Duffing-harmonic Oscillator Research Articles

Solution of Duffing-Harmonic Oscillator by the Power Series Method

Solution of Duffing-Harmonic Oscillator by the Power Series Method

A New Approximate Solution for a Generalized Nonlinear Oscillator

The equivalent linearization method based on weighted averaging is employed to obtain a approximate solution for a generalized nonlinear oscillator given in the form of $$ \ddot{u} + \alpha u + \beta u^{m} + \frac{{\gamma u^{n} }}{{\mu + \delta u^{p} }} = 0. $$ The amplitude–frequency relationship of oscillation is given in a closed-form. Accuracy of the approximate solution is verified by comparing the obtained solution with the numerical solution using the 4th-order Runge–Kutta method. Specifically, the obtained results for some special cases such as cubic-, quintic-, seventh order-Duffing oscillators, Duffing-harmonic oscillator, nonlinear oscillator with fractional nonlinearity are compared with the results achieved by the other approximate methods.

Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator

In this work, the differential equation of motion of the undamped mathematical pendulum and Duffing-harmonic oscillator are discussed by using the variational iteration method. Additionally, common problems of pendulum are classified and Lagrange multipliers are obtained for each type of problem. Examples are given for illustration.

Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator

In this work, the differential equation of motion of the undamped mathematical pendulum and Duffing-harmonic oscillator are discussed by using the variational iteration method. Additionally, common problems of pendulum are classified and Lagrange multipliers are obtained for each type of problem. Examples are given for illustration.

Intelligent computing for Duffing-Harmonic oscillator equation via the bio-evolutionary optimization algorithm

This paper presents a bio-evolutionary metaheuristic approach to study the harmonically oscillating behavior of the Duffing equation. The proposed methodology is an amalgamation of the artificial neural network with the firefly algorithm. A novelty in the activation of neurons of artificial neural network is described using the cosine function with the angular frequency. Chronologically, artificial neural network approximates discretizes the nonlinear functions of the governing problem, which then undergoes an optimization process by the firefly algorithm that then later generates the effective values of the unknown parameters. Generally, the algorithm and implementation of the scheme are assimilated by considering an application of Duffing-harmonic oscillator. Some error measurements, in order to discuss the convergence and accuracy of the scheme, are also visualized through tables and graphs. An effective optimized relationship between the angular frequency and amplitude is derived and its results are depicted in a tabular form. The comparison of the proposed methodology is also deliberated by homotopy perturbation method. Moreover, the geometrical illustration of the trajectories of the dynamic system is also added in the phase plane for different values of amplitude and angular frequency.

Periodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method

Duffing harmonic oscillator is a common model for nonlinear phenomena in science and engineering. This paper presents He´s Energy Balance Method (EBM) for solving nonlinear differential equations. Two strong nonlinear cases have been studied analytically. Analytical results of the EBM are compared with the solutions obtained by using He´s Frequency Amplitude Formulation (FAF) and numerical solutions using Runge-Kutta method. The results show the presented method is potentially to solve high nonlinear oscillator equations.

Frequency analysis of nonlinear oscillations via the global error minimization

AbstractThe capacity and effectiveness of a modified variational approach, namely global error minimization (GEM) is illustrated in this study. For this purpose, the free oscillations of a rod rocking on a cylindrical surface and the Duffing-harmonic oscillator are treated. In order to validate and exhibit the merit of the method, the obtained result is compared with both of the exact frequency and the outcome of other well-known analytical methods. The corollary reveals that the first order approximation leads to an acceptable relative error, specially for large initial conditions. The procedure can be promisingly exerted to the conservative nonlinear problems.

A new analytical technique based on harmonic balance method to determine approximate periods for Duffing-harmonic oscillator

The Duffing-harmonic oscillator is a common model for nonlinear phenomena in science and engineering. In this paper, a new analytical technique has been presented to determine approximate periods of a strongly nonlinear Duffing-harmonic oscillator. Generally, a set of difficult nonlinear algebraic equations appear when harmonic balance method is imposed. The power series solutions of these equations are invalid. The proposed idea avoids this limitation and the necessity of numerically solving such nonlinear algebraic equations with very complex nonlinearities. In this technique, different parameters for the same nonlinear problems are found, for which the power series solution yields desired results. Besides a suitable truncation formula is found in which the solution measures better results than existing solutions. It is remarkable that this procedure is simple and takes less computational effort for determining second and higher order periods of oscillation for such nonlinear problems and shows a good agreement compared with the exact ones.

Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

A numerical method for finding the solution of Duffing-harmonic oscillator is proposed. The approach is based on hybrid functions approximation. The properties of hybrid functions that consist of block-pulse and Chebyshev cardinal functions are discussed. The associated operational matrices of integration and product are then utilized to reduce the solution of a strongly nonlinear oscillator to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. The results are compared with the exact solution and results from several recently published methods, and the comparisons showed proper accuracy of this method.

Analytical solutions of a generalized Duffing-harmonic oscillator by a nonlinear time transformation method

The analytical solutions of nonlinear oscillators obtained from most perturbation or approximate methods usually have poor accuracy near homoclinic/heteroclinic (HH) orbits. In this Letter, we propose a nonlinear time transformation method to overcome such difficulty. In particular, we apply such method with Padé approximation to find analytical solutions of a generalized Duffing-harmonic oscillator having a rational form for the potential energy. For some parametric ranges, HH orbits exist in such an oscillator. For analytical approximation of periodic solution obtained from the present method, it is shown that the relative error of period with respect to the exact period tends to zero when the amplitude of periodic solution tends to either zero or infinity. The relative error is still very small even near to HH orbits. Furthermore, analytical approximate of HH orbits can also be obtained. From the illustrative examples, the phase portraits are in excellent agreement with the exact HH orbits. The results from the present method are compared with the exact solutions and that from the cubication method.

Homotopy analysis approach to Duffing-harmonic oscillator

This paper uses the homotopy analysis for the Duffing-harmonic oscillator. The auxiliary parameter in the deformation equation is numerically determined. The response and the frequency of the Duffing-harmonic oscillator are calculated. The analytical results are validated in numerical simulations.

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic–geometric mean as well as Legendre's formula to approximately obtain this mean are used.

A new analytical approximation to the Duffing-harmonic oscillator

In this paper, a novel analytical approximation to the nonlinear Duffing-harmonic oscillator is presented. The variational iteration method (VIM) is used to obtain some accurate analytical results for frequency. The accuracy of the results is excellent in the whole range of oscillation amplitude variations.

Frequency-amplitude relationship of the Duffing-harmonic oscillator

The variational iteration method, the variational method and the parameter-expanding method are applied to obtain the frequency-amplitude relationship of the Duffing-harmonic oscillator. The obtained results reveal that all the three methods are very effective and convenient.

Application of He's frequency-amplitude formulation to the Duffing-harmonic oscillator

The work presents a derivation of frequency-amplitude of the Duffing-harmonic oscillator from a formulation suggested by Ji-Huan He. The obtained result is valid for all amplitudes, and its maximal error is less than 2.2

Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

A modified He's homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He's homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulae obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.

Determination of the frequency–amplitude relation for a Duffing-harmonic oscillator by the energy balance method

This paper applies He’s energy balance method to determine the frequency–amplitude relation of the Duffing-harmonic oscillator, which gives a good estimate for the angular frequency.

Application of He's Homotopy Perturbation Method to the Duffing-Harmonic Oscillator

Application of He's Homotopy Perturbation Method to the Duffing-Harmonic Oscillator

Solutions of the Duffing-harmonic oscillator by an iteration procedure

A modified iteration procedure is applied to the Duffing-harmonic oscillator. With the procedure, the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained.

Higher accuracy analytical approximations to the Duffing-harmonic oscillator

A new method has been presented for analytically solving the Duffing-harmonic oscillator. The method is obtained by combining Newton's method with the harmonic balance method. By using the method, one obtains linear algebraic equations instead of nonlinear algebraic equations. The complexity of the HB method is greatly simplified. Iteration of procedure yields rapid convergence with respect to exact solution. The results are valid for the complete range of oscillation amplitude, including the limiting cases of amplitude approaching zero and infinity.