Fourth-order Runge-Kutta Method Research Articles

Effect of slip velocity on Newtonian fluid flow induced by a stretching surface within a porous medium

This article aims to examine an unsteady 2-D laminar flow of magnetohydrodynamic fluid caused by an elastic surface immersed in a permeable medium under the influence of thermal radiation and extended heat flux. Thermal conductivity and viscosity both are supposed to vary with temperature. This flow model also includes velocity slip, heat source, and joule heating. The governing equations of the fluid, including momentum and energy equations, of the proposed problem are transfigured into a system of interconnected non-linear ordinary differential equations through similarity transformations. The resultant equations are solved efficiently by employing the shooting technique in combination with the fourth-order Runge-Kutta method. Numerical values and the effect of numerous governing factors on the flow field, temperature distribution, local skin friction coefficient, and Nusselt number are showcased via graphs and tables. The investigation reveals that velocity slip, heat source, and porosity parameters enhance the temperature field while diminishing the velocity field. Furthermore, the velocity slip parameter notably reduces both the coefficient of skin friction and the Nusselt number.

Large sampling intervals for learning and predicting chaotic systems with reservoir computing

Reservoir computing (RC) is an efficient artificial neural network for model-free prediction and analysis of dynamical systems time series. As a data-based method, the capacity of RC is strongly affected by the time sampling interval of training data. In this paper, taking Lorenz system as an example, we explore the influence of this sampling interval on the performance of RC in predicting chaotic sequences. When the sampling interval increases, the prediction capacity of RC is first enhanced then weakened, presenting a bell-shaped curve. By slightly revising the calculation method of the output matrix, the prediction performance of RC with small sampling interval can be improved. Furthermore, RC can learn and reproduce the state of chaotic system with a large time interval, which is almost five times larger than that of the classic fourth-order Runge–Kutta method. Our results show the capacity of RC in the applications where the time sampling intervals are constrained and laid the foundation for building a fast algorithm with larger time iteration steps.

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Peridynamics contact model: Application to healing using phase field theory

This paper presents a contact algorithm in non-ordinary state-based peridynamics (NOSB-PD) framework and its application in healing phase field (HPF) theory. The idea here is to exploit the nonlocality in PD to model the contact between two deformable bodies. The proposed method does not require artificial short-range force or additional degrees of freedom, e.g., Lagrange multiplier, which are often used in the conventional classical and PD contact models. In the regions where contact is likely to take place, neighbor search is performed at every time step and extra particles entering the horizon are identified. These extra particles form artificial bonds which transfer forces only in compression. This feature is essential for rebound simulation. Envisioning surface-to-surface bonding as the opposite process of cracking, we propose an HPF theory to model high velocity impact induced bonding between two material bodies. Here an HPF based bonding criterion is postulated and integrated with the proposed contact algorithm. The solution of the governing equations is obtained using the fourth order Runge-Kutta method. The efficacy of the proposed contact algorithm is demonstrated by simulations of impact and rebound of two plates, two cylinders, and cylinder with rectangular prism and comparing the energy time histories with finite element solutions obtained using ANSYS®. The potential of the proposed HPF theory is demonstrated through simulation of impact of two plates where the plates rebound when the impact velocity is below a critical value, and above that velocity, the plates join with each other.

Analysis of SEIQV Epidemic Model on the Spread of Covid-19 in Jember Regency

Covid-19 is a disease that attacks the respiratory system caused by infection with SARSCoV-2. Efforts to prevent the spread of Covid-19 by vaccination. The spread of disease can be modeled into a mathematical equation. The model used in this study is the SEIQV Model. The disease spread model is analyzed by finding the equilibrium point and the stability of the equilibrium point as well as the basic reproduction number. In the Covid-19 distribution model, a bifurcation analysis is carried out which is needed to determine changes in stability and changes in the number of equilibrium points. Then perform a numerical solution using the fourth-order Runge-Kutta method which is simulated using MATLAB.

