Half power bandwidth method

I have experimental input and output acceleration data for shaker. I want to find the damping ratio from the frf graph using half power bandwidth method. I could not extract it from the graph. Can you help me please?

[Mathieu NOE]

hello

can you share your data and code ?

[Fuad Al-Gaadi]

@Mathieu NOE

Thank you for your response . my code file is attached. I can not upload the data files here because of the size. You can download it from google drive.

Thank you in advance for your help.

https://docs.google.com/spreadsheets/d/1pu5XqAoJZfRe9WeAo0ydIB_1A2kvoj9L8WBGo6-goY4/edit?usp=sharing

[Mathieu NOE]

hello again

a few points that surprised me when doing a preliminary analysis of your data ... how where they generated & collected ?

normally we would like the input (test) signal to covert the entire frequency range with equal energy accross the frequency range (that's why I was expecting either a chirp or random type signal for the input)

here it seems to me that even the "input" is actually somehow an output of the system. The fft graph ob both signals shows almost the same spectral content.

I don't undertsnad the type of signal you are using , maybe you could explain to me how you created the test signal.

the time display will show a signals that are not stationnary in time (the is one period of blank and also a spurious "incident" on the output channel between t = 500 s to 700 s)

the record is very long, no need to do such long records for a good identification. 10% of that is even too large.

input signal = green

output signal= pink

Time display

the fft spectra (both channels) represent a tonal content of multiple harmonics of a tone at 1.95 Hz (??)

FFT display

so in the end , I don't know what I can do with your data , but using other ("good") data I can at least show you a code that works !

see below (and the data itself is the attached mat file)

Result for beam identification

nota the 3rd plot (coherence) - the coherence is the ratio of the output ennergy that is correlated with the input. It should be very close to one in the case

• your system is linear,
• does not contain too much uncorreleted noise on the output signal
• and the input signal has sufficient strength and covers all the frequency range

here in this example you can see the coherence is good (close to 1) in a large fraction of the frequency range.

% Example for Beam identification

data = load('beam_experiment.mat');
x = transpose(data.x); %input
y = transpose(data.y); %output
fs = data.fs;   % sampling frequency

NFFT = 2048;
NOVERLAP = round(0.75*NFFT);  % 75 percent overlap

%% solution 1 with tfestimate (requires Signal Processing Tbx)
% [Txy,F] = tfestimate(x,y,hanning(NFFT),NOVERLAP,NFFT,fs);

%% alternative with supplied sub function 
[Txy,Cxy,F] = mytfe_and_coh(x,y,NFFT,fs,hanning(NFFT),NOVERLAP); 
% Txy = transfer function (complex), Cxy = coherence, F = freq vector

% Bode plots
figure(1),
subplot(3,1,1),plot(F,20*log10(abs(Txy)));
ylabel('Mag (dB)');
subplot(3,1,2),plot(F,180/pi*(angle(Txy)));
ylabel('Phase (°)');
subplot(3,1,3),plot(F,Cxy);
xlabel('Frequency (Hz)');
ylabel('Coh');

%%% damping ratioes for modes
N = 2 ; % number of (dominant) modes to identify
[fn,dr] = modalfit(Txy,F,fs,N,'FitMethod','pp');
T = table((1:N)',fn,dr,'VariableNames',{'Mode','Frequency','Damping'})


%%%%%%%%%%%%%%%%%%%%%%%
function [Txy,Cxy,f] = mytfe_and_coh(x,y,nfft,Fs,window,noverlap)
% Transfer Function  and Coherence Estimate 
    
% compute PSD and CSD
window = window(:);
n = length(x);		% Number of data points
nwind = length(window); % length of window
if n < nwind    % zero-pad x , y if length is less than the window length
    x(nwind)=0;
    y(nwind)=0;  
    n=nwind;
end
x = x(:);		% Make sure x is a column vector
y = y(:);		% Make sure y is a column vector
k = fix((n-noverlap)/(nwind-noverlap));	% Number of windows
					% (k = fix(n/nwind) for noverlap=0)
index = 1:nwind;

Pxx = zeros(nfft,1); 
Pyy = zeros(nfft,1); 
Pxy = zeros(nfft,1); 

for i=1:k
    xw = window.*x(index);
    yw = window.*y(index);
    index = index + (nwind - noverlap);
    
    Xx = fft(xw,nfft);
    Yy = fft(yw,nfft);
    Xx2 = abs(Xx).^2;
    Yy2 = abs(Yy).^2;
    Xy2 = Yy.*conj(Xx);
    Pxx = Pxx + Xx2;
    Pyy = Pyy + Yy2;
    Pxy = Pxy + Xy2;
    
end

% Select first half
if ~any(any(imag([x y])~=0))   % if x and y are not complex
    if rem(nfft,2)    % nfft odd
        select = [1:(nfft+1)/2];
    else
        select = [1:nfft/2+1];   % include DC AND Nyquist
    end

