Derivation of a Damage Sensitive Feature Using the Haar Wavelet Transform this paper, a damage sensitive feature based on the wavelet transform of the vibration

signal is derived. The theoretical aspects of wavelet decomposition of vibration signals are presented. It is shown that the energies of the wavelet coefficients at appropriate scales can be used as damage sensitive features. Expressions for the energies of wavelet coefficients using the Haar wavelet basis function are derived for a single degree and a multidegree of freedom system. It is shown that the energies of the wavelet coefficients extracted at higher scales are functions of the physical parameters of the system and the loading function. Finally, the migration of damage sensitive feature vectors with damage is illustrated for the ASCE Benchmark Structure. DOI: 10.1115/1.3130821 Structural health monitoring SHM has been receiving consid- erable attention in the civil engineering community during the past decade 1-6. Structural health monitoring consists of the following two steps: a diagnosis, which involves damage iden- tification, localization, and quantification; and b prognosis, which includes the structure's residual capacity estimation and residual life forecasting 7. Recent research in damage detection algorithms shows the extensive use of signal processing and pat- tern classification techniques 8. Signal processing techniques were introduced for damage diagnosis by Farrar and co-workers 8-10. Such methods rely on the signatures obtained from the recorded vibration, strain, or other data to extract features that change with the onset of damage. In particular, damage diagnosis algorithms require the tracking of some damage sensitive features. The main focus of this paper is the derivation of a damage sensi- tive feature using the Haar wavelet transform and establishing a closed form relation with the physical parameters of the structural system. In the context of SHM, earlier work was carried out in wavelet based system identification of nonlinear structures by Staszewski 11, Ghanem and Romeo 12 and Kijewski and Kareem 13. Ghanem and Romeo 12 used the discrete wavelet transform to identify the parameters of a time varying system in the presence of noise. Kijewski and Kareem 13 proposed guidelines for se- lection of wavelet central frequencies, its use in modal separation, and in addition developed a padding scheme to account for end effects. Spanos and Failla 14 reviewed the literature on the ap- plications of wavelet transforms to vibration problem. In the con- text of this paper, they investigated system identification and dam- age detection with wavelets. Hou et al. 15 used the wavelet transform to study the sudden jump in the signal when the stiff- ness of the system is changed. In recent studies conducted by the authors 16,17, the autore- gressive moving average models were used to fit the vibration signals obtained from a sensor. However, these models are valid for stationary signals. Thus, in this study, the continuous wavelet transform is used to decompose the signal to account for nonsta- tionarity. It is important to note that although the application of the model developed in this paper is for stationary signals, it can also be applied to nonstationary signals with appropriate modifi- cations. A damage sensitive feature relating to the energies of the wavelet coefficients of the vibration signal is derived. It is shown that the energies of wavelet coefficients of acceleration signals at higher scales are related to parameters of the physical system and thus are particularly suitable for the purpose of damage detection. In order to illustrate this wavelet based method for damage detec- tion, it is applied to the ASCE Benchmark Structure 18. The paper is organized as follows. In Sec. 2, the theoretical aspects of wavelet decomposition of vibration signals are dis- cussed. The continuous wavelet transform of a signal is written in terms of the Fourier transform of the signal and the wavelet basis. In Secs. 3 and 4, this framework is used to derive a closed form expression for the energies of the Haar wavelet for a single degree and multiple degree of freedom systems, respectively. The rela- tionship of the damage sensitive feature to physical parameters of the structure such as mode shapes, stiffnesses, and damping ratios is demonstrated. Finally, in Sec. 5, the damage sensitive feature is used to discriminate between a damaged and an undamaged state using vibration signals obtained from the ASCE Benchmark Structure.