Fourier Series Of Functions Research Articles

Effects of surface roughness in microchannel with passive heat transfer enhancement structures

We present a numerical study of the effects of surface roughness, which is unavoidable due to manufacturing tolerance, on laminar flow and heat transfer in microchannels. Three different channel types namely, square channel, wavy channel and dimpled channel are studied. Surface roughness is modeled as a superposition of waves described by Fourier series and exponential correlation function. Up to 2% relative roughness is introduced to one of the four side walls. We introduce the surface roughness by meshing a smooth channel. The boundary nodes are then displaced to introduce surface roughness. Periodic boundaries are used to study the flow and heat transfer features in the fully developed region. The Reynolds number keeps at 500, and 0.5 W/mm2 heat flux is adopted on the bottom of heat sinks. Results show that roughness could increase the overall flow resistance and Nusselt number. While relative roughness is about 2%, pressure drop increased by 3%, 5.3%, 5.9%, Nusselt number increased by 10%, 6.5%, 19.8% in square channel, wavy channel and dimpled channel respectively. Moreover, local influence of roughness highly depends on the geometric shape of the microchannel. The simulation results are expected to provide guidance to roughness control in microchannel manufacturing.

Structural integrity assessment on cracked composites interaction with aeroelastic constraint by means of XFEM

In this paper, a novel approach in assessing the structural integrity of cracked composite plates under the aeroelastic condition by using XFEM is presented. To the authors’ knowledge, this is the first time that aeroelastic condition is coupled in XFEM to model the crack propagations. Previous researches from the literature had only considered a static crack condition. This research focuses on determining the first failure experienced by the cracked composite plate, either the crack will propagate causing a fracture, or the composite plate will fail due to aeroelastic instability imposed at the critical flutter speed. The proposed scheme is used to solve the limitation in XFEM within Abaqus that only general static and implicit dynamic analysis can be performed. The structure is assumed to interact with minimal gust, and the deflections by time are expressed in the equation of periodic motion based on Fourier Series Function (FSF). The results show that at a particular fibre orientation, once the damaged composite plate is deformed due to the dynamics load at dive speed, it fails due to the crack propagation first instead of the flutter. In contrast, another fibre configuration shows good resistance to crack propagation and fails due to flutter instability.

Density functional theory calculations of generalized stacking fault energy surfaces for eight face-centered cubic transition metals

In this work, we use density functional theory to calculate the entire generalized stacking fault energy (GSFE) surface for eight transition metals with a face-centered cubic structure: Ag, Au, Cu, Ir, Ni, Pd, Pt, and Rh. Analysis of the ⟨112⟩ GSFE curves finds that the displacements corresponding to the unstable stacking fault energy are larger than the ideal value for all eight metals except Ag and Cu. Over the entire surface, Pt is found to not possess well-defined local maxima or minima, suggesting spreading in favor of dissociation of the dislocation core, unlike the other seven metals. Our calculations also reveal that at a large ⟨112⟩ displacement, where atoms on two {111} adjacent planes are aligned, an anomalous local minimum occurs for Ir and Rh. The oddity is explained by relatively large, localized atomic displacements that take place in the two metals to accommodate the alignment that do not occur in the other six metals. In addition to the fully calculated surfaces, we characterize a continuous 11-term Fourier-series function, which provides a particularly excellent representation of the GSFE surfaces for Ag, Au, Cu, Ni, and Pd.

A new Fourier series method and displacement function for in-plane vibration and power flow characteristics analysis of isotropic rectangular plates

