Terms Of Fourier Series Research Articles

Trajectory planning of a closed five-bow-shaped bar linkage based on finite Fourier series

With aim to reduce the energy consumption, a trajectory planning method is presented for a closed five-bow-shaped bar linkage, which can be propelled itself by morphing configuration. The objective herein is to optimize the driving joints trajectories within the global feasible region when the linkage rolls along the ground with a desired acceleration. The driving joint trajectories were represented by finite Fourier series, whose coefficients were solved by genetic algorithm to ensure a minimal energy consumption of the linkage. The impact of the number of terms of finite Fourier series on the energy consumption was also discussed through numerical examples. As a result, the energy consumption based on this strategy had been reduced by 19%, comparing with the constant potential energy strategy. A number of terms between six and eight using to denote the joint trajectories are appropriate, because that a small number of terms is incapable of expressing the joint trajectories accurately, whereas, a large number makes the joints to be subjected to vibration shock. At last, simulation on a virtual model and experiments on a prototype were carried out to verify the effectiveness of the proposed method.

A novel time-varying grey Fourier model for variable amplitude seasonal fluctuation sequences

PurposeA time-varying grey Fourier model (TVGFM(1,1,N)) is proposed for the simulation of variable amplitude seasonal fluctuation time series, as the performance of traditional grey models can't catch the time-varying trend well.Design/methodology/approachThe proposed model couples Fourier series and linear time-varying terms as the grey action, to describe the characteristics of variable amplitude and seasonality. The truncated Fourier order N is preselected from the alternative order set by Nyquist-Shannon sampling theorem and the principle of simplicity, then the optimal Fourier order is determined by hold-out method to improve the robustness of the proposed model. Initial value correction and the multiple transformation are also studied to improve the precision.FindingsThe new model has a broader applicability range as a result of the new grey action, attaining higher fitting and forecasting accuracy. The numerical experiment of a generated monthly time series indicates the proposed model can accurately fit the variable amplitude seasonal sequence, in which the mean absolute percentage error (MAPE) is only 0.01%, and the complex simulations based on Monte-Carlo method testify the validity of the proposed model. The results of monthly electricity consumption in China's primary industry, demonstrate the proposed model catches the time-varying trend and has good performances, where MAPEF and MAPET are below 5%. Moreover, the proposed TVGFM(1,1,N) model is superior to the benchmark models, grey polynomial model (GMP(1,1,N)), grey Fourier model (GFM(1,1,N)), seasonal grey model (SGM(1,1)), seasonal ARIMA model seasonal autoregressive integrated moving average model (SARIMA) and support vector regression (SVR).Originality/valueThe parameter estimates and forecasting of the new proposed TVGFM are studied, and the good fitting and forecasting accuracy of time-varying amplitude seasonal fluctuation series are testified by numerical simulations and a case study.

Bir Direnç ve İki Kapasitörlü Lineer Bir RC Devresi için Cauchy Probleminin Fourier Serisi Cinsinden Çözümü

As it is known, solutions obtained in terms of Fourier series are widely used for more detailed analysis of mathematical models defined for different engineering problems. In this study, for the mathematical analysis of transient events in active RC based linear electronic circuits, the Cauchy problem is defined and solved in terms of Fourier series. For this reason, first of all, the mathematical model needed for the mathematical definition of the investigated problem was created. Based on the defined mathematical model, a quadratic differential equation for the linear RC electric circuit is obtained and the voltage and current changes are analysed for simple special cases using the Fourier series. As a result, mathematical modelling of transient processes occurring in linear RC circuits and mathematical problems defined as the Cauchy problem for such circuits have been solved analytically. It is thought that the results of the research will contribute theoretically and practically to the solution of the problems that arise in the study and design automation of different electronic circuits containing nonlinear circuit elements.

