Range-Kutta Method Research Articles

Misalignment and Rub-Impact Coupling Dynamics of Power Turbine Rotor with Offset Disk

When the dual rotor system of the aircraft engine is operating, the mass eccentricity of the power turbine rotor and the misalignment of the shaft coupling or the bearing will cause too large vibration of the rotor; this vibration leads to the rub-impact between the rotor and the casing. The power turbine rotor from the dual rotor system is taken as the research object in this paper. Considering the misalignment, the resulting rub-impact faults, the imbalance of rotor and the disk offset, the equation of motion for the system is developed according to the Lagrangian Equation, and then the Range-Kutta Method is adopted to solve the equation. The influence of the key parameters such as the rotating speed, the misalignment angle and the rub-impact clearance on the dynamics of the system is studied; the finite element analysis was carried out to validate the correctness of the theoretical modeling method. The results show that the rub-impact increases the stiffness of the system; the Hopf bifurcation occurs in the misalignment and rub-impact coupling system; the vibrational stability near the half of the switching speed slumps with the increase of the misalignment angle; with increasing of the stiffness, the number of the chaotic zone increases, and the range of the chaos is widening; enlarging the rub-impact clearance is beneficial to reduce the degree of the rub-impact system and enhance the stability of the system.

Magnetized Flow of Maxwell Fluid over a Slippery Stretching Reactive Surface with Thermophoretic Deposition

This manuscript investigated mathematically magnetized Maxwell fluid over slippery stretching reactive surface with thermophoretic deposition. Similarity transformation was used to recast partial differential equations modeling flow problem to nonlinear coupled ordinary differential equations which were solved using fourth order Range-Kutta method and Newton-Raphson shooting technique. Numerical results were compared with literature-based results and found to be in good accord. Skin friction coefficient, Nusselt number, Sherwood number, velocity profiles, temperature profiles and concentration profiles which are of importance to engineers, were found to be influenced by thermo-physical parameters governing the dynamics of flow. Their effects were illustrated in tabular form and graphically. The study found that increasing Thermophoretic deposition parameter, Momentum slip parameter and Biot number amplified rate of heat transfer but decreased rate of mass transfer and Skin friction coefficients. Thermal Grashof, Solutal Grashof, and Damkohler numbers reduced skin friction coefficients but increased heat and mass transfer rates.

The Thermal Absorption/Generation on Ferro-fluid Combined Convective Flow Over Curvilinear Porous Surfaces

In the influence of fluid buoyancy forces, the ferrofluid combined convective flow in porous curvilinear surfaces is studied with thermal generation/absorption effect. In the ambient flow conditions, the pressure gradient terms and ferrofluid buoyancy forces are replaced by the free steam velocity . The governing equations of the present problem are converted to ODEs by introducing non-dimensional functions and similarity variable. Boundary conditions of first derivative of velocities and temperature of our problem were constructed by the initial value problem, also the unknown initial conditions are found by shooting methods, and then a set of ODEs is solved numerically by the integration scheme of the six-order Range-Kutta method. The results of the solutions are presented graphically of velocity and thermal profiles with the help of MATLAB for different values of suction parameter and heat absorption parameter . Finally, the comparisons of the results highlight the justification of the numerical calculation accepted in the presence study. The problems in curvilinear surface study of boundary layer flow are complicated in fluid mechanics with applications of natural science and engineering.

Numerical study for bio-convection effects on MHD nano-fluid flow past a porous and extending wedge

We explored the insinuations of bio-convection and thermal radiation on nanofluid transportation across stretching permeable wedge with magnetic force. Appropriate similarity transformation variables are utilized to achieve ordinary differential equations. In order to tackle the non-linearity of these equations, numerical procedure based on shooting technique and Range Kutta method are harnessed on MATLAB platform. Computational and devour is carried out to evaluate the influence of controlling limitations on temperature, velocity, concentration of nanofluids and micro-organisms density. The growing strength of thermophoresis and Brownian motion enhance the fluid temperature. The profile volume fraction show decline against higher values of parameters which are Lewis number, unsteadiness and Brownian motion but opposite trend noted against higher value of Williamson and thermophoresis parameters. The skin friction values rise with the growing values of parameter of wedge angle for the moving wedge. The motile organism profile exhibits decrease against growing strength of Peclet number, bioconvection Lewis number, temperature difference and unsteady parameters while opposite behavior has been noted against wedge angle parameter.

