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} 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} \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}. '

Then

 LaTeX: \begin{alignat}{2} \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} \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} \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.