Search results for: evolutionary equations of electromagnetic theory of time-domain

Authors: M. F. Patricio, P. M. Rosa

Abstract:

In this paper a class of numerical methods to solve linear and nonlinear PDEs and also systems of PDEs is developed. The Differential Transform method associated with the Method of Lines (MoL) is used. The theory for linear problems is extended to the nonlinear case, and a recurrence relation is established. This method can achieve an arbitrary high-order accuracy in time. A variable stepsize algorithm and some numerical results are also presented.

Keywords: Method of Lines, Differential Transform Method.

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Authors: Radouane Elbahjaoui, Hamid El Qarnia

Abstract:

Melting of Paraffin Wax (P116) dispersed with Al₂O₃ nanoparticles in a rectangular latent heat storage unit (LHSU) is numerically investigated. The storage unit consists of a number of vertical and identical plates of nano-enhanced phase change material (NEPCM) separated by rectangular channels in which heat transfer fluid flows (HTF: Water). A two dimensional mathematical model is considered to investigate numerically the heat and flow characteristics of the LHSU. The melting problem was formulated using the enthalpy porosity method. The finite volume approach was used for solving equations. The effects of nanoparticles’ volumetric fraction and the Reynolds number on the thermal performance of the storage unit were investigated.

Keywords: Nano-enhanced phase change material, phase change material, nanoparticles, latent heat storage unit, melting.

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Authors: O. A. Apampa, Y. A. Jimoh, K. A. Olonade

Abstract:

The need to save time and cost of soil testing at the planning stage of road work has necessitated developing predictive models. This study proposes a model for predicting the dry density of lateritic soils stabilized with corn cob ash (CCA) and blended cement - CCA. Lateritic soil was first stabilized with CCA at 1.5, 3.0, 4.5 and 6% of the weight of soil and then stabilized with the same proportions as replacement for cement. Dry density, specific gravity, maximum degree of saturation and moisture content were determined for each stabilized soil specimen, following standard procedure. Polynomial equations containing alpha and beta parameters for CCA and blended CCA-cement were developed. Experimental values were correlated with the values predicted from the Matlab curve fitting tool, and the Solver function of Microsoft Excel 2010. The correlation coefficient (R2) of 0.86 was obtained indicating that the model could be accepted in predicting the maximum dry density of CCA stabilized soils to facilitate quick decision making in roadworks.

Keywords: Corn cob ash, lateritic soil, stabilization, maximum dry density, moisture content.

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Authors: Cha’o-Kuang Chen, Ching-Chang Cho

Abstract:

This paper investigates the natural convection heat transfer performance in a complex-wavy-wall cavity filled with power-law fluid. In performing the simulations, the continuity, Cauchy momentum and energy equations are solved subject to the Boussinesq approximation using a finite volume method. The simulations focus specifically on the effects of the flow behavior index in the power-law model and the Rayleigh number on the flow streamlines, isothermal contours and mean Nusselt number within the cavity. The results show that pseudoplastic fluids have a better heat transfer performance than Newtonian or dilatant fluids. Moreover, it is shown that for Rayleigh numbers greater than Ra=10³, the mean Nusselt number has a significantly increase as the flow behavior index is decreased.

Keywords: Non-Newtonian fluid, Power-law fluid, Natural convection, Heat transfer enhancement, Cavity, Wavy wall.

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Authors: Phavanee Narataruksa, Karn Pana-Suppamassadu, Sabaithip TungkamaniRungrote Kokoo, Prayut Jiamrittiwong

Abstract:

The major focus of this work was to characterize hydrodynamics in a packed-bed with and without static mixer by using Computational Fluid Dynamic (CFD). The commercial software: COMSOL MULTIPHYSICSTM Version 3.3 was used to simulate flow fields of mixed-gas reactants i.e. CO and H2. The packed-bed was a single tube with the inside diameter of 0.8 cm and the length of 1.2 cm. The static mixer was inserted inside the tube. The number of twisting elements was 1 with 0.8 cm in diameter and 1.2 cm in length. The packed-bed with and without static mixer were both packed with approximately 700 spherical structures representing catalyst pellets. Incompressible Navier-Stokes equations were used to model the gas flow inside the beds at steady state condition, in which the inlet Reynolds Number (Re) was 2.31. The results revealed that, with the insertion of static mixer, the gas was forced to flow radially inward and outward between the central portion of the tube and the tube wall. This could help improving the overall performance of the packed-bed, which could be utilized for heterogeneous catalytic reaction such as reforming and Fischer- Tropsch reactions.

