An analytical technique for nonlinear oscillators based on the higher order odd polynomial functions

Abstract

An analytical technique based on the certain odd polynomial functions is introduced for solving nonlinear oscillatory problems, applicable to both autonomous and non-autonomous cases. A set of algebraic equations are found when the proposed technique is applied. The algebraic equations are linear and relatively straightforward for determining the unknown coefficients associated with the solution. An excellent agreement is found between approximated and numerical solutions, which prove that the technique is very efficient and easy to implement. The technique allows to obtain any desired order of approximate solutions. The results are shown in figures.

Introduction

Nonlinear oscillators have a great importance in the area of engineering applications. Many researchers continuously invest their time and effort to explore the behaviour of nonlinear damped forced oscillators. Analytical techniques, numerical techniques, or a combination of both are usually employed to determine the solutions. The classical perturbation methods^1–4 were widely used to find periodic solutions of weakly nonlinear oscillators (e.g. $\ddot{x}+x+\epsilon f\left(x\right)=0$ , where $\epsilon \ll 1$ is a nonlinear parameter). Later, several authors modified to classical perturbation methods^5–9 to solve strongly nonlinear problems where $\epsilon =\mathrm{{\rm O}}\left(1\right)$ . By considering a finite Taylor series as an approximate solution, a semi-analytical technique, namely, the differential transform method (DTM)^10–15 was used for obtaining periodic solution of strongly nonlinear oscillators. However, a lot of terms are essential to obtain a desired solutions but it is a laborious task. Recently, a new analytical technique, similar to DTM^16,17 has been developed to solve strongly nonlinear autonomous oscillators with large oscillations.

Burton and Hamdan^18,19 made a modification on Lindstedt-Poincare method to study conservative system of strongly nonlinear oscillators with weak harmonic excitation. After that, the modified multiple time scale method, ²⁰ the modified harmonic balance method ²¹ were used to obtain periodic solutions of strongly nonlinear damped forced oscillators. However, in the cases of large excitations, the above methods do not yield satisfactory results. A modified Lindstedt-Poincaré method was introduced by Cheung et al. ²² for certain the following strongly nonlinear oscillator

$\ddot{x}+{\omega }_0^{2}x+\epsilon f\left(x,\dot{x}\right)+\epsilon g\left(\nu t\right)=0,x\left(0\right)=0,\dot{x}\left(0\right)=b,$

(1.1)

where $\epsilon \ll {\omega }_0^2$ , ${\omega }_0$ is known as unperturbed frequency, $\epsilon$ is a constant, $f$ is a nonlinear function satisfying $f\left(-x,-\dot{x}\right)=-f\left(x,\dot{x}\right)$ and $g$ represents a periodic forcing term, typically a trigonometric sine or cosine function and $\nu$ is a frequency of external excitation. In the recent past, a new modified Lindstedt-Poincaré method was introduced by Alam et al. ²³ for solving nonlinear damped forced oscillations equation (1.1). The obtained solutions are valid for large excitations. But the mathematical manipulations are laborious. In addition, several authors are continuously giving their knowledge and effort to significantly analyse the nonlinear damped force oscillators.^24–26 In this paper, an analytical technique based on the certain odd polynomial functions is introduced to solve nonlinear oscillatory problems whether it is autonomous or non-autonomous. The technique is straightforward, and the process of determining the solutions is relatively simple.

The method

Introducing a new variable $\tau =\nu t-\theta$ , equation (1.1) is transformed to

${\nu }^2{x}^{″}+{\omega }_0^{2}x+\epsilon f\left(x,\nu x^{\prime }\right)+\epsilon g\left(\tau +\theta \right)=0,x\left(0\right)=0,x^{\prime }\left(0\right)=\frac{b}{\nu }=\beta$

(2.1)

where the primes denote the differentiation with respect to $\tau$ . It is noted that a periodic solution is obtained with a phase difference $\theta$ . ²³ In accordance to the proposed technique $x$ and $\theta$ are expanded in powers of the polynomial function, ${\psi }_n\left(\tau \right)$ as

$x\left(\tau \right)=B_1{\psi }_n\left(\tau \right)+B_3{{\psi }_n}^3\left(\tau \right)+B_5{{\psi }_n}^5\left(\tau \right)+\cdots ,$

(2.2)

and

$\theta ={\theta }_0+{\theta }_2{{\psi }_n}^2\left(\tau \right)+{\theta }_4{{\psi }_n}^4\left(\tau \right)+\cdots ,$

