Hilbert-Huang method is proposed to identify of modal responses of hysteretic multi-degree-of-freedom (MDOF) structures under free vibration. In this study, a response time history is first decomposed into intrinsic mode functions (IMFs) using the empirical mode decomposition (EMD) method. Then, the Hilbert transform is applied to each IMF to obtain the instantaneous amplitude and frequency. Modal analysis of nonlinear structures, which treats the nonlinearity at each time step as a pseudo force, is used to uncouple the equations of motion in order to obtain the modal responses. The Bouc-Wen model is used to simulate the hysteretic behavior of a two-degree-of-freedom shear-beam building model. The IMF components of the displacement responses are compared with the modal responses obtained from the modal analysis, in which three cases are considered for the pseudo force. In Cases 1 and 2, the pseudo force is the hysteretic force without and with its linear component, respectively. The pseudo force in Case 3 is taken in the form that the stiffness matrix becomes the same as that of the linear building model. It is shown that the IMF components, their instantaneous amplitudes and frequencies agree quite well with those of the modal responses obtained from Case 1 of the modal analysis.
The Empirical Mode Decomposition (EMD), which combines with the Hilbert transform (HT), has been used successfully to identify the dynamic characteristics of linear multi-degree-of-freedom structures. In this study, the EMD method is applied to the identification of nonlinear normal modes (NNMs) of nonlinear multi-degree-of-freedom (MDOF) structures. It is shown that the intrinsic mode functions (IMFs), which are obtained by applying the EMD method to the structural response, agree quite well with the nonlinear modal responses obtained from the invariant manifold approach. A two-degree-of-freedom building model with nonlinear stiffness is used for illustration. The EMD method is applied to decompose the measured response of the building model. The resulting IMFs are compared with the corresponding nonlinear modal responses, including their instantaneous frequencies and time-dependent amplitudes from the HT method. The comparison indicates that the resulting IMFs from the EMD method can reveal the nonlinear modal responses. It is seen that this EMD-based technique is fairly accurate to determine the nonlinear stiffness characteristics of the building model. The result suggests that the IMFs can be used to determine the physical dynamic properties of nonlinear structures.
Most structural responses can be considered as the superposition of some monotonic components. These monotonic components contain modal information that can be used for structural damage detection and health monitoring. This paper presents a comparative study of three techniques for signal decomposition and analysis. These techniques are the wavelet transform (WT) technique, the empirical mode decomposition (EMD) technique, and the principle component analysis (PCA) technique. These techniques are all capable of decomposing multi-component signals into a summation of mono-components without resorting to the traditional frequency-domain approach. All three techniques can estimate natural frequencies, damping ratios and mode shapes of a structure from its time-domain vibration responses and hence can be used to monitor structural condition. A numerical study on a three-story shear-beam building frame is performed and presented to show the accuracy of these techniques.
Recently, the empirical mode decomposition (EMD) in combination with the Hilbert spectrum method has been proposed to identify the dynamic characteristics of linear structures. In this study, this EMD and Hilbert spectrum method is used to analyze the dynamic characteristics of a damaged reinforced concrete (RC) beam in the laboratory. The RC beam is 4m long with a cross section of 200mm X 250mm. The beam is sequentially subjected to a concentrated load of different magnitudes at the mid-span to produce different degrees of damage. An impact load is applied around the mid-span to excite the beam. Responses of the beam are recorded by four accelerometers. Results indicate that the EMD and Hilbert spectrum method can reveal the variation of the dynamic characteristics in the time domain. These results are also compared with those obtained using the Fourier analysis. In general, it is found that the two sets of results correlate quite well in terms of mode counts and frequency values. Some differences, however, can be seen in the damping values, which perhaps can be attributed to the linear assumption of the Fourier transform.