Lagrangian: singularity in inverted pendulum EoMs?

AI Thread Summary
The discussion centers on the analysis of an inverted pendulum problem related to bicycle mechanics, where the equations of motion exhibit an unphysical singularity as the angle approaches zero. The equations derived from the Lagrangian formulation initially suggest a problematic singularity due to the assumption of a massless carriage. However, introducing a non-zero mass for the carriage resolves the issue, leading to modified equations of motion that eliminate the singularity. The key takeaway is that the singularity arises from the unrealistic assumption of a light carriage, and incorporating a finite mass provides a more accurate model without singular behavior. The discussion highlights the importance of considering all components' masses in dynamic systems to avoid unphysical results.
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Hi all, I'm doing some analysis of a bicycle mechanics problem and at one point the approximations I'm making mean that the problem reduces to the classic inverted pendulum. I'm very confused, as the equations I've worked out appear to have an unphysical singularity in them, and I can't see where it's coming from.

So, the problem is this: equations of motion of a point mass m on the end of a light rigid rod (length l), the other end of which is attached to a light, frictionless carriage, constrained to move only in the horizontal direction.

So, the problem has two degrees of freedom: x and \theta.

By my reckoning, the position vector of the pendulum bob, counting angle clockwise from the upwards vertical, is:

\mathbf{r}=(x + l \sin\theta, l \cos\theta)

This gives a Lagrangian of:

L = \frac{m \dot{\mathbf{r}}^2}{2} - m g \cos{\theta} = \frac{m \dot{x}^2}{2} + \frac{m l^2 \dot{\theta}^2}{2} + m \dot{x} \dot{\theta} l \cos{\theta} - m g \cos{\theta}

Now, the Euler-Lagrange equations end up as

\ddot{x} + \ddot{\theta} l \cos{\theta} - \dot{\theta}^2 l \sin{\theta} = 0
\ddot{\theta} + \ddot{x} \cos\theta / l - g \sin\theta/l=0

which in turn reduce to:

\ddot{x} = \frac{\dot{\theta}^2 l - g \cos \theta }{\sin\theta}
\ddot{\theta} = \frac{g / l - \dot{\theta}^2 \cos\theta}{\sin\theta}

Do you spot the problem yet? There's a nasty singularity when \theta \to 0.

Can anyone tell me what's going wrong here?
 
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To anyone wondering, the "unphysical" singularity arises from the unphysical assumption of a light carriage. With a carriage of nonzero mass M the EoMs become:

<br /> \ddot{x} = \frac{2 m \sin \theta \left(g \cos \theta - l \dot{\theta}^2\right)}{2 M + m (1 - \cos 2 \theta)}<br />
<br /> \ddot{\theta} = \frac{m l \dot{\theta}^2 \cos \theta-g (m+M)}{- m l \sin \theta - M l / \sin\theta}<br />

With no singularity as long as M \neq 0!
 
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