A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation

Improving the bearing design helps in reducing the negative consequences related to errors in installation, manufacturing, deflections under severe loading conditions, progressive wear of machine elements, and many other aspects. One of the methods of such a design improvement effort is changing the bearing profile along the bearing width to compensate for the reduction in the geometrical gap between the shaft and the bearing inner surface due to the aforementioned causes. Since in all rotating machinery, unbalance usually exists at some level, this paper deals with the response of this modified bearing to unbalanced excitation to evaluate the effectiveness of such geometrical design on the dynamic characteristics of the rotor-bearing system. The numerical solution is performed using the finite difference method by assuming Reynolds boundary conditions to determine the cavitation limits, and the 4th-order Range-Kutta method is used to determine the time responses resulting from the unbalance excitation. The time responses to this type of excitation show that the rotor-bearing with the improved geometrical design is more stable, particularly at high speeds. In addition, this modification leads to an improvement in the lubricant layer thickness and the reduction in the levels of the generated pressure between the surfaces despite the presence of large deviations from the perfectly aligned bearing system. Furthermore, the suggested geometrical design overcomes the problem of asymmetricity in the pressure field resulting from the shaft deviation to a large extent. The results of this work (the enhancement in the level of the film thickness and the improvement in the dynamic response of the system as well as the reduction of the maximum pressure value) extend the range of misalignment in which the rotor bearing systems can operate safely which represents a significant step in designing the rotor-bearing system.

Analysis of viscously dissipated three-dimensional flow of Williamson fluid with nonlinear radiation and activation energy

The main purpose current exploration is to examine the influence of viscous dissipation and nonlinear radiation in three-dimensional (3D) Williamson fluid flow on an exponentially stretching through porous medium with the effect of activation energy along with chemical reaction. By employing suitable transformation, the nonlinear partial equations (PDEs) are transmuted to nonlinear ordinary differential equations (ODEs) which are subsequently solved numerically by using Range-Kutta method with shooting technique by utilization Matlab script. Characteristics of resulting physical parameters appearing in modeling like the Weissenberg number, Prandtl number, nonlinear radiation parameter, porosity parameter, Eckert number and chemical reaction parameter are comprehensively analyzed in graphical forms with an interest in providing the physical meaning to each parameter. The outcomes anticipate that, the thermal profile is improved by the increase in the values of nonlinear radiation parameter due to inner heat. The thermal profile enhances by strengthening of Eckert number but it has opposite trend through Prandtl number. The impact of obtained parameters on skin friction coefficient, mass and heat transfer rates is illustrated by tables. The validity of the present findings is presented by comparing them with the previous investigations.

Modeling and simulation of erbium-ytterbium co-doped optical waveguide amplifiers with dual-wavelength pumping at 980 nm and 1480 nm

With the rapid development of silicon photonic chips and integrated photonic circuits, erbium-doped optical waveguide amplifiers have received more and more attention in order to compensate for the transmission and coupling losses caused by the integration of optical devices on a chip. Pumping wavelength and pumping efficiency directly affect the gain and noise figure of the amplifier. In this paper, we propose an innovative dual-wavelength pumping method based on an erbium-ytterbium co-doped optical waveguide amplifier with simultaneous pumping at 980 nm and 1480 nm. A relaxation method based on the fourth-order Range-Kutta method is used to solve the rate and propagation equations and simulate the gain characteristics of the dual-wavelength pumping method for different ytterbium-erbium ion concentration ratios, erbium ion concentrations and ratio K between the 980 nm and total pump power. From the simulation results, it can be seen that the gain of the dual-wavelength pumping method is higher than that of the single-wavelength pumping methods when the erbium ion concentration exceeds 3 × 1026 m−3. At higher erbium ion concentrations, the dual-wavelength pumping method can provide higher gain for optical waveguide amplifiers, and may be able to become a new choice of pumping method for optical waveguide amplifiers.

