# Homework Help: Critical Angle of an Inverted pendulum

1. Aug 21, 2011

### Poppinfresh

Hi everyone, I’m designing and building an inverted pendulum on a cart for a 3rd year undergrad engineering course. So far i have been going quite well, but i have come across a question and it has me completely stumped. Any help would be great

1. The problem statement, all variables and given/known data
The pendulum on a cart only moves in one direction and has a force F applied to it by a DC motor and gearbox that is controlled by a voltage U. Assume that the controller is maintain the pendulum to some constant angle theta*. Practically the control signal is limited to -Umax<U<Umax, there exists some range -thetacrit<theta<thetacrit, from which the pendulum angle can be maintained for a given cart velocity V. Attempting to control the pendulum beyond this range requires more control effort than is available. Neglect friction.

Express the critical angles +thetacrit and -thetacrit as functions of the cart velocity V.

2. Relevant equations
The acceleration of the cart required to maintain a constant angle is a = mgtan(theta).

3. The attempt at a solution
Needing to have the critical angles for a cart velocity has got me stumped, if the pendulum is attached to the cart, then surely the only way to get a critical angle is to find the acceleration? If you ignore friction, then in the dynamics of the system there is no velocity term present, so how can you find anything as a function of velocity?
I’m sorry if my "attempt" at a solution sounds lazy, i just have no place to start.

Thankyou

EDIT: Nevermind i think i have it, by analysing the DC motor you can find input as a funtion of Current and Velocity, and Current as a function of force. Knowing the acceleration to maintain a constant angle you cam multiply through by cart mass and get the force. Substitute everything, do a bit of algebra and out pops your answer :) Im pretty sure this is right.

Last edited: Aug 21, 2011