Strong Nonlinear Differential Equation Research Articles

Utilization of variable thermal conductivity and diffusion coefficient on non-Newtonian Prandtl model with modified heat and mass fluxes

PurposeThis study aims to analyze the Prandtl fluid flow in the presence of better mass diffusion and heat conduction models. By taking into account a linearly bidirectional stretchable sheet, flow is produced. Heat generation effect, thermal radiation, variable thermal conductivity, variable diffusion coefficient and Cattaneo–Christov double diffusion models are used to evaluate thermal and concentration diffusions.Design/methodology/approachThe governing partial differential equations (PDEs) have been made simpler using a boundary layer method. Strong nonlinear ordinary differential equations (ODEs) relate to appropriate non-dimensional similarity variables. The optimal homotopy analysis technique is used to develop solution.FindingsGraphs analyze the impact of many relevant factors on temperature and concentration. The physical parameters, such as mass and heat transfer rates at the wall and surface drag coefficients, are also displayed and explained.Originality/valueThe reported work discusses the contribution of generalized flux models to note their impact on heat and mass transport.

Effect of an inclined magnetic field on unsteady mixed convective stagnation point flow over a permeable stretching sheet with radiative heat transfer

In this study, we analyzed the time-dependent, 2D, inclined magnetohydrodynamics mixed convection stagnating point flow of nanoparticle toward a permeable stretchable surface with radiative heat effect. Four different types of water-based nanomaterials, namely titanium dioxide copper aluminum oxide and copper oxide are analyzed in this investigation. Numerous engineering disciplines, including energy systems, material processing, thermal management, and environmental engineering, can benefit greatly from research in mixed convection, MHD, and nanofluids. The boundary layer representations of the dominating partial differential equations PDEs are converted into strong nonlinear ordinary differential equations ODEs utilizing similarities approach. The ODEs are computed numerically employing 4th order Runge–Kutta methodology-based shooting scheme. Few special situations, a remarkable alignment is determined between the present study and the outcomes obtained in the existing research. The impact of numerous pertinent variables on the prominent quantities such as velocity, temperature and concentration distributions, drag friction coefficient and heat transport are demonstrated graphically. It is noticed that the velocity and thermal curve decrease as the nanoparticle volume friction factor increases. Moreover, it is analyzed that higher values of nanoparticle concentration diminishes the velocity and temperature distributions.

Analysis of nonlinear vibrations of two-degree-of-freedom systems

In this paper, the motions of two-mass systems with two-degree-of-freedom are investigated by using an analytical approach. The masses are connected by linear and nonlinear springs. The motions of systems are described by systems of two coupled strong nonlinear differential equations. Nonlinear differential equations are transferred into a single equation by using some intermediate variables. An analytical method, the equivalent linearization method, is employed to analyze the free nonlinear vibration of systems. The oscillation systems with different values of the parameters are investigated in this paper. In order to verify the accuracy of the obtained results, the present solutions are compared with those achieved by the Hamiltonian approach and the exact solutions. The comparison results show that the obtained solutions are more accurate than those obtained by the Hamiltonian approach.

Exact Closed-Form Solution for the Oscillator with a New Type of Mixed Nonlinear Restitution Force

This paper shows an oscillator with a spring made of material where the stress is a function not only of strain but also strain rate. The corresponding restitution force is of strong nonlinear monomial type and is the product of displacement and velocity of any order. The mathematical model of the oscillator is a homogenous strong nonlinear second-order differential equation with an integer- or non-integer-order mixed term. In the paper, an analytical procedure for solving this new type of strong nonlinear equation is developed. The approximate solution is assumed as the perturbed version of the exact solution in the form of a sine Ateb function. As a result, it is obtained that the amplitude, period, and frequency of vibration depend not only on the coefficient and order of nonlinearity, but also on the initial velocity. The procedure is tested on two examples: oscillator perturbed with small linear damping and small linear displacement functions. The analytically obtained results are compared with the exact numerical ones and show good agreement. It is concluded that the mathematical model and also the procedure developed in the paper would be convenient for prediction of motion for this type of oscillator without necessary experimental testing.

