Fourth-order Runge-Kutta Method Research Articles

Hydro-magneto-thermal aspects of ternary composite nanomaterial over arbitrarily inclined thin needle influenced by linear and nonlinear slips

Because of accelerated demands of advanced technologies like power station, chemical production and microelectronics, it necessitates the need of novel type of fluids with more heat transfer capability. Due to synergistic effect, ternary composite nanofluids (TCNFs) ensure better thermophysical and Rheology properties thereby acting as better suitable heat transfer fluid in wire coating, metal spinning, aerodynamics, medicine and engineering industries, etc. In view of such relevance, flow and heat transfer aspects of TCNF MWCNT + Al2O3 + TiO2 + water induced by linear and nonlinear slips over arbitrarily inclined moving thin needle are investigated in this study. Thompson and Troian nonlinear slip model is modified by developing it in polar coordinates. Quadratic thermal radiation phenomenon is adopted. Fourth-order Runge–Kutta method is used to obtain requisite numerical solution. Major outcomes indicate that fluid velocity of TCNF whittles down with amplification of magnetic parameter due to the flow induced by either linear or nonlinear slips. Lower value of Reynolds number favoring linear slip leads to effective intensification of nondimensional temperature distributions. Surface viscous drag and heat transfer rate get ameliorated with growth in size of thin needle under the influence of both linear and nonlinear slips.

Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane surrounded by nonlinear elastic foundation

In this article, the nonlinear vibration analysis of a rectangular hyperelastic membrane embedded within the nonlinear elastic foundation exposed to uniformly distributed hydrostatic pressure is investigated. The aim is to study the nonlinear vibration behaviors of a rectangular hyperelastic membrane when both primary and 1:1 internal resonances happen in the system. Also, by considering the system embedded within the nonlinear elastic foundation, the model of the structure has been developed close to the real situation. The structure is made of an isotropic, homogeneous, incompressible, and hyperelastic material. To model the system, the neo-Hookean hyperelastic constitutive law is used. The nonlinear elastic foundations include two linear terms Winkler and Pasternak, and a nonlinear cubic term. The governing equations are obtained with Hamilton’s principle of thin hyperelastic membrane and the assumption of finite deformations. Then, using Galerkin’s method, the equation of motion in the transverse direction, which is considered in two modes, is discretized. The resonant case includes the 1:1 internal resonance between two modes and primary resonance, in which the excitation frequency is close to the natural frequency of the first mode. In this analysis, the responses of nonlinear vibration behavior, including primary and 1:1 internal resonances, are obtained with analytical and numerical methods. The analytical and numerical methods are, respectively, the multiple scales method and the fourth-order Runge–Kutta method. Finally, the effects of the elastic foundation parameters and geometrical characteristics on the nonlinear vibration behavior are examined.

A comparative study on pulsating flow of Au + SWCNT/blood and Au + MWCNT/blood based Jeffrey hybrid nanofluid in a vertical porous channel with entropy generation

In the present exploration, entropy production on hydromagnetic pulsating flow of Au + SWCNT/blood and Au + MWCNT/blood-based Jeffrey hybrid nanofluid in a vertical porous channel with Cattaneo-Christov theory has been inspected. The impacts of Ohmic heating, viscous dissipation, and thermal radiation are considered. The transformation of the governing partial differential equations into a system of ordinary differential equations is made by applying the perturbation method and then numerically solved by the fourth-order Runge-Kutta method aided by bvp4c shooting technique built-in MATLAB software. The effects of various emerging parameters and variables on velocity, temperature, entropy generation, and the Bejan number are displayed through graphs. The consequences of physical parameters on heat transfer rate are prearranged in a table. This analysis demonstrates that the velocity of both hybrid nanofluids is intensified with an upsurge in the Grashof number, while it is decreasing by giving higher values in an applied magnetic field. The temperature of both hybrid nanofluids increases with an increment in the radiation parameter, and Eckert number. The entropy generation is an increasing function of Eckert number and radiation parameter. The heat transfer rate has a higher impact in the case of (Au + MWCNT/blood) hybrid nanofluid as compared to (Au + SWCNT/blood) hybrid nanofluid and mono nanofluid.

