An analytical approach to investigate the response and stability of Van der Pol-Mathieu-Duffing oscillators under different excitation functions

Kimiaeifar, Amin

doi:10.1002/mma.1269

Cited by 17 publications

(3 citation statements)

References 16 publications

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“…When the base function is selected, the auxiliary functions H (t), initial approximations U 0 (t) and the auxiliary linear operators L must be chosen in such a way that the corresponding high-order deformation equations have solutions with the functional form similar to the base functions. This method referred to as the rule of solution expression (Fooladi et al 2009;Kimiaeifar 2010). The linear operator L is chosen as:…”

Section: Energy Balancementioning

confidence: 99%

Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

Ibsen

Barari

Kimiaeifar

2010

Sadhana

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Nonlinear functions are crucial points and terms in engineering problems and the solutions of many important physical problems are centered on finding accurate solutions to these functions. In this paper, a new method called max-min method has been presented for deriving accurate/approximate analytical solution to strong nonlinear oscillators. Furthermore, it is shown that a large class of linear or nonlinear differential equations can be solved without the tangible restriction of sensitivity to the degree of the nonlinear term, adding that the method is quite convenient due to reduction in size of calculations. Results obtained by max-min are compared with Homotopy Analysis Method (HAM), energy balance and numerical solution and it is shown that, simply one term is enough to obtain a highly accurate result in contrast to HAM with just one term in series solution. Finally, the phase plane to show the stability of systems is plotted and discussed.

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Section: Energy Balancementioning

confidence: 99%

Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

Ibsen

Barari

Kimiaeifar

2010

Sadhana

Self Cite

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“…The residuals can be obtained as: The original frequency-amplitude formulation reads [29][30][31][32]]…”

Section: Mathematical Techniquementioning

confidence: 99%

A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

Barari

Kimiaeifar

Nejad

et al. 2012

Shock and Vibration

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Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

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“…Differential equations play a prominent role in our real world, and occur in many scientific disciplines such as physics, chemistry, biology, and economics. Different analytic approaches have also been employed to solve differential equations, so far. In this article, a new algorithm that efficiently solves ODEs, including the Lane–Emden equation of index m , logistic differential equation and Riccati equation is presented.…”

Section: Introductionmentioning

confidence: 99%

A new algorithm for solving differential equations

Hosseini

Biazar

Ansari

et al. 2012

Math Methods in App Sciences

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Different analytic methods have been proposed to solve differential equations, so far. In this paper, a novel analytic method that efficiently solves ODEs is presented. This method requires only the calculation of the first Adomian polynomial, namely A0, and does not need to solve the functional equation in each iteration, as well as provides less computational work than other existing methods. Some important ordinary differential equations including the Lane–Emden equation of index m, the logistic nonlinear differential equation, and the Riccati equation are considered to illustrate the efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

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An analytical approach to investigate the response and stability of Van der Pol-Mathieu-Duffing oscillators under different excitation functions

Cited by 17 publications

References 16 publications

Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

A new algorithm for solving differential equations

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