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How to obtain the parameters using curve fitting for a damped driven harmonic oscillator?
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Rajesh
2016 年 5 月 11 日
Hi,
I am trying to fit my experimental data to analytical solution to determine the parameters involved in the system behavior. I am trying to model a cantilever with a periodically varying force (sawtooth wave function with period 1.655 sec) at its tip. The system behaves as a damped driven harmonic oscillator which can be described by the differential equation:
m(d^2x/dt^2)+b(dx/dt)+kx=f(t)
The analytical solution to this differential equation is of the form:
x(t)=x_h(t)+x_p(t)
x_h(t)= exp(-alpha*time)*xm.*cos(gamma*time+phi) //homogeneous part
x_p(t)=Q*T/(2*m*wo) + {summation n=1 to N [ Q*T*(-1/(m*pi)*1/n*sin(wn*time- delta)/sqrt((wo^2-wn^2)^2+4*alpha^2*wn^2))] } //particular solution
wn=2*pi*n/T; delta=atan((2*wn*alpha)/(wo^2-wn^2));
Q=3.2700e-06 m^3/s in my case (slope of the sawtooth wave); T=1.655s; wo=sqrt(k/m); alpha=b/(2*m); gamma=sqrt(k/m-alpha^2); xi=b/(2*sqrt(m*k)); A=x(t=0); B=1/gamma*(dxdt0+xi*wo*A); xm=sqrt(A^2+B^2); phi=atan(B/A);
I am trying to use lsqcurvefit to fit the data and obtain the values of m, b and k. I get a fit (the blue curve) that goes midway through the data. What I don't understand is why the fit does not capture the oscillatory behavior in the data? nlinfit also gives a similar fit wherein the oscillations are not captured by the fit.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/164080/image.png)
The above is the fit that I obtain. I have attached the matlab code that I have written.How can I obtain a better fit for my data? What other curve fitting algorithm would be more suitable?
Thanks,
Rajesh
4 件のコメント
Star Strider
2016 年 5 月 12 日
Rajesh
2016 年 5 月 12 日
The two problem are quite different. The problem that you have pointed involves the reading from a photodetector in the absence of any external force. It is the free vibration of a capillary cantilvered at one end. The data there is the noise/ power spectrum of the cantilever and it involved finding out the dominant frequencies in the spectrum.
The problem her is one in which a periodically varying force is applied (sawtooth wave) at the tip of the capillary. This makes it behave like a damped oscillator which repeats its behavior over a time period. The two vectors in matlab code corresponds to one such time period. From this dataset I want to fit the equation of a harmonic oscillator subject to an external force and obtain the values of m, b and k. The readings from this problem are attained by image processing of video of the capillary at high frame rate.
Star Strider
2016 年 5 月 12 日
The code I posted would be the same for both problems. It would implement the solution of your differential equation. You would be able to get your coefficients from the coefficients estimated by the fminsearch call (although you could use other curve-fitting functions with my objective function without changing it).
Rajesh
2016 年 5 月 12 日
I took the same approach for this problem but in this case I am not able to capture the oscillations as I would have wanted. I used the same code that you had posted with minor modifications (changed the sample rate and the objective function). The fit that I get when I include 10 sine terms:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/182953/image.png)
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Fs=1000; Fn = Fs/2;
L = size(t,1);
y = detrend(y);
fty = fft(y)/L;
Fv = linspace(0, 1, fix(L/2)+1)*Fn;
Iv = 1:length(Fv);
Wp = 5/Fn;
Ws = 20/Fn;
Rp = 1;
Rs = 25;
[n,Wn] = buttord(Wp,Ws,Rp,Rs);
[b,a] = butter(n,Wn);
[sos,g] = tf2sos(b,a);
freqz(sos, 2048, Fs)
yf = filtfilt(sos,g,y);
yu = max(yf);
yl = min(yf);
yr = (yu-yl);
yz = yf-yu+(yr/2);
zt = t(yz(:) .* circshift(yz(:),[1 0]) <= 0);
per = 2*mean(diff(zt));
ym = mean(y);
fit = @(b,x) b(1) .* exp(b(2).*x) .* (sin(2*pi*x./b(3) + 2*pi/b(4)) + sin(2*pi*x./b(5) + 2*pi/b(6)) + sin(2*pi*x./b(7) + 2*pi/b(8)) + sin(2*pi*x./b(9) + 2*pi/b(10)) + sin(2*pi*x./b(11) + 2*pi/b(12))...
+sin(2*pi*x./b(13) + 2*pi/b(14)) + sin(2*pi*x./b(15) + 2*pi/b(16)) + sin(2*pi*x./b(17) + 2*pi/b(18)) + sin(2*pi*x./b(19) + 2*pi/b(20)) + sin(2*pi*x./b(21) + 2*pi/b(22)))+ b(23);
fcn = @(b) sum((fit(b,t) - yf).^2);
s = fminsearch(fcn, [yr; -1; per; -1; per; -1; per; -1; per; -1; per; -1; per; -1; per; -1; per; -1; per; -1; per; -1; ym]);
xp = linspace(min(t),max(t),819);
figure(1)
plot(t,y,'b')
hold on
plot(t,yf,'y', 'LineWidth',2)
plot(xp,fit(s,xp), 'r', 'LineWidth',1.5)
hold off
xlabel('Time')
ylabel('Amplitude')
legend('Original Data', 'Lowpass-Filtered Data', 'Fitted Curve')
The analytical solution is for a complete time period and to get a good fit I had to excluded the data from both the ends. I am not sure how to obtain the original set of parameters that I wanted to fit since now the time interval is changed. Any suggestions?
Thanks
回答 (0 件)
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