Nonlinear oscillations of a point charge in the electric field of charged ring using a particular He's frequency-amplitude formulation

TL;DR

This paper applies He's frequency-amplitude formulation to accurately analyze nonlinear oscillations of a point charge in a charged ring's electric field, effective for both small and large amplitudes without linearization.

Contribution

It introduces a specific application of He's method with optimized points, improving accuracy over previous approaches for nonlinear oscillations of a point charge.

Findings

Method achieves high accuracy compared to exact solutions.

Effective for both small and large amplitude oscillations.

Outperforms previous methods, especially at large amplitudes.

Abstract

In this paper, He's frequency-amplitude formulation with some choice of location points that improve accuracy is applied to determine the periodic solution for the nonlinear oscillations of a punctual charge in the electric field of charged ring. The results of the present study are valid for small and large amplitudes of oscillation. The present method can be applied directly to highly nonlinear problems without any discretization, linearization or restrictive assumptions. Finally, compared with the exact solution shows that the result obtained is of high accuracy and we will do a comparative study with previous research done on the same problem, we will see that our approach is much better especially for large amplitude values.

Tables

Table 1. Table 1: Comparison between frequencies ω a ​ p ​ p ​ ( A ) subscript 𝜔 𝑎 𝑝 𝑝 𝐴 \omega_{app}(A) and ω e ​ x ​ ( A ) subscript 𝜔 𝑒 𝑥 𝐴 \omega_{ex}(A) for small values of A 𝐴 A .

$A$	${\omega }_{app}\left(A\right)$ Eq. (7)	${\omega }_{ex}\left(A\right)$ Eq. (10)	$\frac{{\omega }_{app}}{{\omega }_{ex}}\left(A\right)$
$1/1000$	$0.9999992500$	$0.9999994375$	$0.9999998124$
$1/100$	$0.9999250066$	$0.9999437540$	$0.9999812515$
$1/10$	$0.9925650290$	$0.9944148226$	$0.9981398169$
$1/2$	$0.8458970108$	$0.8795518875$	$0.9617363373$

Table 2. Table 2: Comparison between frequencies ω a ​ p ​ p ​ ( A ) subscript 𝜔 𝑎 𝑝 𝑝 𝐴 \omega_{app}(A) and ω e ​ x ​ ( A ) subscript 𝜔 𝑒 𝑥 𝐴 \omega_{ex}(A) for large values of A 𝐴 A

$A$	${\omega }_{app}\left(A\right)$ Eq. (7)	${\omega }_{ex}\left(A\right)$ Eq. (9)	$\frac{{\omega }_{app}}{{\omega }_{ex}}\left(A\right)$
$1$	$0.5946035575$	$0.4704457120$	$1.2639153516$
$5$	$0.0868500332$	$0.0831210434$	$1.0448621628$
$10$	$0.0313876621$	$0.0308692709$	$1.0167931144$
$50$	$0.0028275788$	$0.0028231732$	$1.0015605432$
$100$	$0.0009999250$	$0.0009993712$	$1.0005541029$
$1000$	$0.0000316227$	$0.0000316221$	$1.0000175879$

Table 3. Table 3: Relative errors % obtained in the present study for some small and large values of A 𝐴 A

$A$	Relative errors %	A	Relative errors %
$1/1000$	$0.0000181$	$10$	$1.6793114$
$1/100$	$0.0018742$	$20$	$0.6085781$
$1/10$	$0.1860181$	$50$	$0.1560544$
$1/2$	$3.8263660$	$100$	$0.0554103$
$1$	$13.318421$	$1000$	$0.0017588$

