H j

where Gf (a>) and Gyy(«) are the auto-power spectra of exciting force f (t) and structure response y(t), respectively. When the frequency spectrum of input source is smooth and approximately close to white noise with a finite bandwidth, the power spectrum can be treated as a constant C, which reduces Eq. (2) to:

where \H(w)\ equals to the ratio of amplitude of the signal at test point to that at reference point; and is the phase difference between the signal at test point and that at reference point. The signs of mode shape coordinate at each test point are correlated to the phase relation of cross-power spectra: same signs for same phases, opposite signs for opposite phases.

For ancient pagodas with small damping ratios and well-separated modal frequencies, when rn & ¿> under the arbitrary random excitation, the ratio between peak value of auto-power spectrum and that of cross-power spectrum, that is its transfer function, can be approximated by the ratio of mode shapes.

3.4 Model updating and the reference criteria

Due to the uncertainty of structural parameters of the ancient pagoda, dynamic characteristics predicted by the analytical model often differ from field measurements. The proper reference criteria should be provided for the structural parameters identification in the model updating.

From the field testing we can obtain the first n order modal parameters of a structure with N degrees of freedom, i.e., natural frequencies [w2T] = diag(a>j, &>|,..., ofy and mode shapes [fir ] = diag }, {fi2},..., {fin}]. The structural mass matrix [Ma] and stiffness matrix [KA] can be determined through structural analysis (finite element analysis for example). Basically, there are two sets of reference criteria for structural parameters identification (Li Guoqian, Li Jie, 2002): (1) [k2] = [w\], [fi] = [fiT]; and (2) [k2] = [a>\], [M] = [Ma].

Generally, [rn\] measured from the field test is accurate while [fir] is less accurate. On the other hand, [.Ma] obtained from structural analysis is comparatively accurate but [Ka] is less close to the actual values. For the ancient pagodas, because absence of the original constructing data, the field test data is very valuable for the structural parameters identification. To take full advantage of the information from field test, the reference criteria (1) is usually selected as the reference for the analysis model updating.

Many model updating techniques have been developed in the past (Smith M J, Hutton S G. 1992, Farhat C, Hemez F M. 1993, Renken J A. 1995, Alvin K F. 1996, Chung Y T. 1997), however, for a pagoda with complicated architectural details, the 3D F.E. model consists of tens of thousands of meshing units and the degrees of freedom usually, it is still a burdensome task on structural parameters identification. To improve the effect of the model updating, the sensitivity system should be constructed for selection of the structural parameters firstly. Besides, taking the fast analysis advantage of ANSYS program, the conventional trial-error method also can be used to simplify the model updating procedure.

3.5 Sensitivity of dynamic behavior to structural parameter adjustment

The sensitivity-based model updating procedure has been recognized as an effective approach for improving FE models and the correlative researches (Zhang Dewen, Zhang Lingmi, 1992, Jung H. 1992, Friswelli MI, and Mottershead JE. 1995, Dascotte E, et al. 1995) are helpful for the construction of the sensitivity system of ancient pagodas.

Suppose the structural parameters such as mass, stiffness, geometric dimensions and material characteristics described as p,-(i = 1, 2,...,n), and the eigenvalues or eigenvectors are considered as derivative functions of structural parameters, the dynamic characteristics of the pagoda can be expressed as F = F(p1, p2,..., pn). Then the sensitivity of F to structural parameter pi is SFp, = dF(p1, p2,..., pn)/dpt, and the bigger the absolute value of Sppi, the more the sensitivity of model characteristic to structural parameter p,. A little adjustment to pi will often cause a big change in the dynamic characteristics.

According to the structural dynamics, the y-order eigenvalue Xy and eigenvector fi(Y), should satisfy:

By solving the above formula's partial derivative to the i— parameter, the sensitivity of the eigenvalue can be obtained:

And the sensitivity of the eigenvector is:

And the sensitivity of the eigenvector is:

Taking the stiffness parameter of structure as main study object, the sensitivity of the y-order eigenvalue and eigenvector to stiffness kg, respectively, are:

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