scholarly journals On the Analytical Solutions of the Forced Damping Duffing Equation in the Form of Weierstrass Elliptic Function and its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
S. A. El-Tantawy ◽  
Alvaro H. Salas ◽  
M. R. Alharthi
Keyword(s):  
Analytical Solution ◽  
Elliptic Function ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Duffing Equation ◽  
Weierstrass Elliptic Function ◽  
Pendulum Equation ◽  
Rlc Circuit ◽  
Linear Damping ◽  
Forced Term

In this study, a novel analytical solution to the integrable undamping Duffing equation with constant forced term is obtained. Also, a new approximate analytical (semianalytical) solution for the nonintegrable linear damping Duffing oscillator with constant forced term is reported. The analytical solution is given in terms of the Weierstrass elliptic function with arbitrary initial conditions. With respect to it, the semianalytical solution is constructed depending on a new ansatz and the exact solution of the standard Duffing equation (in the absence of both damping and forced terms). A comparison between the obtained solutions and the Runge–Kutta fourth-order (RK4) is carried out. Moreover, some complicated oscillator equations such as the constant forced damping pendulum equation, forced damping cubic-quintic Duffing equation, and constant forced damping Helmholtz–Duffing equation are reduced to the forced damping Duffing oscillator, in which its solution is known. As a practical application, the proposed techniques are applied to investigate the characteristics behavior of the signal oscillations arising in the RLC circuit with externally applied voltage.

scholarly journals An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator

2021 ◽  
Vol 2021 ◽  
pp. 1-8 ◽  
Author(s):  
Alvaro H. Salas ◽  
S. A. El-Tantawy ◽  
Noufe H. Aljahdaly
Keyword(s):  
Elliptic Function ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Duffing Oscillator ◽  
Higher Order ◽  
Strong Nonlinearity ◽  
Classical Dynamics ◽  
Elementary Functions ◽  
Weierstrass Elliptic Function ◽  
Quadratic Damping

The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.

scholarly journals A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Alvaro H. Salas S ◽  
Jairo E. Castillo H ◽  
Darin J. Mosquera P
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Initial Conditions ◽  
Negative Ions ◽  
Nonlinear Problems ◽  
New Approach ◽  
Weierstrass Elliptic Function ◽  
Electronegative Plasma ◽  
The Given

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

scholarly journals The Duffing Oscillator Equation and Its Applications in Physics

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Alvaro Humberto Salas Salas ◽  
Jairo Ernesto Castillo Hernández ◽  
Lorenzo Julio Martínez Hernández
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Duffing Equation ◽  
Soliton Theory ◽  
The Way

In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

scholarly journals On the Approximate Solutions of the Constant Forced (Un)Damping Helmholtz Equation for Arbitrary Initial Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Alvaro H. Salas ◽  
Castillo H. Jairo E ◽  
M. R. Alharthi
Keyword(s):  
Analytical Solution ◽  
Helmholtz Equation ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Quantum Plasma ◽  
Helmholtz Equations ◽  
Adomian Decomposition

This paper presents some novel solutions to the family of the Helmholtz equations (including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2))) which have been reported. In the beginning, equation (1) is solved analytically using two different techniques (direct and indirect solutions): in the first technique (direct solution), a new assumption is introduced to find the analytical solution of equation (1) in the form of the Weierstrass elliptic function with arbitrary initial conditions. In the second case (indirect solution), the solution of the undamping (standard) Duffing equation is devoted to determine the analytical solution to equation (1) in the form of Jacobian elliptic function with arbitrary initial conditions. Moreover, equation (2) is solved using a new ansatz and with the help of equation (1) solutions. Also, the evolution equations (equations (1) and (2)) are solved numerically via the Adomian decomposition method (ADM). Furthermore, a comparison between the approximate analytical solution and approximate numerical solutions using the fourth-order Runge–Kutta method (RK4) and ADM is reported. Furthermore, the maximum distance error for the obtained solutions is estimated. As a practical application, the Helmholtz-type equation will be derived from the fluid governing equations of quantum plasma particles with(out) taking the ionic kinematic viscosity into account for investigating the characteristics of (un)damping oscillations in a degenerate quantum plasma model.

scholarly journals A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Alvaro H. Salas ◽  
Simeon Casanova Trujillo
Keyword(s):  
Differential Equation ◽  
Analytic Solution ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Integrable Case ◽  
New Approach ◽  
One Dimensional ◽  
Weierstrass Elliptic Function ◽  
Quintic Equation ◽  
The One

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

scholarly journals Analytical and Numerical Investigation on the Duffing Oscilator Subjected to a Polyharmonic Force Excitation

2015 ◽  
Vol 45 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Svetlin Stoyanov
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Discrete Time ◽  
Amplitude Modulation ◽  
Duffing Oscillator ◽  
Vibration Signal ◽  
Vibration Diagnostics ◽  
Significant Difference ◽  
Excitation Force ◽  
Force Excitation

Abstract An analytical solution for a specific case of the forced Duffing oscillator is proposed. The excitation force contains two harmonics with significant difference frequencies. This case corresponds to a presence of a defect in the machinery and is in the art of the machinery vibration diagnostics. The results obtained show an amplitude modulation. Therefore, the presence of an amplitude modulation in the vibration signal may be used as an indicator for a malfunction. Analytical solution derived clarifies how the amplitude modulation occurs. Also, a numerical solution is realized and compared with the analytical one. For this, the Duffing equation is solved numerically and then, the spectrograms of vibrations are obtained through a Discrete-time Fourier Transform.

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

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