(2.3)

where ${\psi }_n\left(\tau \right)$ is obtained by

${\psi }_i\left(\tau \right)=\frac{{{\psi }_i}^{*}}{{{\psi }_i}^{*}\left(\frac{\pi }{2}\right)},{{\psi }_i}^{*}={\int }_{\pi /2-x}^{\pi /2}{{\psi }_{i-1}}^{*}d\tau ,i=1,2,\cdots ,n;{\psi }_0=1$

(2.4)

and $B_1$ , $B_3$ , $B_5,\cdots$ and ${\theta }_0$ , ${\theta }_2$ , ${\theta }_4,\cdots$ are unknown constants depend on $\nu$ , $\epsilon$ and ${\omega }_0$ . To get the better approximate solutions, the value of $n$ is considered more than $3$ . In this paper, $n$ is considered to be $7.$

Differentiating $x\left(\tau \right)$ twice with respect to $\tau$ , the following results are found (Appendix A):

$x^{\prime }\left(\tau \right)=\left(1-\frac{300{{\psi }_n}^2}{61{\pi }^2}-\frac{45360{{\psi }_n}^4}{3721{\pi }^4}\right)\left(B_1+3B_3{{\psi }_n}^2+5B_5{{\psi }_n}^4+\cdots \right),$

(2.5)

$\begin{array}{l}{x}^{″}\left(\tau \right)={\left(1-\frac{300{{\psi }_n}^2}{61{\pi }^2}-\frac{45360{{\psi }_n}^4}{3721{\pi }^4}\right)}^2\left(6B_3{\psi }_n+20B_5{{\psi }_n}^3+\cdots \right)\\ -\left(\frac{600{\psi }_n}{61{\pi }^2}+\frac{1440{{\psi }_n}^3}{3721{\pi }^4}+\frac{104064{{\psi }_n}^5}{226981{\pi }^6}\right)\left(B_1+3B_3{{\psi }_n}^2+5B_5{{\psi }_n}^4+\cdots \right)\end{array}$

(2.6)

From equation (2.4), it is found that ${\psi }_n=0$ when $\tau =0$ . Applying the initial condition $x^{\prime }\left(0\right)=\beta$ in equation (2.5), $B_1=\beta$ is obtained. Now substituting $x\left(\tau \right)$ , $x^{\prime }\left(\tau \right)$ and $x^{″}\left(\tau \right)$ together with $\theta$ from equation (2.3), into equation (2.1) and equating the coefficients of ${\left({\psi }_n\right)}^i,i=0,1,2,3,\cdots$ a set of linear algebraic equations are found which provide the values of unknown coefficients $B_3$ , $B_5,\cdots$ and ${\theta }_0$ , ${\theta }_2$ , ${\theta }_4,\cdots$ in terms of $\beta$ , $\nu$ , $\epsilon$ and ${\omega }_0$ .

From equation (2.4), it is also found that the maximum value of ${\psi }_n=1$ when $\tau =\frac{\pi }{2}$ . Certainly, $x\left(\tau \right)$ has the maximum value known as amplitude of the oscillation, say $a>0$ at $\tau =\frac{\pi }{2}$ (If $b>0$ as well as $\beta >0$ , then $a>0$ and if $b<0$ as well as $\beta <0$ , then $x\left(\tau \right)$ becomes minimum, $-a$ at $\tau =\frac{\pi }{2}$ ). Therefore, a relation among $a$ , $\beta$ , $\nu$ , $\epsilon$ and ${\omega }_0$ is obtained from equation (2.2) as

$a=\beta +B_3+B_5+\cdots .$

(2.7)

In general, $\nu$ , $\epsilon$ and ${\omega }_0$ are given. To obtain a periodic solution, $a$ and $\beta$ are to be determined. So, it requires another relation between $a$ and $\beta$ . Multiplying equation (2.1) by $2x^{\prime }$ and then integrating with respect to $\tau$ , it readily becomes ¹⁶

${{\nu }^2\beta }^2+\epsilon F+\epsilon G={\omega }_0^2{a}^2,$

(2.8)

where

$F={\int }_0^{\pi /2}2x^{\prime }fd\tau \hspace{0.17em}\text{and}\hspace{0.17em}G={\int }_0^{\pi /2}2x^{\prime }g\left(\tau +\theta \right)d\tau .$

(2.9)

Thus, the determination of $a$ and $\beta$ is clear. It is noted that equation (2.8) is well known as an energy form of equation (2.1).

