Bayesian Fast Fourier Transform Approach for Modal Updating Using Ambient Data

The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian Fast Fourier Transform approach (BFFTA) for modal updating is presented which uses the statistical properties of the Fast Fourier transform (FFT) to obtain not only the optimal values of the updated modal parameters but also their associated uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important when one plans to proceed with the updating of a theoretical finite element model based on modal estimates. The proposed approach requires only one set of response data in contrast to many of the existing frequency-based approaches which require averaging. It is found that the updated PDF can be well approximated by a Gaussian distribution centred at the optimal parameters at which the posterior PDF is maximized. Examples using simulated data are presented to illustrate the proposed method.

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Optimization Method of the Car Seat Rail Abnormal Noise Problem Based on the Finite Element Method

Shock and Vibration ◽

10.1155/2017/4132092 ◽

2017 ◽

Vol 2017 ◽

pp. 1-13

Author(s):

Huijie Yu ◽

Xinkan Zhang ◽

Chen Zhang

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Natural Frequencies ◽

Joint Probability ◽

Element Model ◽

Joint Probability Distribution ◽

Natural Frequencies And Modes ◽

Car Seat ◽

Finite Element Software ◽

Element Method

The finite element model of the seat rail is established with a spring-damping element to simulate the ball in the rail joint part. The stiffness and damping parameters of the joint part are determined by the combination of finite element method and experiment. Firstly, the natural frequencies and modes of the guide rail are obtained by modal experiment. The stiffness of the spring-damping element is optimized in the finite element software to make the natural frequencies and modes of the system consistent with the experimental ones. Secondly, the dynamic response curve of the key nodes is obtained through sweeping experiment, and the damping of the spring-damping element is optimized in the finite element software to make the nodal response of the system output consistent with the experiment. Then, the gap of the joint part of the car seat rail is studied considering the factors of load and structure randomness. The influence factors of the gap are selected by Hammersley experimental design method. The results show that the gap is normally distributed, and therefore the confidence interval of the gap is obtained. Finally, the joint probability distribution of the gap is obtained under the condition that the load and the structure are all random, which provides the theoretical guidance for determining the reasonable gap of the joint.

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The Application of Probability Theory to Model Updating

Volume 1B: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4168 ◽

1997 ◽

Author(s):

Loukas Papadopoulos ◽

Ephrahim Garcia

Keyword(s):

Model Updating ◽

Test Procedure ◽

Simulated Data ◽

Element Model ◽

Reduction Factor ◽

Mode Shapes ◽

Modal Parameters ◽

Stiffness Reduction ◽

Experimental Errors ◽

Global Stiffness Matrix

Abstract A method is proposed for probabilistically model updating an initial deterministic finite element model using measured statistical changes in natural frequencies and mode shapes (i.e., modal parameters). The approach accounts for variations in the modal properties of a structure (due to experimental errors in the test procedure). A perturbation of the eigenvalue problem is performed to yield the relationship between the changes in eigenvalues and in the global stiffness matrix. This stiffness change is represented as a sum over every structural member by a product of a stiffness reduction factor and a stiffness submatrix. Monte Carlo simulations, in conjunction with the variations of the structural modal parameters, are used to determine the variations of the stiffness reduction factors. These values will subsequently be used to estimate statistics for the corrected stiffness parameters. The effectiveness of the proposed technique is illustrated using simulated data on an aluminum cantilever Euler-Bernoulli beam.

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Data-Driven Modeling of a Commercial Photovoltaic Microinverter

Modelling and Simulation in Engineering ◽

10.1155/2018/5280681 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11

Author(s):

Hayder D. Abbood ◽

Andrea Benigni

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Frequency Domain ◽

Time Domain ◽

Simulated Data ◽

Data Driven ◽

System Level ◽

Operating Modes ◽

System Level Simulation ◽

Data Driven Modeling

We present a data-driven modeling (DDM) approach for static modeling of commercial photovoltaic (PV) microinverters. The proposed modeling approach handles all possible microinverter operating modes, including burst mode. No prior knowledge of internal components, structure, and control algorithm is assumed in developing the model. The approach is based on Artificial Neural Network (ANN) and Fast Fourier Transform (FFT). To generate the data used to train the model, a Power Hardware in the Loop (PHIL) approach is applied. Instantaneous inputs-outputs data are collected from the terminals of a commercial PV microinverter at time domain. Then, the collected data are converted to the frequency domain using Fast Fourier Transform (FFT). The ANNs that are the core of the DDM are developed in frequency domain. The outputs of the ANNs are then converted back to time domain for validation and use in system level simulation. The comparison between measured and simulated data validates the performance of the presented approach.