Efficient Numerical Solution of the Sharma-Tasso-Olver Equation Using Radial Basis Function-Pseudo Spectral Method with Comparative Analysis

In this study, we will solve the Sharma-Tasso-Olver equation by utilizing the radial basis function-pseudo spectral approach. Pseudo spectral techniques are renowned for their exceptional precision but have constraints regarding their ability to handle geometric variations. The integration of Radial Basis Functions (RBF) with pseudo spectral methods offers a solution to surmount this particular limitation. To estimate the differentials with respect to the variable , we will use the radial basis functions, including commonly used ones like multiquadric, inverse multiquadric, and inverse quadric, have been employed. Furthermore, the problem has transformed into the ordinary differential equation (ODE), and the solution to this ODE system has been acquired through the application of the fourth-order Runge-Kutta method. Additionally, a comparative analysis has been conducted between the solutions obtained through the suggested technique and the exact solutions.

ANALYSIS AND OPTIMAL CONTROL OF HIV GROWTH MODEL IN THE BODY WITH ANTIRETROVIRAL THERAPY

Human Immunodeficiency Virus (HIV) is a virus that affects the human immune system. HIV infection causes a decrease in the body's immunity because the virus attacks immune-building cells, especially T-CD4 cells. Currently, there is no treatment that can cure or eliminate HIV, but antiretroviral therapy can be done. This study discusses the growth model of HIV in the body that is given control in an effort to maximize healthy T-CD4 cells. In this model, the infection-free and infected equilibrium points are also discussed and their stability is analyzed. Then the optimal control is solved using the Pontryagin Maximum Principle method and solved numerically using the fourth-order Runge Kutta method. Based on the analysis and simulation results, the system is asymptotically stable around the infection-free equilibrium point and unstable around the infected equilibrium point. Simulation results show that with the control of antiretroviral therapy, the T-CD4 cell population grows significantly which can improve the quality of life of patients. And the growth of HIV in the body can be inhibited until it cannot reproduce itself.

Numerical Investigation of Diffusion thermo and Thermal Diffusion on Mixed Convection flow of Williamson Nanofluid through a vertical cone with porous material in the presence Thermal radiation and Activation Energy

In this paper, analyze the impact of Activation energy, Diffusion thermo and thermal diffusion with heat and mass transfer inherent of over a vertical cone through Thermally Radiant Williamson nanofluid with saturated porous medium under the convective boundary condition in the presence of thermal radiation has been studied. The coefficients of Brownian and thermophoresis diffusions are also taken into consideration. The governing partial differential equations are reduced to a couple of nonlinear ordinary differential equations by using suitable transformation equations; these equations are then solved numerically with the use of the conventional fourth-order Runge Kutta method accompanied by the shooting technique. As a result, the effects of various physical parameters on the velocity, temperature, and nanoparticle concentration profiles as well as on the skin friction coefficient and rate of heat transfer are discussed with the aid of graphs and tables. This study has been directly applied in the pharmaceutical industry, microfluidic technology, microbial improved oil recovery, modelling oil and gas-bearing sedimentary basins, and many other fields. Further, to check the accuracy and validation of the present results, satisfactory concurrence is observed with the existing literature.

Ultra-low frequency vibration energy harvesting of piezoelectric vibration systems with an adjustable device

Energy harvesting from external Ultra-low frequency vibrations is currently attracting a lot of attention. However, for the same system, it is difficult to satisfy the need for more efficient energy harvesting in a variety of external environments. Accordingly, an adjustable gearing device connected to the energy harvesting system is used herein for adapting energy harvesting to a wide range of external environments. The device uses four symmetrical inclined springs. One end of the springs is connected to a gear and the other end to a mass element. The geometrical parameters of the system can be varied by rotating the gears, thus varying the dynamic stiffness of the system. The kinetic equations of the system are derived and their nonlinear terms are Taylor-expanded. The amplitude-frequency response curve of the system is obtained by solving using the harmonic balance method. Multi-color plots of RMS values of induced voltage and electric power are obtained using the fourth-order Runge-Kutta method. The results show that the device has excellent adjustable performance. Adjusting the rotation angle of the gears can change the soft and hard characteristics of the system's dynamic stiffness and the intrinsic frequency, thus improving the system's energy harvesting performance in different environments.