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

This paper presents a modified Runge–Kutta (MRK) scheme for the generalized Benjamin–Bona–Mahony–Burgers equation. The MRK method separate this equation into two parts which apply different finite difference scheme in space, and finally use fourth-order Runge–Kutta method in time.The scheme is conditionally stable and convergent with order of O (тау4+h2), the Comparisons of the existing numerical methods demonstrate that this scheme is capable of high accuracy and stability.

Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach.

Compartment models are implemented to understand the dynamic of a system. To analyze the models, a numerical tool is required. This manuscript presents an alternative numerical tool for the SIR and SEIR models. The same idea could be applied to other compartment models. The result starts with transforming the SIR model to an equivalent differential equation. The Dirichlet series satisfying the differential equation leads to an alternative numerical method to obtain the model's solutions. The derived Dirichlet solution not only matches the numerical solution obtained by the fourth-order Runge-Kutta method (RK-4), but it also carries the long-run behavior of the system. The SIR solutions obtained by the RK-4 method, an approximated analytical solution, and the Dirichlet series approximants are graphically compared. The Dirichlet series approximants order 15 and the RK-4 method are almost perfectly matched with the mean square error less than 2 × 10-5. A specific Dirichlet series is considered in the case of the SEIR model. The process to obtain a numerical solution is done in the similar way. The graphical comparisons of the solutions achieved by the Dirichlet series approximants order 20 and the RK-4 method show that both methods produce almost the same solution. The mean square errors of the Dirichlet series approximants order 20 in this case are less than 1.2 × 10-4.

Nonlinear analysis of vibration attenuation of ring-spring-resonator based meta-isolation system

Ring-spring-resonator (RSR) based nonlinear meta-isolation system (RSR-M) was proposed and its vibration attenuation characteristic was investigated experimentally and theoretically. The force–displacement hysteresis loops of the selected wire rope isolators (WRIs) exhibit softening under compression and hardening under extension, and are identified by a modified mathematical model combined with the normalized Bouc-Wen model. Frequency response curves of conventional isolation system and RSR-M−2 were calculated based on the single-degree-of-freedom (SDOF) and two-degree-of-freedom (TDOF) models, along with the fourth-order Runge-Kutta method. Shaking table experiments were carried out to evaluate the nonlinear dynamic response and vibration attenuation property of the RSR-M, which proves to be in good agreement with the numerical results of acceptable validity. Dynamic hysteresis loops and phase portraits were plotted to study the dynamic responses and stability of the RSR-M. The experimental results show that, no matter vertical or horizontal direction, the RSR-M possesses better vibration attenuation performance compared to conventional isolation system. In general, the meta-isolation vibration system provides a new reference and idea for the practice designing of the isolation system in some sense.

Direct multiple shooting and direct collocation perform similarly in biomechanical predictive simulations

Direct multiple shooting (DMS) and direct collocation (DC) are two standard transcription methods for solving optimal control problems (OCP) in biomechanics and robotics. They have yet to be compared in terms of solution and speed. Through five examples of predictive simulations solved using five transcription methods and 100 initial guesses in the Bioptim software, we showed that no single method outperformed the others systematically. All methods converged to almost the same solution (cost, states, and controls) in all but one OCP, with several local minima found in the latter. Nevertheless, DC based on fourth-order Legendre polynomials provided overall better results, especially regarding dynamics integration accuracy compared to DMS based on a fourth-order Runge–Kutta method. Furthermore, expressing the rigid-body constraints using inverse dynamics was usually faster than forward dynamics. DC with dynamic constraints based on inverse dynamics converged to better and less variable solutions. We recommend starting with this transcription to solve OCPs but keep testing other methods.

Simulation of anchor chain based on lumped mass method

In order to develop a anchoring operation simulation system and improve safety during anchoring operations, a relatively accurate mathematical model of anchoring operations needs to be established. In this paper, the stress condition of anchor chain under environmental and subsea geological conditions is further studied and the stress condition of anchor chain is analyzed based on the previous research. In this paper, a quasi-static model based on catenary method is used as the basis of dynamic analysis, and the dynamic model of anchor chain is established based on the concentrated mass method, which fully considers the influence of anchor chain weight, hydrodynamic force, ocean current and interaction with the seabed. The fourth-order Runge Kutta method was used to solve the model numerically, and a calculation procedure was developed. The accuracy of the model was verified by comparing the calculated results with the experimental results, indicating that the constructed anchor chain dynamics model has a high accuracy.