Detection of bearing faults based on band-pass filters and Fourier interpolation of the load torque

Bearing faults are widely found in mechatronics systems especially those that are required to work with unbalance loads. A highly reliable mechatronics system requires high quality bearings and/or effective bearing fault diagnostic procedure. This paper deals with the detection of rolling bearing faults based on band-pass filters and Fourier interpolation of the load torque. The reference torque, which is the output of the speed controller, is considered to be an approximation of the load torque. The reference torque is band-pass filtered and then interpolated in terms of Fourier series by using a sliding window method. The Fourier coefficients associated with a healthy bearing are served as a baseline and are compared with faulty lines corresponding to faulty bearings to detect the presence of a bearing failure. The proposed diagnostic method can be conducted online and does not require any additional sensors. Furthermore, the proposed method is able to detect single-point defects whose faulty levels are located at level C of the ISO 10816 Part 3. A mechatronic system equipped with artificial bearing faults is built in the laboratory to verify the effectiveness of the proposed method

Inverse problems for the beam vibration equation with involution

This article considers inverse problems for a fourth-order hyperbolic equation with involution. The existence and uniqueness of a solution of the studied inverse problems is established by the method of separation of variables. To apply the method of separation of variables, we prove the Riesz basis property of the eigenfunctions for a fourth-order differential operator with involution in the space ${{L}_{2}}(-1,1)$. For proving theorems on the existence and uniqueness of a solution, we widely use the Bessel inequality for the coefficients of expansions into a Fourier series in the space ${{L}_{2}}(-1,1)$. A significant dependence of the existence of a solution on the equation coefficient $\alpha$ is shown. In each of the cases $\alpha <-1$, $\alpha >1$, $-1<\alpha<1$ representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are written out.

Inverse Problem for a Fourth-Order Hyperbolic Equation with a Complex-Valued Coefficient

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions.

Nonlinear oscillations of a cylindrical shell with a circumferentially discontinuous elastic base

This work investigates the linear and nonlinear vibrations of a functionally graded (FG) transversally excited cylindrical shell and in contact along its length with a circumferentially discontinuous elastic base, a configuration frequently found in practical applications. The elastic medium is modeled as a Winkler foundation. The adopted distribution of the FG material along the thickness is similar to that of a sandwich shell, with one material predominating in the core and the other on both faces. First a finite element analysis (FEA) is performed to obtain the natural frequencies and vibration modes of the FG cylindrical shell and a parametric analysis is conducted to study the influence of the contact region and foundation stiffness on natural frequencies and mode shapes. Even though the problem is a symmetric, the cylindrical shell displays a sequence of symmetric and antisymmetric vibrations modes and these modes may interact due to nonlinear coupling. For much geometry the lowest vibration modes have very close natural frequencies, leading to internal resonances, which influence strongly the nonlinear shell response. To obtain a reliable reduced order model (ROM) for the nonlinear dynamic analysis, Donnell’s nonlinear shallow shell theory is considered. To discretize the nonlinear equations of motion by the Galerkin method, a consistent modal solution is obtained considering nonlinear couplings and interactions. Here, first the displacements are consistently derived from a perturbation procedure proposed in previous papers. To start the perturbation procedure, a seed solution is necessary for the transversal displacement field. For this, first the linear vibration modes obtained from the finite element analysis are expanded in Fourier series in terms of the circumferential coordinate. Then, the essential modes to be used in the ROM are identified using a mathematical procedure that combines Fourier transform and Parseval theorem. The proposed modal solution for the transversal displacements is also used to obtain the modal expansions for the axial and circumferential displacements and their modal amplitudes are determined as a function of the modal amplitudes of the transversal displacements, thus reducing even further the number of unknows. These expansions are finally used to discretize the nonlinear equation of motion in the transversal direction and the resulting equations of motion are solved numerically using continuation software AUTO. A geometry displaying multiple internal resonances is selected and resonance curves, time-responses and phase-portraits are obtained to analyze the nonlinear dynamics of the shell. Depending on the excitation, different bifurcation scenarios are obtained, with important changes in stability and resonant peaks due to modal interaction. Competition between coexisting stable attractors is investigated using basins of attraction, which exhibit a complex topology, often with fractal boundaries.

