Non-linear Pendulum

  • #1
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If I calculate the time period of a non linear pendulum using elliptical integral equation, then how can I find out the angular displacement.
 

Answers and Replies

  • #2
berkeman
Mentor
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  • #3
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The differential equation which represents the motion of a simple pendulum is
36e0d601a33a7562dfb162abd7e58859a40ccff1

If it is assumed that the angle is much less than 1 radian
e631c25b3635b429ab40642a8c2e2c23ff28fa87

and
above equation becomes
a4a67eb92d55d29ed4df138689010456418a3b15


Given the initial conditions θ(0) = θ0 and /dt(0) = 0, the solution becomes

c87ee425bd0a91e0e2b11316a3fa755c474e6e77

The period of the motion, the time for a complete oscillation is
93d0e7e6fb1df1c2541d6fabbfba15924e35cde4

For amplitudes beyond the small angle approximation,time period can be calculated using formula:
9b33d77af48c6b672509da8394e0e7387675db8e

K is the complete elliptic integral of the first kind defined by
7b79d77a3118e9cd02c82ee78498c50d71405646


So, If somehow I am able to calculate the Time period for non linear Pendulum, then how can I plot the angular displacement θ(t) against time.
 
  • #4
Baluncore
Science Advisor
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I'm not certain exactly what you want but there are some interesting articles here on the period of extreme precision pendulum clocks. The articles include useful references.
http://www.leapsecond.com/hsn2006/
This is a discussion of the circular error correction of period using AGM.
http://www.leapsecond.com/hsn2006/pendulum-period-agm.pdf
You may be able to invert the equation in his conclusion.
Unfortunately they appear to be transcendental solutions.

Following that are pendulum simulations that model displacement against time.
http://www.leapsecond.com/hsn2006/pendulum-simulation-1.pdf
Perhaps there is a solution using the model, conversion between PE and KE.
 
  • #5
DrClaude
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There is no simple analytic formulation of ##\theta(t)## for the non-linear pendulum.

Check out https://en.wikipedia.org/wiki/Pendulum_(mathematics). You will find there how the period is found to be an elliptical integral, without having to know ##\theta(t)##. There is also an expression for ##\theta(t)## in terms of a Fourier series.
 
  • #6
DrClaude
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There is no simple analytic formulation of ##\theta(t)## for the non-linear pendulum.

Check out https://en.wikipedia.org/wiki/Pendulum_(mathematics). You will find there how the period is found to be an elliptical integral, without having to know ##\theta(t)##. There is also an expression for ##\theta(t)## in terms of a Fourier series.
 

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