Keywords: Packed Bed, Static Mixer, Computational Fluid Dynamic (CFD).

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Authors: Maharavo Randrianarivony

Abstract:

The length  of a given rational B'ezier curve is efficiently estimated. Since a rational B'ezier function is nonlinear, it is usually impossible to evaluate its length exactly. The length is approximated by using subdivision and the accuracy of the approximation n is investigated. In order to improve the efficiency, adaptivity is used with some length estimator. A rigorous theoretical analysis of the rate of convergence of n to  is given. The required number of subdivisions to attain a prescribed accuracy is also analyzed. An application to CAD parametrization is briefly described. Numerical results are reported to supplement the theory.

Keywords: Adaptivity, Length, Parametrization, Rational Bezier

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Authors: S. Behnia, S. Fathizadeh

Abstract:

Conductivity properties of DNA molecule is investigated in a simple, but chemically specific approach that is intimately related to the Su-Schrieffer-Heeger (SSH) model. This model is a tight-binding linear nanoscale chain. We have tried to study the electrical current flowing in DNA and investigated the characteristic I-V diagram. As a result, It is shown that there are the (quasi-) ohmic areas in I-V diagram. On the other hand, the regions with a negative differential resistance (NDR) are detectable in diagram.

Keywords: Charge transfer in DNA, Chaos theory, Molecular electronics, Negative Differential resistance.

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Authors: J. H. Park, T. H. Lee, S. M. Lee, H. Y. Jung

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In this paper, a generalized synchronization scheme, which is called function synchronization, for chaotic systems is studied. Based on Lyapunov method and active control method, we design the synchronization controller for the system such that the error dynamics between master and slave chaotic systems is asymptotically stable. For verification of our theory, computer and circuit simulations for a specific chaotic system is conducted.

Keywords: Chaotic systems, synchronization, Lyapunov method, simulation.

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Authors: Nafisur Rahman, Habibur Rahman, Syed Najmul Hejaz Azmi

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This paper presents a simple and sensitive kinetic spectrophotometric method for the determination of ramipril in commercial dosage forms. The method is based on the reaction of the drug with 1-chloro-2,4-dinitrobenzene (CDNB) in dimethylsulfoxide (DMSO) at 100 ± 1ºC. The reaction is followed spectrophotometrically by measuring the rate of change of the absorbance at 420 nm. Fixed-time (ΔA) and equilibrium methods are adopted for constructing the calibration curves. Both the calibration curves were found to be linear over the concentration ranges 20 - 220 μg/ml. The regression analysis of calibration data yielded the linear equations: Δ A = 6.30 × 10-4 + 1.54 × 10-3 C and A = 3.62 × 10-4 + 6.35 × 10-3 C for fixed time (Δ A) and equilibrium methods, respectively. The limits of detection (LOD) for fixed time and equilibrium methods are 1.47 and 1.05 μg/ml, respectively. The method has been successfully applied to the determination of ramipril in commercial dosage forms. Statistical comparison of the results shows that there is no significant difference between the proposed methods and Abdellatef-s spectrophotometric method.

Keywords: Equilibrium method, Fixed-time (ΔA) method, Ramipril, Spectrophotometry.

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Authors: Nahid Ghasemi, Morteza Sohrabi, Yasan Soleymani

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The quantified residence time distribution (RTD) provides a numerical characterization of mixing in a reactor, thus allowing the process engineer to better understand mixing performance of the reactor.This paper discusses computational studies to investigate flow patterns in a two impinging streams cyclone reactor(TISCR) . Flow in the reactor was modeled with computational fluid dynamics (CFD). Utilizing the Eulerian- Lagrangian approach, implemented in FLUENT (V6.3.22), particle trajectories were obtained by solving the particle force balance equations. From simulation results obtained at different Δts, the mean residence time (tm) and the mean square deviation (σ2) were calculated. a good agreement can be observed between predicted and experimental data. Simulation results indicate that the behavior of complex reactor systems can be predicted using the CFD technique with minimum data requirement for validation.

