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Abstract

A pendulum system is widely used for vibration attenuation and energy harvesting. This paper considers a pendulum under the attraction of double magnets. The governing equation is obtained by Newton’s second law, and it is simplified to Duffing–Holmes oscillator. Its approximate periodic solution with a good accuracy is obtained with easy by Ji-Huan He’s frequency formulation.

Keywords

Magnetic pendulum periodic motion frequency–amplitude relationship

Introduction

The mathematical pendulum is a mathematical model for many nonlinear vibration systems^11–4, and it is widely used for vibration attenuation in high buildings^55,6 and energy harvesting^7,8. Figure 1 shows a pendulum pounding tuned mass damper for vibration attenuation in a wind turbine, where the viscoelastic boundary is to absorb the vibrating energy, as a result, vibration attenuation is realized⁹. The energy harvesting from pendulum oscillation is to convert the vibrating energy from wind to other energy (e.g. electronic energy, or mechanical energy).^10,11 Due to high nonlinearity, various numerical methods were applied to solve pendulum systems,^12,13 but the numerical solutions can not reveal the explicit relationship between the frequency and the amplitude.

Figure 1.

Vibration attenuation of a wind turbine with a pendulum pounding tuned mass damper.

There are many attractive properties of a pendulum system, for examples, periodic and chaotic properties. In this paper, we will study a pendulum system under action of double magnets, the magnetic pendulum has widely applications, especially in condensed matter physics.^14–17

Mathematical model

The studied system is illustrated in Figure 1, an iron ball is fixed at the end of a flexible wire, which is attracted by two magnets located at a below place of the ball. The geometric parameters are illustrated in Figure 2.

Figure 2.

Pendulum system under action of double magnets.

We assume that the attracting force of the magnet on the ball can be expressed as follows:

F = \frac{k m}{r^{2}}

(1)

where

r

is the distance between the ball and the magnet,

k

is the magnetic constant, and

m

is the ball mass. The governing equation can be obtained by Newton’s second law,

m L \frac{d^{2} θ}{d t^{2}} + m g \sin θ + F_{1} \cos α_{1} + F_{2} \cos α_{2} = 0

(2)

where the magnetic forces can be expressed, respectively, as follows:

F_{1} = \frac{k m}{L_{1}^{2}} = \frac{k m}{(a {}^{2}+ H^{2} + L^{2}) - 2 L (a \sin θ + H \cos θ)}

(3)

F_{2} = \frac{k m}{L_{2}^{2}} = \frac{k m}{(a {}^{2}+ H^{2} + L^{2}) + 2 L (a \sin θ - H \cos θ)}

(4)

The angles of $α_{1}$ and $α_{2}$ can be expressed, respectively, as follows:

α_{1} (θ) = \frac{π}{2} - θ + \arctan \frac{a - L \sin θ}{H - L \cos θ}

(5)

α_{2} (θ) = \frac{π}{2} - θ - \arctan \frac{a + L \sin θ}{H - L \cos θ}

(6)

So equation (2) becomes

\begin{array}{l} m L \frac{d^{2} θ}{d t^{2}} + m g \sin θ + \\ + \frac{k m}{(a {}^{2}+ H^{2} + L^{2}) - 2 L (a \sin θ + H \cos θ)} \cos {\frac{π}{2} - θ + \arctan \frac{a - L \sin θ}{H - L \cos θ}} + \\ + \frac{k m}{(a {}^{2}+ H^{2} + L^{2}) + 2 L (a \sin θ - H \cos θ)} \cos {\frac{π}{2} - θ - \arctan \frac{a + L \sin θ}{H - L \cos θ}} = 0 \end{array}

(7)

When $F_{1} = F_{2} = 0$ , we have the well-known mathematical pendulum

\frac{d^{2} θ}{d t^{2}} + \frac{g}{L} \sin θ = 0

(8)

Using Taylor series, and approximating equation (7) to $O (θ^{3})$ , we can obtain the following equation:

{\begin{matrix} \frac{d^{2} θ}{d t^{2}} + ω_{0}^{2} θ - ε θ^{3} = 0 \\ θ (0) = A, \frac{d θ}{d t} (0) = 0 \end{matrix}

(9)

where

ω_{0}^{2}

and

ε

are collected coefficients of the linear term and the nonlinear term, respectively. The coefficients are too long, for simplicity, we write

