Sine Series Expansion Research Articles

Two-dimensional plane strain consolidation of unsaturated soils considering the depth-dependent stress

In practical engineering, the total vertical stress in the soil layer is not constant due to stress diffusion, and varies with time and depth. Therefore, the purpose of this paper is to investigate the effect of stress diffusion on the two-dimensional (2D) plane strain consolidation properties of unsaturated soils when the stress varies with time and depth. A series of semi-analytical solutions in terms of excess pore air and water pressures and settlement for 2D plane strain consolidation of unsaturated soils can be derived with the joint use of Laplace transform and Fourier sine series expansion. Then, the inverse Laplace transform of the semi-analytical solution is given in the time domain using a self-programmed code based on Crump's method. The reliability of the obtained solutions is proved by the degeneration. Finally, the 2D plots of excess pore pressures and the curves of settlement varying with time, considering different physical parameters of unsaturated soil stratum and depth-dependent stress, are depicted and analyzed to study the 2D plane strain consolidation properties of unsaturated soils subjected to the depth-dependent stress.

Accurate Solution for Orthotropic Beams with Continuously Variable Cross Sections

Orthotropic materials and components are widely used in various engineering fields, such as aerospace, energy, infrastructure, and water conservancy projects. The orthotropic components such as slabs and beams cannot be investigated accurately by using traditional beam theory because it is unable to take more elastic constants into account. The precise stress distribution of orthotropic simply supported beams of any thickness is determined in this article. Meanwhile, all undetermined coefficients of the stress function are derived by performing a Fourier sine series expansion upon the bottom and top surfaces of the beam. It shows that the resolution corresponds well with the numerical analysis findings from the finite element method. It indicates rapid convergence and can obtain high-precision and high stress analysis results, which has a good application prospect in engineering.

Numerical tool for quantifying energy deposition effects on representative hypersonic vehicle structures

This paper presents a methodology to rapidly assess the transient thermal-structural response of a representative skin panel experiencing finite energy deposition over a hypersonic trajectory. The thermal state of the panel is calculated using reference-enthalpy heating with two-dimensional conduction. Panel thermal stress is calculated using a linear Levy sine series expansion model, complemented by a non-linear finite element analysis. Numerical results using the X-43A trajectory as a case study are presented. Energy deposition is observed to have a greater effect on the skin panel when it is already thermally and aerodynamically loaded. The panel is more structurally compromised for energy pulses of longer duration, and/or applied earlier during the X-43A trajectory.

Exact solutions for stochastic Bernoulli–Euler beams under deterministic loading

This study deals with two general solutions for a simply supported linear elastic Bernoulli–Euler beam with a stochastic bending flexibility, subjected to a deterministic loading. Two model problems are considered. The first problem is associated with a trapezoidally distributed load, whereas the second problem treats a sinusoidally distributed load. The importance of the solution for the trapezoidal load lies in its practicality. The derivation of stochastic characteristics for random beams under a sinusoidal load is useful due to the expandability to generally distributed loads by a Fourier sine series expansion. Numerical results are reported for various cases illustrating the effect of stochasticity of the beam’s properties on its flexural response.

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

SummaryBased on the consolidation theory raised by Fredlund, the solutions for the equal‐strain consolidation of unsaturated foundation with the prefabricated vertical drain considering smear effect and drain resistance are analytically formulated in this paper. Firstly, governing equations for excess pore pressures (i.e., excess pore‐air and pore‐water pressures) under the equal‐strain hypothesis are derived with the introduction of radial boundary conditions. Afterwards, the obtained coupled equations are solved by applying general integration, decoupling process, and Fourier sine series expansion. The smear coefficients and factors of drain resistance corresponding to air and water phases are both captured explicitly in the final solutions. Furthermore, the degenerated solutions are employed to verify the reliability of the current solutions. Finally, a parametric study is conducted to study the consolidation characteristics of the proposed foundation model against modeling sizes (S and N), smear coefficients (αa and αw), and drain resistance factors (Ga and Gw).

Semi-analytical solutions to the two-dimensional plane strain consolidation for unsaturated soil with the lateral semi-permeable drainage boundary under time-dependent loading

Abstract Using Fourier sine series expansion and Laplace transform, semi-analytical solutions for the two-dimensional plane strain consolidation problem of unsaturated soils are presented subjected to time-dependent loading with the lateral semi-permeable drainage boundary (LSDB), which is the third boundary and can be used to consider the smear effect of the vertical drain. Then, degenerated and numerical solutions are provided to verify the validity of the proposed solutions. Finally, the consolidation property involving the LSDB and time-dependent loading is depicted and analyzed, and the consolidation behavior is shown to result from the application of the LSDB and loading.

Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load

This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.