Keywords: Impinging streams reactor, Residence timedistribution, CFD, Eulerian-Lagrangian approach

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Authors: Dowluru Ravi Kumar, Bhima Prabhakara Rao

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Propagation of solitons in single-mode birefringent fibers is considered under the presence of third-order dispersion (TOD). The behavior of two neighboring solitons and their interaction is investigated under the presence of third-order dispersion with different group velocity dispersion (GVD) parameters. It is found that third-order dispersion makes the resultant soliton to deviate from its ideal position and increases the interaction between adjacent soliton pulses. It is also observed that this deviation due to third-order dispersion is considerably small when the optical pulse propagates at wavelengths relatively far from the zerodispersion. Modified coupled nonlinear Schrödinger-s equations (CNLSE) representing the propagation of optical pulse in single mode fiber with TOD are solved using split-step Fourier algorithm. The results presented in this paper reveal that the third-order dispersion can substantially increase the interaction between the solitons, but large group velocity dispersion reduces the interaction between neighboring solitons.

Keywords: Birefringence, Group velocity dispersion, Polarization mode dispersion, Soliton interaction, Third order dispersion.

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Authors: Mohammad Hadi Bordbar, Timo Hyppänen

Abstract:

The radiative exchange method is introduced as a numerical method for the simulation of radiative heat transfer in an absorbing, emitting and isotropically scattering media. In this method, the integro-differential radiative balance equation is solved by using a new introduced concept for the exchange factor. Even though the radiative source term is calculated in a mesh structure that is coarser than the structure used in computational fluid dynamics, calculating the exchange factor between different coarse elements by using differential integration elements makes the result of the method close to that of integro-differential radiative equation. A set of equations for calculating exchange factors in two and threedimensional Cartesian coordinate system is presented, and the method is used in the simulation of radiative heat transfer in twodimensional rectangular case and a three-dimensional simple cube. The result of using this method in simulating different cases is verified by comparing them with those of using other numerical radiative models.

Keywords: Exchange factor, Numerical simulation, Thermal radiation.

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Authors: E. Abedi, A. Ahmadi Nadooshan, S. Salehi

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Modeling and vibration of a flexible link manipulator with tow flexible links and rigid joints are investigated which can include an arbitrary number of flexible links. Hamilton principle and finite element approach is proposed to model the dynamics of flexible manipulators. The links are assumed to be deflection due to bending. The association between elastic displacements of links is investigated, took into account the coupling effects of elastic motion and rigid motion. Flexible links are treated as Euler-Bernoulli beams and the shear deformation is thus abandoned. The dynamic behavior due to flexibility of links is well demonstrated through numerical simulation. The rigid-body motion and elastic deformations are separated by linearizing the equations of motion around the rigid body reference path. Simulation results are shown on for both position and force trajectory tracking tasks in the presence of varying parameters and unknown dynamics remarkably well. The proposed method can be used in both dynamic simulation and controller design.

Keywords: Flexible manipulator, flexible link, dynamicmodeling, end point.

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Authors: Farzin Salmasi

Abstract:

Energy dissipation in drops has been investigated by physical models. After determination of effective parameters on the phenomenon, three drops with different heights have been constructed from Plexiglas. They have been installed in two existing flumes in the hydraulic laboratory. Several runs of physical models have been undertaken to measured required parameters for determination of the energy dissipation. Results showed that the energy dissipation in drops depend on the drop height and discharge. Predicted relative energy dissipations varied from 10.0% to 94.3%. This work has also indicated that the energy loss at drop is mainly due to the mixing of the jet with the pool behind the jet that causes air bubble entrainment in the flow. Statistical model has been developed to predict the energy dissipation in vertical drops denotes nonlinear correlation between effective parameters. Further an artificial neural networks (ANNs) approach was used in this paper to develop an explicit procedure for calculating energy loss at drops using NeuroSolutions. Trained network was able to predict the response with R2 and RMSE 0.977 and 0.0085 respectively. The performance of ANN was found effective when compared to regression equations in predicting the energy loss.

