MODAL MODIFICATION OF VIBRATING SYSTEMS: SOME PROBLEMS AND THEIR SOLUTIONS

Abstract

Experimental modal testing data form an incomplete set of natural frequencies and mode shapes. The incompleteness of modal representation is inherent also to continuous systems that are represented by discrete models. Furthermore, current state-of-the-art numerical methods for determining the eigenvalues of large discrete systems can extract only some eigenvalues and eigenvectors associated either with low frequencies or the high-frequency range. Therefore, when predicting the effect of structural modification on the spectrum of a modified structure an error called a ‘truncation error’ exists in the calculations.The paper focuses on the prediction of the error in the spectrum of a modified structure resulting from the incomplete representation of the model. An optimal solution in a Rayleigh–Ritz sense is derived and the truncation error is bounded. Then the inverse problem of determining the structural modification that is needed to assign the spectrum is considered. A mathematical formulation characterising a continuous family of solutions is first given. Then a procedure for determining physical realisable solutions is developed. Finally, it is shown that if both left and right eigenvectors of the system are known then it is possible to solve the inverse structural modification problem without truncation error. The extraction of left eigenvectors from experimental data, however, can be an ill-conditioned problem. Thus, the modification could be sensitive to noise in the measured data, and the truncation error alone might not be the dominant mechanism of error.