The generation of in-plane vibration in plates is an important issue and frequently occurs due to the presence of excitations in the ship’s hull due to turbulent fluid flows, turbulent airflow excitation on aerospace structures, gear system subjected to axial excitation, assemblies housing piezoelectric crystals and sandwiched plates, and so on. The present analysis aims to establish a universal and numerically efficient method for determination of in-plane vibration characteristics of isotropic rectangular plates both for conventional and general boundary conditions. The new in-plane Fourier series and displacement function of the plate have been developed using beam displacement functions in x and y directions, respectively, under in-plane condition. A modified Fourier series assumption for the in-plane beam displacement has been utilised and further developed as plate displacement function. The computational efficiency of the present method is compared in terms of convergence of natural frequency parameter, speed of execution and manual convenience to reduce human errors with the frequently used Fourier series method by various researchers. Rayleigh–Ritz procedure has been applied to determine the in-plane natural frequencies. The mode shapes for few conventional and generally varying boundary conditions have been presented and analysed. The dynamic response has been obtained and analysed in terms of the in-plane mobility and power flow characteristics of the plate under varying boundary conditions. The validity of results obtained by the current method has shown excellent accuracy and faster convergence with the existing results. The present results can provide a benchmark to analyse the dynamic in-plane response of plate systems being used for built-up structures in real engineering applications.

Local bending stiffness identification of beams using simultaneous Fourier-series fitting and shearography

In this paper, we present a novel method for the identification of the local bending stiffness of beams. We use shearography to capture measurements of vibrating beams, so the input data for the identification is the modal slope – the differential of the modal shape. The modal slope is fitted by two Fourier-series functions, one of which is derived from a thin-beam model. The local bending stiffness is identified as the one corresponding with the best match between the measured and the two fitted modal slopes. This identification method, which we call simultaneous Fourier-series fitting, is demonstrated on numerically-generated inputs, as well as on experimental measurements. We use a flat, concave and convex beam, as well as beams with locally varying bending stiffness mimicking local damage to verify the method. It is shown that the method gives accurate results and is robust to noise. Additionally, it has advantageous properties that make it useful and practical: using this method, it is possible to perform the identification from only a sub-region of a beam and even without specifying the boundary conditions.

Vibration analysis of nanorods by the Rayleigh-Ritz method and truncated Fourier series

Abstract A solution procedure applicable for analyzing the free longitudinal vibration of nanorods is presented in this study. The longitudinal displacement of the nanorod is sought as a linear combination of a Fourier series and auxiliary trigonometric functions. The auxiliary functions are introduced to account for all the relevant discontinuities in the boundary displacement and its derivatives. The improved Fourier series represents a residual or conditioned displacement that is of at least C1 class. The basic theory of the nanorod is shown, which included the total potential energy and the kinematic energy of the nanorod system. The Rayleigh-Ritz method is applied to determine the coefficients of the improved Fourier series expansion for the displacements. The solution approach here is referred to as the improved Fourier-Ritz method. The generality, accuracy and efficiency of the presented approach are fully demonstrated and verified through benchmark examples involving classical and elastic boundary conditions.

Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

Temperature Monitoring of High-Speed Railway Bridges in Mountainous Areas

Solar radiation and temperature can cause the deformation of high-speed railway bridges which has an obvious effect on the position of the rails and thus on passenger comfort, especially when the bridge spans a mountainous area. In this study, the temperatures of a track–bridge system and the deflections of the tall piers belonging to a high-speed railway line in China were monitored for over three years. By adopting extreme value theory and a time-series method, the temperatures and temperature gradients of structural components of the bridge were investigated in relation to time. The results show that the monitored temperatures can be broken down into the uniform temperature and the fluctuation temperature, which cause axial deformation and structural deflection, respectively. A uniform temperature curve was fitted using the Fourier series function and then used to calculate the axial deformations for different components of the track–bridge system over a one-year period. Additionally, the extreme temperature differences estimated for a 100-year return period can be used to encapsulate the maximum values of the daily temperature difference. Thus, the proposed temperature model can effectively estimate the structure’s temperature-induced deformations.

