Modal forcing

The diagonal elements of the excitation spectral intensity matrix describes the energy of the corresponding modes. The variations of their identified values are shown in Figures 3.27 and 3.28 for the Kammuri and Nuri, respectively. Note that comparing the spectral intensities of the modal forces for the different modes is meaningless since their values depend on the normalization of the mode shapes as discussed in Section 3.3.1.2. However, the time variation of the spectral intensity of a mode indicates the variation of the excitation energy with respect... 

Figure 3.32 shows the semi-log scatter plot of the modal frequencies and the corresponding spectral intensities of the modal forces. Due to the possible nonlinear characteristics of the structure, the equivalent modal frequencies decreased with an increasing value of the corresponding modal force. There were obvious negative linear correlations for the first three modes. However, for the fourth mode, the identified values with different typhoons followed different trend lines with different slopes. This indicates that the fourth modal frequency is sensitive to some other factors, such as structural interaction properties, or other ambient conditions (temperature, rainfall and humidity, etc.). Figure 3.33 shows a similar plot for the relationship... 

Figure 3.32 Identified modal frequencies versus the corresponding spectral intensities of the modal forces...

The modal forcing f is a linear combination of the components of g so it is also a zero-mean Gaussian vector process. It is stationary with the spectral density matrix function ... 

In the special case of modal identification, the first part of the forcing parameters are the elements of the upper right triangular part (diagonal inclusive) of the Nm x Nm submatrix of S fo of the modal forces (symmetry defines the lower triangular part). 

As observed previously in the examples for the Bayesian spectral density approach, the coefficients of variation for the frequencies in all cases are much smaller than those of the damping ratios, indicating that frequencies are identified much better than damping ratios. An additional result observed, but not tabulated here, is that the modal damping ratios exhibit significant correlation with the corresponding modal forcing spectral intensifies. 

The generalized modal forces due to the impulsive force at the second floor are... 

PSD matrix of modal forces SgC " (assumed constant within the band)... 

The prediction error PSD Sg. This includes contributions from the channel noise as well as modeling errors, e.g., due to unaccounted modes and colored nature of modal force. 

The asymptotic expressions of the posterior coefficient of variation of the modal parameters are summarized in Table 1. In terms of correlation, except for that between the damping ratio C and the modal force PSD S, the correlation between any pair of the modal parameters among /, C, S, Se, < is asymptotically small, at most 0(0-The correlation between C and S is... 

For small damping ratio, the posterior uncertainties of the natural frequency and mode shape are significantly smaller than those of the damping ratio and the modal force PSD. The former are 0(C ) and the latter are... 

Having in hand the modal displacements u,(r) along the structure, one can use the member stiffiiess properties in order to derive the modal force f ,(t) on the member and subsequently the required stresses. Considering all the modal contributions, one can write... 

Seismic Analysis of Concrete Bridges Numerical Modeling, Fig. 22 Modal force distribution, location of the equivalent SDOF systems, and modal parameters for the main transverse modes of the overpass bridge... 

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