- Research
- Open Access
The rapidly convergent solutions of strongly nonlinear oscillators
- M. S. Alam^{1}Email author,
- Md. Abdur Razzak^{1},
- Md. Alal Hosen^{1} and
- Md. Riaz Parvez^{2}
- Received: 26 January 2016
- Accepted: 18 July 2016
- Published: 4 August 2016
Abstract
Based on the harmonic balance method (HBM), an approximate solution is determined from the integral expression (i.e., first order differential equation) of some strongly nonlinear oscillators. Usually such an approximate solution is obtained from second order differential equation. The advantage of the new approach is that the solution converges significantly faster than that obtained by the usual HBM as well as other analytical methods. By choosing some well known nonlinear oscillators, it has been verified that an n-th (n ≥ 2) approximate solution (concern of this article) is very close to (2n − 1)-th approximations obtained by usual HBM.
Keywords
- Nonlinear oscillation
- Harmonic balance method
- Duffing equation
Background
When f(x) is not a simple polynomial function (e.g., pendulum equation, \( l\,\textit{\"{x}} + g\,\,\sin \,x = 0 \)), Eq. (3) is valid for amplitude of oscillation, a < 1. Sometimes the nonlinear function, f depends on both x and \( \dot{x} \) (e.g., \( \textit{\"{x}} + (1 + \dot{x}^{2} )x = 0 \)). In this case, the integral expression of such equations has been taken in the form of Eq. (3).
The modified Lindsted–Poincare method (Cheung et al. 1991; He 2002a, b; Ozis and Yildirim 2007), He’s homotopy perturbation method (Belendez et al. 2007a, b; Belendez 2007), iterative method (Mickens 1987a, b, 2005, 2010; Lim and Wu 2002; Lim et al. 2006; Hu 2006a, b; Guo et al. 2011; Haque et al. 2013), He’s energy balance method (He 2002a, b) etc. are also used to investigate nonlinear oscillators. Though, all these analytical methods (Mickens 1984, 1986, 1987a, b, 1996, 2005, 2010; West 1960; Lim and Wu 2002, 2003; Lim et al. 2005, 2006; Wu et al. 2006; Belendez et al. 2006, 2007a, b; Alam et al. 2007; Hu 2006a, b; Lai et al. 2009; Cheung et al. 1991; He 2002a, b; Ozis and Yildirim 2007; Belendez 2007; Guo et al. 2011; Haque et al. 2013) have been developed for handling nonlinear oscillator Eq. (1), they provide almost similar results for a particular approximation. Recently, HBM has been modified by truncating some higher order terms of the algebraic equations of related variables to the solution [see Hosen et al. (2012) for details] and it measures more correct result than the usual HBM solutions (derived in Wu et al. 2006; Belendez et al. 2006; Alam et al. 2007; Hu 2006a, b; Lai et al. 2009) as well as other solutions derived by several analytical methods (Belendez 2007; Belendez et al. 2007b; Mickens 1987a, b, 2005, 2010; Lim and Wu 2002; Lim et al. 2006; Hu 2006a, b; Guo et al. 2011; Haque et al. 2013; He 2002a, b). However for any approximation, the result (even the solution obtained in Hosen et al. (2012)) is not better than the next higher approximation. Moreover, the modification on HBM used in Hosen et al. (2012) is valid for some nonlinear oscillators especially when f(x) contains a term, x ^{3}. In this article, a new approach (based on the HBM) has been introduced in which the solution rapidly converges toward its exact solution. The trial solution is similar to that of Hosen et al. (2012) and the determination of the related unknowns is also similar. Yet the solution converges faster than the usual solution. Actually an nth (n ≥ 2) approximate solution of Eq. (2) is almost similar to the (2n − 1)-th approximation obtained from Eq. (1). To verify this statement, the second and third approximations have been obtained from the integral expressions of some important nonlinear oscillators. The new solutions are respectively close to the third and fifth approximations determined by usual HBM which are agree with the statement.
Methods
All other expressions x ^{4} − a ^{4}, x ^{6} − a ^{6}, … of Eq. (3) have a factor x ^{2} − a ^{2}; so that a common factor \( {a^{2}} {\sin^{2}} \varphi\) must be cancelled when all these values are substituted in Eq. (3). It is noted that the canceling of the common (i.e., \( {a^{2}} {\sin^{2}} \varphi\)) factor makes the solution better than usual solution. Otherwise the solution does not converge fast. It also makes the solution different from that obtained by the energy balance method.
