MATLAB Examples

Study case: Clamped-free beam

The present example is similar to the one used in [5], where the dynamic response of a 100 m high clamped-free steel beam was studied. Simulated time series are used, where the first 4 eigen-modes have been taken into account. These time series correspond to acceleration response of the beam to white noise excitation. The simulated data are generated with the central difference method [1], and the eigen-modes/eigen-frequencies are generated using [2]. A modal damping ratio of 0.005 for every modes is used. These parameters are considered as the "target modal parameters".

The TDD is an output only modal analysis method inspired from [3] to retrieve the mode shapes and the modal damping ratio from acceleration data. Analysing ambient vibrations data is more challenging than impulse response data due to the lower signal to noise ratio recorded. However, ambient vibrations monitoring is more common for large civil engineering structures. The most challenging task is the estimation of the modal damping ratio (higher sensitivity to measurement noise) which is done for each mode using [4].

Contents

Prep-processing

clear all,close all;clc;
load('beamData.mat')
fn = wn/(2*pi);
Nmodes = numel(fn);
fs = 1/median(diff(t));
rng(1)
[Nyy,N]=size(Az);

Time series and power spectral density visualization

figure
subplot(211)
plot(t(1:1e3),Az(30,1:1e3));
xlabel('t(s)')
ylabel('Acceleration (m/s^2)')
title('Acceleration at the top of the beam')

subplot(212)
pwelch(Az(end,:),[],[],[],1/median(diff(t)))

Minimalist example

tic
[phi_TDD,zeta] = TDD(Az,fs,fn);
toc

% We get a good agreement between the computed and measured modes shapes
figure
for ii=1:size(phi_TDD),
    subplot(2,2,ii)
    hold on;box on;
    h1 = plot(linspace(0,1,size(phi_TDD,2)),phi_TDD(ii,:),'ro','linewidth',1.5);
    h2 = plot(linspace(0,1,size(phi,2)),-phi(ii,:),'k-','linewidth',1.5);
    xlabel('y (a.u.)')
    ylabel(['\phi_',num2str(ii)])
    if ii==1,
        legend('Measured','Target','location','SouthWest')
    end
end

disp('left: target damping. Right: Measured damping')
disp([5e-3*ones(Nmodes,1),zeta(:),])
Elapsed time is 1.173775 seconds.
left: target damping. Right: Measured damping
    0.0050    0.0045
    0.0050    0.0038
    0.0050    0.0054
    0.0050    0.0041

Example with options

We specifiy a longer time for the autocorrelation funciton (Ts = 50 seconds) This only affects the modal damping ratio.

[phi_TDD,zeta] = TDD(Az,fs,fn,'Ts',50);

disp('left: target damping. Right: Measured damping')
disp([5e-3*ones(Nmodes,1),zeta(:),])
left: target damping. Right: Measured damping
    0.0050    0.0069
    0.0050    0.0033
    0.0050    0.0051
    0.0050    0.0039

Comparison with FDD

The frequency domain decomposition (FDD) was introduced by Brinker et al.: BRINCKER, Rune, ZHANG, Lingmi, et ANDERSEN, P. Modal identification from ambient responses using frequency domain decomposition. In : Proc. of the 18*‘International Modal Analysis Conference (IMAC), San Antonio, Texas. 2000

The FDD example in [5] more time consuming because the matrix of cross spectral densities must be computed, and the SVD applied at every frequency step. In a previous version,the FDD was included inside this example. To optimize the clarity, the code related to the FDD has been moved in a different submission% [5]. The acceleration data in the FDD [5] and the TDD are the same to facilitate the comparison between the two methods.

References

[1] https://se.mathworks.com/matlabcentral/fileexchange/53854-harmonic-excitation-of-a-sdof

[2] https://se.mathworks.com/matlabcentral/fileexchange/52075-eigen-value-calculation-of-a-continuous-beam--transverse-vibrations-

[3]Byeong Hwa Kim, Norris Stubbs, Taehyo Park, A new method to extract modal parameters using output-only responses, Journal of Sound and Vibration, Volume 282, Issues 1–2, 6 April 2005, Pages 215-230, ISSN 0022-460X, http://dx.doi.org/10.1016/j.jsv.2004.02.026.

[4] https://se.mathworks.com/matlabcentral/fileexchange/55557-modal-parameters-identification-from-ambient-vibrations--sdof-

[5] https://se.mathworks.com/matlabcentral/fileexchange/57153-automated-frequency-domain-decomposition--afdd-