Keywords: Air bubble, drop, energy loss, hydraulic jump, NeuroSolutions

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Authors: Kelvin Rozier, Vladimir E. Bondarenko

Abstract:

It is known that an analog Hopfield neural network with time delay can generate the outputs which are similar to the human electroencephalogram. To gain deeper insights into the mechanisms of rhythm generation by the Hopfield neural networks and to study the effects of noise on their activities, we investigated the behaviors of the networks with symmetric and asymmetric interneuron connections. The neural network under the study consists of 10 identical neurons. For symmetric (fully connected) networks all interneuron connections aij = +1; the interneuron connections for asymmetric networks form an upper triangular matrix with non-zero entries aij = +1. The behavior of the network is described by 10 differential equations, which are solved numerically. The results of simulations demonstrate some remarkable properties of a Hopfield neural network, such as linear growth of outputs, dependence of synchronization properties on the connection type, huge amplification of oscillation by the external uniform noise, and the capability of the neural network to transform one type of noise to another.

Keywords: Chaos, Hopfield neural network, noise, synchronization

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Authors: Takashi Shimizu, Tomoaki Hashimoto

Abstract:

Recently, crystal growth technologies have made progress by the requirement for the high quality of crystal materials. To control the crystal growth dynamics actively by external forces is useuful for reducing composition non-uniformity. In this study, a control method based on model predictive control using thermal inputs is proposed for crystal growth dynamics of semiconductor materials. The control system of crystal growth dynamics considered here is governed by the continuity, momentum, energy, and mass transport equations. To establish the control method for such thermal fluid systems, we adopt model predictive control known as a kind of optimal feedback control in which the control performance over a finite future is optimized with a performance index that has a moving initial time and terminal time. The objective of this study is to establish a model predictive control method for crystal growth dynamics of semiconductor materials.

Keywords: Model predictive control, optimal control, crystal growth, process control.

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Authors: Wenyu Song, Bingxi Li, Zhongbin Fu, Bo Zhang

Abstract:

Although water only takes a little percentage in the total mass of soil, it indeed plays an important role to the strength of structure. Moisture transfer can be carried out by many different mechanisms which may involve heat and mass transfer, thermodynamic phase change, and the interplay of various forces such as viscous, buoyancy, and capillary forces. The continuum models are not well suited for describing those phenomena in which the connectivity of the pore space or the fracture network, or that of a fluid phase, plays a major role. However, Lattice Boltzmann methods (LBMs) are especially well suited to simulate flows around complex geometries. Lattice Boltzmann methods were initially invented for solving fluid flows. Recently, fluid with multicomponent and phase change is also included in the equations. By comparing the numerical result with experimental result, the Lattice Boltzmann methods with phase change will be optimized.

Keywords: Frozen soil, Lattice Boltzmann method, Phase change, Test rig.

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Authors: B. Bagheri Nakhjavanlo, T. S. Ellis, P.Raoofi, Sh.ziari

Abstract:

This paper presents an application of level sets for the segmentation of abdominal and thoracic aortic aneurysms in CTA datasets. An important challenge in reliably detecting aortic is the need to overcome problems associated with intensity inhomogeneities. Level sets are part of an important class of methods that utilize partial differential equations (PDEs) and have been extensively applied in image segmentation. A kernel function in the level set formulation aids the suppression of noise in the extracted regions of interest and then guides the motion of the evolving contour for the detection of weak boundaries. The speed of curve evolution has been significantly improved with a resulting decrease in segmentation time compared with previous implementations of level sets, and are shown to be more effective than other approaches in coping with intensity inhomogeneities. We have applied the Courant Friedrichs Levy (CFL) condition as stability criterion for our algorithm.

Keywords: Image segmentation, Level-sets, Local fitting binary, Thoracic aorta.

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Authors: Mansour Nikpour, Mohd Zin kandar, Mohsen Ghasemi, Hossein Fallah

Abstract:

Over half of the total electricity consumption is used in buildings. Air-conditioning and electric lighting are the two main resources of electricity consumption in high rise buildings. One way to reduce electricity consumption would be to limit heat gain into buildings, therefore reduce the demand for air-conditioning during hot summer months especially in hot regions. On the other hand natural daylight can be used to reduce the use of electricity for artificial lighting. In this paper effective factors on minimizing heat gain and achieving required day light were reviewed .As daylight always accompanied by solar heat gain. Also interactions between heat gain and daylight were discussed through previous studies and equations which are related to heat gain and day lighting especially in high rise buildings. As a result importance of building-s form and its component on energy consumption in buildings were clarified.

Keywords: High rise buildings, energy demand, day lighting, heat gain.

