How to Curvefit amplitude output of a spring-mass-damper system to find coefficients?

First of all, I'm not sure if the title gives the right idea, and if this is possible. If that's the case, my apologies in advance.

I have a data set that displays the amplitude of a system that shows decaying motion. (Sample data set attached). I want to identify the quadratic damping applied in the system. So, I want to find a polynomial in the form a + b*(amplitude). I don't want a polynomial of nth order either. How can I do this?

Answer 1

[Star Strider]

Try this —

load('sampleData.mat')
% whos

objfcn = @(b,t) b(1) .* exp(b(2).*t) .* cos(2*pi*b(3)*t + b(4)) + b(5);

Lvlm = islocalmax(y, 'MinProminence',0.5);
Per = mean(diff(x(Lvlm)));
Freq = Per;
ymax = max(y);

B0 = [ymax; -1E-2; Freq; randn(2,1)];
[B,fv] = fminsearch(@(b) norm(y - objfcn(b,x)), B0)

B = 5x1

13.4564 -0.0478 -0.2511 0.0046 0.0677

fv = 86.0608

% nnz(Lvlm)

figure
plot(x,y)
hold on
% plot(x(Lvlm), y(Lvlm), 'vr')
plot(x, objfcn(B,x))
hold off
grid
text(25, 12, sprintf('$f(x) = %.3f \\cdot e^{%.3f \\cdot t} \\cdot cos(2 \\cdot \\pi \\cdot %.3f\\ t %+.3f) %+.3f$',B), 'Interpreter','latex', 'FontSize',11)

% xlim([0 50])

.

[Alex Sha]

If taking fitting function as @Star Strider proposed:

b(1) .* exp(b(2).*t) .* cos(2*pi*b(3)*t + b(4)) + b(5);

the results below will be little better:

Sum Squared Error (SSE): 3612.83461221834
Root of Mean Square Error (RMSE): 0.299830523951318
Correlation Coef. (R): 0.990716970140293
R-Square: 0.981520114923963

Parameter	Best Estimate    
---------	-------------    
b1       	-13.9887747654467
b2       	-0.05066160816698
b3       	-199.749779715466
b4       	-3.01726078981019
b5       	0.287608394237128