Fast converging exact power series for the time and period of the simple pendulum

Benacka, Jan

doi:10.1088/1361-6404/aa543f

Cited by 9 publications

(16 citation statements)

References 19 publications

(17 reference statements)

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“…We should note that the series in (3) is the same as in [14]. Whereas the substitution used in [14] appears inspired, we now see that it turns out to be a particular example of the more general method described here. To further demonstrate the general method of expanding about a typical value, we proceed to another example of interest.…”

Section: Simple Pendulummentioning

confidence: 87%

Improving series convergence: the simple pendulum and beyond

Duki

Doerr

2018

Eur. J. Phys.

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A simple and easy to implement method for improving the convergence of a power series is presented. We observe that the most obvious or analytically convenient point about which to make a series expansion is not always the most computationally efficient. Series convergence can be dramatically improved by choosing the center of the series expansion to be at or near the average value at which the series is to be evaluated. For illustration, we apply this method to the well-known simple pendulum and to the Mexican hat type of potential. Large performance gains are demonstrated. While the method is not always the most computationally efficient on its own, it is effective, straightforward, quite general, and can be used in combination with other methods.

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Section: Simple Pendulummentioning

confidence: 87%

Improving series convergence: the simple pendulum and beyond

Duki

Doerr

2018

Eur. J. Phys.

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“…Since the exact periodic solution to the pendulum motion is in terms of special functions, the goal of analytical studies is to provide approximate solutions in terms of elementary functions. 7 Consequently, numerous studies have been carried out to formulate approximate analytical solutions to the undamped oscillations of the simple pendulum (see refs. 1,2,[6][7][8][9][10][11][12][13] and many of the cited articles in these references).…”

Section: Introductionmentioning

confidence: 99%

“…7 Consequently, numerous studies have been carried out to formulate approximate analytical solutions to the undamped oscillations of the simple pendulum (see refs. 1,2,[6][7][8][9][10][11][12][13] and many of the cited articles in these references). Most studies concentrate on finding approximate analytical expressions for the time period in terms of the oscillation amplitude, [6][7][8]10,11 while a few other studies 1,2,9,12,13 have attempted to find approximate solutions for the oscillation histories as well.…”

Section: Introductionmentioning

confidence: 99%

“…The implication is that the existing methods would normally require a complementary method to provide solutions for those amplitudes that they do not cover. For example, most existing methods 6–8,10–13 cannot give accurate results for both the time period and oscillations histories of ‘extremely’ large amplitudes ( 175°<θ0<180°) except for the solution provided by Butikov. 9 However, Butikov’s solution is not applicable for all possible amplitudes of the simple pendulum but limited to extremely large amplitudes.…”

Section: Introductionmentioning

confidence: 99%

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Approximate periodic solution for the large-amplitude oscillations of a simple pendulum

Big‐Alabo

2019

International Journal of Mechanical Engineering Education

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This paper presents approximate periodic solutions to the anharmonic (i.e. not harmonic or non-sinusoidal) response of a simple pendulum undergoing moderate- to large-amplitude oscillations. The approximate solutions were derived by using a modified continuous piecewise linearization method that enabled very accurate solutions to the pendulum oscillations for the entire range of possible amplitudes i.e. [Formula: see text]. The present solution method is very simple and can be used to obtain amplitude-frequency solutions as well as the displacement and velocity histories of the simple pendulum without the need for a complementary method. The purpose of this paper is to present simple and accurate approximate analytical solutions to the large-amplitude oscillations of the simple pendulum that can be applied by undergraduates.

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“…The arithmetic-geometric mean algorithm has been used to derive a sequence of approximate solutions for the period of a simple pendulum [6], and accurate approximate expressions have also been derived for the solution of the equation of motion of a simple pendulum [7]. An approximate analytical formula for the period has been obtained using the power series expansion [8] [9] [10] [11] [12]. An exact solution of the equation of motion of a pendulum using a function has also been proposed.…”

Section: Introduction 1pendulummentioning

confidence: 99%

Pendulum analysis by leaf functions and hyperbolic leaf functions

Shinohara

2019

ams

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The mathematical model representing the equation of motion of a pendulum is nonlinear. Solutions that satisfy the equation cannot be represented by elementary functions, such as trigonometric functions. To solve such problems, it is common to linearize the nonlinear equations and derive approximate numerical solutions and exact solutions. Applying such linearization is limited to cases in which the angle of the pendulum is relatively small. In cases where the angle of the pendulum is large, various methods have been presented that rely on numerical solutions and the exact solutions based on Jacobian elliptic functions, to name one example. On the other hand, the author has been studying a certain type differential equation. The second derivative of a function is equal to the term multiplied by −n(or n) for terms whose function is raised to 2n − 1. Curves based on solving this differential equation are constructed as regular waves with periods. The author has termed the function satisfying this differential equation the leaf function (or the hyperbolic leaf function). In this paper, we attempt to apply this leaf function (or hyperbolic leaf function) to undamped pendulums and overdamped pendulums for large angles.

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Fast converging exact power series for the time and period of the simple pendulum

Cited by 9 publications

References 19 publications

Improving series convergence: the simple pendulum and beyond

Improving series convergence: the simple pendulum and beyond

Approximate periodic solution for the large-amplitude oscillations of a simple pendulum

Pendulum analysis by leaf functions and hyperbolic leaf functions

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