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Falling Chain

Falling Chain. Luu Chau Kayla Chau Jonathan Bernal. On the paradox of the free falling folded chain M.Schagerl A. Steindl W. Steiner H. Troger Dr. Tyler McMillen. Reference. speed=1; % speed of falling chain (1_slow 100_fast) T=1; % time of calculations (secs)

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Falling Chain

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  1. Falling Chain Luu Chau Kayla Chau Jonathan Bernal

  2. On the paradox of the free falling folded chain • M.Schagerl • A. Steindl • W. Steiner • H. Troger Dr. Tyler McMillen Reference

  3. speed=1; % speed of falling chain (1_slow 100_fast) T=1; % time of calculations (secs) n=7; % number of links (must be odd number) frames=5; % number of frames per T M=15; % total mass of the chain L=2; % length of the chain (meters) m=1; % mass attached to end of chain a=.00475; % length of link b=.0025; % width of link e=b/a; % ratio h=L/n; % distance between two joints mu=M/n; % mass of each link g=9.81; % gravity times=linspace(0,T,frames); % number of moments p.156 Initial Condition for Parameters

  4. JOINT

  5. initial=[zeros(1,(n-1)/2) pi/2 ones(1,(n-1)/2)*pi zeros(1,n)]; initial = 0 0 0 1.5708 3.1416 3.1416 3.1416 0 0 0 0 0 0 0 Initialize Condition for Chain

  6. Moment of Inertia Forces acting on joints Reference p.157,162

  7. Iy=((mu*h^2)/12)*(2*a/h)^2*(1+3*e)/(1+e); %moment of inertia Iz=((mu*h^2)/12)*(2*a/h)^2*(1+e)^2; %moment of inertia for i=1:n for j=1:n G(i,j)=(M/mu)*h+n*h-(max(i,j)-0.5)*h; %nxn matrix, equations of motion end if (i/2)==(i-ceil(i/2)) %if “i” is even I(i)=Iz; else %if “i” is odd I(i)=Iy; end end p.157,162 Calculate Moments of Inertia

  8. [t, phi] = ode23(@equation,times,initial); Compute Angles

  9. t = phi = 0 0 0 0 1.5708 3.1416 3.1416 3.1416 0.2500 -0.0442 0.0491 0.2926 0.3162 2.5375 3.3701 3.0859 0.5000 0.0887 0.0322 0.3209 0.0319 -0.1541 0.9014 3.2265 0.7500 -0.5014 1.4469 0.0484 -1.4148 0.3501 0.3505 -0.1128 1.0000 -0.0366 -0.5347 0.2591 -1.1734 2.4062 3.1726 -0.9964 ODE output

  10. Coordinates of joints Reference p.161

  11. for i=1:frames for j=2:n+1 x(i,j)=x(i,j)+h*sum(sin(phi(i,1:j-1))); y(i,j)=y(i,j)-h*sum(cos(phi(i,1:j-1))); end end p. 161 Compute Coordinates of Each Joint

  12. x = 0 0 0 0 0.2857 0.2857 0.2857 0.2857 0 -0.0126 0.0014 0.0838 0.1726 0.3349 0.2702 0.2861 0 0.0253 0.0345 0.1247 0.1338 0.0899 0.3140 0.2897 0 -0.1373 0.1462 0.1600 -0.1222 -0.0242 0.0739 0.0417 0 -0.0105 -0.1561 -0.0829 -0.3463 -0.1546 -0.1635 -0.4034  y = 0 -0.2857 -0.5714 -0.8571 -0.8571 -0.5714 -0.2857 0 0 -0.2854 -0.5708 -0.8444 -1.1159 -0.8808 -0.6025 -0.3172 0 -0.2846 -0.5702 -0.8413 -1.1269 -1.4092 -1.5865 -1.3018 0 -0.2505 -0.2859 -0.5712 -0.6156 -0.8840 -1.1524 -1.4363 0 -0.2855 -0.5314 -0.8075 -0.9181 -0.7062 -0.4206 -0.5759 Output

  13. First frame with corresponding coordinates

  14. xball=0.4; yball=-0.5*g*times.^2; plot(times,yball,'b',times,y(:,n+1),'r') Plot Graph

  15. for i=1:frames plot(x(i,:),y(i,:),'.-') %chain hold on plot(x(i,n+1),y(i,n+1),'o','MarkerFaceColor','r','MarkerSize',8) %end of chain plot(xball,yball(i),'o','MarkerFaceColor','g','MarkerSize',9) %falling object axis([-2 2 -L 0]) mov(i)=getframe; hold off end movie(mov,2,speed) Plot Movie

  16. movie(mov,1,speed) Movie

  17. n equations of motion (for each link) Reference p.162

  18. for i=1:n % right side of equation 4.2 f(i+n)=-h*sum(sin(X(i)-X(1:n)).*X(n+1:2*n).^2.*G(i,:)')-g*sin(X(i))*G(i,i); A(i,1:n)=(h*cos(X(i)-X(1:n)).*G(i,:)')'; % left side of 4.2 A(i,i)=A(i,i)+(I(i)/mu-h^2/4); end re1=A\f(n+1:2*n)'; re = [X(n+1:2*n); re1]; p.162 Function in ODE

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