Mathematics

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17 pages, 6367 KiB

Open AccessArticle

The Discovery of Truncated M-Fractional Exact Solitons and a Qualitative Analysis of the Generalized Bretherton Model

by Haitham Qawaqneh, Khalil Hadi Hakami, Ali Altalbe and Mustafa Bayram

Mathematics 2024, 12(17), 2772; https://doi.org/10.3390/math12172772 - 7 Sep 2024

Viewed by 551

This paper is concerned with the novel exact solitons for the truncated M-fractional (1+1)-dimensional nonlinear generalized Bretherton model with arbitrary constants. This model is used to explain the resonant nonlinear interaction between the waves in different phenomena, including fluid dynamics, plasma physics, ocean [...] Read more.

This paper is concerned with the novel exact solitons for the truncated M-fractional (1+1)-dimensional nonlinear generalized Bretherton model with arbitrary constants. This model is used to explain the resonant nonlinear interaction between the waves in different phenomena, including fluid dynamics, plasma physics, ocean waves, and many others. A series of exact solitons, including bright, dark, periodic, singular, singular–bright, singular–dark, and other solitons are obtained by applying the extended sinh-Gordon equation expansion (EShGEE) and the modified

(G^{'} / G^{2})

-expansion techniques. A novel definition of fractional derivative provides solutions that are distinct from previous solutions. Mathematica software was used to obtain and verify the solutions. The solutions are shown through 2D, 3D, and density plots. A stability process was conducted to verify that the solutions are exact and accurate. Modulation instability was used to determine the steady-state results for the corresponding equation. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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12 pages, 4551 KiB

Open AccessArticle

On Spatial Systems of Bars Spherically Jointed at Their Ends and Having One Common End

by Valentin Răcășan and Nicolae-Doru Stănescu

Mathematics 2024, 12(17), 2680; https://doi.org/10.3390/math12172680 - 28 Aug 2024

Viewed by 521

Abstract

In this paper we consider a system of linear bars, spherically jointed at their ends. For each bar one end is linked to the origin. We discuss the equations from which one obtains the deviation of the origin, and some possible optimizations concerning [...] Read more.

In this paper we consider a system of linear bars, spherically jointed at their ends. For each bar one end is linked to the origin. We discuss the equations from which one obtains the deviation of the origin, and some possible optimizations concerning the minimum displacement of the origin and the minimum force in one bar, which are the main goals of the paper. The optimization is performed considering that for two bars one end is unknown; that is, the angles between the bars and the axes are unknown. It is proved that it is difficult to obtain an analytical solution in the general case, and the problem can be discussed only by numerical methods. A numerical case is also studied and some comments concerning the results are given. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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23 pages, 3421 KiB

Open AccessArticle

Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van der Waals Equation

by Haitham Qawaqneh, Jalil Manafian, Mohammed Alharthi and Yasser Alrashedi

Mathematics 2024, 12(14), 2257; https://doi.org/10.3390/math12142257 - 19 Jul 2024

Cited by 5 | Viewed by 850

Abstract

The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der [...] Read more.

The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der Waals equation explains the phase separation phenomenon. Noncovalent Van der Waals or dispersion forces usually have an effect on the structure, dynamics, stability, and function of molecules and materials in different branches of science, including biology, chemistry, materials science, and physics. Solutions are obtained, including dark, dark-singular, periodic wave, singular wave, and many more exact wave solutions by using the modified extended tanh function method. Using the fractional derivatives makes different solutions different from the existing solutions. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other applications of the Van der Waals equation. The solutions may be useful in distinct fields of science and civil engineering, as well as some basic physical ones like those studied in geophysics. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs by using Mathematica software. The obtained results are newer than the existing results. Stability analysis is also performed to check the stability of the concerned model. Furthermore, modulation instability is studied to study the stationary solutions of the concerned model. The results will be helpful in future studies of the concerned system. In the end, we can say that the method used is straightforward and dynamic, and it will be a useful tool for debating tough issues in a wide range of fields. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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20 pages, 3610 KiB

Open AccessArticle

An Analytical Study of the Effects of Kinematic Parameters on the Motion Stability of a 3-RPR Parallel Manipulator in Singular Configurations

by Yu-Tong Li and Yu-Xin Wang

Mathematics 2024, 12(11), 1771; https://doi.org/10.3390/math12111771 - 6 Jun 2024

Viewed by 734

Abstract

Due to the Jacobian matrix rank reduction near singularities, applying numerical methods to study PMs’ motion stability at singularities is quite difficult. As a result, there is a scarcity of literature on the investigation of PMs’ dynamic behaviors near singularities and the influence [...] Read more.

