Falling ChainFalling ChainLuu Chau

Kayla ChauJonathan Bernal

ReferenceReference

On the paradox of the free falling folded chain M.Schagerl A. Steindl W. Steiner H. Troger

Dr. Tyler McMillen

Initial Condition for ParametersInitial Condition for Parametersspeed=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 TM=15; % total mass of the chainL=2; % length of the chain (meters)m=1; % mass attached to end of chaina=.00475; % length of linkb=.0025; % width of linke=b/a; % ratio h=L/n; % distance between two jointsmu=M/n; % mass of each linkg=9.81; % gravitytimes=linspace(0,T,frames); % number of moments

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JOINT

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 ChainInitialize Condition for Chain

Reference p.157,162 Reference p.157,162

Moment of Inertia

Forces acting on joints

Calculate Moments of InertiaCalculate Moments of InertiaIy=((mu*h^2)/12)*(2*a/h)^2*(1+3*e)/(1+e); %moment of

inertiaIz=((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; endend

p.157,162

Compute Angles Compute Angles

[t, phi] = ode23(@equation,times,initial);

ODE outputODE output

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

Reference p.161Reference p.161

Coordinates of joints

Compute Coordinates of Each JointCompute Coordinates of Each Joint

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))); endend

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OutputOutput

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

First frame with corresponding coordinates

Plot GraphPlot Graph

xball=0.4;yball=-0.5*g*times.^2;

plot(times,yball,'b',times,y(:,n+1),'r')

Plot MoviePlot Moviefor 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 endmovie(mov,2,speed)

MovieMoviemovie(mov,1,speed)

Reference p.162Reference p.162

n equations of motion (for each link)

Function in ODEFunction in ODEfor 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];

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