Damped Duffing Equation Research Articles

Novel Approximations to the Damped Parametric Driven Pendulum Oscillators

The damped parametric driven nonlinear pendulum equation/oscillator (NPE), also known as the damped disturbed NPE, is examined, along with some associated oscillators for arbitrary angles with the vertical pivot. For analyzing and solving the current pendulum equation, we reduce this equation to the damped Duffing equation (DDE) with variable coefficients. After that, the DDE with variable coefficients is divided into two cases. In the first case, two analytical approximations to the damped undisturbed NPE are obtained. The first approximation is determined using the ansatz method while the second one is derived using He’s frequency formulation. In the second case, i.e., the damped disturbed NPE, three analytical approximations in terms of the trigonometric and Jacobi elliptic functions are derived and discussed using the ansatz method. The semianalytical solutions of the two mentioned cases are graphically compared with the 4th-order Runge–Kutta (RK4) approximations. In addition, the maximum error for all the derived approximations is estimated as compared with the RK4 approximation. The proposed approaches as well as the obtained solutions may greatly help in understanding the mysteries of various nonlinear phenomena that arise in different scientific fields such as fluid mechanics, plasma physics, engineering, and electronic chips.

On the oscillations in a nonextensive complex plasma by improved differential transformation method: An application to a damped Duffing equation

In this study, the nonlinear damping oscillations in a complex non-Maxwellian plasma are investigated. For this purpose, the set of fluid equations of the present plasma model is reduced to the Burger-modified Korteweg De Vries equation (BmKdV) equation using a reductive perturbation technique. Using the traveling wave transformation, the BmKdV equation can be reduced to a damped Duffing equation. The numerical solutions to the damped Duffing equation are obtained using multistage differential transformation method (MsDTM). Also, we compared the obtained results to the semi-analytical approximations using the Padé differential transformation (PDTM) method and numerical solution, by the 4th-order Rung Kutta (RK4) method and analytical solution by He’s frequency method. The impact of relevant plasma parameters, namely, negative dust concentrations and ion kinematic viscosity on the profile of dust ion-acoustic oscillations are examined. The suggested mathematical approaches can help many authors for explaining the mystery of their laboratory results. Moreover, the suggested numerical method can be applied for solving higher order nonlinearity oscillations for a long domain.

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

Solution to a Damped Duffing Equation Using He’s Frequency Approach

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt–Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.

ON TWO-SCALE DIMENSION AND ITS APPLICATION FOR DERIVING A NEW ANALYTICAL SOLUTION FOR THE FRACTAL DUFFING’S EQUATION

In this paper, the analytical solution that describes the evolution in time of the fractal damped Duffing equation subjected to external forces of elliptic type is derived using He’s two-scale fractal transform and the elliptic balance method (EBM). This solution predicts the evolution in time of the Duffing equation and unveils qualitative and quantitative system behavior when the values of the fractal parameter varies, and how these affect the frequency, the wavelength, and the oscillation amplitude from the start of the motion. Comparison of the amplitude–time response curves over the selected time-interval with those obtained from numerical simulations confirms the accuracy of the derived analytical solution.

On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method

In this paper, a strongly nonlinear oscillator equation with higher-order nonlinear (cubic) restoring force, namely damped Duffing equation with higher-order nonlinearities, has been solved and analyzed analytically and numerically using some effectiveness and high-accuracy methods. Firstly, the physical oscillator system consisting of a spring–mass moves under cubic resorting force with constant spring is considered for getting damped Duffing equation. After that a semi-analytical solution is obtained in terms of the Jacobian elliptic function cn. Also, the numerical approximate solution using Runge–Kutta-4th (RK4) procedure is carried out and compared to the semi-analytical solution. It is found that the agreement between both solutions is very good especially if the coefficient of the cubic nonlinear term becomes larger than the coefficient of the linear one. However, we made some improvements to the semi-analytical solution using the numerical moving boundary method to get higher accuracy solution. Furthermore, the error analysis is estimated for both the semi-analytical solution, improved semi-analytical solution, and RK4 solution. The method that is used for getting the semi-analytical solution is elementary, i.e., allows to obtain only integrable case without using any Lie group theory. In another meaning, our approach gives exact/analytical solution for the integrable case such as undamped Duffing equation. Moreover, the t-intercepts and amplitude points are estimated precisely. Furthermore, the plasma oscillations are discussed in the framework of damped Duffing equation. Firstly, we reduced the fluid basic equations of pair-ion plasmas with Maxwellian electrons using reductive perturbation technique to a modified korteweg de-Vries Burgers (mKdVB) equation, and by using traveling wave transformation the mKdVB equation is converted to the damped Duffing equation. After that we compared between all mentioned solutions using plasma parameters.

Positive periodic solution for p-Laplacian neutral damped Duffing equation with strong singularities of attractive and repulsive type

This paper is devoted to investigating the following p-Laplacian neutral damped Duffing equation with singularity: \t\t\t(ϕp(u(t)−cu(t−τ))′)′+Pu′(t)+g(u(t))=e(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(\\phi _{p}\\bigl(u(t)-cu(t-\\tau )\\bigr)' \\bigr)'+Pu'(t)+g\\bigl(u(t)\\bigr)=e(t), $$\\end{document} where g has a singularity at u=0. Applying the Manásevich–Mawhin theorem on a continuous case of topological degree, we obtain the existence of a positive periodic solution for this equation.

