An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime

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

doi:10.1119/1.2215616

Cited by 85 publications

(90 citation statements)

References 14 publications

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“…The approximation is more accurate than other simple relations. Good agreement with experimental data is verified [Lima F. M. S., 2006]. In 2007 Siboni studies the period of a pendulum can be accurately determined by an arithmetic-geometric map.…”

Section: Introductionsupporting

confidence: 59%

Study of Experimental Simple Pendulum Approximation Based on Image Processing Algorithms

Kamil¹,

Al-Zuky²,

Al-Tawil³

2011

APR

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In this study we used image processing algorithms to determine pendulum ball motion and present an approach for solving the nonlinear differential equation that governs its movement. This formulae show an excellent agreement with the exact period calculated with the use of elliptical integrals, and they are valid for both small and large amplitudes of oscillation.Also, we reveal some interesting aspects of the study simple pendulum or procedures for determining analytical approximations to the periodic solutions of nonlinear differential equations.

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Section: Introductionsupporting

confidence: 59%

Study of Experimental Simple Pendulum Approximation Based on Image Processing Algorithms

Kamil¹,

Al-Zuky²,

Al-Tawil³

2011

APR

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“…But the phase shadow plot has advantages not only when the system's behavior has a limit cycle, but also to display equilibrium points [Shamolin, 2009], bifurcations [Lima & Arun, 2006;Luo & Guo, 2016] and chaotic behavior [Han et al, 2013]. This statement will be justified in the following subsections.…”

Section: Resultsmentioning

confidence: 99%

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

Luque

Barbancho

Cañete

et al. 2017

Int. J. Bifurcation Chaos

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Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360• . In these cases the use of the phase portrait does not properly depict the system's evolution. Normalized phase portraits or cylindrical phase portraits have been extensively used to overcome the original phase portrait's disadvantages. In this research a new graphic representation is introduced: the phase shadow. Its use clearly reveals the system behavior while overcoming the drawback of the existing plots. Through the paper the method to obtain the graphic is stated. Additionally, to show the phase shadow's expressiveness, a rotating pendulum is considered. The work exposes that the new graph is an enhanced representational tool for systems having equilibrium points, limit cycles, chaotic attractors and/or bifurcations.

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“…Because in what follows we shall need to refer to various quantity algebras, we prefer to reformulate the channel-cochannel duality in its above-displayed adjointness form. 9 We take the two-way map (C, C * ) (shown above) to be a fundamental concept in its own right that captures the information-theoretic essence of instrument-based measurement.…”

Section: Banach Algebra Approach To Physical Measurementmentioning

confidence: 99%

“…For that we need an interpretational framework, 10 which secures a concrete physical meaning of quantities, their data propositions and laws -enabling a validation of propositions using measurement outcomes. Another way to express this is to say that in addition to the foregoing syntactic algebraic quantity and state apparatus, intended for effective theoretical analysis, we need their mathematically concrete realizations, implementations or distinguished representations, which allow us to designate a specific frame or basis, in which all pertinent physical variables and coordinates can be fixed for cal- 9 We shall interpret channel C * ontologically as a forward model of a measurement process that represents the causal direction of information flow from the target system to a measuring instrument. And its dual cochannel C is best thought of epistemically as an inverse model, representing the measurand's reconstruction or estimation operation, acting on its pointer quantity.…”

Section: Banach Algebra Approach To Physical Measurementmentioning

confidence: 99%

“…In one implementation, equally spaced laser beams are directed perpendicularly to the pendulum's plane of motion, with electric sensors on its opposite side. Design specifics of this and related position measurement methods may be found in [9] and the references therein. Since we may know the response of this stroboscopic measuring system without knowing the technical details of its optical and electrical dynamics, we shall assume that the value space Val( p) of pointer quantity p (calibrated in angular or length units) of p is a subset of discrete points in C. This means that since the pendulum's positions are measured at discrete time instants stroboscopically with limited accuracy, we need a spatial discretization of the circle group C of positions, specified by the discrete set C ε = d f εN ∩ C = {0, ε, 2ε, · · · } of possible angular pendulum positions on a circle, where ε > 0 is a fixed discretization parameter, chosen in arc or length units, e.g., one arc second or one millimeter.…”

Section: Discrete Measurement Of Angular Positions Of a Physical Pendmentioning

confidence: 99%

See 1 more Smart Citation

Measurement, Information Channels, and Discretization: Exploring the Links

Domotor¹,

Batitsky²

2009

Measurement Science Review

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The goal of this paper is to present a unified algebraic-analytic framework for (static and dynamic) deterministic measurement theory, which we find to be fully adequate in engineering and natural science applications. The starting point of this paradigm is the notion of a quantity algebra of a measured system and that of a measuring instrument, underlying the causal linkages in classical 'system + instrument' interactions. This approach is then further enriched by providing a superimposed data lattice of measurement outcomes, intended to handle the information flow from the measured system to its measurand's designated instrument. We argue that the language of Banach and von Neumann algebras is ideally suited for the treatment of quantities, encountered in theoretical and experimental science. These algebras and convex spaces of expectation functionals thereon together with information (co)channels between them provide a comprehensive information-theoretic framework for measurement theory. Concrete examples and applications to length and position measurements are also discussed and rigorously framed within the proposed quantity algebra and associated information channel paradigms. In modeling physical systems, investigators routinely rely on the assumption that state spaces and time domains form a continuum (locally homeomorphic to the real line or its Cartesian powers). But in sharp contrast, measurement and prediction outcomes pertaining to physical systems under consideration tend to be presented in terms of small discrete sets of rational numbers. We investigate this conceptual gap between theoretical and finitary data models from the perspectives of temporal, spatial and algebraic discretization schemes. The principal innovation in our approach to classical measurement theory is the representation of interactive instrument-based measurement processes in terms of channel-cochannel pairs constructed between dynamical quantity algebras of a target system and its measurand's measuring instrument.

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An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime

Abstract: A simple approximate formula is derived for the dependence of the period of a simple pendulum on the amplitude. The approximate is more accurate than other simple formulas. Good agreement with experimental data is verified.

Cited by 85 publications

References 14 publications

Study of Experimental Simple Pendulum Approximation Based on Image Processing Algorithms

Study of Experimental Simple Pendulum Approximation Based on Image Processing Algorithms

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

Measurement, Information Channels, and Discretization: Exploring the Links

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