Example

Let us consider a nonlinear damped forced oscillator

$\ddot{x}+{\omega }_0^{2}x+\epsilon \mu \dot{x}+\epsilon x^3=\epsilon E\sin \left(\nu \mathrm{t}\right).$

(3.1)

In terms of $\tau$ , equation (3.1) becomes

${\nu }^2{x}^{″}+{\omega }_0^{2}x+\epsilon \mu {\nu x}^{\prime }+\epsilon x^3=\epsilon E\sin \left(\tau +\theta \right).$

(3.2)

The noted set of linear algebraic equations for $B_3$ , $B_5,\cdots$ and ${\theta }_0$ , ${\theta }_2$ , ${\theta }_4$ , $\cdots$ of this oscillator are obtained as

$\begin{array}{l}\epsilon \beta \mu \nu =\epsilon E{s}_0,-\frac{600\beta {\nu }^2}{61{\pi }^2}+6{\nu }^2{B}_3+\beta {\omega }_0^2=\epsilon E{c}_0,\\ -\frac{300\beta \mu \epsilon \nu }{61{\pi }^2}+3\epsilon \mu \nu B_3=\epsilon E{c}_0{\theta }_2-\frac{1}{2}\epsilon E{s}_0,\\ \epsilon {\beta }^3-\frac{1440\beta {\nu }^2}{3721{\pi }^4}-\frac{5400B_3{\nu }^2}{61{\pi }^2}+{\omega }_0^2{B}_3+20{\nu }^2{B}_5=\frac{100E\epsilon c_0}{61{\pi }^2}-\frac{1}{6}E\epsilon c_0-E\epsilon {\theta }_2{s}_0,\cdots ,\end{array}$

(3.3)

where $c_0=\cos \hspace{0.17em}{\theta }_0$ and $s_0=\sin \hspace{0.17em}{\theta }_0$ . Solving equation (3.3), the coefficients $B_3$ , $B_5,\cdots$ and ${\theta }_0$ , ${\theta }_2$ , $\cdots$ are found as

$\begin{array}{l}{\theta }_0={\sin }^{-1}\frac{\beta \mu \nu }{E},B_3=\frac{100\beta }{61{\pi }^2}-\frac{\beta {\omega }_0^2}{6{\nu }^2}+\frac{\epsilon E{c}_0}{6{\nu }^2},{\theta }_2=-\frac{300\beta \mu \nu s_{e0}}{61E{\pi }^2}+\frac{3\mu \nu B_3{s}_{e0}}{E}+\frac{t_0}{2},\\ B_5=\frac{72\beta }{3721{\pi }^4}+\frac{270B_3}{61{\pi }^2}-\frac{{\omega }_0^2{B}_3}{20{\nu }^2}-\frac{\epsilon {\beta }^3}{20{\nu }^2}-\frac{\epsilon E{\theta }_2{s}_0}{20{\nu }^2}-\frac{E\epsilon c_0}{120{\nu }^2}+\frac{5E\epsilon c_0}{61{\pi }^2{\nu }^2},\\ {\theta }_4=\frac{{\theta }_2}{2}+\frac{5B_5\mu \nu s_{e0}}{E}-\frac{45360\beta \mu \nu s_{e0}}{3721E{\pi }^4}-\frac{900\mu \nu B_3{s}_{e0}}{61E{\pi }^2}-\frac{t_0}{24}+\frac{{100t}_0}{61{\pi }^2},\end{array}$

(3.4)

$B_7=\frac{17344\beta }{1588867{\pi }^6}+\frac{2500B_5}{427{\pi }^2}+\frac{1440B_3}{26047{\pi }^4}-\frac{{\omega }_0^2{B}_5}{42{\nu }^2}-\frac{\epsilon {\beta }^2{B}_3}{14{\nu }^2}+\frac{E\epsilon \hspace{0.17em}{c}_0}{5040{\nu }^2}-\frac{25E\epsilon \hspace{0.17em}{c}_0}{1281{\pi }^2{\nu }^2}-\frac{\epsilon E{\theta }_4{s}_0}{42{\nu }^2},$

where $s_{e0}=\sec \hspace{0.17em}{\theta }_0$ and $t_0=\tan \hspace{0.17em}{\theta }_0$ .

