Solving Double Pendulum Motion w/ Hit on Lower Rod

In summary, the conversation discusses the problem of finding the distance d in a double pendulum system, where the upper rod moves with frequency w and the lower rod with frequency -w after being hit at a distance d from the point that connects the rods. The equations of motion and Lagrange equations are mentioned as tools for solving the problem, with the goal of finding the generalized impulse F, which is the change in generalized momentum. The lagrangian, a function of four variables, is also mentioned, and the conversation ends with a discussion on how to incorporate the distance d into the generalized impulse.
  • #1
JohanL
158
0
If you have have double pendulum of two rods where the rods moves with the same frequency w at the equilibrium position and at the eq. position you hit the lower rod at a distance d from the point that connects the rods. The hit results in a motion where the upper rod moves with frequency w and the lower rod with frequency -w. If the rods lengths are l and their masses m what are the the distance d.

I have no problem to find the equations of motion for the ordinary double pendulum. With the hit i think it gets something like

Lagrange equations = F

Where F is different for theta1 and theta2.
If you find these equations you can solve for which d the resulting motion is w for the upper rod and -w for the lower rod, right?
But how do you write F?

Thank you
 
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  • #2
You need to right down the lagrangian:

L = T - U

Where T is the kinetic energy and U is the potential energy.

In this case, the lagrangian will be a function of four variables:

[tex]L(\theta_1 , \theta_2 , \theta_1 ', \theta_2' )[/tex]

Where primes denote differentiation with respect to time. So, after you have expressed the lagrangian in terms of those four variables, your equations of motion are the Euler-Lagrange equations (one for each position variable, so two total).
 
  • #3
Thank you.
But as i tried to say the problem for me is to find the generalized impulse F.
Because Lagrange equations leads to
F = Generalized impulse = change in generalized momentum
If I find F the problem is solved.
and how do i get the distance d into F?
 

Related to Solving Double Pendulum Motion w/ Hit on Lower Rod

1. How does a double pendulum work?

A double pendulum is a physical system consisting of two pendulums attached to each other. The first pendulum is attached to a fixed point, while the second pendulum is attached to the end of the first one. When the pendulums are set in motion, they exhibit complex and chaotic behavior due to the interaction between the two pendulums.

2. How does hitting the lower rod affect the motion of the double pendulum?

Hitting the lower rod adds energy to the system, causing the pendulums to move in a different way than they would without the hit. This can result in unpredictable and chaotic motion, making it difficult to predict the exact trajectory of the pendulums.

3. Can the motion of a double pendulum be solved mathematically?

Yes, the motion of a double pendulum can be solved using mathematical equations and principles of physics. However, due to the chaotic nature of the system, exact solutions can be difficult to obtain and may require advanced mathematical techniques.

4. How does the length of the pendulum affect its motion?

The length of the pendulum affects its motion by changing the period of the oscillations. A longer pendulum will have a longer period, meaning it takes longer to complete one full swing. This can also affect the stability and predictability of the double pendulum system.

5. What are some real-world applications of studying double pendulum motion?

Studying double pendulum motion can have applications in fields such as physics, engineering, and robotics. It can also be used as a model for understanding more complex systems with chaotic behavior, such as weather patterns and stock market fluctuations.

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