A general and efficient harmonic balance method for nonlinear dynamic simulation

This paper presents a general and efficient harmonic balance (GE-HB) method to expeditiously evaluate the periodic responses of nonlinear systems. By employing the fast Fourier transform (FFT) technique, it rapidly computes harmonic expansion coefficients of the nonlinear terms and their corresponding constant tensor matrices. Notably, the nonlinear part of the Jacobian matrix is just the product of the coefficients and the constant tensor matrices, loop computations are no longer requisite for calculating the Jacobian matrix. This, facilitates accelerated determination of the nonlinear portion of the Jacobian matrix, thus obtaining the periodic responses of the system in an efficient manner. Furthermore, the constant tensor matrices are exclusively contingent upon the harmonic bases of the system's periodic response, devoid of reliance on the equation's specific formulation, facilitates obtaining the Jacobian matrix in a general manner. Simulation results on a 20-degree-of-freedom nonlinear oscillator chain system demonstrate that the efficiency of the GE-HB method is about tenfold that of the harmonic balance-alternating frequency/time domain (HB-AFT) method and twenty-fivefold that of the fourth-order Runge-Kutta (RK4) method. Further, the simulation of a 284-degree-of-freedom nonlinear rotor system exhibits the GE-HB method's computational efficiency surpassing HB-AFT method by over 500 times and Newmark method by over 3000 times. Moreover, in these two examples, the GE-HB method captures all solution branches, including unstable periodic solutions, providing comprehensive insights into periodic responses of the systems. The GE-HB method proposed here has the potential to explore dynamic characteristics for high-dimensional complex nonlinear systems in engineering.

Design and optimization of a two-degrees-of-freedom single-sided vibro-impact nonlinear energy sink for transient vibration suppression of a thin plate

To rapidly suppress the transient vibration response of a thin plate, a two-degrees-of-freedom single-sided vibro-impact nonlinear energy sink (TSSVI NES) is proposed in this paper. Based on the Kirchhoff hypothesis, the coupled equations of motion between the plate and the TSSVI NES are established. The system is modeled by the Galerkin technique and numerically solved using the fourth order Runge-Kutta method. To obtain the optimal parameters of the TSSVI NES, a particle swarm optimization method is adopted. The system response with the TSSVI NES is compared with that of the cubic nonlinear energy sink (CNES) and the single-sided vibro-impact nonlinear energy sink (SSVI NES), showing that the vibration suppression ability of the TSSVI NES is better. The mechanism of rapid response suppression of the TSSVI NES is demonstrated by way of simulations. A parametric study is also conducted to determine the effects of damping, the coefficient of restitution, mass, stiffness, and installation position of TSSVI NES on the efficacy of the vibration control device. The numerical results demonstrate that the performance of the TSSVI NES is robust to the damping, coefficient of restitution, and installation position, but sensitive to mass difference and stiffness difference. Finally, the effect of the TSSVI NES is examined when the plate is excited by seismic loads, demonstrating a good response suppression performance.

Effect of Interstage Torsional Damping on Nonlinear Dynamic Characteristics of Two-Stage Planetary Gear System

Abstract This paper explores the influence of interstage torsional damping on the nonlinear dynamics of a two-stage planetary gear transmission system. A comprehensive nonlinear dynamic model is developed, incorporating interstage torsional damping, stiffness, time-varying meshing parameters, damping, and tooth side clearance. The nonlinear equations are derived and solved using the fourth-order Runge–Kutta method. The investigation reveals that the changes in torsional damping between stages significantly affect the nonlinear dynamic characteristics of the system, especially under load fluctuations. Bifurcation characteristics are analyzed using phase diagrams, Poincaré diagrams, time history diagrams, and frequency–domain spectrograms. The simulation results demonstrate that heightened torsional damping between stages distinctly affects the motion states across excitation frequencies. Notably, it mitigates the unstable motion state caused by high-frequency excitation when tooth side clearance is minimal.

Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods.

Generalized Shannon entropy sparse wavelet packet transform for fault detection of traction motor bearings in high-speed trains

An effective structural health monitoring method of traction motor bearings is a powerful guarantee for the safety operation of high-speed trains. However, it is exceptionally difficult to detect bearing fault characteristics from the vibration signals of traction motor bearings operating at high rotational speeds. In this scenario, a generalized Shannon entropy sparse wavelet packet transform (GSWPT) for fault detection of motor bearings is proposed in this paper. Firstly, a generalized Shannon entropy sparse regularization method is proposed to obtain sparse wavelet reconstruction coefficients by extending the definition of the Shannon information entropy, and the non-convex sparse regularization function is minimized by synergistic swarm optimization algorithm. Then, the wavelet node coefficients are weighted according to the second-order cyclostationarity index of the wavelet packet node to further enhance the sparsity of the reconstructed signal. Moreover, the optimal decomposition level of GSWPT is adaptively selected by the maximum sparsity and cyclostationarity criterion. Particularly, in order to verify the bearing fault detection performance of GSWPT in practical engineering, a bearing fault dynamic model of traction motor in high-speed train was established based on Hertz contact theory and the fourth-order Runge-Kutta method to obtain simulated data under strong Gaussian white noise, and a corresponding test platform was constructed to collect experimental data under different operating conditions. Finally, the applications on the simulated and experimental signals of traction motor bearings in high-speed trains demonstrate that GSWPT significantly outperforms the conventional wavelet packet transform, dual-tree complex wavelet packet transform, blind deconvolution, modal decomposition, and Infogram methods to some extent for fault detection.

Nonlinear vibrations, bifurcations and chaos of piezoelectric composite lattice sandwich plate with four simply supported edges

This paper is devoted to investigate the nonlinear dynamics, bifurcation and chaos of the piezoelectric composite lattice sandwich plate with 1:3 internal resonance. Galerkin method is used to discretize the nonlinear partial differential governing equations of motion into the ordinary differential equations. The multi-scale method is used to obtain the modulation equations of the piezoelectric composite lattice sandwich plate in the polar coordinates and Cartesian coordinates in the primary and parametric resonances, respectively. Using Newton-Raphson method, the bifurcation diagrams of the steady-state equilibrium solution are obtained for the modulation equation with the change of the system parameters. The stability of the steady-state equilibrium solution is analyzed for the modulation equation. The existence of the static bifurcations, such as the saddle-node bifurcation, pitchfork bifurcation and Hopf dynamic bifurcations are found. The complex nonlinear jump phenomena caused by the existence of the multiple nonlinear steady-state solution regions are analyzed in detail. Fourth-order Runge-Kutta method is used to continue tracing the nonlinear periodic solution of the modulation equation in Cartesian coordinates. The time-history diagram, phase portrait and Poincaré map are obtained for the piezoelectric composite lattice sandwich plate in the nonlinear resonance. The dynamic process of the nonlinear periodic solution is described for the modulation equation from the limit cycle oscillation to the chaotic attractor in detail.

Study on the steady state position of ocean towing cables

A study is conducted on the static equilibrium posture of the towing cable in water under a constant speed state in the towing system. Firstly, the underwater force state of the towing body is calculated and then the posture and tension of the towing cable are analyzed and calculated. As the cable is divided into two states which are above water and below water, a differential equation system is established as the boundary. The initial value of the equation can be transformed by the drag and gravity values of the towing body. The program solves the differential equations using the fourth-order Runge Kutta method and draws the attitude curve. Finally, it verifies the theoretical calculation results through experiments.

