Analytical Solution Technique Research Articles

Analysis of the electrical field in viscosity sensors with torsionally vibrating quartz cylinders

The potential and electric fields in viscosity sensors with torsionally vibrating quartz cylinders are calculated for various sensor geometries by means of analytical solution techniques and a numerical integration method for the Laplace equation. Sensor geometries with electrodes on the surface of the cylinder and offset at a distance from the cylinder are investigated for different angular extensions of the electrodes and dielectric constants of the fluid. It is found that the electric field is more favorable for the excitation of the torsional vibration when the electrodes are located at a distance from the cylinder surface rather than on the cylinder. When the electrodes are on the surface of the cylinder, the direction of the piezoelectrically active component of the electric field is reversed in certain regions near the edges of the cylinder, which hinders the excitation of the torsional vibration. As a result of this analysis, a novel viscosity sensor with line conductor electrodes was developed and validated in our laboratory. Furthermore, this article provides a brief review of the available literature on torsionally vibrating quartz cylinder viscometers.

Effect of temperature on dynamic behavior of cracked metallic and composite beam

Euler-Bernoulli beam model is used to analyze the thermal vibration of cracked metallic and laminated composite beam with fully clamped ends. An analytical solution technique is proposed to investigate the natural vibration of cracked beam subjected to axial thermal load. An open and stable crack is considered perpendicular to the beam axis. In this study, three different types of material such as steel, copper and aluminum and their composition are used to analyze the thermal effect on different modes of frequency. The effects of thermal stress on the vibration of cracked metallic and composite fully clamped beam are revealed. The results show that the effect of temperature is significant on the natural frequency of cracked metallic and composite beam.

A Study of Transient Heat Transfer through a Moving Fin with Temperature Dependent Thermal Properties

In this article, heat transfer through a moving fin with convective and radiative heat dissipation is studied. The analytical solutions are generated using the two-dimensional Differential Transform Method (2D DTM) which is an analytical solution technique that can be applied to various types of differential equations. The accuracy of the analytical solution is validated by benchmarking it against the numerical solution obtained by applying the inbuilt numerical solver in MATLAB ($pdepe$). A good agreement is observed between the analytical and numerical solutions. The effects of thermo-physical parameters, such as the Peclet number, surface emissivity coefficient, power index of heat transfer coefficient, convective-conductive parameter, radiative-conductive parameter and non-dimensional ambient temperature on non-dimensional temperature is studied and explained. Since numerous parameters are studied, the results could be useful in industrial and engineering applications.

Investigation the geodesic motion of three dimensional rotating black holes

We study the geodesic equations in the space-time of neutral Brans–Dicke Dilaton black hole in three dimensions and BTZ black hole. We use the process of separation of the Hamilton–Jacobi equation to obtain the constants of motion. The whole analytical solution of the geodesic equations in the space-times of the intended black holes are shown completely. Moreover, the geodesic equations are solved in terms of Weierstrass elliptic functions. Furthermore, with use of the analytical solution and effective potential technique some trajectories around the black holes are classified. Meanwhile, by analytical solution, effective potential and considering the zeroes of underlying polynomials, some possible orbits are plotted. Finally, we compare our results with Cruz et. al. [17] and we indicate the benefits of the analytical method which is applied in this paper.

On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam

The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical and environmental conditions. However, this effect as an internal property of materials has not been studied when the nanobeams include an internal damping feature. To this end, a closed-circuit condition is considered taking converse piezo–flexoelectric behavior. The kinematic displacement of the classical beam using Lagrangian strains, also applying Hamilton’s principle, creates the needed frequency equation. The natural frequencies are measured in nanoscale by the available nonlocal strain gradient elasticity model. The linear Kelvin–Voigt viscoelastic model here defines the inner viscoelastic coupling. An analytical solution technique determines the values of the numerical frequencies. The best findings show that the viscoelastic coupling can directly affect the flexoelectricity property of the material.

Analytical matrix solutions of linear ordinary differential equations with constant coefficients

The article puts forward a modified finite element method based on decomposition and analytical solution techniques. The algorithm is as follows. A complex structure is divided into simple form sub-regions which involve partial differential equations. Next, the equations are decomposed. The decomposed equation solutions are written using analytical solution formulae. Meanwhile, the finite element size of the method proposed is defined only by the value of an averaging interval of required functions, since ordinary differential equation formulae are analytical. The algorithm has been tested by solving rectangular plate and bicurved shallow shell bending problems. The results proved proper convergence to precise values with increasing number of finite elements.