Investigation of vibration characteristics of gear-bearing transmission system considering the effect of profile shifted coefficient

In this paper, a dynamics model of the gear-bearing transmission system with bending and torsion coupling is established, and the influence of gear shifted on the vibration characteristics of the gear system is considered. First, the time-varying mesh stiffness of the profile shifted gear is derived by using the potential energy method. Then, based on the Lagrange equation numerical method, the vibration differential equation of the model is obtained, and the differential equation is solved by the fourth-order Runge-Kutta method. Finally, the influence of the profile shifted coefficient and input torque on the vibration characteristics of the gear transmission system is analyzed by using the time domain graph and the frequency domain graph. The conclusions drawn from this study can help engineers better understand the running performance and vibration characteristics of profile shifted gear.

MHD-driven chemically active and thermally radiative Prandtl hybrid nanofluid flow on stretching device with Ohmic heating, dissipation, and diffusion effects

This study explores the magnetohydrodynamics of reactive hybrid nanofluid flow with the Prandtl fluid model. The flow is past a linear stretching device and its physical characteristic are studied for the novel effects of viscous dissipation, Joule heating, chemical reaction, thermal radiation, and double diffusion. Furthermore, the multiple slip conditions are considered while framing the governing system of equations. The mathematical set of equations is transmuted with similarity invariants and obtained from the nonlinear set of ordinary differential equations. The numerical simulation is carried out via the Runge–Kutta fourth-order method with a shooting scheme. The graphical illustration of numerical outcomes is sketched through MATLAB. The code validation is achieved via comparative analysis with recent publications. It is observed that the applied magnetic field retard the fluid motion and increases the thermal boundary. Dissipation and radiation effects also boost the thermal boundary regime.

Launching Process Analysis of Aircraft Gun with Closed Type Gas Reflection Device

Recoil reduction is one of the most important issues for the design of aircraft gun. Traditional approaches for this purpose will produce back-blast, which constitutes a hazard to fighter planes or attack helicopters. This paper presents a closed type gas reflection device for the aircraft gun aiming to reduce the recoil force without ejecting the propellant gas rearward. The launching process is modeled by coupling the classical interior ballistic model and the flow equations for the closed type gas reflection device. The fourth order Runge-Kutta method is adopted to solve the modified interior ballistic model to obtain the recoil efficiency of the closed type gas reflection device. On this basis, effects of various parameters on the recoil efficiency and muzzle velocity are studied systematically. Finally, shooting experiments are carried out upon a 30mm caliber aircraft gun. The simulated results agree well with the experiment ones. The results show that, by using the closed type gas reflection device, the recoil efficiency of 26.01% is attained without ejecting the propellant gas rearward, nor decreasing the muzzle velocity dramatically.

Fire Control Arithmetic on MRSI of Electromagnetic Guns

In order to improve the firing density and damage efficiency of the electromagnetic gun, the control algorithm of multiple rounds simultaneous impact(MRSI) of the electromagnetic gun was established by controlling the initial velocity of projectile and the time interval of the projectile launch. Under the premise that the electromagnetic gun can effectively improve the strike density and damage efficiency, the allowable distance error and the allowable flight time error are set, the fourth-order Runge Kutta method is used to calculate the outer ballistic. According to Pareto genetic algorithm, the objective function, namely two allowable errors, is optimized to obtain the optimal solution of muzzle velocity and angle of multiple projectiles. The simulation verifies that the electromagnetic gun can achieve multiple rounds simultaneous impact in different ranges, and verifies the feasibility of the algorithm to achieve multiple rounds simultaneous impact, which is of great significance to improve the damage efficiency and fire density of the electromagnetic gun in the future.

