The accurate identification of excitation forces in vibrating systems is an important issue from the aspects of design, control and diagnosis. Since direct measurement of the input forces is often very difficult or almost impossible, inputs are often identified indirectly, i.e., from responses under operational conditions and system characteristics[l-4].
In general, the accuracy in the indirect force identification can be improved in the least squared sense by increasing the number of output responses for estimation of the power specσum matrix. However, the number of output responses to be measured simultaneously can not be increased indefmitely due to hardware limitations. In order to overcome these limitations, estimation of the output power spectrum matrix can be done in a sequential way using reference signals. In this method, two different types of approach can be applied according to whether the number of independent excitation sources is just one or more than one. When there exists a single excitation source, processing of the signals with respect to any reference point by the transmissibility function approach[5] enables correct estimation of the whole output power spectrum matrix. That is, one of the most basic assumptions in the transmissibility function approach is that the structure is excited by a single source. Therefore, it can not be applied to the cases where multiple uncorrelated sources are exciting the structure. In this case, one column (or row) of the output power spectrum matrix is not proportional to the rest of it. That is, the output power spectrum matrix formulated by the transmissibility functions varies depending on the choice of the reference location.
The Principal Component Analysis(PCA) is a method of extracting compact information from a matrix by investigating its dimensionality, which was introduced in multivariate statistics for data reduction[6] and applied for the determination of operational deflection shapes in multi-source environments[7-8]. Since the PCA method enables decomposition of the output power spectrum matrix, it can also be usefully applied to the input identification in the multi-source environments. The basic idea of the PCA approach in the multi-source environments is to decompose the outputs into several principal components which are independent of each other and then, subsequently, to apply the transmissibility function approach between each principal component and the whole response measurements. Thereby the method enables all of the output responses of interest be obtained in a series of measurement by roving the sensors the number of which is less than the number of the total response points. In this study,a method of the indirect force identification using the PCA approach is proposed.
The number of principal components of the output power spectrum matrix is equal to the number of non-zero singular values of the matrix. However, it seldom happens that singular values become perfectly zeroes due to effects of many unknown factors such as the measurement noise. In such a case, the number of the principal components can be determined by ignoring singular values smaller than a given level. Therefore, a technique to estimate the measurement noise level from the measured Frequency Response Functions(FRF) and output power spectrum matrix under some assumptions is presented and another technique to determine the number of the principal components is established using the estimated measurement noise level. The proposed approach is applied to a numerical example to show its validity.
The paper is organized as follows. First, the basic formulation of the indirect force identification is reviewed and the transmissibility function approach for obtaining the output power spectrum matrix is explained in a single-source environment. Then,the PCA approach is proposed and the output noise level estimation process is presented for determining the number of principal components. Finally,the validity of the proposed method is demonstrated using experiments.
