Inverted Pendulum: Relate Force to Angle/Displacement?

[paton51]

Im considering a problem were an inverted pendulum is resting against a ledge an an angle theta from the horizontal. A force is applied to the end causing it to move, This force is dynamic and decrease as the angle increases. The critical point is when the pendulum passes the vertical and cannot return to its resting point falls to the other side.

Does anyone know how i can relate the magnitude of the force to either the angle or displacement of the end of the pendulum?

suggestions welcome.
thanks

 

[Dale]

I would look around the net and see if you can find any information on the control algorithm used by the Segway personal transport. That is essentially an inverted pendulum.

 

[astrokreng]

The relationship between force and angle/displacement in an inverted pendulum can be described by the principles of torque and equilibrium. When a force is applied to the end of the pendulum, it creates a torque which causes the pendulum to rotate about its pivot point. As the angle increases, the distance between the pivot point and the center of mass of the pendulum decreases, resulting in a decrease in the torque. This decrease in torque causes the pendulum to slow down and eventually reach a critical point where it cannot return to its resting point and falls to the other side.

To relate the magnitude of the force to the angle or displacement, you can use the equation τ = Fr, where τ is the torque, F is the force, and r is the distance between the pivot point and the point where the force is applied. This equation shows that as the angle or displacement increases, the torque decreases, and therefore the magnitude of the force must also decrease to maintain equilibrium.

Another way to relate force to angle/displacement is through the concept of energy. As the pendulum moves, it gains kinetic energy and loses potential energy. The force applied can be related to the change in potential energy, which in turn is related to the angle or displacement of the pendulum. This relationship can be described by the equation PE = mgh, where PE is the potential energy, m is the mass of the pendulum, g is the acceleration due to gravity, and h is the height of the pendulum's center of mass.

In summary, the force applied to an inverted pendulum can be related to the angle or displacement through the principles of torque and equilibrium, as well as the concept of energy. By understanding these relationships, you can analyze the behavior of an inverted pendulum and predict its motion.