Lyapunov-Malkin type approach of equilibrium stability in a critical case applied to a switched model of a servomechanism with state delay

A general theorem on equilibrium stability in a critical case is applied to switched strong nonlinear differential equations with time delay on state, characterising electrohydraulic servomechanisms dynamics. Basically, its proof involves the use of the Lyapunov-Malkin approach to stability and multiple complete Lyapunov-Krasovskii functionals. The fulfilment of the equilibrium stability condition of the nonlinear system returns to the asymptotic stability condition of the linearised equations and to the so-called Lyapunov conditions for the latter. The transformation of the nonlinear system into the canonical form specific to the Lyapunov-Malkin theorem, and the verification of the two conditions mentioned above require analytical developments doubled by numerical simulations, since the mathematical models are too complex to be approached only analytically. As a consequence, an important result is obtained, for the first time, regarding the thresholds of admissible delay in preserving the stability of a real world object, vital for the safety of the aircraft.

Prediction of Surface Roughness in Turning Applying the Model of Nonlinear Oscillator with Complex Deflection

This paper deals with prediction of the roughness of a cutting surface in the turning process, applying the vibration data of the system. A new type of dynamic model for a workpiece-cutting tool system, appropriate for vibration simulation, is developed. The workpiece is modelled as a mass-spring system with nonlinear elastic property. The cutting tool acts on the workpiece with the cutting force which causes strong in-plane vibration. Based on the experimentally measured values, the cutting force is analytically described as the function of feed ratio and cutting speed. The mathematical model of the vibrating system is a non-homogenous strong nonlinear differential equation with complex function. A new approximate solution for the nonlinear equation is derived and analytic description of vibration is obtained. The solution depends on parameters of the excitation force, velocity of rotation and nonlinear properties of the system. Increasing the feed ratio at a constant velocity of the working piece, the frequency of vibration decreases and the amplitude of vibration increases; increasing the velocity of working piece for constant feed ratio causes an increase of the frequency and a decrease of the amplitude of vibration. Experiments demonstrate that the analytical solution of the nonlinear vibration model in turning process is in direct correlation with the cutting surface roughness. The predicted surface roughness is approximately (1–2) × 10−3 times smaller than the amplitude of vibration of the nonlinear model considered in this paper.

The modified high-order Haar wavelet scheme with Runge–Kutta method in the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation

In this work, we focus on the modified high-order Haar wavelet numerical method, which introduces the third-order Runge–Kutta method in the time layer to improve the original numerical format. We apply the above scheme to two types of strong nonlinear solitary wave differential equations named as the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation. Numerical experiments verify the correctness of the scheme, which improves the speed of convergence while ensuring stability. We also compare the CPU time, and conclude that our scheme has high efficiency. Compared with the traditional wavelets method, the numerical results reflect the superiority of our format.

Three Dimensional Non-linear Radiative Nanofluid Flow over a Riga Plate

Numerous techniques in designing zones happen at high temperature and functions under high temperature are in a way that involves non-linear radiation. In weakly conducting fluids, however, the currents induced by an external magnetic field alone are too small, and an external electric field must be applied to achieve an efficient flow control. Gailitis and Lielausis, devised Riga plate to generate a crossed electric and magnetic fields which can produce a wall parallel Lorentz force in order to control the fluid flow. It acts as an efficient agent to reduce the skin friction. So, in this paper, we start the numerical investigation on the three-dimensional flow of nanofluids with the inclusion of non-linear radiation past a Riga plate. To this end, the numerical investigation is conducted on the three-dimensional flow of nanofluids with the inclusion of non-linear radiation past a Riga plate. Water (H2O) and Sodium Alginate (NaC6H9O7) are the base fluids, whereas Magnetite (Fe3O4) and Aluminium oxide (Al2O3) are the nanoparticles. The mathematical formulation for Sodium Alginate base fluid is separated through the Casson model. Suitable transformations on governing partial differential equations yield strong non-linear ordinary differential equations. Numerical solutions for the renewed system are constructed by fourth-order Runge-Kutta method with shooting technique. Various deductions for flow and heat transfer attributes are sketched and discussed for various physical parameters. Furthermore, the similarities with existing results were found for the physical quantities of interest. It was discovered, that the temperature ratio parameter and the radiation parameter enhance the rate of heat transport. Moreover, the NaC6H9O7 - Al2O3 nanofluid improves the heat transfer rate. Likewise, H2O-Fe3O4 nanofluid stimulates the local skin friction coefficients.

