I am working through an example of an inverted pendulum on a cart, exactly like this:(adsbygoogle = window.adsbygoogle || []).push({});

L is the length of the massless pendulum arm.

Two degrees of freedom, x and theta. If I write out the kinetic energy as three parts: translational KE from cart + translational KE from pendulum + rotational KE from pendulum, I end up missing a term (compared to writing out the KE in vector form with each mass).

In vector form:

[itex]T = \frac{1}{2}M\dot{r_{1}}\cdot\dot{r_{1}} + \frac{1}{2}m\dot{r_{2}}\cdot\dot{r_{2}}[/itex]

[itex]T = \frac{1}{2}M\dot{x}^{2} + \frac{1}{2}m(\dot{x}^{2}+{l^2}{\dot{\theta}^2}+2l\dot{x}\dot{\theta}cos{\theta})[/itex]

I know above is correct, but I don't know what the term [itex]ml\dot{x}\dot{\theta}cos{\theta}[/itex] represents.

If I write the KE using the other method, I get

[itex]T = \frac{1}{2}M\dot{x}^{2} + \frac{1}{2}m\dot{x}^{2}+\frac{1}{2}m{l^2}{\dot{\theta}^2}[/itex]

where each term represents translational KE from cart, translational KE from pendulum, and rotational KE from pendulum, respectively. I am missing the term from the KE that was obtained using vectors.

I think I'm missing the velocity component of the pendulum mass in the y direction?

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# Inverted Pendulum on Cart: What is this term in the KE?

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