Analisis dan Kontrol Optimal pada Model Dinamik Penyebaran Virus Zika

Zika is a virus from flaviviridae family and flavivirus genus that spread by Aedes Aegypti mosquito and that can cause serious problem disease such as Guillain Barre Syndrome (GBS) and mikrosefalus. In this paper used spreading dynamic zika virus model that consists of two population and divided to sub population Susceptible humans, sub-population Asymtomatic Infected humans, sub-population Infected humans, sub-population Recovered humans, sub-population Susceptible mosquitoes, and sub-population Infected mosquitoes. The model that anlysis by determine the basic reproduction number, the point of disease free and endemic equilibrium, and the stability of each point of equilibrium. Then, do the optimal control using Pontryagin principle with numerical solution given by Range-Kutta method. The simulation results show the decreasing sub-population of infected humans, asymptomatic infected human, and population of mosquitoes after given controls such using condom, treatment, and using indoor residual spray.

Effect of Asymmetry in the Modulation Parameters on Self-Focusing of Asymmetric Finite Airy-Gaussian Laser Beam in Collisionless Plasma

The self-focusing/defocusing of asymmetric finite Airy-Gaussian (AiG) laser beam has been investigated by employing standard Wentzel–Kramers–Brillouin (WKB) and paraxial-ray approximations in a nonrelativistic regime for underdense plasma. The second-order non-linear coupled differential equations have been solved numerically by using the fourth-order Range-Kutta method. The effect of asymmetry in the modulation parameters on the self-focusing/defocusing of the asymmetric finite AiG laser beam in collisionless plasma has been studied. It is observed that the self-focusing/ defocusing of asymmetric finite AiG laser beam is strongly connected with the initial values of the laser and plasma parameters such as modulation parameters and plasma frequency.

Numerical solution of gyrotactic microorganism flow of nanofluid over a Riga plate with the characteristic of chemical reaction and convective condition

Bioconvection is a familiar phenomenon of fluid mechanics which explains the hydrodynamic unsteadiness and suspensions of upward swimming microorganisms. Hydrodynamic unsteadiness is due to coupling in physical properties such as fluid flows and the density of microorganisms. The article simulates the nanoflow of Powell-Eyring fluid through the Riga surface. Numerical code of Range-Kutta method incorporated with shooting technique is generated on Matlab to track the Brownian motion of microorganisms. The contribution of the chemical reaction is also taken into account for expediting thermophoretic force in magnetohydrodynamic (MHD) nanoflow. Parametric study reveals that shear stress at the walls of the Riga surface and motion of the microorganisms are significantly influenced by the magnetic parameter and other dimensionless quantities.

Numerical and perturbation solutions of cross flow of an Eyring-Powell fluid

This communication presents a comparative analysis of two-dimensional cross flow of non-Newtonian fluid with heat and mass transfer is presented in this article. Eyring-Powell fluid is chosen as the main carrier of heat and nano species through a uniform horizontal channel. Effects of suction are also taken into account by placing porous walls. Main source of the flow is the motion of upper plate that moves with a constant velocity in axial direction. Two different nano flows have been formulated by neglecting and, as well as, applying constant pressure gradient, respectively. In addition to this, the analytical solution is validated with the numerical solution. Perturbation technique is employed to obtain a sustainable solution for the highly nonlinear and coupled differential equations. Further, Range-Kutta method with shooting technique is employed to get an approximate solution. It if inferred that both numerical and series solutions display a complete agreement.

