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So I have been trying to derive the equations of motion of the inverted physical pendulum in a cart, but I seem to be confused about the derivation of its Kinetic Energy. I know this physical system is very popular and while I have searched and searched I couldn't find an answer to my question anywhere.

Scheme of the problem

So I divided the kinetic energy into the cart's and pendulum's: $$ T = T_C + T_P $$

The cart's one is pretty straight forward $T_C = 1/2 M \dot{x}^2$, where I am denoting $x$ the horizontal coordinate of the cart's point mass.

My trouble is now with the pendulum's Kinetic Energy. I would assume I would have to sum the translational energy of the pivot point $T_{pivot}=1/2 m \dot{x}^2$ to the rotational energy of the pendulum $T_{rot} = 1/2 I \dot{\theta}^2$, where $I$ is the moment of inertia of the pendulum with respect to the pivot point (Note: the angle $\theta$ i chose is with respect to the upper vertical, unlike in the image up there).

With this I got: $$ \mathcal{L} = \frac{1}{2}(M+m) \dot{x}^2 + \frac{1}{2} I \dot{\theta}^2 - mgl\cos\theta $$

And therefore the equations of motion: $$ (M+m) \ddot{x} = F(t) $$ $$ I \ddot{\theta} - mgl \sin\theta = 0 $$

These equations, though, seem too simple compared to the equations I have seen out there for this problem. I would really appreciate if someone could point out my mistakes.

Qmechanic
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gilbertocunha
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  • Your position vector to the CM is $\vec R= [ x+l\cos(\theta),l\sin(\theta)]^T$ Thus the kinetic energy will be? – Eli May 07 '20 at 14:36
  • @Eli I considered the translational kinetic energy of the pivot and not the CM, maybe that was a wrong assumption. In that case I would just get the $T=1/2 m (\dot{x_{CM}}^2 + \dot{y_{CM}}^2)$ for the pendulum? or would I have to add a rotational kinetic energy? (I was thinking maybe with moment of inertia relative to the center of mass) – gilbertocunha May 07 '20 at 15:38
  • no you have to take the CM velocity for translation and for rotation $I=I_{cm}+m,l^2$ – Eli May 07 '20 at 16:02
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    @Eli Thank you for your help, Eli. So basically I should consider the translational kinetic energy of the CM and the rotational relative to the pivot point? – gilbertocunha May 07 '20 at 16:26

3 Answers3

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First, that the Lagrangian will have a term containing $F(t)$, or you will not get $(M+m)\ddot{x}=F(t)$. Second that if $F(t)$ is an explicit function of $t$, then Lagrangian will also be the explicit function of $t$ and then you have to consider more general form of Euler-Langrangian equation. For that refer to https://physics.stackexchange.com/a/437198/203041.

sslucifer
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  • That function $F(t)$ is a non conservative force (therefore not included in the lagriang, I believe) acting along the x-axis on the cart. I use the generalized forces which I believe would be $F(t)$ for the $x$ coordinate and $0$ for the $\theta$ coordinate. Am I wrong in doing this? – gilbertocunha May 07 '20 at 15:42
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I had the same question and after reading some definitions, I've got the answer: The kinetic energy of a rigid body which has planar motion is always

$$T=T_{Gtranslate}+T_{rotate/G}$$

or

$$T=1/2Mv^2_G+1/2l_G\omega^2$$

where $G$ is the center of mass. So in this pendulum you have to calculate $v_G=\sqrt{\dot{x_G}^2+\dot{y_G}^2}$ and $\omega=\dot{\theta}$ and. Then the kinetic energy will be

$$T=\frac{1}{2}M(\dot{x_G}^2+\dot{y_G}^2)+\frac{1}{2}I_G\dot{\theta}^2 + T_{cart}$$

There is a paper from MIT 2.003SC course which has the same solution: http://bit.ly/PendulumonACart

jng224
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Tieu Binh
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The position vector to the center of mass is

$$\vec R=\left[ \begin {array}{c} x+L\sin \left( \theta \right) \\ L\cos \left( \theta \right) \end {array} \right] $$

from here the velocity of the CM

$$\vec v=\frac{d}{dt}(\vec R)=\left[ \begin {array}{c} {\dot x}+L\cos \left( \theta \right) \dot\theta \\ -L\sin \left( \theta \right) \dot \theta \end {array} \right] $$

so the kinetic energy

$$T=\frac m2\,\vec v\cdot \vec v+\frac{I_{\text{CM}}}{2}\,\dot\theta^2+ \frac M2\,\dot x^2$$

the potential energy is: $$U=-m\,g\,L\sin(\theta)$$

Eli
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