Annotated Translations of Three of the Euler’s Papers on Celestial Mechanics

Annotated translations from Latin of three of the Euler’s papers on celestial mechanics are presented, which fall into the category of three-body problems. The first translation deals with an exact solution of three bodies that move around the common center of mass and always line up. This is considered the first work from which the three collinear Lagrange points could be obtained. The second translation deals with motions of Sun, Earth and Moon in syzygy and Moon libration as well, where, for the first time, Euler introduces an archaic form of a Fourier sine series expansion to describe the Moon’s wagging motion. The last translation relates to a paper that was written with the goal of alleviating astronomical computations of the perturbed motion of the Moon around the Earth by the Sun, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body. Despite showing great analytical skills, Euler gave no indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moon’s orbit.

Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube

This paper presents an analytical formulation of nonlocal elasticity theory for the buckling analysis of simply supported carbon nanotubes with rotational springs at both ends. The lateral displacement function is represented by a Fourier sine series expansion. Stoke's transformation is applied to construct the coe cient matrix of the corresponding systems of linear equations. This matrix gives more exibility in boundary conditions. The accuracy of proposed method is validated for three well-known boundary conditions available in the literature. A very good agreement has been obtained. The present method permits to have more e cient stability matrix for calculating the buckling loads of carbon nanotubes with any desired boundary conditions.

Sine series expansion of associated Legendre functions

The most regularly used mathematical tools for representing the geopotential globally are the spherical harmonics, which consists of the longitude-dependent Fourier transform and of the latitude-dependent associated Legendre functions. While the former is by definition a Fourier series, the latter also can be formed to that. An alternative formulation for the sine series expansion of associated Legendre polynomials has been derived based on well-known recurrence formulae. The resulted formulae are subsequently empirically tested for errors to determine the limitations of its use, and strong dependence on the co-latitude has been found.

Multi-channel optical anti-counterfeiting of micro-mirror arrays

Based on the Fourier series expansion theorem and the optical characteristics of the strong and stationary reflectivity of the sine three-dimensional microstructure palettes, we establish a RGBSm 3D microstructure model of optical variable device that make use of the sine series expansion of the complicated function. In the model, the pixelgram is constituted by the sine series with item number, security password color and three primary colors RGB colors. The purpose of anti-counterfeiting is achieved by the modulation of the item number parameter m of sine series and security password color S.

Equiconvergence of eigenfunction expansions for Sturm-Liouville operators with a distributional potential

The Sturm-Liouville operator in the space under Dirichlet boundary conditions is investigated. It is assumed that , (here, differentiation is used in the distributional sense). The problem of when the expansion of a function in terms of a series of eigenfunctions and associated functions of the operator is uniformly equiconvergent on the whole of the interval with its Fourier sine series expansion is considered. It is shown that such uniform convergence holds for any function in the space .Bibliography: 22 titles.

Maclaurin series-based Lagrangian method for economic dispatch with valve-point effect

Power utilities in general use Lagrangian method to solve economic dispatch (ED) problem for ease of implementation. This is possible only if the cost function of generating units is convex. Owing to valve-point effect exhibited by multi-valve steam turbines, the cost function is non-convex. The Maclaurin series-based Lagrangian method is proposed to solve complicated, non-convex and non-linear ED problems. In this method, the rectified sinusoid function is represented by Maclaurin sine series expansion and is solved using the Lagrangian method. The problem is solved iteratively using lambda iteration method. The proposed methodology is validated using IEEE 30-bus test system. In addition, the effectiveness of the new algorithm is demonstrated with 3, 13 and 40 generator test systems available in the literature. The results obtained substantiate the applicability of the proposed method for solving ED problems with non-smooth cost functions at par with stochastic search techniques generally used.

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = −d 2/dx 2 + q(x) with the Dirichlet boundary conditions in the space L 2[0, π] under the assumption that the potential q(x) belongs to W 2 −1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L 1. We also consider the case of potentials belonging to the scale of Sobolev spaces W 2 −θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W 2 θ [0, π] with 0 < θ < 1/2, then, for any function in the space L 2[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.

On the static buckling of an externally pressurized finite circular cylindrical shell

The static buckling behaviour of an imperfect finite cylindrical shell, stressed by either a lateral or hydrostatic pressure, is here investigated by assuming that the imperfection can be regarded as the first term in the Fourier sine series expansion .The buckling modes are assumed to be in the shape of the imperfection which is in turn given in the shape of the classical buckling mode .Regular perturbation technique in asymptotic expansions of the relevant parameters is used and a simple expression for determining the static buckling load of the structure is determined. It is observed that , this procedure, perhaps more than other ones ,can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The result is strictly asymptotic. JONAMP Vol. 11 2007: pp. 323-332

Initial post dynamic buckling of a quadratic-cubic column pressurized by a sinusoidally slowly varying dynamic load

In this investigation, we determine the dynamic buckling load of an imperfect finite column resting on a mixed quadratic-cubic nonlinear elastic foundation trapped by an explicitly time dependent sinusoidally slowly varying dynamic load .The resultant coefficients are dynamically slowly varying and the formulation contains two small but Mathematically unrelated parameters upon which regular perturbation procedure are used in asymptotic expansions of the variables. The imperfection is assumed in the shape of the mth term in the Fourier sine series expansion with small (in absolute value) Fourier coefficients. A generalization of Lindsted-Poincare procedure is used and the structure under investigation is, on the main, a nonlinear oscillatory system with small perturbations. The results obtained are strictly asymptotic and are valid for small absolute values of the perturbation parameters. Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 199-214