A unique technique for analytical solution of 2-D dual phase lag bio-heat transfer problem with generalized time-dependent boundary conditions

The eigenfunction-based solutions for various heat conduction models (Fourier and non-Fourier) exhibit mismatch at the boundaries due to the discrepancy between the boundary conditions (BCs) and the corresponding eigenfunction profiles. This mismatch leads to spurious oscillations and erroneous solutions near the boundaries. Recently developed finite integral transform (FIT) based analytical solution of the dual phase lag (DPL) bio-heat transfer problem with generalized time-dependent BCs in a 2-D planar skin tissue phantom also encounters the similar mismatch, especially for Dirichlet BCs. The present study aims at removing such mismatch from eigenfunction-based solutions, by the homogenization of generalized time-dependent BCs which is essentially carried out by subtracting an auxiliary function from the temperature of the domain of interest. This novel treatment yields a modified problem in terms of the modified temperature with homogeneous BCs and contains a modified source. For this, the above-mentioned auxiliary functions need to satisfy a set of conditions at the boundaries. However, the functions satisfying such conditions may not be unique. Therefore, an additional condition (pseudo-steady state condition) is imposed with a view to obtaining relatively simple, unique auxiliary functions. The current study also explores the homogenization process for 2-D bio-heat transfer problem for the tissue phantom in conjunction with the use of orthogonal eigenfunction expansion method (OEEM). The present developed solution approach is demonstrated using two test problems comprising of (a) constant surface temperature and (b) sinusoidal heat flux on the surface, respectively. In case of constant surface temperature, the present approach gets rid of large spurious oscillations near the boundary. In case of surface heat flux, temperature prediction by the present approach shows good agreement in the domain compare to that predicted by the FIT approach. Besides that, the flux distribution predicted on the boundary by present approach is a realistic non-zero value, whereas, the FIT approach produces zero heat flux for the same applied non-zero heat flux at the boundary. Moreover, for both the test problems, the developed novel approach facilitates significant reduction in number of terms in the series summation as compared to the corresponding FIT solutions, for the same level of accuracy.

Stochastic modeling and analysis of road–tramway intersections

In the last decades, the socio-demographic evolution of the population has substantially changed mobility demand, posing new challenges in minimizing urban congestion and reducing environmental impact. In this scenario, understanding how different modes of transport can efficiently share (partially or totally) a common infrastructure is crucial for urban development. To this aim, we present a stochastic model-based analysis of critical intersections shared by tram traffic and private traffic, combining a microscopic model of the former with a macroscopic model of the latter. Advanced simulation tools are typically used for such kind of analyses, by playing various traffic scenarios. However, simulation is not an exhaustive approach, and some critical, possibly rare, event may be ignored. For this reason, our aim is instead to adopt suitable analytical solution techniques and tools that can support instead a complete, exhaustive analysis, so being able to take into account rare events as well. Transient analysis of the overall traffic model using the method of stochastic state classes is adopted to support the evaluation of relevant performance measures, namely the probability of traffic congestion over time and the average number of private vehicles in the queue over time. A sensitivity analysis is performed with respect to multiple parameters, notably including the arrival rate of private vehicles, the frequency of tram rides, and the time needed to recover from traffic congestion.

Analytical Study of Transient Heat Transfer in a Triangular Moving Porous Fin with Temperature Dependant Thermal Properties

The present work investigates heat enhancement through a moving porous fin of trangular profile. The interaction between the fluid and the porous media in not locally thermodynamic equilibrium. The transport through the porous medium is derived using Darcy’s model. The calculations are carried out using the two-dimensional Differential Transformation Method (2D DTM), which is an analytical solution technique that can be applied to various types of differential equations. The approximate analytical solutions are validated using the numerical solution obtained via the inbuilt numerical solver in MATLAB (pdepe). The results reveal that the 2D DTM can achieve accurate results in predicting the solution of such problems. The impact of the thermal parameters on temperature distribution is performed using the closed-form solution generated by DTM.

