Lecture 09&10 - Rigid Body Dynamics 3&4

Resources: Modelling and Simulation Lecture Notes.pdf, Chapter 2

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These lectures focus on Lagrange mechanics as a way of describing a moving system.

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Sub-topics

Lagrange Mechanics

• Lagrange Equation
• Generalized Coordinates
• Kinetic Energy
• Potential Energy
• Euler-Lagrange Equation

Examples

Wagon with spring system

We have

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right)-\left(\frac{\partial T}{\partial q}-\frac{\partial V}{\partial q}\right)=\tau$$

We have the following kinetic and potential energy

$$\begin{array}{rl}T& =\frac{1}{2}m{v}^2=\frac{1}{2}m{\dot{q}}^2\\ V& =\frac{1}{2}k\Delta x^2=\frac{1}{2}k{q}^2\end{array}$$

With the following partial derivatives

$$\begin{array}{rl}\frac{\partial T}{\partial \dot{q}}& =m\dot{q}\\ \to \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right)& =m\ddot{q}\\ \frac{\partial T}{\partial q}& =0\end{array}$$

Which gives us

$$\begin{array}{rl}m\ddot{q}+kq& =0\\ \to \ddot{q}& =-\frac{1}{m}kq\end{array}$$

Which is quite easy to solve

Cart with Inverted Pendulum