This research is focused on the analysis of an inverse problem to estimate the applied forces from measured responses indirectly. Accurate estimation of dynamic forces acting on a structure is very important in design, control, and diagnosis. Because of the installation problems, the direct measurements of such forces are often difficult or impossible. Then an indirect force determination with the measurements of the responses on the structure is the only practical way.
In order to analyze the characteristics of indirect force determinations, the condition number and the smallest singular value of the transfer function submatrix are thoroughly investigated with respect to complex variable s. Then an inverse characteristic polynomial is defined as the determinant of dynamic stiffness submatrices, and its properties are studied. From the inverse characteristic polynomial, the existence of the inverse poles, where the inverse of transfer function submatrix does not exist, is proved, and their properties are examined. Also, an inverse transfer function, defined as the ratio of Laplace transform of a force to that of a response, is explicitly expressed with a rational function. Then, using the explicit expression of inverse transfer function, the characteristics of indirect force determination are examined. All the results related to the characteristics of indirect force determination were verified by several experiments.
At certain frequency regions, the identified forces may contain an unacceptable amount of error. In this study, by investigating the characteristics of the inverse transfer function and frequency response function (FRF) error, the indirect force determination errors are analyzed statistically, and the special frequency regions, where force determination error is very large, are searched for each case of the number of response measurements and applied forces. It is shown that the large force determination error near inverse poles is mainly due to the rank deficiency of the FRF submatrix, while the large error in the vicinity of resonance frequencies comes from the larger FRF error.
The accuracy and numerical stability of identified forces depend on the selection of response measurement degrees. In this study, a performance index for the selection of the best measurement degrees is suggested for each case of the number of response measurements and applied forces, and its validity is examined experimentally. Also, in order to simplify the performance index, the concept of orthogonal projector is introduced, and the new measurement degree selection method is proposed.
A regularization process to reduce the identified force errors especially near the inverse poles is proposed. Since the degree of singularity of the FRF submatrix near inverse poles is mainly related to system damping, a regularization method using the additional damping effects is proposed. Also, an optimal additional damping ratio is derived. It is shown that, the greater the order of magnitude of the FRF error is than that of modal damping ratio, the larger the force determination error reduction can be obtained with the suggested regularization procedure. The proposed regularization method is tested by experiments, and its effects examined.
All the proposed ideas are applied to the indirect transmitted force determination of a washing machine. Using the response measurements on the motor mount, the transmitted forces from electric motor are recovered, and the errors of identified force are examined. By comparing the results using the best measurement degrees with those using randomly-selected measurement degrees, the feasibility of the proposed measurement degree selection method is examined.