For this oscillator, equation (2.8) becomes

${{\nu }^2\beta }^2+\epsilon \mu \nu D+\epsilon ES={\omega }_0^2{a}^2+\epsilon a^4/2,$

(3.5)

where

$D={\int }_0^{\pi /2}2{x^{\prime }}^{2}d\tau =2{\beta }^2+4\beta B_3+\frac{18B_3^2}{5}+4\beta B_5-\frac{18144{\beta }^2}{3721{\pi }^4}-\frac{200{\beta }^2}{61{\pi }^2}-\frac{720\beta B_3}{61{\pi }^2},$

(3.6)

and

$S={\int }_0^{\pi /2}2x^{\prime }\hspace{0.17em}\sin \left(\tau +\theta \right)d\tau =c_0\left(\frac{11\beta }{12}+\frac{3B_3}{2}+\frac{50\beta }{61{\pi }^2}+\frac{7\beta {\theta }_2}{15}+\frac{6B_3{\theta }_2}{5}+\frac{2\beta {\theta }_4}{5}\right)$

$+s_0\left(\frac{101\beta }{60}+\frac{7B_3}{5}+2B_5-\frac{40\beta }{61{\pi }^2}-\frac{b{\theta }_2}{2}\right)$

(3.7)

Now substituting $B_3$ , $B_5,\cdots$ into equation (2.7) and then solving together with equation (3.5), the values of $a$ and $\beta$ are obtained for every value of $\nu$ .

In the case of $E=\mu =0$ , equation (3.1) turns into a cubic Duffing oscillator

$\ddot{x}+{\omega }_0^{2}x+\epsilon x^3=0.$

(3.8)

In terms of $\tau$ , equation (3.8) becomes

${\nu }^2{x}^{″}+{\omega }_0^{2}x+\epsilon x^3=0.$

(3.9)

The unknowns $B_3$ , $B_5,\cdots$ are obtained of equation (3.9) as

$\begin{array}{l}{B}_3=\frac{100\beta }{61{\pi }^2}-\frac{\beta {\omega }_0^2}{6{\nu }^2},B_5=\frac{72\beta }{3721{\pi }^4}+\frac{270B_3}{61{\pi }^2}-\frac{{\omega }_0^2{B}_3}{20{\nu }^2}-\frac{\epsilon {\beta }^3}{20{\nu }^2},\\ B_7=\frac{17344\beta }{1588867{\pi }^6}+\frac{2500B_5}{427{\pi }^2}+\frac{1440B_3}{26047{\pi }^4}-\frac{{\omega }_0^2{B}_5}{42{\nu }^2}-\frac{\epsilon {\beta }^2{B}_3}{14{\nu }^2},\end{array}$

(3.10)

and the energy form of equation (3.9) is

${{\nu }^2\beta }^2={\omega }_0^2{a}^2+\epsilon a^4/2.$

(3.11)

Solving equation (2.7) and equation (3.11) with these coefficients $B_3$ , $B_5,\cdots$ , the values of $\nu$ and $\beta$ are obtained for different values of the amplitude $a$ .

Results and discussions

An analytical technique has been introduced for solving nonlinear oscillators, applicable to both autonomous and non-autonomous systems. In this paper, the nonlinear damped force oscillator as well as cubic Duffing oscillators are considered. The algebraic equations containing unknown coefficients are linear. In contrast, corresponding equations for HBM ²¹ are nonlinear. This is a key advantage of the proposed technique. Several methods^20,27,28 have been introduced for solving damped forced oscillators and suitable results yielded only for small excitation. The present technique is suitable for both small as well as large excitations. Cheung et al. ²² expanded ${\nu }^2$ in equation (3.1) as a series in terms of $\alpha$ rather than $\epsilon$ , and shown the frequency-amplitude curve for large excitations, considering only the first harmonic. Although, only the first harmonic was considered, the mathematical calculations are still laborious. The present technique offers a more efficient alternative. Additionally, for various excitation, the frequency against amplitude curves (displayed in Figures 1–4) show a good agreement with numerical results obtained using the Runge-Kutta fourth-order method (RK4).

Figure 1.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $\mu =0.1,{\hspace{0.17em}\omega }_0=1,E=10$ .

Figure 2.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $\mu =0.1,{\hspace{0.17em}\omega }_0=1,E=20$ .

Figure 3.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $\mu =0.2,{\hspace{0.17em}\omega }_0=1,E=1$ .

Figure 4.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $\mu =0.2,{\hspace{0.17em}\omega }_0=1,E=20$ .

To further validate and demonstrate the accuracy of the current solutions, a comparison of time versus displacement for various system parameters is plotted in Figures 5–7 alongside the corresponding numerical results. The comparison reveals that the present solutions align remarkably well with the numerical solutions. In Figure 8, the 3-D error graph of the proposed method corresponding to the RK4 of equation (3.1) for the variations in parameters $E$ and $\nu$ which shows the acceptable errors.