Kinematic errors prediction for multi-axis machine tools’ guideways based on tolerance

In this paper, a systematic approach on how to predict kinematic errors based on tolerance of machine tools’ guideways is introduced. Firstly, the truncated Fourier series function is applied to fit curve of guideways surface. Since geometric profile errors are regarded as a bridge between tolerance and kinematic errors of machine tools’ guideways, the mapping relationship between tolerance and geometric profile errors of machine tools’ guideways is formulated, and the mapping relationship between geometric profile errors and kinematic errors of guideways is established. Then, kinematic errors prediction model based on tolerance of guideways is subsequently proposed. Finally, simulation verification is conducted with this method. Simulation results show the range of the predicted kinematic errors (positioning error, y direction and z direction straightness error, roll error, pitch error, and yaw error) is 17.12 μm, 56.57 μm, 70.71 μm, 28.28 μrad, 141.42 μrad, and 113.14 μrad, respectively. In order to verify the feasibility and effectiveness of the presented method, a measuring experiment is carried out on guideways of a gantry-type five-axis milling machine tools by using a dual-frequency laser interferometer. The measured and identified discrete data can be fitted precisely by Fourier curve fitting method. The fitting results show the range of the measured kinematic errors is 16.96 μm, 59.43 μm, 68.63 μm, 28.65 μrad, 135.40 μrad, and 111.58 μrad, respectively. The maximum residual errors between the predicted and measured values of kinematic errors are 1.67 μm, 5.19 μm, 5.50 μm, 1.87 μrad, 9.81 μrad, and 7.07μrad, respectively. Comparing with the measured results of kinematic errors, residual errors are considerably small and can be neglected. Therefore, there is no doubt that this method is effective enough for predicting kinematic errors and can be used to replace the measurement of kinematic errors. In the design stage of machine tools, this approach is convenient for engineers to derive the distribution of kinematic errors. And its basic idea can be applied to other type of machine tools’ guideways.

On Nonuniform Transient Electromagnetic Field Coupling to Overhead Transmission Lines

Transient electromagnetic fields can couple to overhead power transmission and distribution lines, causing damage to the connected equipment. In real cases, incident electromagnetic fields exhibit different spatial and temporal waveform distributions along the line. Normally, to ensure the accuracy of the simulation results, a large number of observation points are needed along the line, which is difficult to obtain. This paper proposes a new approach to address this problem. The proposed approach is based on Agrawal et al.’s coupling model, in which the amplitude and phase distributions of the horizontal exciting electric field along the line are, respectively, represented by the Fourier series and polynomial functions in the frequency domain using two kinds of least squares methods. As a result, the source terms along the line can be accurately evaluated starting from a limited number of observation points. The proposed approach has been validated experimentally.

A theoretical model for the vibration of fuel rod with multi spans supported by springs

The prediction of fuel rod vibration is of importance in the designing of reactor core. A precise simulation of the vibration amplitude of fuel rods is necessary. It is disappointed that the mode shape equation of a spring-supported fuel rod with multi spans is not available in previous literature, due to the singularity of coefficient matrix. In this work, a theoretical model for the vibration of spring-supported rod with multi spans is established. The fuel rod is assumed to be a homogeneous beam. The flexural displacement of the rod is represented as the combination of Fourier series and auxiliary trigonometric functions in order to overcome the discontinuity and singularity of mode shape. The vibration frequency and mode shape are obtained. The calculation results are in satisfactory with theoretical results. The variation of spring constant and its effect could be simulated.

Two-phase shaping approach to low-thrust trajectories design between coplanar orbits

To rapidly design low-thrust trajectories between coplanar orbits for the orbit transfer and rendezvous problems, an analytic shape-based approach is proposed. Unlike all of the existing analytic shape-based approaches, the trajectory is divided into two phases. The finite Fourier series functions are employed to shape each phase of the trajectories. As a result, the fitting performance for the transfer and rendezvous trajectories by using the proposed method is improved. Consequently, the proposed approach can provide solutions with less fuel consumption and smaller maximum thrust acceleration. In addition, the trajectory safety is also ensured by the proposed method. The effectiveness of the proposed approach is validated by extensive numerical simulations.

