I want to envelope the damped curve with an exponential function passing through the peak points. Can someone please help?

X = A1(exp(-alpha*t)).*cos(wd*t)+ A2*(exp(-alpha*t)).*sin(wd*t)* is the equation of the damped curve.

The parameters are t=0:0.000050:0.1; A1 = 12.87701558; A2 = -12.70012814; alpha = 67.91641; wd = 4770.680551

The equation of the enveloping curve is I = Io * exp(-67.91641*t)

where Io is the peak values of the damped curve

I am attaching the image of damped curve as well as its enveloping curve through peak points.

Answer 1

[Ameer Hamza]

Since it is a second-order system, the envelop parameter Io can be analytically expressed as a function of A1 and A2. The following function follows from the trigonometric identities.

Io = sqrt(A1^2+A2^2);

Try the following code,

t=0:0.000050:0.1; 
A1 = 12.87701558; 
A2 = -12.70012814; 
alpha = 67.91641; 
wd = 4770.680551;

X = A1*(exp(-alpha*t)).*cos(wd*t)+ A2*(exp(-alpha*t)).*sin(wd*t);
Io = sqrt(A1^2+A2^2);
I = Io * exp(-67.91641*t);

plot(X);
hold on;
plot(I, 'Color', 'r')
plot(-I, 'Color', 'r')

[Neha Sinha]

Thank you so much Ameer Hamza.

[Neha Sinha]

I have some values of i vs t (domain - time , untitled - i) . A damped curve will be obtained using these values. Then using all the peak values I have to get the attenuation by the below formula

attenuation = [ln(nth peak)-ln((n-1)th peak)]/t(n) - t(n-1)]

We can get Peaks from the curve.

Then I have to envelope the damped curve

Can you please help ? (Use ctrl + a to copy all the values from the attached file)

[Ameer Hamza]

Try this, this calculates attenuation according to the formula you gave and then calculate envelope by using curve fitting

t = data(:,1);
i = data(:,2);
[peakValues, peakLocation] = findpeaks(i);
tPeak = t(peakLocation);

attenuation = (log(peakValues(2:end)) - log(peakValues(1:end-1)))./(tPeak(2:end) - tPeak(1:end-1));

envelop = @(parameter, t) parameter(1)*exp(-parameter(2)*tPeak);
envelopErrorFcn = @(parameter) sum(abs((envelop(parameter, t) - peakValues)));
solution = fmincon(envelopErrorFcn, [0 0], [], []);

plot(t, i, tPeak, envelop(solution, tPeak), 'r-+', tPeak, -envelop(solution, tPeak), 'r-+')

[Neha Sinha]

Thank you so much

[Neha Sinha]

envelop = @(parameter, t) parameter(1)*exp(-parameter(2)*tPeak);

envelopErrorFcn = @(parameter) sum(abs((envelop(parameter, t) - peakValues)));

solution = fmincon(envelopErrorFcn, [0 0], [], []);

plot(t, i, tPeak, envelop(solution, tPeak), 'r-+', tPeak, -envelop(solution, tPeak), 'r-+')

Can you please explain me what's parameter here and how are these codes working?

[Ameer Hamza]

The line

envelop = @(parameter, t) parameter(1)*exp(-parameter(2)*tPeak);

define the equation you want the data to fit. The parameters are the constant values of your model. Compare it with your equation of I to see what are parameter(1) and parameter(2).

The next line defines an error function,

envelopErrorFcn = @(parameter) sum(abs((envelop(parameter, t) - peakValues)));

We want such values of parameters which minimize the difference between predicted and actual values. In the next step, we can use a numerical optimizer (fmincon) to find the values of parameter.