Examples
Quintic Duffing oscillator
It has already been mentioned that an analytical solution can be obtained either from Eq. (7) or from Eq. (8). The aim of this article is to find approximate solution from Eq. (8) rather than Eq. (7). A third approximate solution (in which u _{1} and u _{2} are non-zero) has been mainly considered. Sometimes a second approximate solution has been considered to compare it with existing solution obtained by several authors.
Comparison the approximate frequencies of Eq. (7) between the present method and the usual HBM method with the exact frequency \( \dot{\varphi }_{Ex} \), obtained by direct numerical integration
a | \( \dot{\varphi }_{Ex} \) | \( \dot{\varphi }_{3(q,Usual)} \) Er (%) | \( \dot{\varphi }_{3(q,Present)} \) Er (%) |
---|---|---|---|
0.5 | 1.0192663 | 1.0192661 | 1.0192663 |
0.00002 | 0.00000 | ||
0.7 | 1.0714295 | 1.0714202 | 1.0714295 |
0.00087 | 0.00000 | ||
1 | 1.26471 | 1.26446 | 1.26469 |
0.020 | 0.002 | ||
2 | 3.16666 | 3.16223 | 3.16639 |
0.140 | 0.008 | ||
3 | 6.80379 | 6.79391 | 6.80382 |
0.145 | 0.000 | ||
4 | 11.9959 | 11.9785 | 11.9963 |
0.145 | 0.003 | ||
5 | 18.7007 | 18.6736 | 18.7014 |
0.145 | 0.003 | ||
10 | 74.6909 | 74.5829 | 74.6941 |
0.145 | 0.004 | ||
50 | 1867.09 | 1864.39 | 1867.17 |
0.145 | 0.004 | ||
100 | 7468.34 | 7457.55 | 7468.66 |
0.145 | 0.004 |
Comparing Eqs. (14) and (15), it is clear that the first four terms of \( \dot{\varphi }_{3(q)}^{2} \) obtained in Eq. (14) are identical to those of its exact result, \( \dot{\varphi }_{Ex(q)}^{2} \). But the result of \( \dot{\varphi }_{3(q)}^{2} \) is different from that of \( \dot{\varphi }_{3(q,Usual)}^{2} \) obtained by the usual HBM [see Eq. (39) of Appendix 1: though the solution is obtained from Eq. (7) containing two higher harmonic terms u _{1} and u _{2}]. We see that first three terms of \( \dot{\varphi }_{3(q,Usual)}^{2} \) are identical to its exact result. It is noted that the first four terms of \( \dot{\varphi }_{3(q,Usual)}^{2} \) would be same those of \( \dot{\varphi }_{Ex(q)}^{2} \), when the solution is derived from Eq. (7) containing four higher harmonic terms u _{1}, u _{2}, u _{3} and u _{4}. Certainly, it is a laborious task to determine five unknown u _{1}, u _{2}, u _{3}, u _{4} and \( \dot{\varphi }^{2} \) for any analytical method.
Cubic Duffing oscillator
We see that first six terms of Eq. (21) are identical to the exact result in Eq. (22), and error occurs slightly in 7th term. It is noted that only the four terms of Eq. (45) [see “Appendix 2” and also (7–10)] are identical to the exact frequency when a third approximate solution is obtained from original equation Eq. (16). On the contrary, six terms of the fifth approximate solution (obtained by usual HBM) would be identical to its exact result \( \dot{\varphi }_{Ex(c)}^{2} \). It has already been mentioned that the derivation of a fifth approximate solution is very laborious.
A strongly nonlinear oscillator containing \( \dot{x}^{2} \)
Comparing Eqs. (29) and (32) to Eq. (30), it is clear that second and third approximations respectively measure four and six terms in correct figures. On the contrary, the usual HBM is able to respectively measure three and four terms in correct figures (see “Appendix 3”). Thus the statement is true for nonlinear oscillator, Eq. (23) [or, Eq. (24)].
It is noted that the series given in Eq. (31) is converge only for the small amplitudes in the region a ≤ 1.
Results and discussions
A new analytical approach based on the HBM has been presented to obtain approximate solutions of some well known nonlinear oscillators. Usually, a harmonic balance solution is obtained from the second order equations. Earlier, He (2002a, b) obtained some approximate solutions (mainly first approximation) for various nonlinear oscillators from corresponding first order differential equations (i.e., energy balance equations). But the new approach (concern of this article) is entirely different from He (2002a, b) technique. In this article, the first order equation is rewritten in such a way that every term is completely divisible by \( {a^{2}} {\sin^{2}} \varphi\) for the proposed solution Eq. (4) (see “Methods” section). For three well known nonlinear problems, it has been verified that the solutions are better than corresponding solutions obtained by usual HBM. Recently, Hosen et al. (2012) have developed a technique based on the same method (i.e., HBM), but their solutions are significantly improved for the quadratic and cubic Duffing oscillators (see Hosen et al. (2012) details). On the contrary, the solution obtained by the new approach is better than usual harmonic solution even for the quintic Duffing oscillator.