Figure 5.

Time versus displacement curves of equation (3.1) between present solution and the corresponding numerical one for $\mu =0.1,{\hspace{0.17em}\omega }_0=1,\nu =6,E=10$ with the initial condition $x\left(0\right)=0,\dot{x}\left(0\right)=35.62212$ .

Figure 6.

Time versus displacement curves of equation (3.1) between present solution and the corresponding numerical one for $\mu =0.1,{\hspace{0.17em}\omega }_0=1,\nu =7,E=20$ with the initial condition $x\left(0\right)=0,\dot{x}\left(0\right)=48.78188$ .

Figure 7.

Time versus displacement curves of equation (3.1) between present solution and the corresponding numerical one for $\mu =0.2,{\hspace{0.17em}\omega }_0=1,\nu =5,E=20$ with the initial condition $x\left(0\right)=0,\dot{x}\left(0\right)=25.32035$ .

Figure 8.

3-D error graph of the proposed method corresponding to the RK4 of equation (3.1) for the variations in parameters $E$ and $\nu$ .

The comparison of the approximated and existing amplitudes corresponding to the exact ones of equation (3.1) for various parameters $\mu =0.2,{\hspace{0.17em}\omega }_0=1,E=20$ are shown in Table 1. The proposed method provides the almost similar results as compared with the existing ones for small frequencies. Notably, the proposed method gives excellent results as compared to the existing results for large frequencies.

Table 1.

Comparison of the approximated and existing amplitudes corresponding to the exact ones of equation (3.1) for various parameters $\mu =0.2,{\hspace{0.17em}\omega }_0=1,E=20$ .

Frequency $\nu$	$a_{exact}$	Cheng et al. 22	Zahangir et al. 23	Present study
		$Er(\%)$	$Er(\%)$	$Er(\%)$
3	4.30342	4.29997 (0.08)	4.29602 (0.17)	4.31492 (0.26)
4	5.17308	5.17961 (0.12)	5.17598 (0.06)	5.18546 (0.24)
5	6.17686	6.19106 (0.22)	6.18330 (0.10)	6.18869 (0.19)
6	7.24929	7.27236 (0.31)	7.25971 (0.14)	7.25914 (0.13)
7	8.35674	8.39161 (0.42)	8.36976 (0.16)	8.36323 (0.08)
8	9.48082	9.53244 (0.54)	9.49665 (0.17)	9.48103 (0.002)
9	10.6063	10.6862 (0.75)	10.6307 (0.23)	10.5985 (0.07)

For the cubic Duffing oscillator equation (3.8), the unknown coefficients $B_3$ , $B_5,\cdots$ in equation (3.10) and the energy expression in equation (3.11) are relatively simple. The frequencies of this oscillator for various amplitudes $a$ can be easily determined and are well approximated by numerical frequencies, as shown in Table 2. Additionally, a comparison of the time versus displacement curve for the oscillator in equation (3.8) is plotted in Figure 9. This comparison demonstrates that the present analytical technique is also applicable to the cubic Duffing oscillator and provides excellent results.

Table 2.

Comparison of the present and exact frequency of the cubic Duffing oscillator for various amplitude.

Amplitude $a$	${\nu }_{exact}$	${\nu }_{present}$
		$Er(\%)$
1	1.317776	1.317396 (0.03)
2	1.976016	1.974730 (0.06)
5	4.357461	4.351904 (0.12)
10	8.533586	8.519908 (0.16)
100	84.72748	84.58856 (0.16)

Figure 9.

Time versus displacement curves of equation (3.8) between present solution and the corresponding numerical one for ${\omega }_0=1,a=100,$ with the initial condition $x\left(0\right)=0,\dot{x}\left(0\right)=7083.38678$ .

Conclusion

An analytical technique is presented for solving nonlinear damped forced oscillators, as well as for force free undamped case. The solution procedure of the present technique is simple. Also, it is suitable for large excitations under various damping conditions. The results obtained in the present technique are surprisingly good agreement with corresponding numerical one. However, the proposed analytical technique should continue to develop for applying to the quadratic nonlinear force oscillators as well as Van der Pol nonlinear force oscillators. To sum up, it is said that, the proposed technique has a great effectiveness for solving nonlinear oscillators, particularly for the nonlinear damped forced oscillatory systems arising in nonlinear science and engineering.