Analytical approaches to fractal oscillator for a pendulum with a rolling wheel

Abstract

This study considers two analytical approaches for solving a fractal oscillator modelling a pendulum incorporated with a rolling wheel. By applying fractal variational principle, the approximated frequency and fractal periodic solution are given. Approximations are also derived by using He’s frequency formulation. The accuracy of both methods is validated through comparisons with numerical solutions obtained via Runge-Kutta method and spreading residue harmonic balance method. Numerical results demonstrate that the fractal variational method outperforms He’s frequency formulation and spreading residue harmonic balance method for the conventional oscillators with large parameters. Sensitivity analysis of the approximated frequencies provides insights into the oscillator’s stability characteristics. Furthermore, numerical behaviour of the fractal periodic solutions with various amplitudes and fractional orders is also investigated.

Keywords

oscillator frequency approximation variational principle He’s frequency formulation

Introduction

The study of fractal calculus has become a hot topic for its wide applications in engineering and scientific areas. By theory of fractal calculus, the linear or nonlinear problems from oscillation and fluid fields can be modelled by using the fractal differential equations.^1–5 Different definitions of fractal derivative have been proposed for the fractal differential equations in the literature, including Chen’s fractal Derivative, He’s fractal derivative and Parvate-Gangal’s fractal derivative and others. Chen proposed a fractal derivative based on Hausdorff derivative for modelling anomalous diffusion.^6,7 He’s fractal derivative was used to simulate the phenomena in hierarchical or porous media.⁸ The dynamical system involving fractal space or time can be modelled by applying Parvate-Gangal’s fractal derivative.⁹ Due to the complex property of the fractal differential equations, it is challengeable to obtain the exact solutions. Therefore, many scholars focused on the approximated or numerical solutions to different fractal differential equations. For the fractal oscillation systems, some efficient approaches have been proposed, including harmonic method,^10,11 variational principle,¹² He’s frequency formulation approach,^13,14 multiple time scales technique,¹⁵ Adams-Bashforth predictor-corrector approach,¹⁶ multi-step differential transformation method^17,18 and homotopy perturbation method¹⁹ and others.

In reference 20, Zheng et al. proposed a fractal oscillator for modelling the oscillation behaviour of a pendulum attached to a rolling wheel. When the fractional order of the fractal derivative is equal to one, it corresponds to the conventional oscillator in reference 21. Moatimid and Amer suggested an approximated oscillator for approximately modelling the behaviour of the corresponding pendulum in reference 21. The approximated frequencies and periodic solutions were presented by using the combining technique based upon homotopy perturbation method and Laplace transform. For improving the accuracy of the oscillation system in reference 21, Zheng et al. proposed the fractal complex transformation coupled with spreading residue harmonic balance method for deriving the approximations and characterizing the fractal oscillation behaviour. Parametric sensitivity analysis of frequency and solution approximations is essential to characterize the oscillator’s nonlinear dynamics. While prior study²⁰ has analyzed amplitude-dependent approximations, the sensitivity of frequency and solution approximations to other critical parameters remains underexplored. Furthermore, existing methods such as spreading residue harmonic balance method exhibit declining accuracy for the considered oscillation system with large parameters, limiting their utility in strongly nonlinear regimes. These two issues are the main motivations of this paper, and we will further consider the fractal oscillator in reference 20 by using some new effective approaches. Variational approach proposed by He is an efficient method for solving different nonlinear oscillators.^12,22 This approach together with fractal derivative can be used to investigate the fractal partial differential equations, such as fractal circular sector oscillator²³ and fractal N/MEMS system^2,12 and others. He’s frequency formulation can be used for analyzing the fractal or fractional oscillation systems.^13,14 We will consider these two methods for solving the fractal pendulum oscillator proposed in reference 20. The key contribution of this study lies in its novel approach to deriving fractal approximations through straightforward computations, along with the sensitivity analysis of both the approximated solutions and corresponding frequencies, thereby assessing the methods’ robustness and applicability. By using fractal variational principle defined by variational derivative, the approximated frequency and fractal periodic solution are given by simple calculation. The approximations to the fractal system are also obtained by using He’s frequency formulation approach. A comparative benchmarking framework incorporating Runge-Kutta method, spreading residue harmonic balance method,^20,24 variational principle and He’s frequency formulation demonstrates the computational efficacy of the developed variational framework and frequency formulation approach. Numerical results illustrate the superior efficacy of the variational approach over both spreading residue harmonic balance method and He’s frequency formulation, particularly under high-value parametric regimes in both conventional and fractal oscillator models. Frequency analysis about the amplitude and other parameters is presented to investigate the numerical properties of the approximated frequencies. These insights will be useful to establish foundational criteria for ensuring the oscillatory stability. Finally, concluding remarks summarize principal findings and outline future research directions.

