Dynamic force identification based on enhanced least squares and total least-squares schemes in the frequency domain

Abstract

Identifying dynamic forces from structural responses is necessary when direct measurement of those dynamic forces is impossible using conventional means. A common approach to address this problem is to determine the frequency response function (FRF) matrix, measure the structural responses, and calculate the dynamic forces based on least-squares (LS) scheme. This approach has been proven to be effective in reducing the random errors that occur in structural response signals. Unfortunately, the accuracy of this approach is often hindered by the inversion of an ill-conditioned FRF matrix at frequencies near the structural resonances. To overcome this inversion instability, two regularization filters, namely the truncated singular value decomposition (TSVD) filter and the Tikhonov filter, are used in conjunction with the conventional LS scheme at specific frequencies. Here a criterion for applying these enhanced LS schemes is proposed to aid in determining when the increase in computational effort is better utilized. Furthermore, a new LS form of the Morozov's discrepancy principle is formulated to aid in selecting the optimum regularization parameter for these filters at each frequency. The accuracy in using conventional LS, TSVD-based LS, and Tikhonov filter-based LS schemes are compared analytically and numerically in this paper. It is found that for small-sized FRF matrices, the Tikhonov filter-based LS scheme tends to work better than the TSVD filter-based LS scheme. Since these approaches can only deal with the random errors in the measured structural responses, a total least-squares (TLS) scheme that can also address errors associated with the FRF matrix is proposed in this research. Numerical simulations demonstrate that under certain conditions, the TLS scheme is more effective in reducing the impact of these errors.