Equations

$$\frac{d^{2}u}{d{t}^2}+\frac{u}{(1+u^2{)}^{3/2}}=0,$$

$$u(0)=A,u^{\prime }(0)=0.$$

$$u^{\prime \prime }+u$$

$$u^{\prime \prime }+{\omega }^{2}u$$

$$R_1(t)$$

$$R_2(t)$$

$${\omega }^2=\frac{R_2(t_2)-{\omega }^2{R}_1(t_1)}{R_2(t_2)-R_1(t_1)},$$

$$R_1(0)$$

$$R_2(0)$$

$$\begin{split}\omega^{2}(A)&=\frac{R_{2}(0)-\omega^{2}R_{1}(0)}{R_{2}(0)-R_{1}(0)}\\ &=\frac{\Big{(}\frac{A}{(1+A^{2})^{3/2}}-A^{2}\omega^{2}\Big{)}-\omega^{2}\Big{(}\frac{A}{(1+A^{2})^{3/2}}-A\Big{)}}{\Big{(}\frac{A}{(1+A^{2})^{3/2}}-A^{2}\omega^{2}\Big{)}-\Big{(}\frac{A}{(1+A^{2})^{3/2}}-A\Big{)}}\\ &=\frac{A(1-\omega^{2})}{A(1-\omega^{2})(1+A^{2})^{3/2}}=\frac{1}{(1+A^{2})^{3/2}}.\end{split}$$

$${\omega }_{app}(A)=\frac{1}{\sqrt[4]{(1+A^2{)}^3}}.$$

$$u_{app}(t)=A\cos \left(\frac{1}{\sqrt[4]{(1+A^2{)}^3}}t\right).$$

$${\omega }_{ex}(A)=\frac{2\pi }{{\int }_0^1\frac{4Adu}{\sqrt{(1+A^2{u}^2{)}^{-1/2}-(1+A^2{)}^{-1/2}}}}.$$

$${\omega }_{ex}(A)\approx 1-\frac{9}{16}{A}^2+\frac{411}{1024}{A}^4-\frac{5147}{16384}{A}^6+\mathcal{O}(A^7).$$

$${\omega }_{app}(A)=\frac{1}{\sqrt[4]{(1+A^2{)}^3}}\approx 1-\frac{3}{4}{A}^2+\frac{21}{32}{A}^4-\frac{77}{128}{A}^6+\mathcal{O}(A^7).$$

$$\frac{{\omega }_{app}(A)}{{\omega }_{ex}(A)}\simeq 1-\frac{3}{16}{A}^2+\frac{153}{1024}{A}^4-\frac{17613}{16384}{A}^6+\mathcal{O}(A^7).$$

$$\underset{A\to 0^{+}}{\lim }\frac{{\omega }_{app}(A)}{{\omega }_{ex}(A)}=1,$$

$${\int }_0^1\frac{4\sqrt{u}du}{\sqrt{1-u}}=2\pi .$$

$$\begin{split}\lim\limits_{A\to+\infty}\frac{\omega_{app}(A)}{\omega_{ex}(A)}&=\lim\limits_{A\to+\infty}\frac{\frac{1}{\sqrt[4]{(1+A^{2})^{3}}}}{\dfrac{2\pi}{\Big{(}\displaystyle\int_{0}^{1}\frac{4Adu}{\sqrt{(1+A^{2}u^{2})^{-1/2}-(1+A^{2})^{-1/2}}}\Big{)}}}\\ &=\lim\limits_{A\to+\infty}{\dfrac{\displaystyle\int_{0}^{1}\frac{4Adu}{\sqrt{(1+A^{2}u^{2})^{-1/2}-(1+A^{2})^{-1/2}}}}{2\pi\sqrt[4]{(1+A^{2})^{3}}}}\\ &=\frac{1}{2\pi}\int_{0}^{1}\frac{4\sqrt{u}du}{\sqrt{1-u}}=1.\end{split}$$

$${\omega }_{ex}(A)\approx \frac{\sqrt{2}}{A^{3/2}}.$$

$${\omega }_{EBM}(A)=\frac{2}{A^{3/2}}\sqrt{\sqrt{2}-1}$$

$$\underset{A\to \infty }{\lim }\frac{{\omega }_{HBM}(A)}{{\omega }_{ex}(A)}=\mathrm{0.9101797.}$$

$${\omega }_{R-G}(A)=\frac{1.3337481}{A^{3/2}}$$

$$\underset{A\to \infty }{\lim }\frac{{\omega }_{R-G}(A)}{{\omega }_{ex}(A)}=\mathrm{0.9431023.}$$

$${\omega }_{HBM}(A)=\frac{1.39197}{A^{3/2}}$$

$$\underset{A\to \infty }{\lim }\frac{{\omega }_{HBM}(A)}{{\omega }_{ex}(A)}=\mathrm{0.984273.}$$

Full text

1 Introduction

Nonlinear vibration arises everywhere in science, engineering and other disciplines, since most phenomena in our world today, are essentially nonlinear and are described by nonlinear equations. It is very important in applications to have a version of the frequency (or period) to have a better understanding of the phenomena modeled through differential equations that contain terms with high nonlinearities, and a simple mathematical method is very useful for practical applications.

