Falling Chain Falling Chain Luu Chau Kayla Chau Jonathan Bernal
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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