Inverted Pendulum: Relating Force to Angle/Displacement

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So, the magnitude of the force needed is directly related to the angle of the pendulum and its displacement from the vertical position. In summary, the force needed to counter the torque created by the weight of the pendulum at an angle theta is dependent on the angle and displacement of the pendulum. Suggestions are welcome on how to relate the magnitude of the force to these factors.
  • #1
paton51
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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
 
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  • #2
The weight at the end of the pendulum produces a torque. This is obviously mgr*cos(theta), where r is the length of the pendulum. To counter this torque, the minimum force needed is mg*cos(theta) if the force is directed perpendicularly to the rod.
 
  • #3


I would like to commend you for considering such a problem and seeking to understand the relationship between force and angle/displacement in an inverted pendulum system. This is a complex and interesting area of study that has practical applications in fields such as robotics and control systems.

To answer your question, the force applied to the end of the pendulum can be related to the angle or displacement using the principles of dynamics and energy conservation. In order to fully understand this relationship, it is important to consider the physical properties of the pendulum, such as its mass, length, and center of mass.

One approach to relate the force to angle or displacement is to use the equations of motion for a simple pendulum. These equations take into account the mass, length, and angle of the pendulum, as well as the force applied to it. By solving these equations, you can determine the relationship between the force and the angle or displacement of the pendulum.

Another approach is to consider the conservation of energy in the system. As the pendulum moves, its potential energy (due to its position) is converted into kinetic energy (due to its motion). By analyzing the energy at different points in the pendulum's motion, you can determine the relationship between the force and the angle or displacement.

It is also worth noting that the relationship between force and angle/displacement may not be linear. As you mentioned, the force decreases as the angle increases, which may indicate a non-linear relationship. In this case, mathematical modeling and simulations can help to understand the exact nature of this relationship.

In conclusion, there are multiple ways to relate the magnitude of force to the angle or displacement of an inverted pendulum. I would suggest exploring different approaches and using mathematical tools to analyze the system in order to gain a better understanding of this relationship. Best of luck in your research!
 

What is an inverted pendulum?

An inverted pendulum is a physical system where a pendulum is attached to a moving base such that the pendulum is inverted or upside down. This means that the pendulum’s center of mass is above its pivot point, unlike a traditional pendulum where the center of mass hangs below the pivot point.

How is force related to angle/displacement in an inverted pendulum?

In an inverted pendulum, the force applied to the base can cause the pendulum to move or rotate. This force creates a torque on the pendulum, which is directly related to the angle or displacement of the pendulum. The greater the force, the greater the torque and the larger the angle/displacement of the pendulum.

What factors affect the stability of an inverted pendulum?

The stability of an inverted pendulum is affected by several factors, including the length of the pendulum, the mass of the pendulum, and the force applied to the base. A longer pendulum with a heavier mass and a smaller force applied to the base will result in a more stable inverted pendulum.

How does the concept of center of mass relate to an inverted pendulum?

The center of mass is a point in an object where all of its mass can be considered to be concentrated. In an inverted pendulum, the center of mass is located above the pivot point, making it unstable. The force applied to the base must be carefully controlled in order to maintain the center of mass above the pivot point and keep the pendulum from falling over.

What are some real-world applications of an inverted pendulum?

Inverted pendulums have many practical applications, including balancing robots, segways, and self-balancing scooters. They are also used in engineering and control systems as a way to study and understand the principles of balance and control. Inverted pendulums are also commonly used in academic settings to teach physics and engineering concepts.

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