Effect of Variable Thermal Conductivity Calibration Functions on Fin Temperature Distribution

In this article, we introduce a new thermal conductivity calibration function in modeling heat transfer through extended surfaces. The variable thermal conductivity functions are studied on a stand alone basis and further compared to one another. The calculations are carried out using the Variational Iteration Method (VIM) which is an analytical solution technique. The series solutions are bench-marked against the numerical results obtained by applying the Runge-Kutta fourth order method coupled with shooting technique. The effects of some physical parameters such as the thermogeometric fin parameter and thermal conductivity gradient, on temperature distribution are illustrated and explained.

Analysis of transient heat transfer in radial moving fins with temperature-dependent thermal properties

In this article, thermal analysis of radial moving fins of hyperbolic and rectangular profiles is performed. A time-dependent nonlinear differential equation modeling heat transfer in a radial moving fin is considered. The thermal conductivity and heat transfer coefficient are modeled as linear and power-law functions of temperature, respectively. The analytical solutions are generated using the differential transform method which is an analytical solution technique that can be applied to various types of differential equations. The accuracy of the analytical solutions is validated by benchmarking against the numerical solutions obtained by applying the inbuilt numerical solver in Maple (pdsolve). A good agreement is observed between the analytical and numerical solutions. The effects of embedding parameters on the dimensionless temperature and heat transfer rate are analyzed. The results show that the fin rapidly dissipates heat to the surrounding fluid with an increase in the speed of the moving fin, that is, an increase in the values of the Peclet number. The results also show that the heat transfer through a rectangular fin is more efficient when compared to a fin of hyperbolic profile.

Closed-form solution of optimal control problem of a fractional order system

A formulation and an analytical solution technique of fractional optimal control problem (FOCP) is presented in this paper. The performance index (PI) considered is a conformable fractional integral (CFI) function and is a function of both the state and the control variables. Dynamic behaviour of the system is described by conformable fractional differential equation (FDE). The necessary conditions of optimality and the general transversality condition in terms of Hamiltonian are obtained using variational approach. Both the fixed and free final end point conditions have been considered. An analytical solution technique is presented for solving the conformable FOCPs. To validate the formulation and solution scheme numerical examples are presented.

Diffraction of TE polarized electromagnetic waves by a nonlinear layer: saturated and Kerr nonlinearities

The diffraction of a TE wave by a dielectric layer filled with a nonlinear medium is considered. The layer is located between two half-spaces with constant permittivities. Two types of nonlinearities (nonlinearity with saturation and Kerr nonlinearity) are considered and compared. In this paper, we show the difference between solvability results of the problem in these cases. Namely the diffraction problem with saturated nonlinearity has a unique solution under some additional conditions. At the same time, the diffraction problem in the case of Kerr nonlinearity has an infinite number of solutions. The analytical and numerical solution techniques are developed. Numerical results are presented.

Accurate approximate semi-analytical solutions to the Boussinesq groundwater flow equation for recharging and discharging of horizontal unconfined aquifers

The Boussinesq equation is usually used to describe one-dimensional unconfined groundwater movement. Solutions of this equation are important as they provide useful insights regarding the water table response to stream level variations and allow us to quantify the exchange flow between the stream and the aquifer. Due to the nonlinearity of the Boussinesq equation, the solutions are generally obtained using numerical methods. However, for certain classes of initial and boundary conditions there are both exact and approximate analytical solution techniques. This work focuses on the latter approach. A new mathematical technique for approximate solutions of the Boussinesq equation describing flow in horizontal unconfined aquifers induced by sudden change in boundary head is presented. The method applies to the problems of recharging and dewatering of an unconfined aquifer, and approximate solutions to both problems are derived. The solutions were obtained by introducing an empirical function with four parameters which might be obtained using a numerical fitting procedure. Results based on this technique were found to be easily calculated and to be in good agreement with those obtained using numerical calculation based on Runge-Kutta approach. A benchmark between the proposed solutions and five existing approximate analytical solutions shows that the present solutions are the most accurate approximate solutions among those tested. Applications of the solutions are presented in the context of estimating aquifer hydraulic parameters.

