The Construction of Physical Models from Eigendata

Recent developments and new results associated with Inverse Vibration Problems are reported. The derived methods may be applied to COnStluCt experimental finite element and finite difference models of certain basic vibratory systems from data which is USUdlY available from modal testing. The advantages and limitation of the methods are discussed. A stiffness matrix B mass matrix E matrix in Eq. (21 " diag(h;) hi length of the i-th element K diaglki) k. stifeness of the i-th element I mi mass of the i-th element u, u* modal vectors ". v= ET", 3"' A. A* eigenvalue The problem of reconstructing the physical ;;;zmeters of vibratqry,systems of a certain , known a priory, 1s called an inverse vibration problem. The basic, we11 known, results are: (i) the reconstruction of a mass-spring model of an axially vibrating rod from two set of spectra, corresponding to the fixed-free and fixed-fixed end conditions, and (ii) the reconstruction of a mass-springrod model of a transversely Vibrating beam from spectral sets corresponding to three different end conditions. These results are expounded in detail in Gladwell Ill. Recently, Gladwell and Zhu [2] have shown how to reconstruct the physical parameters of a two, or thxe, dimensional lattice from particular spectra. Ram and Caldwell I31 have solved the problem of constructing multiconnected mass-spring system from constrained spectra, and it is shown, furthermore, in [4] that the simply connected mass-spring system may be reconstructed from its spectrum and the spectrum obtained by attaching a simple oscillator to the system. In these methods the construction is based on eigenvalue data and does not incorporate modal vector data such as is usually available from measurements. Hence, while the above mentioned results are of theoretical interest, their application in sys tern identification for continuous systems is limited for the following reasons. The model order n is determined by the number of illC?.SS"~~d eigenvalues. Since only a SI?ldll number of natural frequencies may be available experimentally, we cannot obtain the large order model that is required for aCC"rL=LCy purposes. Moreover, as 1s well known, the asymptotic behavior of the eigenvalues of the discrete model is different from the behavior of the corresponding continuous system. Thus, not surprisingly, the discrete model derived from the measured natural frequencies of the continuous system leads to results which are not sufficiently useful in practice. In order to circumvent this difficulty a new methodology has been introduced. As Ram and Gladwell (51 have shown. a natural frequency and tWO modal vectors are all that is needed to reCO"?.tr"Ct capart from a scale factor) a