Study case: Clamped-free beam
The present example is similar to the one used in , 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 , and the eigen-modes/eigen-frequencies are generated using . 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  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 .
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);
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)))
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
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
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  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% . The acceleration data in the FDD  and the TDD are the same to facilitate the comparison between the two methods.
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.