Research on axial nonlinear vibration characteristics of ball screw feed system considering segmented restoring force

Taking the ball screw feed system as the research object, the two degree of freedom axial dynamic model of the system is constructed firstly. Based on Hertz contact theory, when the ball screw pair adopts variable lead self-preloading to eliminate the assembly clearance between the ball and raceway, considering the nonlinear segmented axial elastic recovery force generated by the uneven contact deformation between the ball and raceway under the action of external excitation force, further derive the nonlinear dynamic equation group of the system. Next, the fourth-order Runge-Kutta method was used to numerically solve the equation system, obtaining the system’s two and three-dimensional phase diagrams, Poincaré sections, time-domain waveform diagrams, frequency spectra, and bifurcation diagrams. Then, the effects of damping constant, initial contact angle between ball and raceway in ball screw pairs, and number of balls on the system’s response characteristics were analyzed, and the influence of external excitation forces on system stability was further studied. Finally, it was verified through experiments that the axial vibration of the system is indeed nonlinear vibration, providing a theoretical basis for the study of the dynamic characteristics of the ball screw feed system.

Thermal design calculation method of the whole coil closed cooling tower with condensation inside the tube

In this paper, the heat exchange process of a closed cooling tower with condensation inside the tube is analyzed. Combined with the calculation method of a traditional closed cooling tower, the thermal model and mathematical model of the whole coil closed cooling tower with condensation inside the tube are established. The fourth-order Runge-Kutta (RK4) method is used to write a program to solve the differential equation. A case is calculated to solve the heat exchange area and spray water temperature. Its feasibility for design reference is verified.

Accelerating simulations of Li-ion battery thermal runaway using modified Patankar–Runge–Kutta approach

Reliable and accurate simulations of the thermal runaway of Li-ion batteries are crucial for designing thermal management system. However, the stiff nature of thermal abuse kinetics makes it computationally challenging to obtain numerical solutions. To address this, we employed modified Patankar–Runge–Kutta (MPRK) methods for thermal runaway simulations. The explicit nature of these methods leads to computationally efficient simulations, while their ability to maintain positive physical quantities ensures robust integrations. We first investigated the performance of the second- and third-order MPRK methods, along with the well-known fourth-order Runge–Kutta method, for zero-dimensional thermal runaway simulations. Numerical tests revealed the superiority of the third-order MPRK method in terms of robustness and accuracy. To further improve its accuracy during a rapid temperature rise, we introduced an adaptive time-stepping method. This adaptive third-order MPRK method was then implemented in Ansys Fluent for three-dimensional simulations of thermal runaway propagation. Benchmark tests demonstrated an increase of 10 times in the computational speed compared with Fluent’s built-in solver, which was achieved by allowing significantly larger thermal time steps without compromising accuracy. The robust and efficient computational method proposed in this study will pave the way for future studies on multiphysics simulations of thermal runaway phenomena.

Numerical Simulation on the Two-Degree-of-Freedom Flow-Induced Vibration of a Submerged Floating Tunnel under Current

The submerged floating tunnel (SFT) is a novel form of transportation infrastructure for crossing deeper and wider seas. One of the primary challenges in designing SFTs is understanding their hydrodynamic response to complex environmental loads. In order to investigate the two-degree-of-freedom (2-DOF) flow-induced vibration (FIV) response of SFTs under current, a two-dimensional (2D) numerical model was developed using the Reynolds-averaged Navier–Stokes (RANS) method combined with the fourth-order Runge–Kutta method. The numerical results were validated by comparing them with the existing literature. The study then addressed the effects of coupled vibration and structural parameters, i.e., the mass ratio and natural frequency ratio, on the response and wake pattern of SFTs, numerically. The results indicated that coupled vibration had a significant impact on the SFT response at reduced velocities of Urwx ≥ 4.4. A decrease in mass ratio (m* < 1) notably amplified the 2-DOF vibration amplitudes of SFTs at Urwx ≥ 4.4, particularly for in-line vibration. Similarly, a decrease in natural frequency ratio (Rf < 1) significantly suppressed the in-line vibration of SFTs at Urwx ≥ 2.5. Therefore, for the design of SFTs, careful consideration should be given to the effect of mass ratio and natural frequency ratio on in-line vibration.