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Unique Method for Generating Design Earthquake Time History Seeds

Volume 8: Seismic Engineering ◽

10.1115/pvp2008-61244 ◽

2008 ◽

Author(s):

R. E. Spears

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Random Phase ◽

Time History ◽

Energy Curve ◽

Cumulative Energy ◽

Original Curve ◽

Time Histories ◽

Design Earthquake ◽

New Time

A method has been developed which takes a single seed earthquake time history and produces multiple similar seed earthquake time histories. These new time histories possess important frequency and cumulative energy attributes of the original while having a correlation less than 30% (per the ASCE/SEI 43-05 Section 2.4 [1]). They are produced by taking the fast Fourier transform of the original seed. The averaged amplitudes are then pared with random phase angles and the inverse fast Fourier transform is taken to produce a new time history. The average amplitude through time is then adjusted to encourage a similar cumulative energy curve. Next, the displacement is modified to approximate the original curve using Fourier techniques. Finally, the correlation is checked to ensure it is less than 30%. This process does not guarantee that the correlation will be less than 30% for all of a given set of new curves. It does provide a simple tool where a few additional iterations of the process should produce a set of seed earthquake time histories meeting the correlation criteria.

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Updating direct methods

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273318016443 ◽

2019 ◽

Vol 75 (1) ◽

pp. 142-157 ◽

Cited By ~ 1

Author(s):

Carmelo Giacovazzo

Keyword(s):

Probability Distribution ◽

Prior Information ◽

Structural Model ◽

Joint Probability ◽

Direct Methods ◽

Distribution Functions ◽

Joint Probability Distribution ◽

Conditional Distributions ◽

Interatomic Distances ◽

Different Types

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two-, three- and four-phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures.

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On the suitability of using the diagonal Gaussian approximation in the calculation of the magnitude-based likelihood function

Acta Crystallographica Section D Structural Biology ◽

10.1107/s205979832100262x ◽

2021 ◽

Vol 77 (5) ◽

pp. 663-673

Author(s):

Vladimir Y. Lunin

Keyword(s):

Fourier Transform ◽

Probability Distribution ◽

Likelihood Function ◽

Gaussian Approximation ◽

Joint Probability ◽

Joint Probability Distribution ◽

Structure Factors ◽

Preliminary Model ◽

Computational Procedures ◽

The Fourier Transform

Statistical likelihood maximization is currently one of the main tools in computational procedures in biological crystallography. In these procedures, the likelihood function is calculated, as a rule, within the framework of a diagonal Gaussian approximation (DGA) of the joint probability distribution of the real and imaginary parts of a set of structure factors. This approximation assumes pairwise uncorrelated values of various structure-factor components. In this paper, exact formulas are derived for pairwise correlations of structure factors, and conditions under which these correlations can be considered to be negligible are discussed. It is shown that in the case where the probability distribution of the atomic coordinates is related to the region of the molecule or its domains, the correlation of the structure factors of reflections s and w is determined mostly by the magnitudes of the Fourier transform of the probability distribution calculated at the points 2s, 2w, s − w and s + w. However, in the case where the probability distribution describes small corrections to the coordinates of the existing preliminary atomic model, the correlation is determined by the values of the structure factors of the preliminary model that correspond to the 2s, 2w, s − w and s + w reflections rather than by the Fourier transform of the probability distribution. Test cases demonstrate that the practice of using the DGA for calculation of the likelihood when based on sets containing neighbouring reflections may be unjustified in some crystallographic applications, especially in single-particle studies.

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