Numerical and scale analysis of Eyring-Powell nanofluid towards a magnetized stretched Riga surface with entropy generation and internal resistance

The aim of current investigation is to report the impact of convective boundary conditions, heat generation, viscous dissipation, porous medium and activation energy on Eyring-Powell nanofluid over Riga plate. We have used the Darcy-Forchheimer model to characterized the effects of porous medium. The Riga plate describes as an electromagnetics actuator that is made of permanent magnets and recurring behavior of electrodes pointed on plane of surface. Mathematical formulation is derived with viscous dissipation, heat generation and activation energy. The numerical code of shooting method is developed based on Range-Kutta Method of order four to get the fast convergence and accurate results. It is analyzed that the temperature field shows an increasing behavior by varying the values of Eckert number (Ec) radiation parameter (Rd). The Eckert number and modified Hartman number Q accelerate both velocity and temperature profiles, respectively. The nanoparticle concentration profile increases with Eckert number. The walls shear stress is increases and decreasing behavior for the values of modified Hartmann number and Eyring-powell parameter.

The effect of negative Poisson’s ratio on the low-velocity impact response of an auxetic nanocomposite laminate beam

In this paper, an investigation on the low-velocity impact (LVI) response of a shear deformable beam laminated by carbon nanotube reinforced composite (CNTRC) layers is performed. The composite beam is “auxetic” due to the negative out-of-plane Poisson’s ratio (NPR) through special symmetric stacking sequences of layers that are designed based on the Classical Laminate Theory. To study the effect of the out-of-plane NPR on the LVI response of the composite beam, a newly defined Hertz model is developed. The motion equations of Karman type for the CNTRC laminate beam are derived in the framework of the Reddy beam theory and solved by means of a two-step perturbation approach while the dynamic equation of the impactor is built on Newton’s Law. Since temperature-dependent material properties of both carbon nanotube (CNT) and matrix are employed, the thermal influence on the LVI behavior is also investigated. Moreover, a piece-wise method is employed herein to investigate the effect of functionally graded (FG) patterns of the CNT reinforcements on the impact response. Numerical results elucidating the effects of temperature, FG distribution, and CNT volume fraction on the out-of-plane Poisson’s ratio and impact response of the beam are obtained by using a Range–Kutta method and discussed in details.

Numerical Solution of Gray Scott Reaction-Diffusion Equation using LagrangePolynomial

In this paper, numerical solution of one dimensional Gray-Scott equation is obtained using Lagrange Polynomial with Differential Quadrature Method (DQM). After using DQM the considered equation converted into set of ODEs which further can be solved using Range-Kutta method. The efficiency and accuracy of the method is to be checked by using L ∞ and L 2 errors. The obtained numerical results are shown with help of 2D and 3D figures.

Simulation of cavitation of spherically shaped hydrogen bubbles through a tube nozzle with stenosis

PurposeThe purpose of this study is to investigate the monodisperse cavitation of bubbly mixture flow for water and hydrogen mixture flows through a nozzle having a stenosis on the wall.Design/methodology/approachTwo flow regions, namely, quasi-statically stable and quasi-statically unstable increase in the bubble radius, are considered. Different oscillating periods of bubbles in downstream corresponding to various values of Reynolds number are taken into account. The Range–Kutta method is used to tackle nonlinear coupled system of governing equations.FindingsIt is observed that for the larger values of Reynolds number, the void fraction at the upstream section, even at small values, yields instabilities at the downstream. Consequently, owing to sudden increase in the velocity, the bubbles strike the wall with high speed that eventually remove the existing stenosis. This process can be considered as an effective cardiac surgery for arteries with semi-blockage.Originality/valueOriginal research work and to the best of author’s knowledge, this model is reported for the first time.

Numerical Modeling of Mass Loss and Temperature Profiles in the Thermal Decomposition of Bagasse in a Fixed Rectangular Oven

This work deals with the numerical study of the pyrolysis process in the thermochemical conversion of bagasse in an inert atmosphere and a two-dimensional numerical code (2D) of the reactions involved in the pyrolysis of bagasse of the sugarcane located in Morocco. The finite difference and the Range-Kutta method of second order have been used to simulate the pyrolysis of bagasse in a fixed rectangular oven of the bagasse. Transport modeling is made using homogenized equations at the Darcy scale. The thermal imbalance between the solid and the gas phase under the implementation of a temperature model of two fields has been revealed by previous studies, one for gas and one for solid. During this study, the aim has been to model the degradation of bagasse by pyrolysis. It is a very complex phenomenon that usually precedes the stage of heterogeneous combustion. The results provided by the numerical code developed are almost in agreement with the ones obtained by the experimental study presented in literature. This code is quite flexible; it can take into account different boundary conditions and allows changing easily the order of spatial discretization.