ω_{0}^{2}

as follows:

\begin{array}{l} ω_{0}^{2} = \frac{g}{L} - \frac{k \sin (\arctan \frac{a}{H - L} - \frac{π}{2})}{L (H^{2} - 2 H L + L^{2} + a^{2})} \cdot (\frac{L}{(\frac{a^{2}}{{(H - L)}^{2}} + 1) (H - L)} + 1) \\ + \frac{k \sin (\arctan \frac{a}{H - L} + \frac{π}{2})}{L (H^{2} - 2 H L + L^{2} + a^{2})} \cdot (\frac{L}{(\frac{a^{2}}{{(H - L)}^{2}} + 1) (H - L)} + 1) \\ - \frac{2 a k \cos (\arctan \frac{a}{H - L} - \frac{π}{2})}{{(H^{2} - 2 H L + L^{2} + a^{2})}^{2}} + \frac{2 a k \cos (\arctan \frac{a}{H - L} + \frac{π}{2})}{{(H^{2} - 2 H L + L^{2} + a^{2})}^{2}} \end{array}

(10)

When $F_{1} = F_{2} = 0$ , the coefficients for this case are $ω_{0}^{2} = \frac{g}{L}$ , $ε = \frac{g}{6 L}$ . Equation (9) is generally called as the Duffing–Holmes oscillator.^18,19

Periodic solution

We consider a generally oscillator in the form

\frac{d^{2} θ}{d t^{2}} + f (θ) = 0

(11)

It has periodic solution when^20,21

\frac{f (θ)}{θ} > 0

(12)

The square of the frequency can be obtained by He’s frequency formulation²²

ω^{2} = {\frac{d f (θ)}{d θ} |}_{θ = A / 2}

(13)

where A is the amplitude.

Due to its extreme simplicity and remarkable effectiveness, He’s frequency formulation has attracted rocketing attention, and various modifications were appeared in literature, see for examples, Refs. [23–25]. He and Liu²⁶ gave a rigorous mathematical deduction, and it is also valid for fractal oscillators.^27,28

In our study, $f (θ) = ω_{0}^{2} θ - ε θ^{3}$ , so we have

ω = \sqrt{{(ω_{0}^{2} - 3 ε θ^{2})}_{θ = A / 2}} = \sqrt{ω_{0}^{2} - \frac{3}{4} ε A^{2}}

(14)

For a periodic solution, it requires

ω_{0}^{2} - \frac{3}{4} ε A^{2} > 0

(15)

A < \frac{1.1547 ω_{0}}{ε^{1 / 2}}

(16)

As an example, we consider the following case:

{\begin{matrix} θ^{″} + θ - θ^{3} = 0 \\ θ (0) = A, θ^{'} (0) = 0 \end{matrix}

(17)

For equation (17), its approximate solution is as follows:

θ (t) = A \cos ω t

(18)

where

ω = \sqrt{1 - \frac{3}{4} A^{2}}

(19)

The periodic solution requires

A < 1.1547

(20)

For periodic solution of equation (17), it requires that $(θ - θ^{3}) / θ \geq 0$ , that is $θ \leq 1.0$ . That means the critical value of A is $A^{*} = 1.0$ , when $A < 1.0$ , there is a periodic solution. Figure 3 shows the accuracy of the approximate solution of equation (18) for various amplitudes when $A < 1.0$ .

Figure 3.

Comparison of the approximate solution with the exact one when $A = 0.2$ (a), $A = 0.5$ (b), $A = 0.8$ (c), and $A = 0.9$ (d).

Exact period

In order to show the accuracy of the approximate solution obtained by He’s frequency formulation, we search for the exact period of equation (17) for comparison.

Introducing a new variable y defined as follows:

\frac{d θ}{d t} = y

(21)

We can convert equation (17) to the following form:

y \frac{d y}{d θ} + f (θ) = 0

(22)

where

f (θ) = θ - θ^{3}

, or equivalently in the form

y d y + f (θ) d θ = 0

(23)

Equation (23) leads to the following:

\frac{1}{2} y^{2} + F (θ) = h

(24)

where

h = F (θ_{0})

θ_{0} = θ (0) = A

\frac{d F (θ)}{d θ} = f (θ)