Fractal pendulum oscillation model

A pendulum attached to a rolling wheel that is restricted by a lightweight spring was considered in references 20, 21, see Figure 1. For clarity the considered pendulum is assumed to have a length l and a lumped mass m, and its angular displacement is denoted by θ. Here k and r are the stiffness of the spring and the radius of the rolling wheel, respectively.

Figure 1.

Pendulum model.

In order to obtain the equation of motion of the pendulum, we consider the kinetic energy as follows:

T = \frac{m}{2} (r^{2} + l^{2} + 2 r l \cos θ) {(θ^{'})}^{2} .

(1)

The potential energy for this nonlinear system can be given by

V = \frac{1}{2} k {(r θ)}^{2} - m g l \cos θ .

(2)

Then we can formulate the Lagrangian operator L = T − V as follows:

L = \frac{m}{2} (r^{2} + l^{2} + 2 r l \cos θ) {(θ^{'})}^{2} - \frac{1}{2} k {(r θ)}^{2} + m g l \cos θ .

(3)

The Euler-Lagrange equation is defined by

\frac{d}{d t} (\frac{\partial L}{\partial θ^{'}}) - \frac{\partial L}{\partial θ} = 0 .

(4)

Then by (4), the pendulum oscillator can be formulated as

(r^{2} + 1 + 2 r \cos θ) θ^{″} - r {(θ^{'})}^{2} \sin θ + k r^{2} θ + \sin θ = 0 .

(5)

Zheng et al. considered (5) in a fractal time space by using He’s fractal derivative,²⁰ that is

(r^{2} + 1 + 2 r \cos θ) \frac{d^{2 α} θ}{d t^{2 α}} - r {(\frac{d^{α} θ}{d t^{α}})}^{2} \sin θ + k r^{2} θ + \sin θ = 0,

(6)

with the fractal initial conditions defined by

θ (0) = A, \frac{d^{α} θ}{d t^{α}} (0) = 0 .

(7)

Here the fractal derivative proposed by He^8,25–27 is defined in a fractal time space with a fractal dimension α

\frac{d^{α} θ}{d t^{α}} = Γ (1 + α) \lim_{\begin{matrix} t - t_{0} \to △ t \\ △ t \neq 0 \end{matrix}} \frac{θ (t) - θ (t_{0})}{{(t - t_{0})}^{α}} .

(8)

The fractal order α can be used to illustrate the exceptional index of time scale law. We remark that the function θ is differential about t^α, and the chain rule holds for the fractal operator defined by He’s fractal derivative.

Variational approach for fractal oscillator

In reference 22, He proposed a variational approach by using the variational derivative, which was extensively used for solving nonlinear oscillators. By variational approach,^12,22 we have the following variational formulation for (6)

J (A, ω) = \int (- \frac{1}{2} (r^{2} + 1 + 2 r \cos θ) {(\frac{d^{α} θ}{d t^{α}})}^{2} + \frac{1}{2} k r^{2} θ^{2} - \cos θ) d t^{α} .

(9)

We suppose that the fractal periodic solution exists for (6), and can be given by

θ = A \cos (ω t^{α}),

(10)

where A is an amplitude, and ω is an unknown frequency.

By substituting (10) into (9), it follows that

J (A, ω) = \int (- \frac{1}{2} (r^{2} + 1 + 2 r \cos (A \cos τ)) A^{2} ω \sin^{2} τ + \frac{1}{ω} (\frac{1}{2} k r^{2} A^{2} \cos^{2} τ - \cos (A \cos τ))) d τ .

We can obtain the following equation by using the stationary condition ∂J/∂A = 0

\begin{align} \frac{\partial J}{\partial A} & = \int (- (r^{2} + 1 + 2 r \cos (A \cos τ) - A r \cos τ \sin (A \cos τ)) A ω \sin^{2} τ \\ + \frac{1}{ω} (k r^{2} A \cos^{2} τ + \sin (A \cos τ) \cos τ)) d τ = 0 . \end{align}

(11)

By (11), we have the following formulation

\begin{align} ω^{2} & = \frac{\int_{0}^{\frac{T}{4}} (k r^{2} A \cos^{2} τ + \sin (A \cos τ) \cos τ) d τ}{\int_{0}^{\frac{T}{4}} ((r^{2} + 1 + 2 r \cos (A \cos τ) - A r \cos τ \sin (A \cos τ)) A \sin^{2} τ) d τ} \\ = \frac{\int_{0}^{\frac{π}{2}} (k r^{2} A \cos^{2} τ + \sin (A \cos τ) \cos τ) d τ}{\int_{0}^{\frac{π}{2}} ((r^{2} + 1 + 2 r \cos (A \cos τ) - A r \cos τ \sin (A \cos τ)) A \sin^{2} τ) d τ} \\ = \frac{k r^{2} A + 2 B e s s e l J (1, A)}{A (1 + r^{2} + 2 r B e s s e l J (0, A))}, \end{align}

(12)

where BesselJ (0, A) and BesselJ (1, A) are Bessel functions of first kind. The approximated frequency can be followed from (12)

ω = \sqrt{\frac{k r^{2} A + 2 B e s s e l J (1, A)}{A (1 + r^{2} + 2 r B e s s e l J (0, A))}} .

