Modal expansion

This is the propagation equation that is actually solved numerically, but let us express it in terms of field rather than in terms of modal expansion coefficients. From a modal expansion, the transverse part of the electric field is... 

Suppose we have a solution at z = 0, and want to propagate it over a distance corresponding to an integration step Az. Although we wrote the equation in terms of field, it is more natural to solve numerically for the modal expansion coefficients to eliminate fast oscillatory terms. We therefore express the field through the expansion coefficients A which are actually the native solver variables ... 

The modal expansion of effective earthquake forces is shown in Fig. P13.1b. 

S3 = r3in03 = 0.0972m The modal expansion of ml is shown next ... 

The modal expansion of the spatial distribution ml of the effective forces is shown in the following figure. 

The modal expansion of the spatial distribution of effective forces is shown in Fig. PI3.17b. The effective forces in the third mode are all zero, implying that this mode will not be excited by horizontal ground motion. 

Calculation of the (evolutionary) power spectal density of any state variable of engineering importance is performed by a simple numerical integration. Thus, the whole problem is shifted to the efficient determination of the sure frequency response function. The hybrid problems are treated by the substructure synthesis method applying modal expansions of the hydrodynamic pressure. The large water body is analyzed by the boundary element method (BEM), the elastic structure by the finite element method (FEM) when vibrating in vacuum. 

The elastic displacement vector of the neutral axis, [ iUe(P), 0e(P) ] , can be defined with respect to the modal or nodal coordinates depending on the discretization modal expansion or finite elements method. 

This new form allows us to deal with the finite elements model as well as with the modal expansion model. 

This entry presents the authors recent work on data-driven output-only modal identification and damage detection of stmctures. It is found that exploiting the sparse essences of modal expansion and damage information can efficiently and effectively address some challenging problems in ouqjut-only modal identification and damage detection via BSS. A series of novel algorithms are developed with experimental and real-world structure examples for demonstrations. 

In the above section, the sparse wavelet ICA is introduced for identification of the hidden spikelike features that typically indicate structural damage information. In the following, the sparse properties of modal expansion are further exploited and lead to a new output-only modal identification method STFT-ICA within the BSS fi amework which can handle even highly damped structures, as detailed in the following. 

The BSS problem has a similar pursuit with the output-only modal identification issue. The close similarity is implied between the modal expansion Eq. 11 and the BSS model Eq. 1. If the system responses are fed as mixtures into the BSS model, then the target of output-only identifying 4> and q(0 in Eq. 11 can be solved by blind recovery of A and s(0 using those BSS techniques dependent on the independence assumption since q(0 typically with incommensurable fi-equencies are independent on the modal coordinates. 

To address the aforementioned issues of ICA, it is proposed by Yang and Nagarajaiah (2013a) to transform the time-domain modal expansion Eq. 11 to the time-frequency domain where the target modal responses have sparse representations, using the short-time Fourier transform (STFT) prior to the ICA estimation. 

This study shows that properly exploiting the sparse essences of modal expansion and damage information could efficiently and... 

Generalized coordinate analysis Modal expansion Mode acceleration method Mode displacement method Mode superpositimi method Multi-degree-of-freedom (MDOF) system decoupling Order reduction Response series truncation... 

R.H.T. Bates, Modal expansions for electromagnetic scattering from perfectly conducting cylinders of arbitrary cross-sections, Proc. lEE 115, 1443 (1968)... 

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