ANALISIS KESTABILAN DAN SOLUSI NUMERIK PADA MODEL SEIR UNTUK PENYAKIT TUBERKULOSIS

One of the infectious diseases that can be modelled into the SEIR model is Tuberculosis (TB), this is because TB has a bacterial incubation period, so it is at this time that a person enters the exposed subpopulation. TB is divided into two types, namely latent TB and active TB. This study aims to explain the formation of the SEIR Model for the Spread of Tuberculosis, determine the equilibrium point and Basic Reproductive Numbers on the SEIR Model for the Spread of Tuberculosis, analyze the stability of the SEIR Model for the spread of Tuberculosis at the equilibrium point, and make numerical simulations. The result of this research is the formation of a mathematical model on the spread of Tuberculosis, and from the model obtained two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. Then the basic reproduction number ( ) was found through the Next Generation Matrix. Furthermore, the stability analysis was carried out at the disease-free equilibrium point and it was found that the local asymptotic stable model with , while at the endemic equilibrium point it was found that the local asymptotic stable model with . Numerical simulations are presented to show numerical solutions and strengthen the explanation of the stability analysis of the model using the fourth-order Runge-Kutta method with parameters that meet the stability requirements.

Dynamical analysis of an inverted pendulum with positive position feedback controller approximate uniform solution

The inverted pendulum is controlled in this article by using the nonlinear control theory. From classical analytical mechanics, its substructure equation of motion is derived. Because of the inclusion of the restoring forces, the Taylor expansion is employed to facilitate the analysis. An estimated satisfactory periodic solution is obtained with the aid of the modified Homotopy perturbation method. A numerical technique based on the fourth-order Runge–Kutta method is employed to justify the previous solution. On the other hand, a positive position feedback control is developed to dampen the vibrations of an IP system subjected to multi-excitation forces. The multiple time scale perturbation technique of the second order is introduced as a mathematical method to solve a two-degree-of-freedom system that simulates the IP with the PPF at primary and 1:1 internal resonance. The stability of these solutions is checked with the aid of the Routh–Hurwitz criterion. A set of graphs, based on the frequency response equations resulting from the MSPT method, is incorporated. Additionally, a numerical simulation is set up with RK-4 to confirm the overall controlled performance of the studied model. The quality of the solution is confirmed by the match between the approximate solution and the numerical simulation. Numerous other nonlinear systems can be controlled using the provided control method. Illustrations are offered that pertain to implications in design and pedagogy. The linearized stability of IP near the fixed points as well as the phase portraits is depicted for the autonomous and non-autonomous cases. Because of the static stability of the IP, it is found that its instability can be suppressed by the increase of both the generalized force as well as the torsional constant stiffness of the spring. Additionally, the presence of the magnetic field enhances the stability of IP.

A Numerical Investigation of the Effects of Groove Texture on the Dynamics of a Water-Lubricated Bearing–Rotor System

This paper aims to investigate the combined effects of working condition and structural parameters of groove texture on the dynamic characteristics, stability and unbalance response of a water-lubricated hydrodynamic bearing–rotor system to avoid instability and excessive vibration of the rotor. The Navier–Stokes equation, standard K-ε model with enhanced wall treatment and Zwart–Gerber–Belamri cavitation model are considered using the commercial software Fluent to calculate the stiffness and damping coefficients of a groove-textured, water-lubricated bearing based on the dynamic mesh method; the critical mass to express the stability and the unbalance response solved by the fourth order Runge–Kutta method of the rotor are calculated based on dynamic equations. The results indicate that shallower and longer groove textures can improve the direct stiffness along the load direction kyy, weaken the stiffness in the orthogonal direction kxx, improve stability and decrease the unbalance response amplitude of the water-lubricated bearing–rotor system at a greater rotational speed and smaller eccentricity ratio; however, the impact of grooves on damping parameters is not as great as it is on stiffness—there exists an optimum groove width to achieve a best dynamic performance.

Dynamical analysis of a vertical excited pendulum using He’s perturbation method

The current paper investigates the nonlinear dynamics of an excited pendulum by a crank-shaft-slider mechanism (CSSM), in which it is restricted to move vertically. He’s method of homotopy is employed to obtain the analytical solution of the governing differential equation. The numerical solution of this equation is obtained using the fourth-order Runge-Kutta method (RKM). The comparison between both solutions demonstrates a high level of consistency, which confirms the precision of the gained analytic solution. The graphical representations of these solutions are presented to reveal the influence of various parameters on the investigated motion.

