# Inverted pendulum

#### roger

Hi, I'm not sure this is the correct place for this.

I am taking my first Physics course in about 40 years, Computational Physics with the Garcia textbook, "Numerical Methods for Physics".

Problem 21 attached deals with an inverted pendulum.

I have included a scan of the problem and my Matlab m file program.

Using the Verlet method to calculate the next theta and omega with a timestep of .01 seconds, the only stable oscillation seems to occur when A0 and Td are both equal to 1. This seems to contradict what the problem is saying about A0 >> g.

I'm not really sure what this problem is trying to show since we are just plugging in different values for A0 and Td trying to get a stable oscillation around theta = 180 degrees.

I've looked up inverted pendulum and it seems to be a fascinating problem but I'm totally confused as to the point of this problem and what it is trying to show.

Can anyone give me insight as to why A0=Td=1 is the only stable oscillation, and what the point is of showing that?

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#### DrPhil

Hi, I'm not sure this is the correct place for this.

I am taking my first Physics course in about 40 years, Computational Physics with the Garcia textbook, "Numerical Methods for Physics".

Problem 21 attached deals with an inverted pendulum.

I have included a scan of the problem and my Matlab m file program.

Using the Verlet method to calculate the next theta and omega with a timestep of .01 seconds, the only stable oscillation seems to occur when A0 and Td are both equal to 1. This seems to contradict what the problem is saying about A0 >> g.

I'm not really sure what this problem is trying to show since we are just plugging in different values for A0 and Td trying to get a stable oscillation around theta = 180 degrees.

I've looked up inverted pendulum and it seems to be a fascinating problem but I'm totally confused as to the point of this problem and what it is trying to show.

Can anyone give me insight as to why A0=Td=1 is the only stable oscillation, and what the point is of showing that?
I don't speak Matlab, but the syntax looks straightforward - if I get a chance, I will look at your program, because something is definitely wrong with your conclusion. The first thing I notice is a lack of UNITS in your comments - for instance, I would like to know that g/L has units of 1/s^2. [I will have to look up what the Verlet method is!]

Initially, oscillation about theta=0 is always stable, and I would expect theta to lock in with (2pi t/T_d) with a phase difference depending on A_0 and T_d.

For stability at theta=180°, The driving acceleration A_0 has to be (?) great enough that the force of gravity never takes over - or the amplitude of the oscillation has to be small and out of phase .. when the pendulum starts to tip, then the pivot point must move under it far enough that the tip moves the other way.

Saying A_0 = T_d (=1) can't be relevant, because the units of A_0 are m/s^2 and the units of T_d are s. Numbers with different units are never "equal." You might check the units through your program - have you constrained L and theta0 too much?

EDIT, after starting to look at program
I don't like forcing the amplitude to be 1°
accel must be angular acceleration, units 1/s^2
Formula for L is wrong: L = g/g_over_L = 9.8 m, not 1/9.8

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#### roger

Thanks

Thank you for your quick response.

Yes I was very confused by the units since this equation was just introduced in this problem. There were no units given for anything. We were just supposed to pick values for A0 and Td so as to stabilize the oscillations.

I will check on the mistake of calculating L. I'll bet that was where things broke down.