Fourier-based Model Research Articles

Fourier-Based Vibration Model of Electrical Machines Considering Nonideal Orthogonality Between Electromagnetic Forces and Structural Modes

This paper proposes a Fourier-based model of vibration synthesis, improving both precision and speed. For the first time, the nonideal orthogonality between electromagnetic (EM) forces and structural modes is theoretically proposed and modelled by spatial harmonics. Different from the discrete vectors used in previous literature, the proposed method achieves a one-dimension (1-D) continuous model of EM forces and modes. On the basis, the vectors of modes and forces on tooth surfaces are decomposed by fast Fourier transform (FFT), so that the vibration response is synthesized based on Fourier coefficients. Further, the proposed model is applied to a 12-slot/10-pole (12s/10p) interior permanent-magnet (IPM) machine. The spatial harmonics of mode vectors are elaborated to identify the key orders of EM forces before the vibration response is completed. Then, the proposed model is compared with the coupled-field finite element analysis (FEA). The advantages are verified in terms of computational accuracy, speed, and sensitivity to mesh quality. Finally, the feasibility of the proposed model is verified by both coupled-field finite element analysis (FEA) and experiment.

Thermospheric Mass Density Modelling during Geomagnetic Quiet and Weakly Disturbed Time

Atmospheric drag stands out as the predominant non-gravitational force acting on satellites in Low Earth Orbit (LEO), with altitudes below 2000 km. This drag exhibits a strong dependence on the thermospheric mass density (TMD), a parameter of vital significance in the realms of orbit determination, prediction, collision avoidance, and re-entry forecasting. A multitude of empirical TMD models have been developed, incorporating contemporary data sources, including TMD measurements obtained through onboard accelerometers on LEO satellites. This paper delves into three different TMD modelling techniques, specifically, Fourier series, spherical harmonics, and artificial neural networks (ANNs), during periods of geomagnetic quiescence. The TMD data utilised for modelling and evaluation are derived from three distinct LEO satellites: GOCE (at an altitude of approximately 250 km), CHAMP (around 400 km), and GRACE (around 500 km), spanning the years 2002 to 2013. The consistent utilisation of these TMD data sets allows for a clear performance assessment of the different modelling approaches. Subsequent research will shift its focus to TMD modelling during geomagnetic disturbances, while the present work can serve as a foundation for disentangling TMD variations stemming from geomagnetic activity. Furthermore, this study undertakes precise TMD modelling during geomagnetic quiescence using data obtained from the GRACE (at an altitude of approximately 500 km), CHAMP (around 400 km), and GOCE (roughly 250 km) satellites, covering the period from 2002 to 2013. It employs three distinct methods, namely Fourier analysis, spherical harmonics (SH) analysis, and the artificial neural network (ANN) technique, which are subsequently compared to identify the most suitable methodology for TMD modelling. Additionally, various combinations of time and coordinate representations are scrutinised within the context of TMD modelling. Our results show that the precision of low-order Fourier-based models can be enhanced by up to 10 % through the utilisation of geocentric solar magnetic coordinates. Both the Fourier- and SH-based models exhibit limitations in approximating the vertical gradient of TMD. Conversely, the ANN-based model possesses the capacity to capture vertical TMD variability without manifesting sensitivity to variations in time and coordinate inputs.

Fourier Series Approximation of Vertical Walking Force-Time History through Frequentist and Bayesian Inference

The increased ambition of architects coupled with advancements in structural materials, as well as the rapidly increasing pressure on civil engineering sector to reduce embodied carbon, have resulted in longer spans and more slender pedestrian structures. These structures often have one or more low natural frequencies in the range of human walking accompanied with low modal masses and damping ratios. Thus, they are prone to excessive and often resonant vibrations that may compromise the serviceability limit state. Principally the uncertainty in prediction of the vibration serviceability limit state mainly originates from unreliable estimates of pedestrian loading. The key rationale behind this situation is the limited mathematical characterisation featuring in current design codes and guidelines pertinent to pedestrian-induced loading. The Fourier approximation is typically used to describe individual walking forces. Historically, such models are based on limited experimental data and deterministic mathematical descriptions. Current industry used load models featured in design codes and guidelines have been shown to incorporate inherent bias through limited intra-subject variation and poor correlation with real walking loads. This paper presents an improved Fourier model of vertical walking force across multiple harmonics, presented in a Bayesian and Frequentist statistical parameterisation. They are derived using the most comprehensive dataset to date, comprising of over ten hours of continuous vertical walking force signals. Dissimilar to previous Fourier models, the proposed models attempt to encapsulate the surround energy leakage around harmonic integers with a singular value. The proposed models provide consistently lower force amplitudes than any previous model and is shown to be more representative of real walking. The proposed model provides a closer approximation of a structural acceleration than any other similar Fourier-based model. The proposed model provides further evidence to combine the so called high and low frequency load models.

