Fourier analysis of nonlinear pendulum oscillations

Singh, Inderpreet; Arun, P.; Lima, F. M. S.

doi:10.1590/1806-9126-rbef-2017-0151

Cited by 3 publications

(4 citation statements)

References 20 publications

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“…There in that work, I did show that such harmonic approximation is practically indistinguishable from the exact solution θ(t) for amplitudes below π/4 rad, deviating significantly from it only for amplitudes above π/2 rad. This 'loss of harmonicity' has been confirmed by numerical Fourier analysis in a very recent work [24]. On searching for better periodic approximations, I have noted that the elliptic function sn(u; k) which composes the exact solution established in Eq.…”

Section: Periodic Approximate Solutionsmentioning

confidence: 69%

“…3. Then, I have found it natural to take the periodicity of the exact solution into account to derive an analytical Fourier series expansion, giving continuity to a numerical treatment I have developed (with co-authors) in a very recent work [24]. This task revealed many complexities due to the inverse sine function present in the exact solution, our Eq.…”

Section: Discussionmentioning

confidence: 99%

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Simple but accurate periodic solutions for the nonlinear pendulum equation

Lima

2018

Rev. Bras. Ensino Fís.

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Despite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn(u; k) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes.

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Section: Periodic Approximate Solutionsmentioning

confidence: 69%

Section: Discussionmentioning

confidence: 99%

Simple but accurate periodic solutions for the nonlinear pendulum equation

Lima

2018

Rev. Bras. Ensino Fís.

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“…Fourier series analysis of a simple pendulum is used in the determination of transition from linear to nonlinear oscillations [11]. In previous works, the response plot [12], the phase portrait [13], the Poincaré map [10], the power spectrum [14], the Lyapunov exponent [15] and the bifurcation diagram [6] have been used to detect chaos in dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Energy dissipation and behavioral regimes in an autonomous double pendulum subjected to viscous and dry friction damping

Sawant¹,

Shrikanth²

2021

Eur. J. Phys.

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The nonlinear dynamics of a system of a compound double pendulum are experimentally analyzed and numerically verified. The experiments are carried out for a combination of initial angles in the range 0 ⩽ θ 0, ϕ 0 ⩽ 180°, with zero initial angular velocity and with an increment of 1°. The motion of the pendula is captured in the video and the individual pendula are tracked using image processing. The energy dissipation in the system is evaluated using the Rayleigh dissipation function to include non-conservative forces as well, as a lumped parameter in the Lagrangian equation. Furthermore, the experimental and the numerical results are brought to agreement by tuning the dissipation factors. The linear, nonlinear and chaotic behavior of the system is classified in the coordinate plane. The behavioral regime represents the dependency of the nature of the system on the total energy and the difference in instantaneous amplitudes of the pendula. Uncertainty in the behavior of the system is observed in the transition region from the nonlinear to the chaotic regime due to the sensitivity of the system to the initial conditions. The dependence of the dissipation factors on the angular displacements is represented as a surface plot, which highlights the dominance of viscosity in the higher amplitude oscillations and the dominance of dry friction in the lower amplitude oscillations. This phenomenon is established by observing the nature of decay in the response plots of the pendula.

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“…Deformation of an Elastica. The transverse deformation of a thin elastic extensional rod subjected to an axial loading and clamped at its ends is governed by the nonlinear equation[32]…”

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confidence: 99%

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

et al. 2021

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In the past decade, various types of wavelet-based algorithms were proposed, leading to a key tool in the solution of a number of numerical problems. This work adopts the Chebyshev wavelets for the numerical solution of several models. A Chebyshev operational matrix is developed, for selected collocation points, using the fundamental properties. Moreover, the convergence of the expansion coefficients and an upper estimate for the truncation error are included. The obtained results for several case studies illustrate the accuracy and reliability of the proposed approach.

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Fourier analysis of nonlinear pendulum oscillations

Cited by 3 publications

References 20 publications

Simple but accurate periodic solutions for the nonlinear pendulum equation

Simple but accurate periodic solutions for the nonlinear pendulum equation

Energy dissipation and behavioral regimes in an autonomous double pendulum subjected to viscous and dry friction damping

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

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