Second-Order State Propagation for Continuous-Thrust Relative Motion in $J_{2}$-Perturbed Elliptic Orbits

The second-order analytical state propagation under an arbitrary continuous thrust is proposed for relative motion in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$J_{2}$</tex-math></inline-formula> -perturbed elliptic chief orbits. The mean relative orbital elements are used to describe the relative motion. The second-order relative motion equations are obtained by performing the Taylor expansion of the nonlinear relative motion equations about the chief orbit. The original thrust acceleration is approximated by a truncated Fourier series in terms of eccentric anomaly of the chief orbit. Based on the first-order state propagation and the thrust acceleration Fourier-series expansion, the second-order propagation of the mean relative orbital elements is obtained. Numerical simulations show that the state propagation errors of the second-order solution are reduced by 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sim$</tex-math></inline-formula> 2 orders of magnitude when compared with the first-order solution.

The heat conductivity mechanism of the IR heating propagation for surfaces decontamination in protected ground

Modern greenhouses usually have flat concrete surfaces with communications for subsurface and above surface heating. It is proposed to disinfect surfaces with infrared radiation. We solve the corresponding heat equation. Field tests of the temperature field in protected ground have been performed to verify the mathematical model validity and to determine the empiric coefficients. The numerical investigation of the solution when the quantity of the Fourier series terms changes from 100 to 200 gives the heating time τ = 15 …16 min. Using these theoretical and experimental investigations one may conclude that the sufficient heating of a surface at the depth up to 5 cm needs a relatively long time of 15 …16 min and the surface will be strongly overheated during this time. We recommend using infrared heating for the decontamination purposes, but only in the case of thin layers of the ground or of the surface. The temperature necessary for the surface contamination and elimination of different pathogenic microorganisms, fungi, rots etc. emerging in the protected ground is achieved at a relatively small-time interval of 30…40 c. We think that the infrared radiation is a powerful and advanced method of sanitation of thin layers and surfaces.

Transmission strings: a technique for spatially mapping exoplanet atmospheres around their terminators

ABSTRACT Exoplanet transmission spectra, which measure the absorption of light passing through a planet’s atmosphere during transit, are most often assessed globally, resulting in a single spectrum per planetary atmosphere. However, the inherent 3D nature of planetary atmospheres, via thermal, chemical, and dynamical processes, can imprint inhomogeneous structure and properties in the observables. In this work, we devise a technique for spatially mapping the atmospheres of exoplanets in transmission. Our approach relaxes the assumption that transit light curves are created from circular stars occulted by circular planets, and instead we allow for flexibility in the planet’s sky-projected shape. We define the planet’s radius to be a single-valued function of angle around its limb, and we refer to this mathematical object as a transmission string. These transmission strings are parametrized in terms of Fourier series, a choice motivated by these series having adjustable complexity, generating physically practical shapes, while being reducible to the classical circular case. The utility of our technique is primarily intended for high-precision multiwavelength light curves, from which inferences of transmission spectra can be made as a function of angle around a planet’s terminator, enabling analysis of the multidimensional physics at play in exoplanet atmospheres. More generally, the technique can be applied to any transit light curve to derive the shape of the transiting body. The algorithm we develop is available as an open-source package, called harmonica1.

Bandgap mechanism analysis of elastically restrained periodic cylindrical shells with arbitrary periodic thickness

A unified model is established to investigate the wave propagation characteristics of cylindrical shells with arbitrary periodic variable thickness distribution and general boundary conditions. The finite periodic shell including any number of unit cells is considered in a unified pattern using energy principle and Rayleigh-Ritz procedure. System governing equation is formulated based on Love's shell theory, and an improved Fourier series expansion is employed to construct longitudinal, circumferential and radial vibration displacements, in which the unity auxiliary terms are used to overcome differential discontinuity of vibration displacements at elastic boundaries. Arbitrary periodic thickness variation information is condensed into Fourier expansion coefficients. Validity of the proposed model is verified through the comparison with results calculated from WFEM and FEM. High precision and excellent convergence can be observed with only several truncated terms of Fourier series. Results show that the bandgap characteristics is greatly dominated by the variation of shell thickness distributions, which is closely associated with the resonant frequency. Moreover, the translational spring stiffness has a more obvious influence on the bandgap characteristics comparing with the rotational spring stiffness. The mechanism of bandgap characteristics is explained, and the influences of the number of unit cells and boundary reflection on the bandgaps are also investigated.