Fourier-Based Modeling of an Induction Machine Considering the Finite Permeability and Nonlinear Magnetic Properties

Performance prediction of induction machines (IMs) is highly dependent on the accuracy of the material characteristics and geometry of the machine during the modeling stage. To reduce the complexity of an IM model, normally the slotting effects are neglected. Likewise, the permeability of the stator and rotor cores are assumed to be infinite leading to an ideal set of partial differential equations (PDEs) for homogenous and isotropic materials. In this paper, the permeability of the stator and rotor cores are assumed not to be infinite and the slotting effects are taken into consideration to propose a more accurate and realistic model of an IM to reduce the discrepancies between the performance expectations and actual results. Fourier-based (FB) magnetic field approach is used to fulfill this aim via anisotropic layer theory (ALT) enabling the proposed model to include distortion of magnetic flux in the slotted regions. Air-gap flux density, core losses, efficiency and torque of an IM are predicted via the FB model and are validated through the finite element analysis (FEA) and experimental studies.

Optimizing linear phased array transducers for non-destructive evaluation of thin layers

AbstractIn this work, a numerical method is proposed in order to achieve design optimisation of phased array (PA) probes for the special application of defects detection in thin films. This approach relies on an extended Fourier-based model that was adapted to predict the two-dimensional ultrasonic displacement field taking place in a thin plate under individual excitation of PA probe elements which have arbitrary orientation with respect to the examined part surface. Excitation is applied through a fluid couplant and is operated at scheduled delays that are managed to enable emission of constructive pulses. This gives the possibility to steer sound waves towards a direction and to focalize the beam in a selected point. An optimisation algorithm based on the concept of pattern search that does not require evaluation of a gradient was used to find the best match in the multidimensional analysis space of possibilities including the elements orientation angles, the elements lengths, the inter-elements distances and work frequency. Optimisation was performed with the objective to maximize the displacement amplitude at the focal point while minimizing simultaneously the effect of beam side lobes. The results obtained by this approach reveal that focalisation can be achieved with enhanced features in comparison with previous algorithms assuming linear elements that are parallel to the surface of the plate.

Two-Dimensional Fourier-Based Modeling of Electric Machines—An Overview

An increasing need for fast and reliable models has led to a continuous development of Fourier-based (FB) analytical modeling. This paper presents an overview of the techniques that are currently available in FB modeling for electric machines. By coupling that overview to the most relevant literature related to the subject, an interesting starting point is provided for anyone who wants to use or improve FB models. The following seven aspects of FB models are discussed in detail: 1) the magnetic potential (scalar or vector potential); 2) the coordinate system and the solution of the partial-differential equations for each magnetic potential and for each coordinate system; 3) the way in which time dependence is accounted for; 4) the implementation of the source terms; 5) the possibilities to account for slotted structures; 6) the modeling of eccentricity; and 7) the post-processing computation of physical quantities, such as flux density, electromotive force, torque, losses, and eddy currents in conductive objects. Furthermore, this paper gives the closed form solution of the Laplace, Poisson, and Helmholtz equations in each coordinate system. In addition, this paper tackles other important features of FB models such as computational time reduction and coupling the machine model to an electric circuit.

Hourly ambient air humidity fluctuation evaluation and forecasting based on the least-squares Fourier-model

Air humidity is one important measurement parameter involved in a wide range of fields of science and engineering. This study addresses Fourier-series-based evaluation and prediction of hourly ambient air humidity fluctuation. It is shown that hourly ambient humidity fluctuation displays a significant 24-hour cyclical variation. Then on the basis of daily periodic extension for hourly ambient air humidity fluctuation, a finite Fourier-series-based evaluation model is introduced for describing its hourly fluctuation. The Fourier-based analysis in its conventional form, however, cannot be straightly applicable to prediction. Then in combination with the least-squares method for coefficient-seeking based on finite observations of hourly air humidity, the extended version of the Fourier-series model in the least-squares is proposed for forecasting. The proposed Fourier-series-based method is applied to experiments of evaluating and forecasting hourly air humidity fluctuations at different monitoring stations with satisfying results. The experimental results, further mathematical analysis of the conjugate symmetry property and spectrum characteristic analysis for the Fourier-series involved in the Fourier-based model indicate that daily 24-hour air humidity fluctuation is well described by about 12-term harmonics and the extended Fourier-series forecasting model predicts the hourly humidity fluctuation best fitting with less than 6-term harmonics.

Computational-Time Reduction of Fourier-Based Analytical Models

An increasing interest in both efficient electric machines and more extensive control strategies demands evermore faster simulation tools. In that context, Fourier-based models, which combine low computational times with a high accuracy, have already proven their value. However, even Fourier-based models may encounter problems related to CPU usage. To cope with these problems the authors present a number of simple techniques to reduce the computational effort of Fourier-based models for synchronous machines. The techniques are based on simplifying the studied geometry and a qualitative analysis of the machine's time- and spatial-harmonic content. The proposed techniques are validated and a benchmark test has shown that a great reduction in computational burden can be achieved without significant loss of accuracy. The preliminary harmonic analysis gives, by far, the largest reduction of the computational burden.

