30 Z. Zhang et al. 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 10 20 30 Order 40 50 60 Singular Values x 10-8 Fig. 4.5 The singular value plot 40 a b 20 0 -20 Force/N -40 -60 -80 40 20 0 -20 Force/N -40 -60 -80 0 0.02 0.04 0.06 0.08 0.1 Time/s 0.12 0.14 0.16 0.18 Identified True Identified True 0.2 0 0.02 0.04 0.06 0.08 0.1 Time/s 0.12 0.14 0.16 0.18 0.2 Fig. 4.6 Time histories of the estimated and measured sinusoidal force: (a) 60Hz and (b) 80Hz 4.5 Conclusions An efficient inverse method combining the subspace identification algorithm and the homotopy method has been successfully used to estimate dynamic forces acting on structures from measured dynamic responses. The data-driven subspace algorithm is utilized to facilitate the estimation of state space model of the system. Meanwhile, the homotopy method based on the L-curve criterion is employed to discretize the ill-posed problems due to the inversion process. The feasibility of the present method is verified with laboratory experiments on a scale model of a civil ship. The comparisons of the measured and estimated results have demonstrated the effectiveness of the proposed technique for the identification of excitation forces. The present approach is expected to be extended to force identification problems of the nonlinear systems in the future work.
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