Diffraction of a TE‐Polarized Wave by a Nonlinear Goubau Line

AbstractThe diffraction of a cylindrical wave by a nonlinear metal‐dielectric waveguide filled with nonlinear medium is considered. Two widely used types of nonlinearities (Kerr nonlinearity and nonlinearity with saturation) are considered. The problem is to find amplitudes of the reflected and the transmitted fields when the amplitude of the incident field is known. The analytical and numerical solution techniques are developed. Numerical results are presented.

Stable difference scheme for a nonlocal boundary value heat conduction problem

Abstract In this paper, a new finite difference method to solve nonlocal boundary value problems for the heat equation is proposed. The most important feature of these problems is the non-self-adjointness. Because of the non-self-adjointness, major difficulties occur when applying analytical and numerical solution techniques. Moreover, problems with boundary conditions that do not possess strong regularity are less studied. The scope of the present paper is to justify possibility of building a stable difference scheme with weights for mentioned type of problems above.

Flexural Motion of Elastically Supported Rectangular Plates under Concentrated Moving Masses and Resting on Bi-Parametric Elastic Foundation

Vibration analysis of plate-type structural member subjected to travelling masses and under elastically supported boundary conditions is considered in the present study. The specific aim of this work is to investigate the effects of elastically restrained boundary conditions and other vital structural parameters on the dynamic characteristics of structure-mass systems. An approximate analytical solution technique is employed to treat this problem. In particular, a technique based on separation of variables is used to reduce the governing equation of motion to a sequence of second order ordinary differential equations. The Struble’s technique and the integral transformations are thereafter employed to obtain solutions of the second order ordinary differential equations. Approximate analytical solutions for structure-load force and structure-load mass systems are obtained and discussed. Influence of some vital structural parameters on the dynamical system is established. Results show that, the effect of the shear modulus is more greatly pronounced than that of the foundation modulus; hence, it could be dangerous to rely on the design based on Winkler foundation which neglects completely the effect of shear modulus. Comparison was made between the dynamic deflection of the structure for the moving force and moving mass models. Effects of the mass ratio U0 on the dynamic stability of dynamical system are also established.

Identification of constitutive parameters for high temperature deformation of pressure tube of Indian PHWR considering multi-axial state of stress

In case of severe accidents like loss of coolant accident with failure of emergency core cooling system, the pressure tubes get heated quickly causing severe deformation. In this paper, an analytical solution technique has been used for calculating the radial deformation behaviour of pressure tubes. The analysis is carried out for different cases when the pressure tube is subjected to different values of internal pressure and heating rates. In the absence of specimen level creep test data of Indian pressure tube, data of Canadian pressure tube of similar material has been used. Due to observed differences between experimental and analytical values of radial deformation with time, the actual parameters have been calibrated from component level experiments using an optimization algorithm. Both uniaxial and bi-axial states of stress have been considered. Finite element method has also been used to calculate radial deformation of pressure tube and compared with experiment.

Photovoltaic system using Lambert W function-based technique

This article presents a successful simulink model by using the photovoltaic (PV) product along with DC/DC buck converter, lithium ion battery, maximum power point tracker (MPPT) along with loads. The current-voltage (I-V) and power-voltage (P-V) characteristics of a Kyocera KC200GT product is validated using single diode PV model in conjunction with Lambert W function. Fuzzy logic algorithm is incorporated in MPPT for high tracking accuracy at steady state and better performance to the varying atmospheric conditions. The complete model is simulated under two testing scenarios i.e., sunny and cloudy day conditions. Exact analytical solution technique by using Lambert W function is followed by expressing I-V and P-V characteristic of PV module by converting into discrete equations. The proposed Lambert W function based mathematical model is accurate and there is a good agreement between calculated and simulated outputs.

Analysis of nano droplet dynamics with various sphericities using efficient computational techniques

Motion of a vertically falling nano droplet in incompressible Newtonian media with initial velocity is investigated. The instantaneous velocity and acceleration are carried out by using the variational iteration method (VIM) and homotopy perturbation method (HPM), which are analytical solution techniques. The obtained results are compared with Runge–Kutta method in order to verify the accuracy of the proposed methods. The results show that, the analytical solutions are in good agreement with each other and with the numerical solution. Also, the effects of sphericity (φ) on the velocity and acceleration profiles of the nano droplet are explained. Moreover, the results demonstrate that the VIM-Pade and HPM-Pade are very effective in generating analytical solutions for even highly nonlinear problems.