a | \( \dot{\varphi }_{Ex} \) | \( \dot{\varphi }_{3(c,Usual(trunc))} \) (Hosen et al. 2012) Er (%) | \( \dot{\varphi }_{3(c,Usual)} \) Er (%) | \( \dot{\varphi }_{3(c,Present)} \) Er (%) |
---|---|---|---|---|
0.5 | 1.0891582 | 1.0891582 | 1.0891582 | 1.0891582 |
0.00000 | 0.00000 | 0.00000 | ||
0.7 | 1.1676370 | 1.1676370 | 1.1676374 | 1.1676370 |
0.00000 | 0.00004 | 0.00000 | ||
1 | 1.31778 | 1.31778 | 1.31778 | 1.31778 |
0.000 | 0.000 | 0.000 | ||
2 | 1.97602 | 1.97601 | 1.97607 | 1.97602 |
0.000 | 0.003 | 0.000 | ||
3 | 2.73849 | 2.73847 | 2.73862 | 2.73849 |
0.000 | 0.005 | 0.000 | ||
4 | 3.53924 | 3.53921 | 3.53946 | 3.53926 |
0.000 | 0.006 | 0.000 | ||
5 | 4.35746 | 4.35741 | 4.35777 | 4.35748 |
0.001 | 0.007 | 0.000 | ||
10 | 8.53359 | 8.53347 | 8.5343 | 8.53363 |
0.002 | 0.008 | 0.000 | ||
50 | 42.3730 | 42.3724 | 42.3767 | 42.3732 |
0.002 | 0.009 | 0.000 | ||
100 | 84.7275 | 84.7262 | 84.7349 | 84.7279 |
0.002 | 0.009 | 0.000 |
Comparison the approximate frequencies of Eq. (23) between the present method and the usual HBM method with the exact frequency \( \dot{\varphi }_{Ex} \), obtained by direct numerical integration
a | \( \dot{\varphi }_{Ex} \) | \( \dot{\varphi }_{3(q,Usual)} \) Er (%) | \( \dot{\varphi }_{3(q,\Pr esent)} \) Er (%) |
---|---|---|---|
1.8 | 1.52154 | 1.52669 | 1.52180 |
0.339 | 0.017 | ||
2.0 | 1.67047 | 1.68325 | 1.67091 |
0.765 | 0.026 | ||
2.2 | 1.84092 | 1.86926 | 1.84103 |
1.540 | 0.006 | ||
2.4 | 2.03064 | 2.0876 | 2.028 |
2.805 | 0.130 | ||
2.6 | 2.236 | 2.34108 | 2.22324 |
4.700 | 0.571 | ||
2.8 | 2.45251 | 2.63242 | 2.41143 |
7.336 | 1.675 |
From these six figures, we see that the present method provides good agreement with the corresponding numerical solution.
From Eq. (33), it is observed that the approximate frequency, \( \dot{\varphi } \) is undefined at a = 2. It is a big shortcoming of usual HBM.
It has already been mentioned that the series Eq. (31) is mainly converged in the region a ≤ 1, but the series also gives significant better result for obtaining approximate frequency even the amplitude increases up to a = 2.8 (see Table 3). On the contrary, the solution is only valid for in the region a ≤ 1 while the amplitude increases, the solutions are deviated from the numerical solution (see Fig. 3b). Comparing the approximate frequency obtained by usual HBM with the exact approximate frequency determined numerically, it is shown from Table 3 that the relative error of the approximate value is less than 5, 8 % for a < 2.6 and a < 2.8, respectively while the relative error of the approximate frequency obtained in present method is less than 0.6, 2 % for a < 2.6 and a < 2.8, respectively. Therefore, the present method is faster than usual HBM.
Conclusion
Based on HBM, a new technique has been presented for solving a class of nonlinear oscillators. In the case of small values of amplitude, it has been verified that the fourth-order approximate frequency obtained by usual HBM is almost same as the third-order approximate frequency obtained by new method. For the case of large values of amplitude, the approximate frequencies obtained by new method not only gives better results than usual HBM but also gives nicely close to their exact results. Therefore, the results obtained in this paper are much better than those obtained by the usual HBM. The method also proved that it is a powerful mathematical tool for solving nonlinear oscillators.
Declarations
Authors’ contributions
MSA, MAR, MAH, MRP prepared the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the reviewers for their helpful comments/suggestions in improving the manuscript.
Competing interests
The author declares that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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