Complete Analysis of the Nonlinear Pendulum for Amplitudes in All Regimes Using Numerical Integration

Hafez, Youssef I.

doi:10.17654/ijnmasep2015_053_076

Cited by 2 publications

(3 citation statements)

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“…The absolute values of the percentage deviations 100 T i − T agm /T agm of the period approximations from the exact period are shown for some intermediate equations in figures 4(a) and (b)). The two extra terms in the polynomial expansion due to Fernandes [54] clearly improve the accuracy over the Hafez approximation [14].…”

Section: Medium Amplitude Approximationsmentioning

confidence: 94%

“…However, for some applications a simple approximate formula for the variation of the period with amplitude is useful and such formulae continue to be published. Some of the approximate formulae have previously been compared [11][12][13][14] and the present paper presents a more comprehensive review. The formulae, together with their Taylor series expansions, and their percentage deviations Δ(30 • ) and Δ(90 • ) from the exact period at amplitudes of 30 • and 90 • are listed in table 1.…”

Section: Introductionmentioning

confidence: 99%

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Review of approximate equations for the pendulum period

Hinrichsen¹

2020

Eur. J. Phys.

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Precision measurements of the pendulum period as a function of the amplitude can now be made with a variety of instruments including MEMs gyro/accelerometers, and thus theoretical expressions are required for comparison. Unfortunately exact solution of the pendulum equation involves elliptic integrals, which cannot be expressed in terms of elementary functions, and therefore a wide variety of approximations have been published. These range from simple single-term formulae to more sophisticated equations, which apply to a wider range of amplitudes, to an iterative procedure for calculating the precise period. The published approximations are compared as Taylor series expansions, and graphically to indicate their accuracy and their regions of applicability.

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Section: Medium Amplitude Approximationsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Review of approximate equations for the pendulum period

Hinrichsen¹

2020

Eur. J. Phys.

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show abstract

“…Closed-form solutions are available for the single undamped pendulum [2] and the damped pendulum [3], and have also been found for the periodic and the quasiperiodic motion of the double pendulum [4]. Several numerical studies have been documented on the behavior of single-pendulum [5], double-pendulum [6,7] and triple-pendulum systems [8]. A double pendulum exhibits chaotic behavior in a four-dimensional phase space, which has been experimentally reported for a system of compound double pendula with slender bars and asymmetrically oriented square plates as its elements [9,10].…”

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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Complete Analysis of the Nonlinear Pendulum for Amplitudes in All Regimes Using Numerical Integration

Cited by 2 publications

References 0 publications

Review of approximate equations for the pendulum period

Review of approximate equations for the pendulum period

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

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