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Authors: Alaa A. Osman, Amgad M. Bayoumy, Ismail El baialy, Osama E. Abdellatif, Essam E. Khallil

Abstract:

In this paper, a numerical simulation of a finned store separating from a wing-pylon configuration has been studied and validated. A dynamic unstructured tetrahedral mesh approach is accomplished by using three grid sizes to numerically solving the discretized three dimensional, inviscid and compressible Euler equations. The method used for computations of separation of an external store assuming quasi-steady flow condition. Computations of quasi-steady flow have been directly coupled to a six degree-offreedom (6DOF) rigid-body motion code to generate store trajectories. The pressure coefficients at four different angular cuts and time histories of various trajectory parameters and wing pressure distribution during the store separation are compared for every grid size with published experimental data.

Keywords: CFD Modelling, Quasi-steady Flow, Moving-body Trajectories, Transonic Store Separation, Moving-body Trajectories.

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Authors: Z. Hormozi Moghaddam, M. Mokhtari-Dizaji, M. Movahedin, M. E. Ravari

Abstract:

Mechanical index (MI) is used for quantifying acoustic cavitation and the relationship between acoustic pressure and the frequency. In this study, modeling of the MI was applied to provide treatment protocol and to understand the effective physical processes on reproducibility of stem cells. The acoustic pressure and MI equations are modeled and solved to estimate optimal MI for 28, 40, 150 kHz and 1 MHz frequencies. Radial and axial acoustic pressure distribution was extracted. To validate the results of the modeling, the acoustic pressure in the water and near field depth was measured by a piston hydrophone. Results of modeling and experiments show that the model is consistent well to experimental results with 0.91 and 0.90 correlation of coefficient (p<0.05) for 1 MHz and 40 kHz. Low intensity ultrasound with 0.40 MI is more effective on the proliferation rate of the spermatogonial stem cells during the seven days of culture, in contrast, high MI has a harmful effect on the spermatogonial stem cells. This model provides proper treatment planning in vitro and in vivo by estimating the cavitation phenomenon.

Keywords: Ultrasound, mechanical index, modeling, stem cell.

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Authors: Francesco Marotti de Sciarra, Raffaele Barretta

Abstract:

Starting from nonlocal continuum mechanics, a thermodynamically new nonlocal model of Euler-Bernoulli nanobeams is provided. The nonlocal variational formulation is consistently provided and the governing differential equation for transverse displacement is presented. Higher-order boundary conditions are then consistently derived. An example is contributed in order to show the effectiveness of the proposed model.

Keywords: Bernoulli-Euler beams, Nanobeams, nonlocal elasticity.

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Authors: C.Ramesh, V.Vaidehi

Abstract:

Ultra-wide band (UWB) communication is one of the most promising technologies for high data rate wireless networks for short range applications. This paper proposes a blind channel estimation method namely IMM (Interactive Multiple Model) Based Kalman algorithm for UWB OFDM systems. IMM based Kalman filter is proposed to estimate frequency selective time varying channel. In the proposed method, two Kalman filters are concurrently estimate the channel parameters. The first Kalman filter namely Static Model Filter (SMF) gives accurate result when the user is static while the second Kalman filter namely the Dynamic Model Filter (DMF) gives accurate result when the receiver is in moving state. The static transition matrix in SMF is assumed as an Identity matrix where as in DMF, it is computed using Yule-Walker equations. The resultant filter estimate is computed as a weighted sum of individual filter estimates. The proposed method is compared with other existing channel estimation methods.

Keywords: Channel estimation, Kalman filter, UWB, Channel model, AR model

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Authors: Y. Mohseniahouei, K. Abdella, M. Pollanen

Abstract:

In this paper, we explore the applicability of the Sinc- Collocation method to a three-dimensional (3D) oceanography model. The model describes a wind-driven current with depth-dependent eddy viscosity in the complex-velocity system. In general, the Sinc-based methods excel over other traditional numerical methods due to their exponentially decaying errors, rapid convergence and handling problems in the presence of singularities in end-points. Together with these advantages, the Sinc-Collocation approach that we utilize exploits first derivative interpolation, whose integration is much less sensitive to numerical errors. We bring up several model problems to prove the accuracy, stability, and computational efficiency of the method. The approximate solutions determined by the Sinc-Collocation technique are compared to exact solutions and those obtained by the Sinc-Galerkin approach in earlier studies. Our findings indicate that the Sinc-Collocation method outperforms other Sinc-based methods in past studies.