Periodic Heat Transfer in Convective Fins Based on Dual-Phase-Lag Theory

Heat transfer enhancement through extended surfaces is crucial in modern engineering applications such as microelectromechanical systems and electronic components. Conduction heat transfer is the only way to achieve this objective when the mixing augmentation is not possible. This paper investigates the effects of non-Fourier thermal conduction in convective straight fins with arbitrary constant cross section under periodic boundary conditions by introducing the exact analytical solution for the dual-phase-lag heat conduction. The corresponding analytical approach is developed through the Laplace transform method and inversion theorem. To help advance the understanding of fin thermophysical behavior, a generalized model of conduction heat transfer equation is used for studying all of the interpretations to Fourier-based model, namely, parabolic thermal diffusion, hyperbolic thermal wave, and dual-phase-lag models. Heat flux and temperature gradient relaxation times are the characteristics of the dual-phase-lag model, and the simulation results are strictly a function of these two parameters. Therefore, fin temperature distributions are presented for flux precedence and gradient precedence heat flow regimes. Calculations are performed to investigate the influence of temperature gradient relaxation time on the hyperbolic heat conduction characteristics of non-Fourier fins.

Voltage Sources in 2D Fourier-Based Analytical Models of Electric Machines

The importance of extensive optimizations during the design of electric machines entails a need for fast and accurate simulation tools. For that reason, Fourier-based analytical models have gained a lot of popularity. The problem, however, is that these models typically require a current density as input. This is in contrast with the fact that the great majority of modern drive trains are powered with the help of a pulse-width modulated voltage-source inverter. To overcome that mismatch, this paper presents a coupling of classical Fourier-based models with the equation for the terminal voltage of an electric machine, a technique that is well known in finite-element modeling but has not yet been translated to Fourier-based analytical models. Both a very general discussion of the technique and a specific example are discussed. The presented work is validated with the help of a finite-element model. A very good accuracy is obtained.

Editorial: Special issue on time series analysis in the biological sciences

Editorial: Special issue on time series analysis in the biological sciences

A Fast Loss and Temperature Simulation Method for Power Converters, Part II: 3-D Thermal Model of Power Module

This paper describes the development and implementation of an analytical 3-D thermal model for fast and accurate thermal simulation of power device modules in electrothermal converter simulation. A Fourier-based solution is used to solve the 3-D heat equation. The solution can describe the variation of temperature through the whole inverter power module structure as a function of time. The model can simulate thermal interactions resulting from multiple heat sources. The thermal model is extremely fast to simulate compared to finite-element (FEM) approaches. The new model has been implemented in MATLAB/Simulink in order to cosimulate with the converter model which is in the same form. The model has been validated against the computational fluid dynamics (CFD) software package FLOTHERM and shows good agreement. The required aspects of 3-D heat diffusion are captured successfully by the Fourier-based model.

Mathematical moiré models and their limitations

In a previous recent paper we discussed the limitations of the Fourier-based model in the prediction of visible moiré effects. Here, we extend the discussion to other mathematical models that are also being used in the moiré theory: the classical indicial equations method for modelling the moiré fringes between periodic or curvilinear gratings, and two methods that are used to model different facets of the moiré effects between correlated aperiodic layers. We discuss the limitations of each model, explain their significance, and suggest possible ways to overcome them.

Comparison of model and human observer performance for detection and discrimination tasks using dual‐energy x‐ray images

Model observer performance, computed theoretically using cascaded systems analysis (CSA), was compared to the performance of human observers in detection and discrimination tasks. Dual-energy (DE) imaging provided a wide range of acquisition and decomposition parameters for which observer performance could be predicted and measured. This work combined previously derived observer models (e.g., Fisher-Hotelling and non-prewhitening) with CSA modeling of the DE image noise-equivalent quanta (NEQ) and imaging task (e.g., sphere detection, shape discrimination, and texture discrimination) to yield theoretical predictions of detectability index (d') and area under the receiver operating characteristic (Az). Theoretical predictions were compared to human observer performance assessed using 9-alternative forced-choice tests to yield measurement of Az as a function of DE image acquisition parameters (viz., allocation of dose between the low- and high-energy images) and decomposition technique [viz., three DE image decomposition algorithms: standard log subtraction (SLS), simple-smoothing of the high-energy image (SSH), and anti-correlated noise reduction (ACNR)]. Results showed good agreement between theory and measurements over a broad range of imaging conditions. The incorporation of an eye filter and internal noise in the observer models demonstrated improved correspondence with human observer performance. Optimal acquisition and decomposition parameters were shown to depend on the imaging task; for example, ACNR and SSH yielded the greatest performance in the detection of soft-tissue and bony lesions, respectively. This study provides encouraging evidence that Fourier-based modeling of NEQ computed via CSA and imaging task provides a good approximation to human observer performance for simple imaging tasks, helping to bridge the gap between Fourier metrics of detector performance (e.g., NEQ) and human observer performance.