Modelling and analysis of the nonlinear string-mass structure of the vibration absorber

ABSTRACT In this paper a nonlinear string-mass structure of the vibration absorber is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquake. The considered string-mass structure contains a translator movable mass connected with two strings. Due to nonlinear geometric properties of the system the motion of the mass is described with a strong nonlinear second order differential equation. In the paper the approximate procedure for solving of the nonlinear equation of motion is developed. Based on the solution the influence of the string preloading force, slider mass and friction force on the vibration property of the string-mass system is investigated. It is concluded that variation of the preloading string force may be applied as a control parameter for vibration absorption and as the regulator of vibration decay time.

Non-Darcian three-dimensional flow of Fe3O4/Al2O3 nanoparticles with H2O/NaC6H9O7 base fluids past a Riga plate embedded in a porous medium

This paper is mainly concerned with the investigation of steady, three-dimensional flow of H2O/NaC6H9O7 base fluids with Fe3O4/Al2O3 nanoparticles past a Riga plate implanted in a non-Darcian porous medium with internal heat generation/absorption effects. Water (H2O) and Sodium Alginate (NaC6H9O7) are the base fluids, whereas Magnetite (Fe3O4) and Aluminium oxide (Al2O3) happens to be the nanoparticles. The numerical plan for non-Newtonian fluid Sodium Alginate is isolated through the Casson model. Suitable transformations on governing equations yield strong non-linear differential equations. The non-linearity of the basic equations and additional mathematical difficulties led to use numerical method. Plots for the outcomes of various related variables are furnished. The local skin friction coefficient and the local Nusselt number have been expounded. The results end up to signify that the local skin friction coefficient and the local Nusselt number are diminished for larger values of porosity parameter and the local inertia coefficient.

Analytical investigation on the dynamics of the smooth and discontinuous oscillator

In this paper, the dynamics of a nonlinear smooth and discontinuous oscillator, modeled as a string–mass structure, is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquakes. The considered string–mass structure contains a translator movable mass connected with two strings. The motion of the mass is oscillatory and perpendicular to the string’s position. Usually, in strings preloading forces act. Due to geometric and physical properties of the system, the restitution force of the string is nonlinear. The model of the mass motion is a strong nonlinear second-order differential equation. The nonlinearity is of power type, and the order of nonlinearity is any positive real number. An exact solution of the truly nonlinear equation is introduced in the form of the Ateb function (inverse Beta function). Based on the exact solution, the approximate solving procedure of the nonlinear equation of motion is developed. The method is suitable for dynamic analyses of the system. The influence of the preloading force on the nonlinear vibrations of the string–mass system is considered. It is concluded that variation of the string force has influence on the velocity of the amplitude decrease in the system.

Higher-order approximate periodic solution for the oscillator with strong nonlinearity of polynomial type

In this paper the harmonic balance method (HBM) is adopted for solving a special group of oscillators with strong nonlinear damping and elastic forces. The nonlinearity is of polynomial type. The motion is described with a strong nonlinear differential equation, whose approximate solution is assumed as a suitable sum of trigonometric functions. To find the most convenient combination of trigonometric functions as the probe function is the most important part of this investigation. Introducing the procedure of equating the terms with the same order of the trigonometric functions to zero, the problem is transformed into solving a system of nonlinear algebraic equations. Solving these equations the parameters of the solution up to high-order approximation are obtained. In the paper, the suggested solving procedure is applied for two nonlinear oscillator problems: free vibrations of a restrained uniform beam carrying an intermediate lumped mass and of a particle on a rotating parabola. The obtained approximate analytic solutions are compared with the already published results and with the numerically obtained solution. The solution up to third-order approximation is calculated. It is proved that the HBM with the suggested function gives more accurate results than the previous applied ones. Besides, the difference between the third-order approximate analytical solution and the numerical one is negligible. The method works well for different, even high, values of initial amplitudes.