Due to the Jacobian matrix rank reduction near singularities, applying numerical methods to study PMs’ motion stability at singularities is quite difficult. As a result, there is a scarcity of literature on the investigation of PMs’ dynamic behaviors near singularities and the influence of kinematic parameters on the motion stability of PMs. To address the research gap related to the above issues, based on the Gerschgorin perturbation method, Hurwitz exact approach, and the Lyapunov dynamic stability theory, the influence of kinematic parameters and external loads on a PM’s motion stability at singularities is studied for the first time. The theoretical analysis results reported in this paper reveal many previously undiscovered features beyond those derived from previous numerical methods, and indicate the limitations of some widely accepted statements. For example, increasing the angular speed of the movable platform can expand the range of the external loads that meet the motion stability at singular configurations. The prevailing notion in prior research that PMs are unable to support external loads in the direction of the gained DoF at singular configurations is only partially accurate. This pioneering research establishes a theoretical foundation for exploring a new real-time approach to avoid dynamic singularities by fully exploiting the influence mechanisms of kinematic parameters on PMs’ dynamic stability at singularities. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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18 pages, 424 KiB

Open AccessArticle

Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System

by Remus-Daniel Ene and Nicolina Pop

Mathematics 2024, 12(9), 1308; https://doi.org/10.3390/math12091308 - 25 Apr 2024

Cited by 1 | Viewed by 708

Abstract

Mathematical models and numerical simulations are necessary to understand the functions of biological rhythms, to comprehend the transition from simple to complex behavior and to delineate the conditions under which they arise. The aim of this work is to investigate the R [...] Read more.

Mathematical models and numerical simulations are necessary to understand the functions of biological rhythms, to comprehend the transition from simple to complex behavior and to delineate the conditions under which they arise. The aim of this work is to investigate the R

\ddot{o}

ssler-type system. This system could be proposed as a theoretical model for biological rhythms, generalizing this formula for chaotic behavior. It is assumed that the R

\ddot{o}

ssler-type system has a Hamilton–Poisson realization. To semi-analytically solve this system, a Bratu-type equation was explored. The approximate closed-form solutions are obtained using the Optimal Parametric Iteration Method (OPIM) using only one iteration. The advantages of this analytical procedure are reflected through a comparison between the analytical and corresponding numerical results. The obtained results are in a good agreement with the numerical results, and they highlight that our procedure is effective, accurate and usefully for implementation in applicationssuch as an oscillator with cubic and harmonic restoring forces, the Thomas–Fermi equation and the Lotka–Voltera model with three species. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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24 pages, 401 KiB

Open AccessArticle

Dynamics of Non-Autonomous Stochastic Semi-Linear Degenerate Parabolic Equations with Nonlinear Noise

by Xin Liu and Yanjiao Li

Mathematics 2023, 11(14), 3158; https://doi.org/10.3390/math11143158 - 18 Jul 2023

Viewed by 1115

Abstract

In the present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on a bounded or unbounded domain and driven by a nonlinear noise. Since the theory of pathwise random dynamical systems cannot be applied directly to [...] Read more.

In the present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on a bounded or unbounded domain and driven by a nonlinear noise. Since the theory of pathwise random dynamical systems cannot be applied directly to the equation with nonlinear noise, we first establish the existence of weak pullback mean random attractors for the equation by applying the theory of mean-square random dynamical systems; then, we prove the existence of (pathwise) pullback random attractors for the Wong–Zakai approximate system of the equation. In addition, we establish the upper semicontinuity of pullback random attractors for the Wong–Zakai approximate system of the equation under consideration driven by a linear multiplicative noise. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

16 pages, 789 KiB

Open AccessEditor’s ChoiceArticle

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

by Daniel Condurache

Mathematics 2023, 11(6), 1366; https://doi.org/10.3390/math11061366 - 11 Mar 2023

Cited by 1 | Viewed by 1435

Abstract

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative [...] Read more.

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

14 pages, 582 KiB

Open AccessEditor’s ChoiceArticle

Optimal Homotopy Asymptotic Method for an Anharmonic Oscillator: Application to the Chen System

by Remus-Daniel Ene and Nicolina Pop

Mathematics 2023, 11(5), 1124; https://doi.org/10.3390/math11051124 - 23 Feb 2023

Viewed by 1491

Abstract

The aim of our work is to obtain the analytic solutions for a new nonlinear anharmonic oscillator by means of the Optimal Homotopy Asymptotic Method (OHAM), using only one iteration. The accuracy of the obtained results comes from the comparison with the corresponding [...] Read more.

The aim of our work is to obtain the analytic solutions for a new nonlinear anharmonic oscillator by means of the Optimal Homotopy Asymptotic Method (OHAM), using only one iteration. The accuracy of the obtained results comes from the comparison with the corresponding numerical ones for specified physical parameters. Moreover, the OHAM method has a greater degree of flexibility than an iterative method as is presented in this paper. Based on these results, the analytically solutions of the Chen system were obtained for a special case (just one analytic first integral). The chaotic behaviors were excluded here. The provided solutions are usefully for many engineering applications. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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16 pages, 3739 KiB

Open AccessArticle

Brushless Operation of Wound-Rotor Synchronous Machine Based on Sub-Harmonic Excitation Technique Using Multi-Pole Stator Windings

by Muhammad Humza, Tanveer Yazdan, Qasim Ali and Han-Wook Cho

Mathematics 2023, 11(5), 1117; https://doi.org/10.3390/math11051117 - 23 Feb 2023

Cited by 1 | Viewed by 2381

Abstract

This paper presents a topology for the brushless operation of a wound-rotor synchronous machine based on the subharmonic excitation technique by using two sets of multi-pole windings on the armature as well as on the rotor. The armature windings consist of a four-pole [...] Read more.

