,y2,x,y3,x,y4) 2.6 n 0,1,2,3 x=0.8:0.2:15; y1=sqrt(pi/2./x).*bessely(1/2,x); y2=sqrt(pi/2./x).*bessely(3/2,x); y3=sqrt(pi/2./x).*bessely(5/2,x); y4=sqrt(pi/2./x).*bessely(7/2,x); plot(x,y1,x,y2,x,y3,x,y4) axis([ ]) grid on
32 3 (GUI) 1. guide Guide guide, GUI Open Existing GUI Create New GUI 2.,
33 1. 2. M M 3. (pushbutton),,,. String 4. (togglebutton),.,, 5. (radiobutton) tring. Value 1,. Value 0, Value 0 6. (checkbox) tring. Value 1, Value (edit) String 8. (text) String 9. (slider),,,. Macx Min 10. (frame),, 11. (listbox), Min Max Value
34 12. (popupmenu),,,,, Value String 13. (axes) 3. 1., 2. 3.,, fig, M. M, GUI 4. 1., ; ; ;.
35 String FontSize (Property Inspector), Manubar none figure, Label Tag Callback prof edit huasq ex close gcf, huasq. fig huasq. m huasq.m huasq 2.,,,, guide
36 M, popval switch popaval surf contour,. vlist cell vlist eval,, M,, M % M function varargout = huasq(varargin) if nargin == 0 % LAUNCH GUI fig = openfig(mfilename, reuse ); set(fig, Color,get(0, defaultuicontrolbackgroundcolor )); handles = guihandles(fig); guidata(fig, handles); if nargout > 0, varargout{1} = fig; end elseif ischar(varargin{1}) % INVOKE NAMED SUBFUNCTION OR CALLBACK try if (nargout) [varargout{1:nargout}] = feval(varargin{:}); % FEVAL s else
37 feval(varargin{:}); % FEVAL switchyard end catch disp(lasterr); end end % function varargout = radiobutton1_callback(h, eventdata, handles, val1=get(handles.radiobutton1, max ); set(handles.radiobutton1, value, val1); val2=get(handles.radiobutton2, min ); set(handles.radiobutton2, value,val2); hsq1 % % function varargout = radiobutton2_callback(h, eventdata, handles, val1=get(handles.radiobutton1, min ); set(handles.radiobutton1, value, val1); val2=get(handles.radiobutton2, max ); set(handles.radiobutton2, value,val2); hsq2 % % function varargout = togglebutton1_callback(h, eventdata, handles, grid % function varargout = checkbox1_callback(h, eventdata, handles, var title( ) % function hsq1 % t=-3:0.1:3; y1=0.5*(exp(t)-exp(-t)); plot(t,y1) %
38 function hsq2 % t=-3:0.1:3; y2=0.5*(exp(t)+exp(-t)); plot(t,y2) % M function varargout = text2_callback(h, eventdata, handles, varargi function varargout = popupmenu1_callback(h, eventdata, handles, va function varargout = listbox1_callback(h, eventdata, handles, vara % function varargout = slider1_callback(h, eventdata, handles, varar val2=num2str(get(handles.slider1, value )); % set(handles.edit2, string, val2); % function varargout = edit2_callback(h, eventdata, handles, varargi val1=str2double(get(handles.edit2, string )); % set(handles.slider1, value, val1); % function varargout = pushbutton1_callback(h, eventdata, handles, v edit huat % function varargout = pushbutton2_callback(h, eventdata, handles, v close gcf
39 % function varargout = pushbutton3_callback(h, eventdata, handles, v popval=get(handles. popupmenu1, value ); % val1=str2double(get(handles.edit2, string )); % [X,Y]=meshgrid(-5:0.2:val1); switch popval case 1 set(handles. text2, string, Z=X.*(-X.^2-Y.^2) ); Z=X.*exp(-X.^2 - Y.^2); case 2 str2= R=sqrt((2*X).^2+(2*Y).^2)+eps; Z=sin(R)./R; set(handles. text2, string,[str2,sprintf( \n )]); % R=sqrt((2*X).^2+(2*Y).^2)+eps; Z=sin(R)./R; end vlist=get(handles.listbox1, value ); % 1,2 liststr={ mesh(x,y,z),, contour(x,y,z) }; % cla eval(liststr{vlist}); % hold on; % : ρ < a u = xy, u(ρ = 0) = 0. u t = a 2 u xx u(0, t) = 0, u x (l, t) = 0 u(x, 0) = u 0 x/l
绳拉线轴的运动
绳拉线轴的运动 物理系 2001 级程鹏学号 01261094 如图所示, 一绳拉动线轴在水平面上做无滑滚动 线轴与地面的接触点为瞬心 根据在惯性系中刚体对瞬心的角动量定理得, 当绳与地面的夹角大于角 A 时, 力 F 对瞬心的力矩使线轴逆时针转动, 反之当绳与地面的夹角小于角 A 时, 线轴顺时针转动 源程序如下 : %%program gz.m f=1; R=5;r=2; m=10;l=8; theta=pi/6;
%第一次作业
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