Keywords: Boundary Value Problems, Differential Equations, Sinc Numerical Methods, Wind-Driven Currents

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Authors: S. Mahmoodzadeh, J. Shahrabi, M. Pariazar, M. S. Zaeri

Abstract:

In this article, by using fuzzy AHP and TOPSIS technique we propose a new method for project selection problem. After reviewing four common methods of comparing alternatives investment (net present value, rate of return, benefit cost analysis and payback period) we use them as criteria in AHP tree. In this methodology by utilizing improved Analytical Hierarchy Process by Fuzzy set theory, first we try to calculate weight of each criterion. Then by implementing TOPSIS algorithm, assessment of projects has been done. Obtained results have been tested in a numerical example.

Keywords: Fuzzy AHP, Project Selection, TOPSIS Technique.

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Authors: G. P. Abdel Rahim, M. María Guadalupe Moreno Armenta, Jairo Arbey Rodriguez

Abstract:

We investigated the structure and electronic properties of the compound Mg1-xBixO with varying concentrations of 0, ¼, ½, and ¾ x bismuth in the the cesium chloride (CsCl), zinc-blende (ZnS), nickel arsenide (NiAs) NaCl (rock-salt) and WZ (wurtzite) phases. We calculated. The calculations were performed using the first-principles pseudo-potential method within the framework of spin density functional theory (DFT).

Keywords: DFT, Mg1-xBixO, pseudo-potential, rock-salt and wurtzite.

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Authors: Muhibul Haque Bhuyan, Quazi Deen Mohd Khosru

Abstract:

This paper presents a threshold voltage model of pocket implanted sub-100 nm n-MOSFETs incorporating the drain and substrate bias effects using two linear pocket profiles. Two linear equations are used to simulate the pocket profiles along the channel at the surface from the source and drain edges towards the center of the n-MOSFET. Then the effective doping concentration is derived and is used in the threshold voltage equation that is obtained by solving the Poisson-s equation in the depletion region at the surface. Simulated threshold voltages for various gate lengths fit well with the experimental data already published in the literature. The simulated result is compared with the two other pocket profiles used to derive the threshold voltage models of n-MOSFETs. The comparison shows that the linear model has a simple compact form that can be utilized to study and characterize the pocket implanted advanced ULSI devices.

Keywords: Linear pocket profile, pocket implantation, nMOSFET, threshold voltage, short channel effect (SCE), reverse short channeleffect (RSCE).

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Authors: B. Farhadinia

Abstract:

In this article, it is considered a class of optimal control problems constrained by differential and integral constraints are called canonical form. A modified measure theoretical approach is introduced to solve this class of optimal control problems.

Keywords: control problem, Canonical form, Measure theory.

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Authors: I. Algul, G. Akgun, H. Kurtaran

Abstract:

Dynamic analysis of composite doubly curved panels with variable thickness subjected to different pulse types using Generalized Differential Quadrature method (GDQ) is presented in this study. Panels with variable thickness are used in the construction of aerospace and marine industry. Giving variable thickness to panels can allow the designer to get optimum structural efficiency. For this reason, estimating the response of variable thickness panels is very important to design more reliable structures under dynamic loads. Dynamic equations for composite panels with variable thickness are obtained using virtual work principle. Partial derivatives in the equation of motion are expressed with GDQ and Newmark average acceleration scheme is used for temporal discretization. Several examples are used to highlight the effectiveness of the proposed method. Results are compared with finite element method. Effects of taper ratios, boundary conditions and loading type on the response of composite panel are investigated.

Keywords: Generalized differential quadrature method, doubly curved panels, laminated composite materials, small displacement.

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Authors: Hamioud Saida, Khalfallah Salah

Abstract:

In this study, a spectral element method (SEM) is employed to predict the free vibration of a Euler-Bernoulli beam resting on a Winkler foundation with elastically restrained ends. The formulation of the dynamic stiffness matrix has been established by solving the differential equation of motion which was transformed to frequency domain. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Numerical comparisons and examples are performed to show the effectiveness of the SEM and to investigate the effects of various parameters, such as the springs at the boundaries and the elastic foundation parameter on the vibration frequencies. The obtained results demonstrate that the present method can also be applied to solve the more general problem of the dynamic analysis of structures with higher order precision.

Keywords: Elastically supported Euler-Bernoulli beam, free-vibration, spectral element method, Winkler foundation.

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