% single-mass-spring data analysis by Tiernan Fogarty

x=rfcut2(:,2); % load position data
xd=x % keep rough data
m=length(x);
x=rfcut2(11:m,2); % cut the first few values to start at the max of a period
vshift=(max(x)+min(x))/2; % calculate vert shift
x=x-vshift; % shift position values to be centered at zero
td=rfcut2(:,1);
t=rfcut2(11:m,1); % cut the first few values to start at the max of a period

amp=(max(x)-min(x))/2; % compute amplitude
phi=22; % start time (phase shift)

k=200*(real(acos(x(1:7)/amp))./(t(1:7)-phi)).^2; % estimate k from data points
kest=mean(k); % k-estimate is average of k values from first half-period
modx=amp*cos(sqrt(kest/200)*(t-phi)); % compute model (estimate) data given k estimate

figure(1) % plot data vs model for initial k-estimate
subplot(2,1,1)
plot(t,x,t,modx,'r')
title('Initial Estimate for k')
legend('Data', 'Model','Location','SouthEast')
ylim([-.12 .12])

diff=x-modx;
subplot(2,1,2)
plot(t,diff)
title('Difference between Data and Model')
ylim([-.2 .2])

% revise k estimate via least squares minimization
KFunction = @(ke,t) amp*cos((sqrt(ke/200))*(t-phi));
[o1] = lsqcurvefit(KFunction,[kest],t,x);
kest=o1;


modx=amp*cos(sqrt(kest/200)*(t-phi)); % recompute model with new k estimate

figure(2) % view data, model, and the difference between model and data 
subplot(2,1,1)
plot(t,x,t,modx,'r')
title('Least Squares Fit for k')
legend('Data', 'Model','Location','SouthEast')
ylim([-.12 .12])

diff=x-modx;
subplot(2,1,2)
plot(t,diff)
title('Difference between Data and Model')
ylim([-.2 .2])

figure(3)
plot(td,xd,'*',td,xd)
title('Data points')