Approximations for the period of the simple pendulum based on the arithmetic-geometric mean

Carvalhaes, Claudio G.; Suppes, Patrick

doi:10.1119/1.2968864

Cited by 55 publications

(86 citation statements)

References 24 publications

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“…10 convergence of the arithmetic-geometric means is quadratic, an agreement of about 2 n digits between the means is expected after n iterations [24].…”

Section: A Closed-form Expression For the Approximate Frequency In Tementioning

confidence: 99%

“…To do this we take into account that the complete elliptic integral of the first kind K(m) (Eq. (22)) cannot be expressed in terms of elementary functions, but can be numerically evaluated with high precision by a simple procedure based on the arithmetic-geometric mean because the arithmetic-geometric mean is the basis of Gauss' method for the calculation of elliptic integrals [24][25][26]. Because the…”

Section: A Closed-form Expression For the Approximate Frequency In Tementioning

confidence: 99%

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An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

et al. 2009

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The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this paper predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closedform expression for the approximate frequency is obtained in terms of elementary functions.To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.

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“…10 convergence of the arithmetic-geometric means is quadratic, an agreement of about 2 n digits between the means is expected after n iterations [24].…”

Section: A Closed-form Expression For the Approximate Frequency In Tementioning

confidence: 99%

Section: A Closed-form Expression For the Approximate Frequency In Tementioning

confidence: 99%

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

et al. 2009

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“…The nonlinear differential equation for a simple pendulum can be solved exactly and the period and periodic solution expressions involve the complete elliptic integral of the first kind and the Jacobi elliptic functions, respectively [2,4,5]. For this reason, several approximation schemes have been developed to investigate the situation for large amplitude oscillations of a simple pendulum, and several approximations for its largeangle period have been proposed (a summary of most of them can be found in [3,[6][7][8][9][10]) that differ in complexity and domains of validity: some are intuitive, others use ingenuous strategies, while yet others are based on relatively sophisticated procedures [11]. These approximate expressions for the period of a simple pendulum may be obtained in three ways, and in some cases the same results are reached.…”

mentioning

confidence: 99%

Approximate expressions for the period of a simple pendulum using a Taylor series expansion

Beléndez¹,

Garde²,

Márquez³

et al. 2011

Eur. J. Phys.

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An approximate scheme for obtaining the period of a simple pendulum for large amplitude oscillations is analysed and discussed. When students express the exact frequency or the period of a simple pendulum as a function of the oscillation amplitude, and they are told to expand this function in a Taylor series, they always do so using the oscillation amplitude as the variable, without considering that if they change the variable (in this paper to the new variable m), a different Taylor series expansion may be done which is in addition more accurate than previously published ones. Students tend to believe that there is one and only one way of performing a Taylor series expansion of a specific function. The approximate analytical formula for the period is obtained by means of a Taylor expansion of the exact frequency taking into account the KiddFogg formula for the period. This approach based on the Taylor expansion of the frequency about a suitable value converges quickly even for large amplitudes. We believe that this method may be very useful for teaching undergraduate courses on classical mechanics and helping students understand nonlinear oscillations of a simple pendulum.

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“…The system nonlinearities become apparent in the energy dependent natural frequency ω ϑ (ϑ E ). No analytic solution exists for the natural frequency, but it can be calculated numerically by the arithmetic-geometric mean [18]. The nonlinear nature of simple pendulums is also visible in its phase space.…”

Section: A the Abstract Simple Pendulummentioning

confidence: 99%

Fundamental dynamics based adaptive energy control for cooperative swinging of complex pendulum-like objects

Donner

Christange

Buss

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-We present an adaptive energy-based controller for cooperative swinging of complex pendulum-like objects by two agents. The complex pendulum-like object has two oscillation degrees of freedom of which one is to be controlled to a goal energy and the other one has to be damped. The closed-loop fundamental dynamics of a simple pendulum are extended to the two-agent pendulum. Based on the fundamental dynamics, a simple adaptive mechanism identifies the natural frequency of the system. An amplitude factor is adapted such that an agent behaves as a leader or a follower. A leader knows the goal energy and controls the system according to desired reference dynamics. A follower either imitates the leader's energy flow or estimates the leader's goal energy. Properties as stability of both leader-follower structures are analyzed analytically based on the fundamental dynamics assumption. The applicability of the control approaches to a complex pendulum-like object is shown in simulation.

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Approximations for the period of the simple pendulum based on the arithmetic-geometric mean

Cited by 55 publications

References 24 publications

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

Approximate expressions for the period of a simple pendulum using a Taylor series expansion

Fundamental dynamics based adaptive energy control for cooperative swinging of complex pendulum-like objects

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