Abstract
A novel approach about iterative homotopy harmonic balancing is presented to determine the periodic solution for a strongly nonlinear oscillator. This approach does not depend upon the small/large parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. Importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. With this procedure, the higher-order approximate frequency and corresponding periodic solution can be obtained easily. Comparison of the obtained results with those of the exact solutions shows the high accuracy, simplicity, and efficiency of the approach. The approach can be extended to other nonlinear oscillators in engineering and physics.
1. Introduction
Considerable attention has been paid to the study of nonlinear problems not only in all areas of engineering but also in physics and other disciplines, since world is nonlinear essentially and most phenomena are modeled by nonlinear differential equations. However, in general, it is very difficult to get exact solutions for those problems. Thus, to find approximate analytic solutions to these nonlinear problems has been the desire of many researchers for a long time.
During the past few decades, there are many analytical methods used to find approximate solutions of various types of nonlinear equations, such as the variational iteration method [1–4], He’s parameter-expansion method [5–7], harmonic balance method [8–12], homotopy perturbation method [13–17], energy balance method [18, 19], and delta-perturbation [20].
Among these analytical methods, the iterative homotopy harmonic balancing [21] (denoted as IHHB for abbreviation) is a novel and effective technique for solving some strongly nonlinear oscillators. This approach incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. By constructing a parameter which is considered as a “small parameter” in the harmonic balance process, the IHHB does not depend upon the large/small parameter assumption and can get high-order analytical approximations easily. Hence, the application of IHHB can be found in various nonlinear problems [21–23].
In this paper, we consider the following strongly nonlinear oscillator with nonpolynomial term [6]where the over-dot denotes the derivative with respect to the time and is the amplitude of the oscillation.
When , , (1) becomes a truly nonlinear oscillator introduced by Mickens [12] and has been studied by many other investigators [11, 16, 17]. When , , , (1) reduces to a well-known oscillator namely Duffing equation. By simple analysis [6], we know that (1) has periodic solution when .
To get higher-order analytical and accurate approximations for the frequency and solution of this oscillator, IHHB solving is set up. More importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. Excellent agreement of the approximate solution and frequency with the exact ones has been demonstrated and discussed. As can be seen, the results presented in this paper reveal that IHHB is effective and convenient for some nonlinear oscillators with nonpolynomial terms.
2. Basic Ideas of Iterative Homotopy Harmonic Balancing Approach
Consider the following second-order systems:where the over-dot denotes the derivative with respect to the time and is the amplitude of the oscillations. For convenience, we assume .
By introducing a new independent variable , then (2) becomeswhere prime denotes the derivative with respect to and is an angular frequency to be determined.
Let the periodic solution of (3) exist and assume that it can be expressed by such a set of base functions
According to (4), the initial approximate periodic solution satisfying initial conditions should bewhere is an unknown constant to be determined later.
Here, we will use the residual to improve the accuracy. Substituting (5) into (3), the initial residual can be obtained:
If , then happens to be the exact solution. However, in general, such case will not appear for nonlinear problems.
According to harmonic balance method, no secular term in (6) requires eliminating contributions proportional to , through which the unknown constant can be determined. Then, the zero-order approximation solution is obtained of the formwith initial residual
In the following, we introduce a bookkeeping parameter with values in the interval , denote the , as , , and then expand and in a series of in which
Obviously, when , is the zeroth-order approximation and when , is the required approximate solution of (3). Generally, (9) provides the higher-order approximation to the exact solution. For example, the first-order analytical approximation turns out to be
Substituting (11) into (3) and equating the coefficients of the yield
By considering (4) and harmonic balance method, letSubstituting (13) into (12), we consider the following equation:in this way, the initial residual is introduced into (14) to improve the accuracy.
According to harmonic balance method, the right-hand side of (14) should not contain the terms , . Letting their coefficients be zeros yields two linear equations with two unknowns and , through which the two unknowns can be solved easily. We now get the first-order approximationwith the residualwhere and are given by (7) and (13), respectively.
With the process going on, the higher-order approximation can be obtained similarly. For example, for the th-order approximation, we may assume
To determine the unknown parameters and , we substitute (9) into (3) and collect the coefficients of and finally we get
Substituting and into (3), one yields the following residual:
Let us consider the equationand eliminate the contributions proportional to and in the right-hand side of (20); we can solve the unknown and from those equations.
Then, the -order approximations to the frequency and periodic solution can be obtained in the formwhere and are given by (10).
3. Solution Procedure
By introducing a new time scale , then (1) becomeswhere prime denotes the derivative with respect and is an angular frequency to be determined.
By following (7), the initial approximation with the initial conditions can be written intoHere, it is possible to do the following Fourier series expansion:in which
Substituting (23) and (24) into (22) yieldsin which
Eliminating contributions proportional to in (26), that is, solving equation , giveswhich is the same as the one by Xu [24]. Till now we get the zeroth-order analytical approximation (23) and the initial residual
To obtain the first-order analytical solution, we substitute (11) and (13) into (12), expand the function into a Fourier seriesin whichand finally getin which
According to (16), we have
Setting the coefficients of , to zero in the right-hand side of (34) gives
Solving (35) yieldsThus, we obtain the first-order approximationwith the residualwhere , , and are determined by (28), (36), and (37), respectively.
With the procedure going on, similarly, we can get the high-order approximation. For example, the second-order approximation can be obtained as follows:in whichThe variables , , , , and are presented in Appendix.
4. Results and Discussion
In order to illustrate and verify the efficiency and correctness of the presented approach for this strongly nonlinear oscillator, we consider some special cases.
Case 1. If , , , (1) reduces toThen, from (41), the second-order approximation of the frequency can be obtained:The exact frequency [16] is . Therefore, it can be easily proved that the maximal relative error is less than .
Hence, from (40), the second-order approximate solution can be expressed as follows:which agrees very well with the exact solution [6] as shown in Figure 1.
To further illustrate and verify the accuracy of the presented approach in this case, we present the comparison between the approximate and exact frequencies for the second-order approximation by using different methods in Table 1. It is clear that, for the second-order approximation, the result obtained in this paper is better than those obtained previously by other authors.
Case 2. If , , , (1) reduces to a Duffing equationIn this case, the first- and second-order approximate periods obtained by (38) and (41) for different values of are shown in Table 2. The relative errors () are defined as . It can be observed that the approximate periods have a good adjustment with the exact ones.
To verity results, Figure 2 shows the comparisons of the second-order analytical solutions obtained by (40) with the exact ones for , 1, 10, and when . It can be seen from this figure that our analytical results are very close to the exact ones for the wide range of initial amplitude in this case.
Case 3. If , , , we can obtain the following nonlinear oscillator:From Table 3 and Figure 3, we can see that the accuracy of the results obtained in this paper is in excellent agreement with exact ones for the wide range of initial amplitude .
5. Conclusions
An iterative homotopy harmonic balance approach has been presented and applied to deduce the accurate approximations to the angular frequency and periodic solution of a strongly nonlinear oscillator. The high-order analytical approximations of the frequency and solution of this oscillator are obtained. Excellent agreement between approximate results by this approach and the exact ones has been demonstrated and discussed; the discrepancy between the second-order approximate results and exact ones is very very low. And, we can see that the approach considered here is very simple in its principle and has great potential to be applied to other nonlinear oscillators.
Appendix
Consider
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The second author is supported by the National Science Foundation of China (11471118).