(13)

By (10) and (13), we can obtain the following fractal periodic solution to (6)

θ = A \cos (\sqrt{\frac{k r^{2} A + 2 B e s s e l J (1, A)}{A (1 + r^{2} + 2 r B e s s e l J (0, A))}} t^{α}) .

(14)

He’s frequency formulation for fractal oscillator

Consider He’s frequency formulation approach^13,14 for the following nonlinear fractal oscillator

\frac{d^{2 α} θ}{d t^{2 α}} + f (θ) = 0, θ (0) = A, \frac{d^{α} θ}{d t^{α}} (0) = 0 .

(15)

According to (6), f (θ) can be formulated as

f (θ) = (r^{2} + 2 r \cos θ) \frac{d^{2 α} θ}{d t^{2 α}} - r {(\frac{d^{α} θ}{d t^{α}})}^{2} \sin θ + k r^{2} θ + \sin θ .

(16)

By He’s frequency formulation suggested in the existing literatures,^13,14 one can obtain

ω^{2} = \frac{f (θ)}{θ} |_{θ = N A, \frac{d^{α} θ}{d t^{α}} = - \sqrt{1 - N^{2}} ω A, \frac{d^{2 α} θ}{d t^{2 α}} = - N A ω^{2}}

(17)

By (17), we have the following formulation

ω^{2} = \frac{k r^{2} N A + \sin (N A)}{N A (1 + r^{2} + 2 r \cos (N A)) + r (1 - N^{2}) A^{2} \sin (N A)} .

(18)

The approximated frequency from (18) can be given by

ω = \sqrt{\frac{k r^{2} N A + \sin (N A)}{N A (1 + r^{2} + 2 r \cos (N A)) + r (1 - N^{2}) A^{2} \sin (N A)}} .

(19)

By (10) and (19), we have the following fractal periodic solution to (6)

θ = A \cos (\sqrt{\frac{k r^{2} N A + \sin (N A)}{N A (1 + r^{2} + 2 r \cos (N A)) + r (1 - N^{2}) A^{2} \sin (N A)}} t^{α}) .

(20)

The parameter N was set as $\sqrt{3} / 2$ suggested by He^14,28 and Feng.²³ When $N = \sqrt{3} / 2$ , it follows from (20) the following fractal periodic solution

θ = A \cos (\sqrt{\frac{2 \sqrt{3} k r^{2} A + 4 \sin (\sqrt{3} A / 2)}{2 \sqrt{3} A (1 + r^{2} + 2 r \cos (\sqrt{3} A / 2)) + r A^{2} \sin (\sqrt{3} A / 2)}} t^{α}) .

(21)

Numerical results

In this section, we test the numerical behaviour of the approximated solutions and frequencies to (6). In order to illustrate the efficiency of the proposed approaches, we compare the approximations obtained by variational principle (Var) and He’s frequency formulation (He) with the solutions by spreading residue harmonic balance method (SRHBM) and Runge-Kutta method (RK). Two different cases including the conventional oscillator (α = 1) and the fractal oscillator with various fractional orders will be investigated.

We first test the numerical behaviour of (6) with α = 1, where k = 10, r = 20 and different A are set in this case. The notations θ_RK, θ_Var, θ_He and θ_SRHBM are used to denote the approximated solutions by RK, Var, He and SRHBM, respectively. We use the log error of the approximation $\hat{θ}$ for measuring its accuracy

E r r_{\hat{θ}} = \log ∣ \frac{θ_{R K} - \hat{θ}}{θ_{R K}} ∣ .

(22)

Figure 2 shows the numerical comparisons of different solutions resulting from the tested methods. The left and right sides of Figure 2 with A = π/2 and fixed k = 10 and r = 20 indicate that the approximations given by three methods agree well with the numerical solutions by RK. Numerical results of the approximations to (5) with A = π are shown in Figure 3. The error curves in the right side of Figure 3 implies that variational approach performs better than He’s formulation and SRHBM. When A = 2π, the approximated frequency by SRHBM is not real. Thus, we only compare the effectiveness of variational principle and He’s formulation method. Numerical results in Figure 4 show the improvement of variational approach over He’s frequency formulation. Figure 5 shows the approximated solutions and errors with A = π, k = 5 and r = 20. The results of the approximations with A = π, k = 10 and r = 15 are plotted in Figure 6. We can find the proposed methods work well when different parameters are used.