Silencing performance analysis of a membrane cavity with different edge conditions

The current study reports the application of Galerkin and tailored-Galerkin procedures to model the vibrational response of a membrane bounded cavity connected to the elastic plates with different types of edge conditions. The plates contain clamped or pin-jointed type of edge conditions on finite edges, while the membranes are assumed to contain fixed, free, or spring-like edge conditions. Both Galerkin and tailored-Galerkin techniques require the priori solutions to determine the displacements of bridging membranes. In the first approach, the priori solution is expressed in terms of Fourier series which changes by varying the conditions on the edges of membranes. However, in late approach, a unique description of displacements can address a variety of edge conditions. The accuracy of these approaches is confirmed through the satisfaction of conserved power identity and through the reconstruction of matching condition with the truncated form of the solutions. In the modeled configuration, the vibrational energy propagates along the walls as well as through the fluid, and is affected by the variation of edge conditions. Moreover, the role of edge conditions is significant for structure-borne radiation only and is negligible for fluid-borne radiations.

Transport in Astrophysics: II. Diffusion with Advection in Expanding Nebulae

The structure across an expanding shell in which drift and diffusion redistribute material requires careful consideration in order to correlate the surface brightness with the physical parameters. A new solution in terms of Fourier series is suggested for 1D diffusion in the presence of a drift velocity. The astrophysical parameters are chosen in agreement with the astronomical data for the planetary nebula A39. The new solution is then inserted into the existing theory for the astrophysical image which allows dealing with the intensity of radiation emitted in a spherical layer.

Влияние пространственной неоднородности подложек и электрического поля на динамику зажатой капли

This article investigates the forced oscillations of an incompressible liquid drop under the action of an inhomogeneous alternating electric field. The drop is surrounded by an incompressible fluid of a different density and is clamped between two inhomogeneous parallel plates. In equilibrium, the drop has the shape of a round cylinder bounded in the axial direction by these plates. The external electric field acts as an external force causing the contact line to move. To describe the motion of the contact line, a modified Hocking boundary condition is used: the velocity of the contact line is proportional to the deviation of the contact angle from its equilibrium value and the rate of fast relaxation processes, the frequency of which is proportional to the doubled frequency of the electric field. The use of this equation makes it possible to qualitatively describe the experimental dependence of the contact angle as a function of stress, in contrast to the Young-Lippmann equation. The solution of the problem is represented as a Fourier series in terms of eigenfunctions of the Laplace operator. The resulting system of inhomogeneous equations for unknown amplitudes is solved numerically. Graphs of the amplitude-frequency characteristics and the dynamic shape of the drop are plotted for various values of the problem parameters. The wetting parameter depends on the polar angle  , i.e. the coefficient of interaction between the plate and the fluid (contact line) is a function of the coordinates.

Three-Dimensional Solution for the Vibration Analysis of Functionally Graded Rectangular Plate with/without Cutouts Subject to General Boundary Conditions.

Functionally graded materials (FGMs) structures are increasingly used in engineering due to their superior mechanical and material properties, and the FGMs plate with cutouts is a common structural form, but research on the vibration characteristics of FGMs plate with cutouts is relatively limited. In this paper, the three-dimensional exact solution for the vibration analysis of FGMs rectangular plate with circular cutouts subjected to general boundary conditions is presented based on the three-dimensional elasticity theory. The displacement field functions are expressed as standard cosine Fourier series plus auxiliary cosine series terms satisfying the boundary conditions in the global coordinate system. The plate with circular cutout is discretized into four curve quadrilateral sub-domains using the p-version method, and then the blending function method is applied to map the closed quadrilateral region to the computational space. The characteristic equation is obtained based on the Lagrangian energy principle and Rayleigh–Ritz method. The efficiency and reliability of proposed method are verified by comparing the present results with those available in the literature and FEM methods. Finally, a parametric study is investigated including the cutout sizes, the cutout positions, and the cutout numbers from the free vibration characteristic analysis and the harmonic analysis. The results can serve as benchmark data for other research on the vibration of FGMs plates with cutouts.