Recently, many different methods have been developed to obtain the approximate solutions of nonlinear systems including the harmonic balance method Nay-1 ; Mic ; Bel1 ; Bel-0 , the energy balance method Yil ; Khan-1 , the Hamiltonian approach Yil-1 ; He-x0 , the use of special functions Zu-1 ; Zu-2 , the max-min approach He-x ; Ze , the variational iteration method Gan ; He-1 ; He-2 ; He-3 ; Waz-1 and homotopy perturbation Bel-1 ; Bel-2 ; Gan-1 ; Gan-2 ; He-4 ; He-5 ; He-6 . An excellent study, in which many of these techniques can be found in detail to solve nonlinear problems of oscillatory type can be seen in He-8 .

In dimensionless form, the equation of motion of a punctual charge placed at a point on the ring axis, is given by the following nonlinear differential equation Bel-0 ; Bel-00

[TABLE]

with the initial conditions

[TABLE]

Where $u$ is de position of the probe charge in units of the radius of the ring, $t$ is the time measured in terms of the natural frequency of the system.

The main purpose of the present work is to show how to apply the frequency-amplitude formulation with a particular choice of location points, developed recently by Ji-Huan He based on an ancient Chinese algorithm in He-x1 ; He-x2 ; He-9 ; He-8 , and after improved by himself and other authors in He-10 ; He-11 ; Cai ; Ren , to obtain the approximate periodic solution of Eq.(1).

2 He’s frequency-amplitude formulation

In order to utilize He’s amplitude frequency formulation, we choose two trial functions $u_{1}(t)=A\cos(t)$, $u_{2}(t)=A\cos(\omega t)$, which are respectively, the solutions of the following linear differential equations:

[TABLE]

where $\omega$ is assumed to be the frequency of the nonlinear oscillator equation (1). Substituting $u_{1}(t)$ and $u_{2}(t)$ into equation (1), we obtain, respectively, the following residuals

[TABLE]

According to the method improvements made in Cai , the approximate frequency can be obtained as follows:

[TABLE]

where $t_{1}$ and $t_{1}$ are location points. The phase of the residuals is determined by the location points.

Recently, in Ren an improvement in the manner of choosing these points was proposed. Such improvement works very well in the solution of certain problems, this is not the case of the problem studied here, as we will see in the conclusions and discussions section. In He-8 ; He-x1 the phase of the residuals was chosen as [math] for simplicity, i.e. $t_{1}=0$ and $t_{2}=0$. The corresponding residuals are as follows:

[TABLE]

According to Eqs. (4) and (5), the frequency-amplitude formulation is

[TABLE]

Therefore, the analytical approximate frequency $\omega$ as a function of $A$ is

[TABLE]

From equation (7) we obtain the following approximate periodic solution to (1)

[TABLE]

3 Exact and approximate solutions

The nonlinear oscillator described by Eq. (1) is a conservative system. Integrating Eq. (1) and using the initial conditions Eq. (2), the exact angular frequency as a function of $A$ was obtained in Bel-0

[TABLE]

In order to compare the approximate frequencies obtained in the present work and the exact ones given by Eq. (9) we consider the two cases. First we consider small amplitudes and we will see that our approximation is very good, in fact we will see that $\omega_{app}(A)$ of Eq. (7) tends to be $\omega_{ex}(A)$ when $A\to+0$. On the other hand we take into account large amplitudes and show that the approximation obtained by this method given by Eq. (7) tends to be exactly $\omega_{ex}(A)$ when $A\to+\infty$.

3.1 Case I: $0<A<1$

For small values of the amplitude A it is possible to tconsider the following approximation for $\omega_{ex}(A)$, as has been seen in Bel-0

[TABLE]

By taking into account our approximation made through He’s frequency-amplitude formulation Eq. (7) and $\omega_{ex}(A)$ from Eq. (10) we can calculate the Table 1 for small values of $A$.

