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

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Jake il 7 Mar 2024
Commentato: Alex Sha il 8 Mar 2024
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?
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### Risposta accettata

Star Strider il 7 Mar 2024
Try this —
% 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])
.
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Alex Sha il 8 Mar 2024
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

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