Periodic solutions of discontinuous damped Duffing equations

In this paper, we are concerned with the problem of periodic solutions for a class of second order periodically forced nonlinear damped Duffing equations with a discontinuity line x′′+cx′+g(x)=e(t), where g(x) is discontinuous and e∈C0(R,R) is T-periodic. We first show that the periodic solutions can only be harmonic and subharmonic solutions. Secondly, note that g(x) is an abstract function, so the explicit solutions cannot be computed and then known theory cannot be applied directly. Therefore, we give geometric properties of solutions for the discontinuous forced damped Duffing equation. Then by the Poincaré–Bohl theorem we investigate the existence and uniqueness of crossing harmonic and subharmonic solutions of the equation.

Analysis of motion of inverted pendulum with vibrating suspension axis at low-frequency excitation as an illustration of a new approach for solving equations without explicit small parameter

In the classical papers (see, e.g. P.L. Kapitsa, Pendulum with vibrating axis of suspension. Usp. Fiz. Nauk 44 1 (1954) 7–20 (in Russian)) motion of pendulum with vibrating suspension axis was considered in the case when frequency of external loading is much higher than the natural frequency of the pendulum in the absence of this loading. The present paper is concerned with the analysis of inverted pendulum׳s motion at unconventional values of parameters. Case when frequency of external loading and the natural frequency of the pendulum in the absence of this loading are of the same order is studied. Vibration intensity is assumed to be relatively low. A new modification of the method of direct separation of motions (MDSM) is proposed to study the corresponding equation which in the considered case does not contain a small parameter explicitly. The aim is to obtain solutions of this equation in the stability domain. It is revealed that in the considered range of parameters not only the effective stiffness of the system changes due to the external loading, but also its effective mass. Applicability of the proposed approach for solving non-linear equations without small parameter is demonstrated; as an illustration, a damped Duffing equation is considered.

A frequency domain analysis of the effects of nonlinear damping on the Duffing equation

The nonlinearly damped Duffing equation has been studied using a variety of approximation and numerical methods by a number of authors. In this paper, the Duffing equation is examined analytically for the first time using the output frequency response function (OFRF) approach. The theoretical analysis focuses on the effects of the nonlinear stiffness and nonlinear damping on the output spectra as well as the output energy spectra over different frequency ranges. The studies reveal that nonlinear damping has two significant implications for the design of vibration isolation systems. First, nonlinear viscous damping is shown to be more effective in suppressing the resonant peak of a Duffing system than linear damping because the high frequency transmissibility is hardly affected. Second, nonlinear damping can be used in conjunction with nonlinear stiffness to achieve better vibration isolation. Simulation studies are also provided to validate and demonstrate these theoretical findings.

Harmonic and subharmonic solutions for superlinear damped Duffing equation

This paper is devoted to the existence of harmonic and subharmonic solutions for a damped Duffing equation. Our proofs are based on a twisting theorem due to W.Y. Ding.

An effective approach for approximate analytical solutions of the damped Duffing equation

This paper deals with the purely analytic solutions to the damped Duffing equation. It is shown that the employed homotopy method yields uniformly convergent solutions. Optimum values of the convergence control parameter of the computed homotopy series are calculated from the square residual error. The obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed ones, which are further confirmed analytically by the absolute error formula.

A qualitative study of the damped duffing equation and applications

In this paper, we analyze the damped Duffing equation by means of qualitative theory of planar systems. Under certain parametric choices, the global structure in the Poincaré phase plane of an equivalent two-dimensional autonomous system is plotted. Exact solutions are obtained by using the Lie symmetry and the coordinate transformation method, respectively. Applications of the second approach to some nonlinear evolution equations such as the two-dimensional dissipative Klein-Gordon equation are illustrated.

Application of Jacobian Elliptic Functions to the Analysis of the Steady-State Solution of the Damped Duffing Equation with Driving Force of Elliptic Type

In this work, Jacobian elliptic functions are applied to obtain the approximate steady-state solution of the damped Duffing equation with driving force of elliptic type. We assume that the damping coefficient and the nonlinear term need not be small. Numerical comparison between the approximate solution and the numerical integration indicates that our proposed approximate steady-state solution captures well the amplitude-time vibrational response. We have also found that for a certain range of values of the driving frequency ω f and for a stiff spring, there are one, three, or five possible motions for the elliptic excitation amplitude-frequency response curve.

Simple shearing of an incompressible, viscoelastic quadratic material

Viscoelastic constitutive equations of the differential type and of the rate type are applied in the study of the mechanical response of rubberlike materials in a simple shearing deformation superimposed on a specified static longitudinal stretch. Creep, recovery and stress relaxation processes are formulated for a class of incompressible, isotropic viscoelastic quadratic materials. Equations characterizing the creep and recovery processes for materials of differential type are derived, and their exact solutions are obtained. For materials of the rate type, the stress relaxation process is independent of the elastic material response functions. A coupled system of equations for the creep and recovery processees in a material of rate type are presented ; these must be solved numerically. The finite amplitude, damped, free vibration of a rigid body supported by incompressible, viscoelastic quadratic simple shear mountings of the differential type is investigated. The motion of the body is governed by a damped Duffing equation whose solution is discussed by the averaging method.