Comparison of Mathematical Methods to Obtain Concentration and Temperature of Newtonian Fluids in Tubular Reactors

In several areas of engineering, it is possible to put real problems in mathematical functions; when we represent a problem with variables in the form of function, we were able to extract various information from it. This paper compared two different mathematical methods, being the finite difference method and the Fourth Order Range-Kutta method, to analyze the concentration and temperature of the water flow inside a tubular reactor. These results were compared with the analytical and experimental results of the problem, demonstrating that the Fourth Order Range-Kutta method was more advantageous than the finite difference method.

Cutting Tool Prognostic using Markov Model

Objectives: Prognosis of a tool is essential for assigning a proper Condition-based Maintenance program for it. Therefore the objective of the present study is to investigate the reliability of a tool using a stochastic Markov Model. Methods/ Statistical Analysis: This work proposes a stochastic Markov model for estimating the Remaining Useful Life of the turning tool. In this study, a Mild Steel workpiece was machined to a certain length on a lathe machine using a high-speed steel tool and the Flank Wear Width (FWW) were recorded in every 20 minutes interval. This experiment was conducted for stable feed, stable speed and uniform depth of cut until the failure value of the tool flank wear was achieved, i.e. 0.3 mm. Findings: A state based model is developed considering four different degraded stages of the tool. The degradation rates among the states are obtained from the recorded experimental data. The appropriate equations for the four-state Markov model were derived, which show the possibilities of physical changes in the context of time for each level. The set of equations is solved analytically in MATLAB software using Range-Kutta method. After solving these equations, it was concluded that this system is 43% reliable for a 300-minute period and 41% is reliable for 500 minutes time period. Application/Improvements: It helps in preventing any kind of production loss. The remaining lifespan of a tool can be predicted by carefully analyzing the data gathered from the trend exploration method such as health monitoring with time. Keywords: Condition based Maintenance, Cutting Tool, Markov Model, Prognosis, Remaining Useful Life

Optimization Design of Vibratory Jaw Crusher with Double Cavities Based on MATLAB

The vibratory jaw crusher with double cavities was designed for reducing energy consumption by more crushing and less grinding. The dynamic model is established based on its structural characteristics, and the standard state equations are built by using Lagrange method. Based on MATLAB, the trajectory diagrams of the moveable jaws in the two cavities are obtained by Range-Kutta method. The influence of dynamic characteristics on yield and grain size is gained ,and the optimization design idea for the crusher is put forward.

PEMODELAN DAN SIMULASI KATALITIK KONVERTER PACKED BED UNTUK MENGOKSIDASI JELAGA PADA GAS BUANG KENDARAAN BERMESIN DIESEL

Modelling and Simulation of Packed Bed Catalytic Converter for Oxidation of Soot in Diesel Powered VehiclesFlue Gas. Diesel vehicle is used in Indonesia in very big number. This vehicle exhausts pollutants especially diesel sootthat can be reduces by using a catalytic converter to convert the soot to CO2. To obtain the optimal dimension ofcatalytic converter it is needed a model that can represent the profile of soot weight, temperature and pressure along thecatalytic converter. In this study, a model is developed for packed bed catalytic converter in an adiabatic conditionbased on a kinetic study that has been reported previously. Calculation of developed equations in this model usesPolymath 5.X solver with Range Kutta Method. The simulation result shows that temperature profile along catalyticconverter increases with the decrease of soot weight, while pressure profile decreases. The increase of soot weight inentering gas increases the needed converter length. On the other hand, the increase of catalyst diameter does not affectto soot weight along converter and temperature profile, but results a less pressure drop. For 2.500 c diesel engine,packed bed catalytic converter with ellipse's cross sectional of 14,5X7,5 cm diagonal and 0,8 cm catalyst particlediameter, needs 4,1 cm length.