Simple Pendulum Model using Lagrange Equations of Motion


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Simple Pendulum Model using Lagrange Equations of Motion
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1 Simple Pendulum Model using Lagrange Equations of Motion

Simple Pendulum

A common problem in physics and controls is the simple pendulum. The system is easily described with a few parameter:

  • M: Mass at end of pendulum
  • l: Length of pendulum
  • θ: Angular position of pendulum

Start with the Lagrange function (see Lagrange Equation of Motion for Conversative Forces & Lagrange Equations of Motion for NonConservative Forces for more information on Lagrange Equations of Motion) below

LaTeX: L=K-U Lagrange Function

Define Kinetic Energy

LaTeX: K=\frac{1}{2}Ml^2\dot \theta^2 Kinetic Energy

and Potential Energy

LaTeX: U=Mgl\left(1- \cos \theta \right). Potential Energy

Substitute the Kinetic Energy and Potential Energy equations into the Lagrange function. The results is

LaTeX: \begin{alignat}{2}
<p>L & = & K-U \\
& = & \frac{1}{2}Ml^2 \dot \theta^2 - Mgl\left(1- \cos \theta \right)
\end{alignat}. '

Now that we've defined L we can move on to the Lagrange Equation below

LaTeX: \frac{d}{dt}\left( \frac{\partial L}{\partial \dot \theta} \right)-\frac{\partial L}{\partial \theta}=0. Lagrange Equation

To make things simple we will determine LaTeX: \frac{\partial L}{\partial \dot \theta} and LaTeX: \frac{\partial L}{\partial \theta}. Let's begin with

LaTeX: \begin{alignat}{2}
<p>\frac{\partial L}{\partial \dot \theta} & = & \frac{\partial}{\partial \dot \theta}\left(\left(\frac{1}{2}Ml^2\right)\dot \theta^2+Mgl \cos \theta - Mgl\right) \\
& = & \left(\frac{1}{2}Ml^2\right)\frac{\partial}{\partial \dot \theta}\left(\dot \theta^2\right) \\
& = & Ml^2 \dot \theta
\end{alignat}. '


LaTeX: \begin{alignat}{2}
<p>\frac{\partial L}{\partial \theta} & = & \frac{\partial}{\partial \theta}\left(\left(\frac{1}{2}Ml^2\right)\dot \theta^2+Mgl \cos \theta - Mgl\right) \\
& = & \frac{\partial}{\partial \theta}\left(Mgl \cos \theta \right) \\
& = & -Mgl \sin \theta
\end{alignat}. '

Put these back into the Lagrange Equation to get

LaTeX: \begin{alignat}{2}
<p>\frac{d}{dt}\left( \frac{\partial L}{\partial \dot \theta} \right)-\frac{\partial L}{\partial \theta} & = & 0 \\
& = & \frac{d}{dt}\left(Ml^2 \dot \theta\right)+Mgl \sin \theta \\
& = & Ml^2 \ddot \theta + Mgl \sin \theta \\
\end{alignat}. '

And ultimately we end up with a final Simple Pendulum model via the Lagrange Equations of Motion of

LaTeX: \begin{alignat}{2}
<p>\ddot \theta & = & \frac{-Mgl \sin \theta}{Ml^2} \\
& = & -\frac{g}{l} \sin \theta
\end{alignat} '

2 Notes

  • Zak, Stanislaw H. Systems and Control. Oxford University Press, New York, 2003. ISBN 0195150112.

2.1 References