This paper presents a topology for the brushless operation of a wound-rotor synchronous machine based on the subharmonic excitation technique by using two sets of multi-pole windings on the armature as well as on the rotor. The armature windings consist of a four-pole three-phase main winding and a two-pole single-phase additional winding, responsible for the generation of fundamental and subharmonic components of magnetomotive force (MMF), respectively. The rotor contains four-pole field winding and two-pole excitation winding. From the generated air gap MMF, the additional winding is responsible for induction in excitation winding, which feeds DC to the field winding through a rotating rectifier without the need of brushes. Then, the interaction of the magnetic field from the main and the field windings produces torque. The proposed topology is analyzed using 2D finite element analysis (FEM). From the analysis, the generation of the subharmonic component of MMF is verified, which helps in achieving the brushless operation of the wound-rotor synchronous machine. Furthermore, the performance of the proposed brushless multi-pole topology is compared with the existing dual three-phase winding multi-pole topology in terms of current due to induction, output torque, torque ripples, and efficiency. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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16 pages, 2298 KiB

Open AccessArticle

Dynamics of the Vibro-Impact Nonlinear Damped and Forced Oscillator under the Influence of the Electromagnetic Actuation

by Nicolae Herisanu, Bogdan Marinca and Vasile Marinca

Mathematics 2022, 10(18), 3301; https://doi.org/10.3390/math10183301 - 12 Sep 2022

Cited by 4 | Viewed by 1679

Abstract

The main objective of the present work is to find an approximate analytical solution for the nonlinear differential equation of the vibro-impact oscillator under the influence of the electromagnetic actuation near the primary resonance. The trigger of vibro-impact regime is due to Hertzian [...] Read more.

The main objective of the present work is to find an approximate analytical solution for the nonlinear differential equation of the vibro-impact oscillator under the influence of the electromagnetic actuation near the primary resonance. The trigger of vibro-impact regime is due to Hertzian contact. The optimal auxiliary functions method (OAFM) is utilized to give an analytical approximate solution of the problem. The influences of static normal load and electromagnetic actuation near the primary resonance are completely studied. The main novelties of the proposed procedure are the presence of some new adequate auxiliary functions, the introduction of the convergence-control parameters, the original construction of the initial and of the first iteration, and the freedom to choose the method for determining the optimal values of the convergence-control parameters. All these led to an explicit and accurate analytical solution, which is another novelty proposed in the paper. This technique is very accurate, simple, effective, and easy to apply using only the first iteration. A second objective was to perform an analysis of stability of the model using the multiple scales method and the eigenvalues of the Jacobian matrix. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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15 pages, 2990 KiB

Open AccessArticle

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

by Richárd Horváth, Livija Cveticanin and Ivona Ninkov

Mathematics 2022, 10(17), 3214; https://doi.org/10.3390/math10173214 - 5 Sep 2022

Cited by 1 | Viewed by 1608

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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15 pages, 3897 KiB

Open AccessArticle

New Closed-Form Solution for Quadratic Damped and Forced Nonlinear Oscillator with Position-Dependent Mass: Application in Grafted Skin Modeling

by Livija Cveticanin, Nicolae Herisanu, Ivona Ninkov and Mladen Jovanovic

Mathematics 2022, 10(15), 2706; https://doi.org/10.3390/math10152706 - 31 Jul 2022

Cited by 1 | Viewed by 1580

Abstract

The paper deals with modelling and analytical solving of a strong nonlinear oscillator with position-dependent mass. The oscillator contains a nonlinear restoring force, a quadratic damping force and a constant force which excites vibration. The model of the oscillator is a non-homogenous nonlinear [...] Read more.

The paper deals with modelling and analytical solving of a strong nonlinear oscillator with position-dependent mass. The oscillator contains a nonlinear restoring force, a quadratic damping force and a constant force which excites vibration. The model of the oscillator is a non-homogenous nonlinear second order differential equation with a position-dependent parameter. In the paper, the closed-form exact solution for periodic motion of the oscillator is derived. The solution has the form of the cosine Ateb function with amplitude and frequency which depend on the coefficient of mass variation, damping parameter, coefficient of nonlinear stiffness and excitation value. The proposed solution is tested successfully via its application for oscillators with quadratic nonlinearity. Based on the exact closed-form solution, the approximate procedure for solving an oscillator with slow-time variable stiffness and additional weak nonlinearity is developed. The proposed method is named the ‘approximate time variable Ateb function solving method’ and is applicable to many nonlinear problems in physical and applied sciences where parameters are time variable. The method represents the extended and adopted version of the time variable amplitude and phase method, which is rearranged for Ateb functions. The newly developed method is utilized for vibration analysis of grafted skin on the human body. It is found that the grafted skin vibration properties, i.e., amplitude, frequency and phase, vary in time and depend on the dimension, density and nonlinear viscoelastic properties of the skin and also on the force which acts on it. The results obtained analytically are compared with numerically and experimentally obtained ones and show good agreement. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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