Vibration analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates

The intent of this paper is to firstly perform the vibration analysis of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates with arbitrary boundary conditions by means of the semi-analytical method which is proposed by the author’s team. In the material model, the distribution of the carbon nanotubes is uniform or functionally graded along with the thickness direction of structures and four types of the CNTs distribution are studied in this paper. The refined rule of mixtures approach containing the efficiency parameters is adopted to determine the properties of the composite media. The admissible displacement functions of the FG-CNTRC circular, annular and sector plates are uniformly expanded as the modified Fourier series which embodies a standard cosine Fourier series and several auxiliary functions which are introduced to eliminate the limit of the boundary conditions. On this foundation, the first-order shear deformation elasticity theory is employed to construct the energy expression of the FG-CNTRC circular, annular and sector plates. Then the Ritz-variational energy method is used to decide the natural frequencies and the associated mode shapes. To examine the convergence, accuracy, stability and efficiency of the computational model, the comprehensive studies are conducted. Based on that, some crucial parametric studies covering the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs and boundary conditions are also investigated in detail.

Approximation of Electrocardiograms with Help of New Mathematical Methods

The problem of approximation of electrocardiograms has a great practical importance and it is constantly in the field of research. Solution of this problem allows us to automate and computerize the process of diagnosis and to detect timely deviations from normal characteristics of cardiograms of heartbeat in the early stages of appearance and development of a disease. There are different methods for approximation of cardiograms. All these methods have their drawbacks. The article deals with new methods of approximation of cardiograms, which do not have any disadvantages of the known methods. The developed mathematical models are based on new methods of approximation of step functions, and these methods can significantly reduce errors of the approximation of cardiograms. In this paper new methods of approximation of step functions with estimation of errors of the approximation are supposed. The proposed methods do not have any drawbacks of traditional expansions of step functions in Fourier series and can be used in problems of mathematical modeling of a wide class of processes and systems, in particular, for solution of medical and biological problems.

Vibration Characteristics Analysis of Cylindrical Shell‐Plate Coupled Structure Using an Improved Fourier Series Method

A simple yet accurate solution procedure based on the improved Fourier series method (IFSM) is applied to the vibration characteristics analysis of a cylindrical shell‐circular plate (S‐P) coupled structure subjected to various boundary conditions. By applying four types of coupling springs with arbitrary stiffness at the junction of the coupled structure, the mechanical coupling effects are completely considered. Each of the plate and shell displacement functions is expressed as the superposition of a two‐dimensional Fourier series and several supplementary functions. The unknown series‐expansion coefficients are treated as the generalized coordinates and determined using the familiar Rayleigh‐Ritz procedure. Using the IFSM, a unified solution for the S‐P coupled structure with symmetrical and asymmetrical boundary conditions can be derived directly without the need to change either the equations of motion or the expressions of the displacements. This solution can be verified by comparing the current results with those calculated by the finite‐element method (FEM). The effects of several significant factors, including the restraint stiffness, the coupling stiffness, and the situation of coupling, are presented. The forced vibration behaviors of the S‐P coupled structure are also illustrated.

An exact solution for the free-vibration analysis of functionally graded carbon-nanotube-reinforced composite beams with arbitrary boundary conditions

We present an exact method to model the free vibration of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) beams with arbitrary boundary conditions based on first-order shear deformation elasticity theory. Five types of carbon nanotube (CNT) distributions are considered. The distributions are either uniform or functionally graded and are assumed to be continuous through the thickness of the beams. The displacements and rotational components of the beams are expressed as a linear combination of the standard Fourier series and several supplementary functions. The formulation is derived using the modified Fourier series and solved using the strong-form solution and the weak-form solution (i.e., the Rayleigh–Ritz method). Both solutions are applicable to various combinations of boundary constraints, including classical boundary conditions and elastic-supported boundary conditions. The accuracy, efficiency and validity of the two solutions presented are demonstrated via comparison with published results. A parametric study is conducted on the influence of several key parameters, namely, the L/h ratio, CNT volume fraction, CNT distribution, boundary spring stiffness and shear correction factor, on the free vibration of FG-CNTRC beams.

