Lagrangian of inverted pendulum

In summary, the conversation discusses an inverted pendulum and how to obtain the Lagrangian and equation of motion for it, taking into account both kinetic and potential energies. It is noted that the Euler-Lagrange equation remains the same even though the Lagrangian is not time-independent. The equation of motion is derived by taking the derivatives of the Lagrangian and setting them equal to each other.
  • #1
adartsesirhc
56
0
An inverted pendulum consists of a particle of mass [tex]m[/tex] supported by a rigid massless rod of length [tex]l[/tex] . The pivot [tex]O[/tex] has a vertical motion given by [tex]z=Asin\omega t[/tex]. Obtain the Lagrangian and find the differential equation of motion.


I'm not sure how to obtain the kinetic and potential energies. For the potential energy, would it just be
[tex]V=mglcos\theta+Asin\omega t[/tex]?

And is the kinetic energy
[tex]T=\frac{1}{2}m(l^{2}\dot{\theta}^{2}+A^{2}\omega^{ 2}cos^{2}\omega t)[/tex]?

Since the Lagrangian wouldn't be time-independent, would this in any way affect the Euler-Lagrange equation, or would it remain the same?

Thanks, all.
 
Physics news on Phys.org
  • #2
Yes, the potential energy is given by V=mglcosθ+Asinωt and the kinetic energy is given by T=12ml2θ˙2+A2ω2cos2ωt. The Euler-Lagrange equation remains the same even though the Lagrangian is not time-independent. The equation of motion can be obtained by taking the derivative of the Lagrangian with respect to θ and setting it equal to the negative of the derivative of the Lagrangian with respect to θ˙. This yields the equation of motion: mglsinθ+2ml2θ˙2θ¨+A2ω2cos2ωtsinθ−A2ω3sinωtcosθ=0.
 
  • #3


The Lagrangian is a function that describes the dynamics of a system in terms of its coordinates and their time derivatives. In the case of an inverted pendulum, the coordinates are the angle of the pendulum, denoted by theta, and the vertical displacement of the pivot point, denoted by z.

To obtain the Lagrangian, we first need to find the kinetic and potential energies of the system. The potential energy of the particle is given by the sum of its gravitational potential energy and the potential energy due to the vertical motion of the pivot point. So, the potential energy can be written as:

V = mglcos(theta) + mAzsin(omega*t)

The kinetic energy of the particle is given by the sum of its translational and rotational kinetic energies. The translational kinetic energy is simply 1/2*m*(z')^2, where z' is the time derivative of z. The rotational kinetic energy is given by 1/2*m*l^2*(theta')^2, where theta' is the time derivative of theta. So, the total kinetic energy can be written as:

T = 1/2*m*(z')^2 + 1/2*m*l^2*(theta')^2

Now, the Lagrangian is defined as the difference between the kinetic and potential energies, L = T - V. Therefore, the Lagrangian of the inverted pendulum can be written as:

L = 1/2*m*(z')^2 + 1/2*m*l^2*(theta')^2 - mglcos(theta) - mAzsin(omega*t)

To find the differential equation of motion, we need to use the Euler-Lagrange equation, which is given by:

d/dt(dL/d(theta')) - dL/d(theta) = 0

Since the Lagrangian does not explicitly depend on time, the Euler-Lagrange equation remains the same. We can rewrite the equation as follows:

d/dt(m*l^2*theta') + mgl*sin(theta) = -m*A*omega^2*cos(omega*t)

This is the differential equation of motion for the inverted pendulum, which can be solved to determine the behavior of the system over time. I hope this helps clarify the process of obtaining the Lagrangian and the differential equation of motion for an inverted pendulum.
 

1. What is the Lagrangian of an inverted pendulum?

The Lagrangian of an inverted pendulum is a mathematical function that describes the dynamics of the system. It takes into account the kinetic and potential energy of the pendulum and is used to derive the equations of motion.

2. How is the Lagrangian of an inverted pendulum derived?

The Lagrangian of an inverted pendulum is derived using the principles of Lagrangian mechanics, which is a mathematical framework for analyzing the motion of systems. It involves defining the generalized coordinates and using the Lagrangian function to find the equations of motion.

3. What factors affect the Lagrangian of an inverted pendulum?

The Lagrangian of an inverted pendulum is affected by the length of the pendulum, the mass of the pendulum, the angle of the pendulum with respect to the vertical, and the forces acting on the pendulum.

4. How is the Lagrangian of an inverted pendulum used to model its motion?

The Lagrangian of an inverted pendulum is used to derive the equations of motion for the system. These equations can then be solved to predict the motion of the pendulum under different conditions.

5. What are the applications of the Lagrangian of an inverted pendulum?

The Lagrangian of an inverted pendulum has various applications in engineering, physics, and robotics. It can be used to design control systems for stabilizing the pendulum, study the stability of the system, and analyze the behavior of similar systems.

Similar threads

  • Introductory Physics Homework Help
Replies
9
Views
636
  • Introductory Physics Homework Help
Replies
5
Views
731
  • Introductory Physics Homework Help
Replies
20
Views
973
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
23
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
512
  • Introductory Physics Homework Help
Replies
17
Views
277
  • Introductory Physics Homework Help
Replies
3
Views
877
  • Introductory Physics Homework Help
Replies
19
Views
722
Back
Top