Figure 2.

Numerical results of (5) with A = π/2.

Figure 3.

Numerical results of (5) with A = π.

Figure 4.

Numerical results of (5) with A = 2π.

Figure 5.

Numerical results of (5) with A = π, k = 5 and r = 20.

Figure 6.

Numerical results of (5) with A = π, k = 10 and r = 15.

Different with the existing analysis in reference 20, we investigate the sensitivity analysis of the approximated frequencies with respect to different parameters. Surfaces of the approximated ω_Var and ω_He are plotted in the left and right sides of Figure 7, respectively, where 0 ≤ A ≤ 2π and 0 ≤ k ≤ 10 are used. The approximated frequencies with different A and r are shown in Figure 8. Figure 9 gives the surfaces of approximated frequencies with 0 ≤ k ≤ 10 and 0 ≤ r ≤ 20. For further understanding the monotonic property of the approximations about different parameters, we also consider the analysis of ω_Var and ω_He about the amplitude A. Figure 10 shows that the approximated $\hat{ω}$ is concave with respect to the amplitude A. The frequency functions ω_Var and ω_He are monotonic increasing about the variable k, see Figure 11. By Figure 12, the approximated $\hat{ω}$ is also concave about the variable r. When the parameter r is small, the approximated frequency increases rapidly as the value of r increases. Afterwards, the variation of the approximated frequency becomes smoother as r increases.

Figure 7.

Surfaces for ω with various A and k.

Figure 8.

Surfaces for ω with various A and r.

Figure 9.

Surfaces for ω with various k and r.

Figure 10.

Approximated frequency ω by Var and He with different A.

Figure 11.

Approximated frequency ω by Var and He with different k.

Figure 12.

Approximated frequency ω by Var and He with different r.

We then consider the fractal oscillator (6) with various amplitudes used in the fractal case. Figure 13 shows the fractal oscillation behaviour of (6) with A = π/2, k = 10 and r = 20. Precisely, the left, middle and right sides of Figure 13 gives the oscillation results of the fractal solutions with α = 0.9, α = 0.6 and α = 0.3, respectively. By Figure 13, the vibrations of the fractal solutions are basically consistent with the variation of fractal order. The numerical behaviour of (6) with A = π is provided in Figure 14, which indicates that the vibrational behaviour of the three solutions exhibits diffusive phenomenon as the decrease in fractional order occurs. Figure 15 gives the fractal oscillation results of the fractal solutions by variational principle and He’s formulation. The vibration behaviour of two fractal solutions is also similar. Furthermore, Figures 16 and 17 show the oscillation behaviour of the fractal solutions with different k and r. We can conclude from the previous figures that the periodicity of vibration gradually weakens as the fractal order decreases.

Figure 13.

Oscillator behaviour of (6) with A = π/2 (Left: α = 0.9, Middle: α = 0.6, Right: α = 0.3).

Figure 14.

Oscillator behaviour of (6) with A = π (Left: α = 0.9, Middle: α = 0.6, Right: α = 0.3).

Figure 15.

Oscillator behaviour of (6) with A = 2π (Left: α = 0.9, Middle: α = 0.6, Right: α = 0.3).

Figure 16.

Oscillator behaviour of (6) with A = π, k = 5 and r = 20 (Left: α = 0.9, Middle: α = 0.6, Right: α = 0.3).

Figure 17.

Oscillator behaviour of (6) with A = π, k = 10 and r = 15 (Left: α = 0.9, Middle: α = 0.6, Right: α = 0.3).

Conclusions

This paper proposed two efficient approaches for solving the fractal oscillator model derived from a pendulum with a rolling wheel. The key contributions lied in the development of novel solution methods, and numerical investigation and sensitivity analysis of the oscillation behaviour. The approximated frequencies and fractal periodic solutions were given by using fractal variational principle and He’s formulation method, respectively. Numerical comparisons with some existing methods were presented to show the effectiveness of the proposed methods for both conventional and fractal oscillation system. Fractal variational principle performs well when large parameters are used. Sensitivity analysis of the approximated frequencies indicated the monotonic or concave property about various parameters. Theoretical and numerical results will provide valuable insights for assessing the stability of the oscillation system. However, the practical utility and physical interpretation of the obtained fractal periodic solutions require deeper investigation, and the optimal selection of parameter N within He’s frequency formulation is also an open question.^29,30 For the strongly nonlinear oscillation systems, we need to consider the combination approach based on the proposed methods and the approximation technique. Future work will address these aspects and extend the proposed methodologies to broader classes of fractal or fractional oscillators.^31–34

Footnotes

Acknowledgements

The authors appreciated the constructive remarks and suggestions of the anonymous referees, which helped to improve this article.

ORCID iD

Junfeng Lu

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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