Three different hybrid nanometrial performances on rotating disk: a non-Darcy model

The most essential uniqueness of hybrid nanoliquid mentions to creation of two variation sorts of scattered nanoparticles in a base liquid. Along these lines, when particle materials have been chosen correctly, they can improve the optimistic features of each other and cover the drawbacks of only a solitary material. The article examines the behaviour of three different hybrid nanoparticles flow in a porous region which is characterized by Forchheimer medium across a rotating disk. Heat transport performance is analysed through convective condition. Strong ordinary nonlinear differential equations are obtained by applying suitable transformation. System of nonlinear equations is solved numerically via RKF method. Repercussion of pertinent parameters on radial velocity, tangential velocity and temperature distribution are elaborated. In addition drag coeffcient and local Nusselt number are physically interpreted. Result shows that both radial velocity and tangential velocity decreases for larger values of stretching parameter and also increment of nanoparticle volume faction increases the local Nusselt number.

Periodic Motion in an Excited and Damped Cubic Nonlinear Oscillator

This paper investigates the steady-state periodic motion in the excited and damped one-degree-of-freedom Duffing oscillator. The oscillator is of the pure cubic type. The excitation is periodical and described by the product of two Jacobi elliptic functions. The mathematical model of the oscillator is a nonhomogeneous second-order strong nonlinear differential equation. The paper develops a procedure for obtaining the steady-state solution of the equation. Conditions for the existence of the steady-state motion of the oscillator are obtained. The influence of the excitation and of the damping on the steady-state motion is analyzed. The paper also investigates the transient to the steady-state motion in the parameter perturbed systems. An analytical method based on the time variable amplitude and the time variable phase is developed. The analysis of the obtained results shows that the damping parameter is an adequate control parameter for the steady-state motion of the oscillator. Analytically obtained results are compared with numerically obtained ones. The difference between solutions is negligible.

Finite Elements Method Coupled with Delaunay Triangulation Method Applied on a Silicon Corner Diode

This work considers a two dimensional numerical device simulation system using a novel digitizing scheme based on finite elements coupled with the Delaunay triangulation method which allowed an optimal mesh involved in nonrectangular devices as a corner diode. A grid was generated automatically according to the specified device geometry standing on the Delaunay triangulation process. The solution of the problem consists in the resolution of three strong nonlinear partial differential equations (PDE) which are, in occurrence, two dimensional Poisson and continuity equations. Modeled voltage, free carriers distribution and I-V characteristics were extracted when the PN junction resolution was based on second-order Bezier curves. The results are presented using a Delaunay triangulation meshing mathematical tool.

Impact of mechanism vibration characteristics by joint clearance and optimization design of its multi-objective robustness

Joint clearances and friction characteristics significantly influence the mechanism vibration characteristics; for example: as for joint clearances, the shaft and bearing of its clearance joint collide to bring about the dynamic normal contact force and tangential coulomb friction force while the mechanism works; thus, the whole system may vibrate; moreover, the mechanism is under contact-impact with impact force constraint from free movement under action of the above dynamic forces; in addition, the mechanism topology structure also changes. The constraint relationship between joints may be established by a repeated complex nonlinear dynamic process (idle stroke - contact-impact - elastic compression - rebound – impact relief - idle stroke movement - contact-impact). Analysis of vibration characteristics of joint parts is still a challenging open task by far. The dynamic equations for any mechanism with clearance is often a set of strong coupling, high-dimensional and complex time-varying nonlinear differential equations which are solved very difficultly. Moreover, complicated chaotic motions very sensitive to initial values in impact and vibration due to clearance let high-precision simulation and prediction of their dynamic behaviors be more difficult; on the other hand, their subsequent wearing necessarily leads to some certain fluctuation of structure clearance parameters, which acts as one primary factor for vibration of the mechanical system. A dynamic model was established to the device for opening the deepwater robot cabin door with joint clearance by utilizing the finite element method and analysis was carried out to its vibration characteristics in this study. Moreover, its response model was carried out by utilizing the DOE method and then the robust optimization design was performed to sizes of the joint clearance and the friction coefficient change range so that the optimization design results may be regarded as reference data for selecting bearings and controlling manufacturing process parameters for the opening mechanism. Several optimization objectives such as x/y/z accelerations for various measuring points and dynamic reaction forces of mounting brackets, and a few constraints including manufacturing process were taken into account in the optimization models, which were solved by utilizing the multi-objective genetic algorithm (NSGA-II). The vibration characteristics of the optimized opening mechanism are superior to those of the original design. In addition, the numerical forecast results are in good agreement with the test results of the prototype.

