Modal damping

The resonant beam test technique forms the basis of the ASTM Standard E756-83 for measuring the viscoelastic properties of damping materials. Fundamentally, the beam test requires that the resonant frequencies of a metal-beam, mounted in cantilever fashion, be determined as a function of temperature and frequency the beam is then coated with a polymer and the resonant frequencies and corresponding modal damping of the composite beam are determined as a function of temperature and frequency. From these two data sets, the vibration damping properties of the polymer can be evaluated. The ASTM Standard provides the necessary equations to obtain the complex modulus data from the collected test data and also provides guidelines for the proper choice of the specimens (1.21. The principal difference between the beam test and the other methods used here is that the beam test calculates the material properties from the test results on the metal beam and the composite beam whereas the... 

In the special case of modal updating, assume that only the lowest Nm modes contribute significantly to the model response and only the modal parameters of these modes are to be identified. Then, the uncertain structural parameters include the modal fi-equencies the modal damping ratios and the components of the measured DOFs of the mode shapes except the ones which are used for normalization, m = 1, 2,..., Nm, thus, there are a total number of Nm(No + 1) unknown modal parameters in modal identification. 

Figure 3.33 Identified modal damping ratios versus the corresponding spectral intensities of the modal forces...

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. 

By means of various methods (Ewins 1984), the respective frequency response Xij a)) can be approximated by the superposition of several (n) adapted harmonic oscillators (so-called modal harmonic oscillators). Their parameters, the modal eigenfrequency co , the associated eigenvector as well as the modal damping 6 , correspond to the modal matrices Mm, Dm, and Cm (Gawronski 2004), but they are not comparable with a real mass, damping, or stiffness. Those modal harmonic oscillators only describe an... 

Sometimes, for sake of conservatism, a condition is artificially imposed that the damping, for each vibration mode of the soil-structure, does not exceed a eertain pereentage of the critieal damping (e.g. 10 per eent). It has to be considered, in these evaluations, that the overall modal damping values should be weighted with the vibration energies relevant to the various parts of the soil-structure. 

It is our aim to characterize the modal behaviour of the turboset in order to avoid resonance. The variable of interest is a quantity characterizing sensitivity to unbalance at nominal speed. This quantity can be readily determined from the eigenfrequencies [Pg.1692]

Modal damping and eigenfrequencies can be represented by defined in (1). In terms of design criterion, Af shovtld remain lower than a critical value, Ms, during the nominal regime of the mrboset (Martinal 2000). Therefore, we have to check that ... 

The frequency response function Hj m) depends on the mass, damping and stiffness matrices it is a function of the natural frequencies >/, the modal damping ratios /, and the normalized mode shapes given by ... 

In order to measure the sensitivity of each parameter, a variation of 20% has been considered for both stiffness parameters and modal damping ratios. In Table 2 the values of the error 8/ are listed. It is seen that a variation of the parameter 0i implies the lowest values of 8/ among the stiffness parameters this is also the case for 3, 4, Is, and among the damping ratios. Therefore, a larger uncertainty is to be expected when these parameters are identified. 

The considered uncertain parameters are M R the parameters of the white noise in the ground motion record, grouped in the vector Z the stiffness parameters [0i 02 03 04 05 0g] and the modal damping ratios [ i 3 4 5 The selected struc-... 

The equivalent modal damping concept and its use in seismic design of steel structures... 

When all modes manage to satisfy the criteria, the transfer function modulus will attain a very smooth shape corresponding to a linear elastic MDOF structure. Then, Eq. (2) is numerically solved in order to obtain the modal damping values of the equivalent linear elastic MDOF structure. For the calculation of the modal damping ratios one uses a) the modulus of the transfer function at the peaks, b) the frequencies that correspond to these peaks and c) the participation factors as calculated from the classical eigenvalue problem involving the mass and initial stiffness of the given structure. 

The equivalent modal damping concept in seismic design -407... 

Figure 4 Design values for equivalent damping for 1.5% IDR and damage Of, = Oy (3rd mode). Table 2 Modal damping f as function of period T, IDR and damage.

The proposed seismic design scheme can be viewed as an improvement to the force based method of current codes with equivalent modal damping values playing the role of the strength reduction factor. 

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