Solving y form equation (24), we have

\frac{d θ}{d t} = y = \pm \sqrt{2 F (θ_{0}) - 2 F (θ)}

(25)

or equivalently in the form

d t = \pm \frac{d θ}{\sqrt{2 F (θ_{0}) - 2 F (θ)}}

(26)

Its period can be calculated as follows:

T = 4 \int_{0}^{θ_{0}} \frac{d θ}{\sqrt{2 F (θ_{0}) - 2 F (θ)}} = 2 \sqrt{2} \int_{0}^{θ_{0}} \frac{d θ}{\sqrt{\int_{0}^{θ_{0}} f (s) d s}} = 2 \sqrt{2} \int_{0}^{A} \frac{d θ}{\sqrt{\int_{0}^{A} f (s) d s}}

(27)

For $f (θ) = θ - θ^{3}$ , so we have

T = 2 \sqrt{2} \int_{0}^{A} \frac{d θ}{\sqrt{\int_{0}^{A} f (s) d s}} = 2 \sqrt{2} \int_{0}^{A} \frac{d θ}{\sqrt{\int_{0}^{A} (s - s^{3}) d s}} = 4 \int_{0}^{A} \frac{d θ}{\sqrt{A^{2} - \frac{1}{2} A^{4} - (θ^{2} - \frac{1}{2} θ^{4})}}

(28)

The above equation can be simplified by introducing a transform, $θ = A \cos t$ ,

\begin{array}{l} T = 4 \int_{0}^{\frac{π}{2}} \frac{A \sin t d t}{\sqrt{A^{2} - \frac{1}{2} A^{4} - (A^{2} \cos^{2} t - \frac{1}{2} A^{4} \cos^{4} t)}} = 4 \int_{0}^{\frac{π}{2}} \frac{\sin t d t}{\sqrt{1 - \frac{1}{2} A^{2} - (\cos^{2} t - \frac{1}{2} A^{2} \cos^{4} t)}} \\ = 4 \int_{0}^{\frac{π}{2}} \frac{d t}{\sqrt{1 - \frac{1}{2} A^{2} (1 + \cos^{2} t)}} = 4 \int_{0}^{\frac{π}{2}} \frac{d t}{\sqrt{1 - A^{2} + \frac{1}{2} A^{2} \sin^{2} t}} \\ = 4 \int_{0}^{\frac{π}{2}} \frac{d t}{\sqrt{(1 - A^{2}) + (1 - A^{2}) \frac{A^{2}}{2 (1 - A^{2})} \sin^{2} t}} = \frac{4}{\sqrt{1 - A^{2}}} \int_{0}^{\frac{π}{2}} \frac{d t}{\sqrt{1 + k \sin^{2} t}} \end{array}

(29)

where

k = \frac{A^{2}}{2 (1 - A^{2})}

For different values of A, the exact period can be calculated according to equation (29), see Table 1.

Table 1.

Comparison of the exact period with the approximate one obtained by He’s frequency formulation.

A	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
Exact period	6.3069	6.3797	6.5073	6.7005	6.9783	7.3750	7.9571	8.8765	10.6192
Approximate period	6.3069	6.3796	6.5066	6.6979	6.9706	7.3539	7.9004	8.7132	10.0291
Relative error	0.00%	0.00%	0.01%	0.04%	0.11%	0.29%	0.71%	1.84%	5.56%

From Table 1, we can also see the high accuracy of the approximate period obtained by He’s frequency formulation. The error becomes large when A tends to 1 because A = 1 is the singularity point.

Discussion and conclusion

A pendulum system leads always to a chaotic motion, and the chaos synchronization of gyroscope systems play an important role in control theory.^29–32 This paper considers a pendulum system under the magnetic attraction. An approximate periodic solution is obtained by He’s frequency formulation, comparison of the approximate solution with the exact one reveals the mathematical simplify and the physical insight. Considering He’s frequency formulation is much simpler and more effective than other analytical methods, for example, the variational iteration method,³³ the variational method,^34,35 and the homotopy perturbation method,^36,37 we can predict that it can be widely used in engineering, and it has opened the path for a new way to fast and accurate insight into the basic properties of a complex vibration system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work is supported by Shaanxi Provincial Education Department (No. 20JK0716).

ORCID iD

Hongjin Ma

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The periodic solution of a pendulum under magnetic action

Abstract

Keywords

Introduction

Mathematical model

Periodic solution

Exact period

Discussion and conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References