# An analytical solution to the equation of motion for the damped nonlinear pendulum

Kim Johannessen

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### Resumé

An analytical approximation of the solution to the differential equation
describing the oscillations of the damped nonlinear pendulum at large angles
is presented. The solution is expressed in terms of the Jacobi elliptic functions
by including a parameter-dependent elliptic modulus. The analytical solution
is compared with the numerical solution and the agreement is found to be
very good. In particular, it is found that the points of intersection with the
abscissa axis of the analytical and numerical solution curves generally differ
by less than 0.1%. An expression for the period of oscillation of the damped
nonlinear pendulum is presented, and it is shown that the period of oscillation
is dependent on time. It is established that, in general, the period is longer than
that of a linearized model, asymptotically approaching the period of oscillation
of a damped linear pendulum.
Originalsprog Engelsk European Journal of Physics 35 3 13 0143-0807 https://doi.org/10.1088/0143-0807/35/3/035014 Udgivet - 2014

### Citer dette

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title = "An analytical solution to the equation of motion for the damped nonlinear pendulum",
abstract = "An analytical approximation of the solution to the differential equationdescribing the oscillations of the damped nonlinear pendulum at large anglesis presented. The solution is expressed in terms of the Jacobi elliptic functionsby including a parameter-dependent elliptic modulus. The analytical solutionis compared with the numerical solution and the agreement is found to bevery good. In particular, it is found that the points of intersection with theabscissa axis of the analytical and numerical solution curves generally differby less than 0.1{\%}. An expression for the period of oscillation of the dampednonlinear pendulum is presented, and it is shown that the period of oscillationis dependent on time. It is established that, in general, the period is longer thanthat of a linearized model, asymptotically approaching the period of oscillationof a damped linear pendulum.",
author = "Kim Johannessen",
year = "2014",
doi = "10.1088/0143-0807/35/3/035014",
language = "English",
volume = "35",
journal = "European Journal of Physics",
issn = "0143-0807",
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}

I: European Journal of Physics, Bind 35, Nr. 3, 2014.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - An analytical solution to the equation of motion for the damped nonlinear pendulum

AU - Johannessen, Kim

PY - 2014

Y1 - 2014

N2 - An analytical approximation of the solution to the differential equationdescribing the oscillations of the damped nonlinear pendulum at large anglesis presented. The solution is expressed in terms of the Jacobi elliptic functionsby including a parameter-dependent elliptic modulus. The analytical solutionis compared with the numerical solution and the agreement is found to bevery good. In particular, it is found that the points of intersection with theabscissa axis of the analytical and numerical solution curves generally differby less than 0.1%. An expression for the period of oscillation of the dampednonlinear pendulum is presented, and it is shown that the period of oscillationis dependent on time. It is established that, in general, the period is longer thanthat of a linearized model, asymptotically approaching the period of oscillationof a damped linear pendulum.

AB - An analytical approximation of the solution to the differential equationdescribing the oscillations of the damped nonlinear pendulum at large anglesis presented. The solution is expressed in terms of the Jacobi elliptic functionsby including a parameter-dependent elliptic modulus. The analytical solutionis compared with the numerical solution and the agreement is found to bevery good. In particular, it is found that the points of intersection with theabscissa axis of the analytical and numerical solution curves generally differby less than 0.1%. An expression for the period of oscillation of the dampednonlinear pendulum is presented, and it is shown that the period of oscillationis dependent on time. It is established that, in general, the period is longer thanthat of a linearized model, asymptotically approaching the period of oscillationof a damped linear pendulum.

U2 - 10.1088/0143-0807/35/3/035014

DO - 10.1088/0143-0807/35/3/035014

M3 - Journal article

VL - 35

JO - European Journal of Physics

JF - European Journal of Physics

SN - 0143-0807

IS - 3

ER -