A harmonic approach to global ocean tide analysis based on TOPEX/Poseidon satellite

The constant and harmonic parts of the global ocean tide are modeled by up to nine major tidal constituents, namely, S2, M2, N2, K1, P1, O1, Mf, Mm, and Ssa. Our computations start with the Fourier sine and cosine series expansion for the tidal constituents, including the constant Mean Sea Level (MSL). Although the frequencies of the tidal constituents are considered known, the coefficients of the sine and cosine functions are assumed to be unknown. Subsequently, the coefficients of the sine and cosine functions, as well as the constant part of the Fourier expansion, were expanded into spherical harmonics up to degree and order n, where n corresponds to the number of linearly independent spherical harmonic base functions needed to model the tidal constituents, determined via independent columns of the Gram matrix. The unknown coefficients of the spherical harmonic expansions are computed using sea level observations within cycles #1–#350 of the TOPEX/Poseidon satellite altimetry over 11 years of its mission. A set of orthonormal base functions was generated for the marine areas covered by TOPEX/Poseidon observations from the spherical harmonics using a Gram-Schmidt orthogonalization process. These were used for modeling the dominant tidal constituents. The computed models based on orthonormal base functions for the nine tidal constituents and the constant part of the Fourier expansion, were tested numerically for their validity and accuracy, proving centimeter accuracy.

The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect

We introduce a new local sine transform that can completely localize image information both in the space domain and in the spatial frequency domain. This transform, which we shall call the polyharmonic local sine transform (PHLST), first segments an image into local pieces using the characteristic functions, then decomposes each piece into two components: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace's equation, biharmonic equation, etc.) given the boundary values (the pixel values along the boundary created by the characteristic function). Subsequently this component is subtracted from the original local piece to obtain the residual. Since the boundary values of the residual vanish, its Fourier sine series expansion has quickly decaying coefficients. Consequently, PHLST can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. Combining this ability with the quickly decaying coefficients of the residuals, PHLST is also effective for image approximation, which we demonstrate using both synthetic and real images. In addition, we introduce the polyharmonic local Fourier transform (PHLFT) by replacing the Fourier sine series above by the complex Fourier series. With a slight sacrifice of the decay rate of the expansion coefficients, PHLFT allows one to compute local Fourier magnitudes and phases without revealing the edge effect (or Gibbs phenomenon), yet is invertible and useful for various filtering, analysis, and approximation purposes.

Fourier NUBS method to express ship hull form

This article presents a method of numerical expression to draw ship hull forms. Recent developments in research into ship hull optimization need a method to express ship hull form as precisely as possible with a small number of design parameters. This method is based on a combination of the Fourier–sine series expansion and nonuniform B-spline (NUBS) interpolation. The merit of the combination is that it removes the wiggles which are often found when generating a rectangular-type curve, such as the midship section of tanker ship, with a simple Fourier series expansion. Here, the procedure is explained, and then a tanker ship hull is generated. Through the discussion, we show the effectiveness of the method.

Development of one-energy group, two-dimensional, frequency dependent detector adjoint function based on the nodal method

One-energy group, two-dimensional computer code was developed to calculate the response of a detector to a vibrating absorber in a reactor core. A concept of local/global components, based on the frequency dependent detector adjoint function, and a nodalization technique were utilized. The frequency dependent detector adjoint functions presented by complex equations were expanded into real and imaginary parts. In the nodalization technique, the flux is expanded into polynomials about the center point of each node. The phase angle and the magnitude of the one-energy group detector adjoint function were calculated for a detector located in the center of a 200×200 cm reactor using a two-dimensional nodalization technique, the computer code EXTERMINATOR, and the analytical solution. The purpose of this research was to investigate the applicability of a polynomial nodal model technique to the calculations of the real and the imaginary parts of the detector adjoint function for one-energy group two-dimensional polynomial nodal model technique. From the results as discussed earlier, it is concluded that the nodal model technique can be used to calculate the detector adjoint function and the phase angle. Using the computer code developed for nodal model technique, the magnitude of one energy group frequency dependent detector adjoint function and the phase angle were calculated for the detector located in the center of a 200×200 cm homogenous reactor. The real part of the detector adjoint function was compared with the results obtained from the EXTERMINATOR computer code as well as the analytical solution based on a double sine series expansion using the classical Green's Function solution. The values were found to be less than 1% greater at 20 cm away from the source region and about 3% greater closer to the source compared to the values obtained from the analytical solution and the EXTERMINATOR code. The currents at the node interface matched within 1% of the average value at the interface. The phase angle varied from 0.1° to 0.4° compared to 0.2° calculated using the point reactor zero power transfer function.