Nonlinear static/dynamic behaviors of composite curved panels with piezoelectric patches in supersonic airflow

The nonlinear aeroelastic static deflection and dynamic response of the composite curved panel with rectilinear- and curvilinear-fiber lay-up configurations under supersonic flow are investigated. The geometrical nonlinear effect of the composite curved panel is depicted by the Von-Karman strain and the aerodynamic load is incorporated through the nonlinear third-order piston theory. The Newton-Raphson method is employed to determine the static deflection of the composite curved panel and the four-order Runge-Kutta method is for the dynamic response, respectively. After examining the accuracy of the established mechanical model through numerical comparisons with existing literatures, the parametric studies focusing on panel curvature, high-order piston aerodynamic force, fiber angle orientation, temperature rise, locations and applied voltages of the piezoelectric patch on nonlinear static deformation and dynamic response are investigated. The results are deliberated in detail, and it is observed that the panel curvature and fiber angle orientation significantly affect the aeroelastic static deflection and dynamic response of the curved panel under thermal environment. Moreover, high-order aerodynamic force enlarges the static deflection and dynamic response of the curved panels with a large curvature whereas the impact of applied voltages of the piezoelectric patch will be weaker.

On the vertical effect of the Non-linear behavior of track–bridge interaction in simply supported railway bridges

The present paper is devoted to studying the influence of the nonlinear behavior of the ballasted track on the dynamic response of a simply supported, non-skewed single ballasted track railway bridge acted by moving trains at constant speeds. Based on a double-beam model the rail track and the bridge deck are modeled as a simply supported Euler–Bernoulli upper beam and lower beam, respectively, which are connected between them through a vertical nonlinear continuous layer with viscous damping, which schematizes the mechanism of the vertical track–bridge interaction. The moving trains are modeled as a series of moving vertical forces (Moving Load Model). The nonlinear viscoelastic layer model describing the behavior of the rail bed is assumed to be a cubic-plus-bilinear type foundation; in particular, a bilinear constitutive law is introduced by the fact that this connection layer is characterized by two stiffness coefficients in compression and in tension at vertical zero deflection. The nonlinear partial differential equations that govern the transverse motion of the coupled system are determined. The Galerkin method and the fourth-order Runge–Kutta method are used to discretize and determine numerically the time–history response of the considered bridge. The obtained numerical results at the first order have shown that the dynamics of the coupled ballasted track–bridge system is governed by a Duffing like oscillator. The effect of the nonlinear behavior of the ballasted track cannot be neglected and yields decreased critical speeds and decreasing resonant amplitudes.

Elasto-plastic analysis of foundations during emergency shutdown due to blast loading

Behaviour of machine foundations is investigated in this study under harmonic loading with varying frequency and pulse loading. Owing to excessive vibrations from the pulse, the machine encounters an emergency shutdown where the machine’s exciting frequency reduces to zero exponentially. The analysis is performed considering an elastic plastic single degree of freedom system with hardening and softening behaviour. Two cases are analysed depending on the time of application of pulse loading to the foundation system. For both the cases, the governing equations of motion are acquired and solved numerically using the fourth order Runge-Kutta method. Using these solutions, displacement-time graphs are obtained for a variety of parameters such as time constant, hardening/softening index, mass of machine and the foundation block, stiffness, pulse load magnitude, damping ratio and decay coefficient to study their influence on the behaviour of the machine-foundation system. The study helps in predicting the dynamic response of foundations under the emergency shutdown conditions due to pulse loading. Further, these also help in taking the decision if there exists a need for vibration barriers to have the displacements well within the permissible limits.

Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system

Nonlinear Single degree of freedom (SDOF) systems are crucial for understanding the behavior of various real-world systems, designing and analyzing mechanical and dynamical systems. In this article we have modified the SDOF model by introducing the nonlinearity as well as damping with/without external force and applied the homotopy perturbation method (HPM) to explain its various phenomena. We have compared the analytical solutions obtained via the HPM to the provided numerical results by the fourth-order Runge-Kutta (RK4) method for verifying the precision and validity of the solutions. The presentation and explanation of the dynamic results can add a new dimension to the research field by influencing the importance of nonlinear SDOF systems.