Analytical study on a Duffing-harmonic oscillator

Tiwari, Shashi Bhushan; Rao, B. Nageswara; Swamy, N. Shivakumar; Sai, K.S.; Nataraja, H. R.

doi:10.1016/j.jsv.2004.11.001

Cited by 32 publications

(17 citation statements)

References 5 publications

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“…It is important to point out that the exact behaviour of the approximate frequency when A tends to zero is not obtained when other approximate methods as used including the harmonic balance method [17][18][19], the homotopy perturbation method [15,16], the energy balance method [20], the variational iteration method [21], a modified iteration procedure [22] or the Ritz procedure [23].…”

Section: Comparison With the Exact And Other Approximate Solutionmentioning

confidence: 99%

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

et al. 2009

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The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. 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 paper predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closedform 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.

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Section: Comparison With the Exact And Other Approximate Solutionmentioning

confidence: 99%

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

et al. 2009

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“…Using a single-term approximate solution ( ) = cos( ) to (36) and the Ritz procedure [40], Tiwari et al [41] obtained an approximate angular frequency as follows:…”

Section: Resultsmentioning

confidence: 99%

“…The computed results for the HFM frequency HFM with exact frequency ex [32], HBM frequency HBM [31], EBM frequency EBM [35], and Tiwari's frequency Tiw [41] are listed in Tables 1 and 2. Table 2 shows that the maximum percentage error between HFM and exact frequency ex is 0.118%.…”

Section: Resultsmentioning

confidence: 99%

“…The merit of this method is that the system of equations obtained for the solution does not need to consider collocation points; this means that the system of equations is obtained directly. A comparative study between HBM [31], EBM [35], Tiwari's method [41], and the proposed method was discussed in Section 4. The obtained results showed that the HFM is accurate, capable, and effective technique for the solution of the Duffing-harmonic oscillator.…”

Section: Resultsmentioning

confidence: 99%

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Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

Heydari

Loghmani

Hosseini

et al. 2014

Journal of Difference Equations

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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.

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“…Lim and Wu [2] proposed an analytical approach based on the combination of the linearization and the harmonic balance to yield the approximate expressions of the frequency as well as the periodic response. Tiwari et al [3] obtained the same frequency and calculated the response based on the combination of the harmonic balance and the Ritz procedure. Hu and Tang [4] derived the same frequency in [2][3] by using the first-order harmonic balance approach via the first Fourier coefficient.…”

Section: Introductionmentioning

confidence: 99%