Approximation of Fourier series in terms of functions in $L_p$ Spaces for $0

Many results introduced for the absolutely convergence of Fourier series in terms of absolutely continuous functions. Here we study the convergence of Fourier series in terms of $p$-integrable functions series.

RPWM selective voltage harmonic elimination method for single‐phase five‐level MPUC inverter based on cell division

The conventional selective harmonic elimination method of random pulse width modulation (RPWM) is only suitable for the two-level inverter. Furthermore, when the harmonics below 6 kHz are selectively eliminated, it is easy to cause a low average switching frequency of the inverter, which leads to the increase of load current distortion rate. This paper proposes an RPWM selective voltage harmonic elimination method for a single-phase five-level MPUC (modified packed U-cells) inverter based on cell division. The MPUC inverter is supposed to be divided into three series cells, and the carrier stacking strategy is used to realize the coordinated control between each cell. In the process of RPWM, the average switching frequency can be controlled within a certain range by selecting the random number. This method is divided into normal mode and master–slave mode. In the normal mode, the selective harmonic elimination is realized by the method of cancelling the front and back terms of Fourier series in each cell. In the master–slave mode, the master cell is mainly used for fundamental power output, and the slave cell is mainly used for RPWM selective harmonic elimination. The effectiveness of the proposed method is verified by the simulation and experimental results.

Equivalent series theorem and obtaining some new summable numerical series using fast expansion polynomials

A theorem on equivalent uniformly converging series is proved, which allows one to find a set of new spectra of summable series. The classification of number series into three classes is given. By expanding special polynomials in Fourier series in terms of various fundamental system of functions, it is possible to summarize a wide class of new numerical series that have a convenient classical form of the first class. These polynomials are borrowed from the fast expansions method of the authors of this article.

Mixed boundary value problems in power-law functionally graded circular annulus

The present work proposes a semi-analytical technique for solution of mixed boundary value problem in functionally graded circular annulus wherein shear modulus varies radially in power-law form while Poisson’s ratio is constant. The technique relies on two main steps. In the first step, corresponding to terms in periodic Fourier series applied individually as traction along the annulus surface, stress and displacement field in the annulus is computed harnessing Airy stress functions approach. In the second step, leveraging the strain–displacement relations in polar co-ordinates, mixed boundary conditions are rendered in terms of displacement all along the annulus surface. Assuming the unknown traction along the annulus surface in terms of periodic Fourier series with finite terms, the modified displacement boundary condition, family of solution from the first step and orthogonality of sine and cosine functions is used to generate a system of simultaneous linear equations for the series coefficients. Knowing the coefficients of periodic Fourier series, stress and displacement field can be computed everywhere in the annulus. The first step is digitized in terms of MAPLE functions, exhaustively validated through traction distribution comprising of normal, shear traction on part of the boundary and pair of equal and diametrically opposite point load along the boundary. The second step is corroborated through a problem where the inner surface is subjected to a specified traction distribution and mixed boundary conditions exist along the outer surface in the form of complete constraint along a part and traction-free condition elsewhere.

APPLYING INVERSE WOMERSLEY PROBLEM FOR INDIRECT DATA INTO GOOD QUALITY INPUT DATA FOR MATHEMATICAL MODELS OF BLOOD FLOW IN HUMAN

The aim of this investigation is to convert experimental flow rate profiles of human cardiovascular system (CVS) (Hwang and Normann [7], Tsangaris and Stergiopulos [11] and Milnor [15]) i.e. indirect input data to direct input data (pressure gradient) using implicit technique of finite difference in inverse Womersley problem. Further pressure gradient profiles have been presented into mathematical form in terms of Fourier series and shown through Figures. Fourier coefficients of pressure gradient profiles are given through Tables. The computer program has been prepared using the finite difference implicit technique and is presented through Appendix.