Parameter identification of pedestrian's spring-mass-damper model by ground reaction force records through a particle filter approach

The spring-mass-damper (SMD) model with a pair of internal biomechanical forces is the simplest model for a walking pedestrian to represent his/her mechanical properties, and thus can be used in human-structure-interaction analysis in the vertical direction. However, the values of SMD stiffness and damping, though very important, are typically taken as those measured from stationary people due to lack of a parameter identification methods for a walking pedestrian. This study adopts a step-by-step system identification approach known as particle filter to simultaneously identify the stiffness, damping coefficient, and coefficients of the SMD model's biomechanical forces by ground reaction force (GRF) records.After a brief introduction of the SMD model, the proposed identification approach is explained in detail, with a focus on the theory of particle filter and its integration with the SMD model. A numerical example is first provided to verify the feasibility of the proposed approach which is then applied to several experimental GRF records. Identification results demonstrate that natural frequency and the damping ratio of a walking pedestrian are not constant but have a dependence of mean value and distribution on pacing frequency. The mean value first-order coefficient of the biomechanical force, which is expressed by the Fourier series function, also has a linear relationship with pacing frequency. Higher order coefficients do not show a clear relationship with pacing frequency but follow a logarithmic normal distribution.

Vibration analysis of the functionally graded carbon nanotube reinforced composite shallow shells with arbitrary boundary conditions

The aim of this paper is to firstly present the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) shallow shells with arbitrary boundary conditions. The first-order shear deformation theory and the artificial spring boundary technique are introduced to achieve the general theoretical modeling and arbitrary boundary conditions, respectively. Then according to the energy variational principle, the energy expression of the FG-CNTRC shallow shells is obtained. Furthermore, the admissible displacement functions of the FG-CNTRC shallow shells are chosen as an improved Fourier series which combines the standard double cosine Fourier series and several auxiliary functions which are introduced to remove any potential discontinuity of the displacement function and its derivatives at the edges. In the current framework, the vibration analysis of the FG-CNTRC shallow shells with arbitrary boundary conditions can be conducted by a unified model, and the calculation core code derived from the MATLAB software does not need changing while the boundary conditions or geometric parameters change. The good accuracy, reliability and efficiency of the current approach are validated by several numerical examples. Simultaneously, a comprehensive parametric investigation concerning the effects of elastic restraint parameters, shear deformation and rotary inertia, shallowness and material properties is performed.

A semi-analytical method for vibration analysis of functionally graded carbon nanotube reinforced composite doubly-curved panels and shells of revolution

The object of this paper is to present a novel semi-analytical method and its associated applications for linear vibration analyses of functionally graded carbon nanotube reinforced composite (FG-CNTRC) doubly-curved panels and shells of revolution on with arbitrary boundary conditions. Distribution of the carbon nanotubes through the thickness of the structures may be uniform or functionally graded and four types of the CNTs distribution are considered in this paper. Properties of the composite media are determined by a refined rule of mixtures approach which contains the efficiency parameters. The translation and rotation displacements of the doubly-curved structures are uniformly expressed as the superposition of a standard cosine Fourier series and several auxiliary functions introduced to eliminate all potential discontinuities of the original displacement function and its derivatives at the edges. Based on that, the energy expression of the FG-CNTRC doubly-curved panels and shells of revolution is examined where the first-order shear deformation elasticity theory is considered. Lastly, to solve the natural frequencies as well as the associated mode shapes by means of the Ritz-variational energy method. Unlike other existing methods, the proposed method is capable of handling various combinations of boundary constraints in a unified fashion, including free, simply-supported, clamped and elastic-supported boundary conditions. Comprehensive studies on the convergence, accuracy, stability and efficiency of the method are derived via the comparison with existing results reported in publications. The parametric studies concerning the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs as well as boundary restraint parameters on free vibration of FG-CNTRC doubly-curved panels and shells of revolution is also investigated in detail.

Periodic Doubling and Chaotic Attractor in the Love Model with a Fourier Series Function as External Force

Periodic Doubling and Chaotic Attractor in the Love Model with a Fourier Series Function as External Force