Critical Angle of an Inverted pendulum

[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

Homework Statement

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.

Homework Equations

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

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 :) I am pretty sure this is right.

 

[KataKoniK]

Hello,

It's great to hear that you are working on an inverted pendulum project for your engineering course. It sounds like a challenging and interesting project!

Based on the information provided, I believe the critical angles +thetacrit and -thetacrit can be expressed as functions of the cart velocity V by considering the forces acting on the pendulum-cart system.

As you mentioned, the acceleration of the cart required to maintain a constant angle is a = mgtan(theta). This means that the force required to maintain this acceleration is F = ma = mg tan(theta).

Since the force F is provided by the DC motor and gearbox, we can assume that it is proportional to the input voltage U. This means that we can write F = kU, where k is a constant.

Substituting this into the equation above, we get kU = mg tan(theta).

Now, let's consider the cart velocity V. We know that the cart is moving in one direction, so the velocity V is related to the acceleration a by V = at, where t is the time it takes for the cart to travel a certain distance.

Substituting this into the equation above, we get kU = mg tan(theta) = mat.

Since we are looking for the critical angles as functions of the cart velocity V, we can rearrange this equation to solve for theta. This gives us theta = arctan(kU/mgt). This means that the critical angles can be expressed as +thetacrit = arctan(kU/mgt) and -thetacrit = -arctan(kU/mgt).

I hope this helps and makes sense. Good luck with your project! Let me know if you have any other questions.