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CentDiff.m

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+function [y] = CentDiff(F,M,K,C,dt,x0,v0)
+% [y] = CentDiff(F,M,K,C,dt,x0,v0)solves numerically the equation of motion
+% of a damped system
+% 
+% 
+% INPUT
+% F : vector  -- size: [1x N] -- Time series representinf the time history of the load. 
+% M : scalar  -- size: [1 x 1] -- Modal mass
+% K : scalar  -- size: [1 x 1] -- Modal stifness
+% C : scalar  -- size: [1 x 1] -- Modal damping
+% dt : scalar  -- size: [1 x 1] -- time step
+% x0 : scalar  -- size: [1 x 1] -- initial displacement
+% v0 : scalar  -- size: [1 x 1] -- initial velocity
+% 
+% OUTPUT
+% y: time history of the system response to the load
+% 
+% author: E. Cheynet  - UiB - last updated: 14-05-2020
+% 
+
+%%
+% Initialisation
+N = size(F,2);
+% preallocation
+y = zeros(size(F));
+
+% initial acceleration
+a0 = M\(F(1)-C.*v0-K.*x0);
+% initialisation of y (first 2 values).
+y0 = x0-dt.*v0+dt^2/2*a0;
+y(:,1) = x0;
+A = (M./dt.^2+C./(2*dt));
+B = ((2*M./dt.^2-K).*y(:,1)+(C./(2*dt)-M./dt.^2).*y0+F(:,1));
+y(:,2) = A\B;
+
+% For the rest of integration points
+for ii=2:N-1
+    A = (M./dt.^2+C./(2*dt));
+    B = ((2*M./dt.^2-K).*y(:,ii)+(C./(2*dt)-M./dt.^2).*y(:,ii-1)+F(:,ii)); 
+    y(:,ii+1) = A\B;
+end
+
+end
+

NExT.m

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+function [R,t] = NExT(y,dt,Ts,method)
+%
+% [R] = NExT(y,ys,T,dt) implements the Natural Excitation Technique to
+% retrieve the free-decay response (R) from the cross-correlation
+% of the measured output y.
+%
+% 
+% Synthax
+% 
+%   [R] = NExT(y,dt,Ts,1) calculates R with cross-correlation
+%   calculated by using the inverse fast fourier transform of the
+%   cross-spectral power densities without zero-padding(method = 1).
+%
+%   [R] = NExT(y,dt,Ts,2) calculate the R with cross-correlation
+%   calculated by using the unbiased cross-covariance function (method = 2)
+%
+% Input
+%   y: time series of ambient vibrations: vector of size [1xN]
+%   dt : Time step
+%   method: 1 or 2 for the computation of cross-correlation functions
+%   T: Duration of subsegments (T<dt*(numel(y)-1))
+% 
+% Output
+% 
+% R: impusle response function
+% t: time vector asociated to R
+% 
+% Author: E. Cheynet - UiB - last modified 14-05-2020
+
+%%
+if nargin<4, method = 2; end % the fastest method is the default method
+if ~ismatrix(y), error('Error: y must be a vector or a matrix'),end
+
+
+[Nyy,N]=size(y);
+if Nyy>N
+    y=y';
+    [Nyy,N]=size(y);
+end
+
+% get the maximal segment length fixed by T
+M = round(Ts/dt);
+switch method
+    case 1
+        clear R
+        for ii=1:Nyy
+            for jj=1:Nyy
+                y1 = fft(y(ii,:));
+                y2 = fft(y(jj,:));
+                h0 = ifft(y1.*conj(y2));
+                R(ii,jj,:) = h0(1:M);
+            end
+        end
+        % get time vector t associated to the R
+        t = linspace(0,dt.*(size(R,3)-1),size(R,3));
+        if Nyy==1
+            R = squeeze(R)'; % if Nyy=1
+        end
+    case 2
+        R = zeros(Nyy,Nyy,M+1);
+        for ii=1:Nyy
+            for jj=1:Nyy
+                [dummy,lag]=xcov(y(ii,:),y(jj,:),M,'unbiased');
+                R(ii,jj,:) = dummy(end-round(numel(dummy)/2)+1:end);
+            end
+        end
+        if Nyy==1
+            R = squeeze(R)'; % if Nyy=1
+        end
+        % get time vector t associated to the R
+        t = dt.*lag(end-round(numel(lag)/2)+1:end);
+end
+% normalize the R
+if Nyy==1
+R = R./R(1);
+else
+end
+

RDT.m

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+function [R,t] = RDT(y,ys,T,dt)
+%
+% [R] = RDT(y,ys,T,dt) returns the free-decay response (R) by
+% using the random decrement technique (RDT) to the time serie y, with a
+% triggering value ys, and for a duration T
+%
+% INPUT:
+%   y: time series of ambient vibrations: vector of size [1xN]
+%   dt : Time step
+%   ys: triggering values (ys < max(abs(y)) and here ys~=0)
+%   T: Duration of subsegments (T<dt*(numel(y)-1))
+% OUTPUT:
+%   R: impusle response function
+%   t: time vector asociated to R
+% 
+% Author: E. Cheynet - UiB - last modified 14-05-2020
+
+%%
+
+if T>=dt*(numel(y)-1)
+    error('Error: subsegment length is too large');
+else
+    % number of time step per block
+    nT = round(T/dt); % sec
+end
+
+if ys==0
+    error('Error: ys must be different from zero')
+elseif or(ys >=max(y),ys <=min(y)),
+    error('Error:  ys must verifiy : min(y) < ys < max(y)')
+else
+    % find triggering value
+    ind=find(diff(y(1:end-nT)>ys)~=0)+1;
+
+end
+
+% construction of decay vibration
+R = zeros(numel(ind),nT);
+for ii=1:numel(ind)
+    R(ii,:)=y(ind(ii):ind(ii)+nT-1);
+end
+
+% averaging to remove the random part
+R = mean(R);
+% normalize the R
+R = R./R(1);
+% time vector corresponding to the R
+t = linspace(0,T,numel(R));
+end
+

expoFit.m

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+function [zeta] = expoFit(y,t,wn,optionPlot)
+% [zeta] = expoFit(y,t,wn) returns the damping ratio calcualted by fitting
+% an exponential decay to the envelop of the free-decay response.
+%
+% Input:
+%   y: envelop of the free-decay response: vector of size [1 x N]
+%   t: time vector [ 1 x N]
+%   wn: target eigen frequencies (rad/Hz) :  [1 x 1]
+%  optionPlot: 1 to plot the fitted function, and 0 not to plot it.
+% 
+% Output
+%  zeta: modal damping ratio:  [1 x 1]
+% 
+% author: E. Cheynet  - UiB - last updated: 14-05-2020
+% 
+
+%%
+% Initialisation
+guess = [1,1e-2];
+% simple exponentiald ecay function
+myFun = @(a,x) a(1).*exp(-a(2).*x);
+% application of nlinfit function
+coeff = nlinfit(t,y,myFun,guess);
+% modal damping ratio:
+zeta = abs(coeff(2)./wn);
+
+% alternatively: plot the fitted function
+if optionPlot== 1, plot(t,myFun(coeff,t),'r'); end
+end
+