Modal frequency

The bending of pitch as a function of amplitude can be modeled explicitly in additive and modal models, by controlling the modal frequencies with a time-... 

In most deterministic identification methods, important observable quantities are measured (e.g., modal frequencies of a building for stiffness identification) and the uncertain model parameters are obtained by minimizing a goodness-of-fit/error function of these measurements... 

The adaptive MCMC simulation method is applied to updating the robust reliability for a two-story structural frame, depicted in Figure 2.21. The bay width and story height are 5.0 m and 2.5 m, respectively. The Young s modulus and mass density are taken to be 200 GPa and 7800 kg/m, respectively. The beams have a cross-sectional area of 0.01 vn5 and a moment of inertia of 6.0 X 10 m but they are 0.02 m and 1.5 x 10 m" for the columns. As a result, the structure has modal frequencies of 5.20 and 15.4 Hz. The structure is assumed to have 1.0% of critical damping for all modes. A simple model with two degrees of freedom is used in the system identification in order to estimate the inter-story stiffnesses and to assess the reliability of the structure. Specifically, the stiffness matrix is given by ... 

Two cases are considered for the available dynamic data that correspond to locally identifiable and unidentifiable cases. For the first case, the modal data T> consists of the identified modal frequencies for both modes of the frame. Rather than performing modal identification on simulated time histories, noisy versions of the modal frequencies are generated and the measurements are = 5.5 Hz and = 14.9 Hz. 

If it is assumed that only one sensor (at either the first or second floor) was used during the modal testing, only the modal frequencies can be identified. In this case, the two stiffness parameters are locally identifiable and the normalized modal goodness-of-fit function is given... 

The second case assumes that only the first modal frequency is observed because, for example, the second mode is not excited significantly during the modal testing. In this case. 

Figure 2.27 shows the updated failure probability of the frame. Assessment of the impact of damage on the reliability of the structure can be performed even though there are infinitely many most probable parameter estimates in this case and it is also seen from the similarity of Figures 2.24 and 2.27. The loss of information about the second modal frequency does not have much effect on the updated failure probability because the first mode dominates the response of this frame. Even the individual values of 0i and 02 cannot be identified, different combinations... 

Temperature variation induces both geometric and material effects to a structure. As a result, the modal frequencies of a structure are influenced by the temperature and it is more important... 

It remains a quadratic function but the coefficients are uncertain. Equation (2.164) is applicable to all modal frequencies for Timoshenko beam models. Therefore, it is proposed to bridge the squared fundamental frequency and the ambient temperature by a quadratic function, and the coefficients can be estimated by Bayesian analysis. 

The problem of parametric identification for mathematical models using input-output or output-only dynamic measurements has received much attention over the years. One important special case is modal identification, in which the parameters for identification are the small-amplitude modal frequencies, damping ratios, mode shapes and modal participation factors of the lower modes of the dynamical system. In other words, the model class in modal identification is the class of linear modal models. Many time-domain and frequency-domain methodologies have been formulated for input excitation and output response measurements [24,48,75,81,187],... 

The modal frequencies of the building are also identified. They correspond to the equivalent linear system of the building since this structure may exhibit nonlinear behavior under this level of excitation. Figure 3.29 shows the variation of the modal frequencies with the associated plus and minus three standard derivations ( 3a) confidence intervals during the typhoon Kammuri. This interval includes a probability of 99.7% for the equivalent modal frequency falling in this range since the posterior PDF is approximately Gaussian. When the intensity of the excitation... 

Figure 3.29 Variation of the identified modal frequencies (Kammuri)...

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...

In the special case of modal updating, assume that only the lowest Nm modes contribute significantly to the response and only the modal parameters of these modes are to be identified. Then, the structural parameters are the modal frequencies modal damping ratios and the elements of the first mode shapes except those elements which are equal to unity for normalization purposes, m = 1,2,Nm- Thus, there are a total number of Nm No + 1) unknown structural modal parameters. 

Figure 4.14 Contours of the marginal updated joint PDF of the first two modal frequencies...

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

In this chapter, a Bayesian model updating method using incomplete modal data is presented with applications to structural health monitoring. As reported in the literature [18,51,52,267], the realistic assumption is made that only the modal frequencies and partial mode shapes of some modes are measured system mode shapes are also introduced, which avoid mode matching between the measured modes and those of the dynamical model. The novel feature... 

In the next section, the proposed updating approach is presented which provides estimates of the system modal frequencies and system mode shapes, as well as estimates of the stiffness model parameters, based on incomplete modal data. Examples with a twelve-story building and a three-dimensional braced frame wiU be used to demonstrate the method with applications to structural health monitoring. 

The prior PDF p(k, 0, C) implies that, given a class of dynamical models and before using the dynamic test data, the most probable values of X and 0 are those that minimize the Euclidean norm (2-norm) of the error in the eigen equation for the dynamical model. This implies that the prior most probable values of X and 0 are the squared modal frequencies and mode shapes of a dynamical model, but these values are never explicitly required. This prior PDF will have multiple peaks because there is no implied ordering of the modes here. 

The mode shapes are usually measured with incomplete components, i.e., with missing DOFs but the modal frequencies are measured with relatively high accuracy. Therefore, the sequence of optimization starts from computing the missing components of the mode shapes. First set the updated model parameters at their nominal values ... 

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