In addition, considering that for small values of $A$

[TABLE]

Hence the ratio between $\omega_{ex}(A)$ and $\omega_{app}(A)$ can also be calculated as the following series

[TABLE]

In the exact limit we have that

[TABLE]

and hence, $\omega_{app}(A)\to\omega_{ex}(A)$ as $A\to 0^{+}$.

In Fig. 1 we plot the comparison between exact solution $u_{ex}=A\cos(\omega_{ex}(A))$ for $A=1/2$. Even though around $A=1/2$ the error is large, as one can see in Table 1, the approximation is acceptable.

3.2 Case II: $A\geq 1$

For large amplitudes we calculate the following values with the approximation made through He’s frequency-amplitude formulation Eq. (7) and the exact result $\omega_{ex}(A)$ given by Eq. (9)

In this case, we calculate again the ratio between both frequencies approximated by exact ones, as in DQ . In our calculations we need to take into account that the following integral do converge

[TABLE]

In the asymptotic limit for the ratio is calculated as follows

[TABLE]

Therefore, $\omega_{app}(A)\to\omega_{ex}(A)$ as $A\to+\infty$.

In Fig. 2 we compare the exact solution $u_{ex}=A\cos(\omega_{ex}(A))$ for $A=10$ and the corresponding approximation. We can appreciate that the semianalytical solution is quite acceptable.

From Tables 1, 2 and Eqs. (13), (15), it can be observed that Eq. (7) yield excellent analytical approximate frequencies for both, very small and very large values of oscillation amplitude $A$.

4 Conclusions and Discussions

In this section we study the precision and validity of the solution method employed in the present work. We compare our results with the exact solution obtained in Bel-0 , all numerical calculations have been made with the help of the software MATHEMATICA. A comparative analysis of the error given by the present method and the error of the best approximation obtained by the Harmonic balancing approach Bel-0 , is made.

The exact solution for the frequency $\omega_{ex}(A)$ is given by Eq. (9) and we can obtain an asymptotic expansion for large amplitudes as was done in Bel-0 , which is given by

[TABLE]

There have been obtained a good approximation for $\omega(A)$ in Yil through the Energy Balance Method (EBM), which conclusion is

[TABLE]

where, considering Eq. (16) we can obtain the large amplitude limit of the ratio of frequencies

[TABLE]

From Eq. (18) one can see that the relative percentage error tends to $8.9\%$ for $A\to\infty$.

In addition, the problem was studied in Vali with the same method. The main difference with the present work is that in Vali the points were chosen following Reng-Gui Ren and Geng-Cai Cai , in such work the best approximation obtained was the following

[TABLE]

where, by considering Eq. (16) they found

[TABLE]

From Eq. (20) it is observed that the relative percentage error tends to $5.6\%$ in the limit $A\to\infty$.

The best known approximation was obtained by the Harmonic Balance Method (HBM) in Bel-0 , whose approximation is

[TABLE]

from where

[TABLE]

The above, is the best known approximation so far, with which we have a percentage relative error of only $1.57\%$ in the limit $A\to\infty$.

By considering the data of tables 1 and 2, we calculate the percentage relative error for small and large amplitudes $A$, the corresponding values are shown in Table 3.

In the present work, from Eq. (15) we obtain the limit $\lim\limits_{A\to+\infty}\frac{\omega_{app}(A)}{\omega_{ex}(A)}=1$, From where the percentage relative error tends to zero when $A\to\infty$ as we can see in the figure 3, which improves the best known approximation $\omega_{HBM}(A)$. Also, in the figure 3 it can be seen that for $0<A<1$ the harmonic balance method used in Bel-0 is a better approximation for the solution of the present problem.

Additionally, we must clarify that the center of the ring presents instability and therefore the oscillations of the charge around this point are not possible. To solve this problem it could be possible to consider that the punctual charge is on a finite conducting wire placed along the axis of the ring as it is done as in Luc , this is also considered in Bel-00 .

Finally, it is concluded from the previous discussion that the alternative approach presented in this paper for the selection of location points in He’s frecuency-amplitude formulation, gives a high precision solution for the nonlinear oscillations of a ponit charge in the electric field of charged ring, specially for large values of $A\gg 1$ or very small $A\ll 1$.