Thermal radiation on unsteady electrical MHD flow of nanofluid over stretching sheet with chemical reaction

This paper focuses on the effects of suction as well as thermal radiation, chemical reaction, viscous dissipation and Joule heating on a two-dimensional natural convective flow of unsteady electrical magnetohydrodynamics (MHD) nanofluid over a linearly permeable stretching sheet. One significant aspect of this study is that electric field employed in revised Buongiorno model has been introduced in view of enhancement of thermal conductivity and consequently better convective heat transfer. The constitute governing equations have been converted into strong non-linear ordinary differential equations by employing suitable transformations and these transformed equations are solved by the Implicit finite difference. From this study, it is found that the presence of magnetic field and suction slows down the fluid motion while it enhances for higher values of an electric field which tends to firmness sticky effect. It is also found that enhancing thermal radiation leads to an increase in nanofluid temperature. The Nusselt number increases with both Brownian motion and unsteadiness parameters.

3D free convective MHD flow of nanofluid over permeable linear stretching sheet with thermal radiation

This paper mainly focuses on the influence of transverse magnetic field as well as thermal radiation on three-dimensional free convective flow of nanofluid over a linear stretching sheet. One remarkable aspect of this study is that a new micro-convection model namely Patel model has been introduced in view of enhancement of thermal conductivity and hence more heat transfer capability of nanofluid. The non-linear partial differential equations have been converted into strong non-linear ordinary differential equations by employing suitable transformations and these transformed equations are solved by Runga-Kutta method of fourth order along with Shooting technique as well as Secant method for better approximation. From this study, it is found that the presence of magnetic field slows down the fluid motion while it enhances the fluid temperature leading to a reduction in heat transfer rate from the surface. It is also found that enhancing thermal radiation parameter causes a reduction in heat transfer rate.

Negative effective mass in acoustic metamaterial with nonlinear mass-in-mass subsystems

In this paper the dynamics of the nonlinear mass-in-mass system as the basic subsystem of the acoustic metamaterial is investigated. The excitation of the system is in the form of the Jacobi elliptic function. The corresponding model to this forcing is the mass-in-mass system with cubic nonlinearity of the Duffing type. Mathematical model of the motion is a system of two coupled strong nonlinear and nonhomogeneous second order differential equations. Particular solution to the system is obtained. The analytical solution of the problem is based on the simple and double integral of the cosine Jacobi function. In the paper the integrals are given in the form of series of trigonometric functions. These results are new one. After some modification the simplified solution in the first approximation is obtained. The result is convenient for discussion. Conditions for elimination of the motion of the mass 1 by connection of the nonlinear dynamic absorber (mass - spring system) are defined. In the consideration the effective mass ratio is introduced in the nonlinear mass-in-mass system. Negative effective mass ratio gives the absorption of vibrations with certain frequencies. The advantage of the nonlinear subunit in comparison to the linear one is that the frequency gap is significantly wider. Nevertheless, it has to be mentioned that the amplitude of vibration differs from zero for a small value. In the paper the analytical results are compared with numerical one and are in agreement.

Procedures for determination of elastic curve of simply and multiple supported beams

In this paper two procedures for determination of the elastic curve of the simply and multiple supported beams are developed. Determination of the elastic curve is complex as it requires to solve a strong nonlinear differential equation with given boundary conditions. For numerical solution the initial guess of the slope at the end of the beam is necessary. Two procedures for obtaining of the initial guess are developed: one, based on transformation of the supported beam into a clamped-free one, and second, on the linearization of the problem. Procedures are applied for calculating of elastic curve of a simply supported beam and a beam with three supports. Obtained results are compared. Advantages and disadvantages of both methods are discussed. It is proved that both suggested procedures give us technically accurate results.