# double differentiation and double integration of sine wave

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ilyas mohamed on 13 Nov 2019
Commented: ilyas mohamed on 13 Nov 2019
hi there
i would like to know if we differentitate position to get velocity and acclereation, do we get same results and graphs going back form accleration to velocity and position using integration?
when differentiating my position to get velocity and acclereation i did work out all three of them by hand, quite sure it is right
here are the codes i have used
this is the differentation of my position and the codes for the plot
clc
close all
t=linspace(0,5,100) % time in seconds and number of points
a=0.5 % amplitude
f=5 % frequency
y=a*sin((2*pi)/f*t) % our posititon
plot(t,y,'b-');
title('position')
y2=(a*2*pi/f)*cos(2*pi/f*t) % velocity differentiated form position
plot(t,y2,'r-');
title('velocity')
y3=(-a*2*pi/f)^2*sin(2*pi/f*t) % accleration differetiated from velocity
plot(t,y3,'g-');
title('accleration')
ylabel('amplitude')
xlabel('time(s)')
this is integration of my acclereation to plot my velocity and position graphs
clear all
close all
clc
a=0.5 %amplitude
f=5 % frequency
t=linspace(0,5,100);
acc=(-a*2*pi/f)^2*sin(2*pi/f*t); % accleration obtained from differentation
figure(1)
plot(t,acc,'g-')
xlabel('Time (sec)')
ylabel('Amplitude (m/sec^2)')
title('acceleration')
velocity=cumtrapz(t,acc); % intgrated velocity from accleration using trapizoidal
figure (2)
plot(t,velocity,'r-')
xlabel('Time (sec)')
ylabel('Amplitude (m/sec)')
title('velocity')
position=cumtrapz(t,velocity); %intgrated velocity from accleration using trapizoidal
figure(3)
plot(t,position,'b-')
xlabel('Time (sec)')
ylabel('amplitude(m)')
title('position')
when i plot my differentitated graphs going from position to accleration, then integrate accleration to position, i dont obtain the same graphs, this is an experiment for findig positon through intgration using simple sine wave.
i would really appreciate any help
thank you

Walter Roberson on 13 Nov 2019
y3=(-a*2*pi/f)^2*sin(2*pi/f*t) % accleration differetiated from velocity
is incorrect. That results in a positive leading factor that includes but the true differentiation only has a and is negative,
y3 = -4*a.*pi^2./f.^2 .* sin(2*pi./f .* t)

James Tursa on 13 Nov 2019
"... when differentiating my position to get velocity and acclereation i did work out all three of them by hand, quite sure it is right ..."
Well, no, unfortunately it isn't. The "a" doesn't get powers as a result of the differentiation ... it simply stays as a factor out front. And the minus sign doesn't get powers either, it simply appears outside of that. So,
y3=-a*(2*pi/f)^2*sin(2*pi/f*t); % accleration differetiated from velocity
:
acc=-a*(2*pi/f)^2*sin(2*pi/f*t); % accleration obtained from differentation
Then remember that integration is ambiguous up to a constant. So in addition to your numerical integration with cumtrapz( ), you will need to supply initial velocity and position. E.g.,
velocity0 = a*(2*pi/f)*cos(2*pi/f*t(1)); % initial velocity
position0 = a*sin((2*pi)/f*t(1)); % initial position
:
velocity=cumtrapz(t,acc)+velocity0; % intgrated velocity from accleration using trapizoidal
:
position=cumtrapz(t,velocity)+position0; %intgrated velocity from accleration using trapizoidal

#### 1 Comment

ilyas mohamed on 13 Nov 2019
thank you guys so mcuh james tursa and walter roberson with combination of both your answers i managed to get the correct answer but i have one more question.
how can you add noise to this data then and filter it
thank you very much