    Pxx = Pxx(select);
    Pyy = Pyy(select);
    Pxy = Pxy(select);
    
else
    select = 1:nfft;
end
Txy = Pxy ./ Pxx;                   % transfer function estimate 
Cxy = (abs(Pxy).^2)./(Pxx.*Pyy);    % coherence function estimate 

f = (select - 1)'*Fs/nfft;

end

[Mathieu NOE]

If you take the code I showed you and use your raw data then the result is juts "horrible"

see the coherence plot , almost zero everywhere => this is because the signals are not appropriate for modal analysis. We need better data than this .

dataset = xlsread('FRF_0402.xlsx','Sheet1','A1:C1046248'); %reading data
%%Creating and filling data variables
y=dataset(:,3); %Response Acceleration on the top of structure
x=dataset(:,2); %Input acceleration in the shaking table plate
t=dataset(:,1); %time vector

dt = mean(diff(t));
fs = 1/dt;

NFFT = 2048*8;
NOVERLAP = round(0.75*NFFT);  % 75 percent overlap

%% solution 1 with tfestimate (requires Signal Processing Tbx)
% [Txy,F] = tfestimate(x,y,hanning(NFFT),NOVERLAP,NFFT,fs);

%% alternative with supplied sub function 
[Txy,Cxy,F] = mytfe_and_coh(x,y,NFFT,fs,hanning(NFFT),NOVERLAP); 
% Txy = transfer function (complex), Cxy = coherence, F = freq vector

% Bode plots
figure(1),
subplot(3,1,1),plot(F,20*log10(abs(Txy)));
ylabel('Mag (dB)');
subplot(3,1,2),plot(F,180/pi*(angle(Txy)));
ylabel('Phase (°)');
subplot(3,1,3),plot(F,Cxy);
xlabel('Frequency (Hz)');
ylabel('Coh');

% %% damping ratioes for modes
% [fn,dr] = modalfit(FRF66db,f1',fs,10,'FitMethod','pp');
% T = table((1:10)',fn,dr,'VariableNames',{'Mode','Frequency','Damping'})

%%%%%%%%%%%%%%%%%%%%%%%
function [Txy,Cxy,f] = mytfe_and_coh(x,y,nfft,Fs,window,noverlap)
% Transfer Function  and Coherence Estimate 
    
% compute PSD and CSD
window = window(:);
n = length(x);		% Number of data points
nwind = length(window); % length of window
if n < nwind    % zero-pad x , y if length is less than the window length
    x(nwind)=0;
    y(nwind)=0;  
    n=nwind;
end
x = x(:);		% Make sure x is a column vector
y = y(:);		% Make sure y is a column vector
k = fix((n-noverlap)/(nwind-noverlap));	% Number of windows
					% (k = fix(n/nwind) for noverlap=0)
index = 1:nwind;

Pxx = zeros(nfft,1); 
Pyy = zeros(nfft,1); 
Pxy = zeros(nfft,1); 

for i=1:k
    xw = window.*x(index);
    yw = window.*y(index);
    index = index + (nwind - noverlap);
    
    Xx = fft(xw,nfft);
    Yy = fft(yw,nfft);
    Xx2 = abs(Xx).^2;
    Yy2 = abs(Yy).^2;
    Xy2 = Yy.*conj(Xx);
    Pxx = Pxx + Xx2;
    Pyy = Pyy + Yy2;
    Pxy = Pxy + Xy2;
    
end

% Select first half
if ~any(any(imag([x y])~=0))   % if x and y are not complex
    if rem(nfft,2)    % nfft odd
        select = [1:(nfft+1)/2];
    else
        select = [1:nfft/2+1];   % include DC AND Nyquist
    end

    Pxx = Pxx(select);
    Pyy = Pyy(select);
    Pxy = Pxy(select);
    
else
    select = 1:nfft;
end
Txy = Pxy ./ Pxx;                   % transfer function estimate 
Cxy = (abs(Pxy).^2)./(Pxx.*Pyy);    % coherence function estimate 

f = (select - 1)'*Fs/nfft;

end

[Fuad Al-Gaadi]

Thank you @Mathieu NOE.

I want to explain to you how the data were collected. The experiment was done using a small shaking table. The input and response acceleration signals were measured using sensors fixed on the shaking table and the top of the structure as shown in the attached figure.1.

the sensor marked with pink measures the input signal (excitation)

the sensor marked with green measures the output signal ( response acceleration).

I don't know how to get better data. Maybe it is necessary to do data filtering.

If you have an idea please tell me.

Thank you in advance

[Mathieu NOE]

hello again

thanks for the explananations - so this is a mock up of a multi stage building as far I can tell.

excuse me for this stupid question , but are you sure you are measuring / collecting the data in the direction of mechanical excitation (just double check ? )

are you suing 3 axis accelerometers ? are you sure which axis is stored in your data ?

what is the signal send to your shaker ?

are you sure that the blocks of concrete are rigidely attached to the "floors"; if they have a tendency so stick / slip; this will create non linearities in your model and reduce the quality of your data (beside that you can't really tell about what are the correct boundary conditions)