Conversion of acceleration to velocity and displacement

Question

Shz713 2016-1-21

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https://ww2.mathworks.cn/matlabcentral/answers/264594-conversion-of-acceleration-to-velocity-and-displacement

评论： Saiprasad Gore 2018-2-16

I have seen two different versions for converting acceleration to velocity and displacement and vice versa. 1- Some websites mentioned (Velocity = Acceleration/-i*w) where omega is the frequency in (radians/sec) = 2*pi*f with f in Hz. and (Disp=Acc/-w^2).

i is sqrt(-1)

2-While in some forum it is written (Velocity = Acceleration/i*w) and (Disp=Acc/w^2).

So which one is correct? I noticed that if the fourier transform of time displacement is taken, then the first sentence is correct. However, when the inverse of fourier transform for continuous time displacement is taken, then the second sentence is correct.

I appreciate if some experienced mates help me out.

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Shz713 2016-1-21

在 MATLAB Online 中打开

I checked; I reckon that for using Omega Arithmetic method, I need to differentiate the inverse of transformed function. So the second sets of equations are correct. based on that, I have written a script to convert the acceleration time records to velocity and displacement and obtain their spectral densities. based on data, all of them are showing correct peaks, but i am not sure about phase information (whether I have to check or not) and units after conversion. Would you comment on it? Cheers

    % code
clear;
signal = load ('d2_Sin.txt'); 
fs = 50;                                 %sampling frequency Hz
ts= 1/fs;                                %intervals between data points/distance between data points
N = length (signal);                        %number of data points 
tmax= (N-1)*ts;
t = 0:ts:tmax;                           %time-step size
bin_num = 0:N-1;                        
fbins = bin_num*fs/N;                    %shows frequencies in Hz 
N_2 = ceil (N/2);                         %keeping only positive side
Acc = fft (signal);
Velo = zeros(size(N));
Disp = zeros(size(N));
for i=1:N
   if fbins(i)>= 1 %      filter the value at this frequency ~= 0
    Velo(i)=Acc(i)/(2*pi*fbins(i)*sqrt(-1));%Velocity frequency domain
    Disp(i)=Acc(i)/(2*pi*fbins(i))^2; %Displacement frequency domain
   else
       Velo(i)= 0;
       Disp(i)= 0;
   end
end
%%reverse Fourier Transform to switch back to time domain
Acc_time = ifft(Acc);
Velo_time = ifft(Velo);
Disp_time = ifft(Disp);
%---------[Spectral Density]---------%
Acc_psd = (Acc.* conj(Acc))/N;                        % power spectral density (PSD)
[Acc_welch,Fy] = pwelch(signal,N,[],N,fs);            %PSD using Welch method
Velo_psd = zeros(size(N));
Disp_psd = zeros(size(N));
Velo_welch = zeros(size(Acc_welch));
Disp_welch = zeros(size(Acc_welch));
for i=1:N
   if fbins(i)>= 1       
    Velo_psd(i)=Acc_psd(i)/(2*pi*fbins(i))^2;
    Disp_psd(i)=Acc_psd(i)/(2*pi*fbins(i))^4; 
   else
       Velo_psd(i)= 0;
       Disp_psd(i)= 0;
   end
end
for i=1:size(Acc_welch)
   if Fy(i)>= 1       
    Velo_welch(i)=Acc_welch(i)/(2*pi*Fy(i))^2;
    Disp_welch(i)=Acc_welch(i)/(2*pi*Fy(i))^4; 
   else
       Velo_welch(i)= 0;
       Disp_welch(i)= 0;
   end
end
%Plotting output in frequency domain
figure('Name','Measurements Conversion in Hz'); subplot (3,1,1);plot(fbins(1:N_2),abs(Acc(1:N_2)),'r'); axis tight;
xlabel('Frequency (Hz)');ylabel('Acceleration (m/s^2) '); title('Acceleration,Velocity and Displacement');
subplot (3,1,2);plot(fbins(1:N_2),abs(Velo(1:N_2)),'k');axis tight;ylabel('Velocity (m/s)');
subplot (3,1,3);plot(fbins(1:N_2),abs(Disp(1:N_2)),'b'); axis tight;ylabel('Displacement (m)');
%
figure('Name','Spectral densities using Fourier Transform'); subplot (3,1,1);plot(fbins(1:N_2),10*log10 (abs((Acc_psd(1:N_2)))),'r'); axis tight;
xlabel('Frequency (Hz)');ylabel('(m/sec^2)^2/Hz'); title('PSD using Fourier Transform for Acceleration,Velocity and Displacement');
subplot (3,1,2);plot(fbins(1:N_2),10*log10 (abs((Velo_psd(1:N_2)))),'k');axis tight;ylabel('Velocity (m/sec)^2/Hz)');
subplot (3,1,3);plot(fbins(1:N_2),10*log10 (abs((Disp_psd(1:N_2)))),'b'); axis tight;ylabel('Displacement (m)^2/Hz');
end

Shz713 2016-3-2

The input can be anything, in this case is recorded due to impact force on a rectangular object.

Saiprasad Gore 2018-2-16

在 MATLAB Online 中打开

@Shz713

After using the code provided by you. The velocity and displacement are obtained in complex number. How do I get both V and D in real numbers. I tried taking abs() of them. But the result didnt match when I tried to check it with the ground motion data obtained from peer berkeley website.

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