# Equation of motion of a classic inverted pendulum in free fall

I was thinking in this interesting problem:

Suppose we have this inverted pendulum: But without this control force $$F$$ and the system would by loose from a height $$h_0$$, with initial velocity $$0$$ and with the pendulum angle equals $$\theta$$, and would do a free fall. What would be the equation of motion of this system?

• Hi, what have you done so far in this problem? Sep 14, 2021 at 17:55

Lets think about it in the reference frame of the falling masses and ignore air resistance. By removing the control force $$F$$ and the force of gravity (by letting the whole system fall freely) there are no longer any forces left. That means that if $$m$$ has initial velocity 0 (with respect to $$M$$) it will continue to have velocity 0; i. e. it will not move with respect to $$M$$. From the non-falling perspective, that means that your whole system just accelerates downward uniformly and the angle $$\theta$$ does not change.

Set cart displacement $$x$$ and rod angle $$\theta$$

1. No external forces applied horizontally so the combined center of mass remains fixed in the horizontal location.

$$M x + m (x - \ell \sin \theta) = 0$$

or in terms of velocities

$$M \dot{x} + m ( \dot{x}- \ell \dot{\theta} \cos \theta) = 0$$

2. Conservation of energy applies which means

$$\underbrace{m \ell \cos \theta}_\text{PE} + \underbrace{ \frac{1}{2} m \dot{x}^2 + \frac{1}{2} (m \ell^2) \dot{\theta}^2}_\text{KE} + \underbrace{ \frac{1}{2} M \dot{x}^2}_\text{KE} = \text{(const)}$$

Use those two expressions in order to solve for $$\dot{\theta}$$ and $$\dot{x}$$ for each angle $$\theta$$.

• I think that the center of mass x direction is ${\frac {m \left( x+l\sin \left( \theta \right) \right) +Mx}{m+M}}$ ?
– Eli
Sep 14, 2021 at 19:39

The position vector to the mass m is:

$$\mathbf R_m(t)=\left[ \begin {array}{c} x \left( t \right) +l\sin \left( \theta \left( t \right) \right) \\ l\cos \left( \theta \left( t \right) \right) \end {array} \right]$$

and to the mass M

$$\mathbf R_M(t)=\begin{bmatrix} x(t) \\ 0 \\ \end{bmatrix}$$

from here you can obtain the kinetic and potential energy and with EL the equation of motions