Obtaining Lagrangian of complicated pendulum

In summary, the student is trying to figure out how to fix a problem with his equations of motion for a pendulum, but does not know how to start. He asks if there is a way to simplify the equations, and asks if Mathematica can help.
  • #1
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I have to create a simulation of the pendulum shown in the .pdf at the bottom of the page. The 3 rods are free to rotate around their pivots in a plane. The two edge rods are connected as close to their edges as possible. There is no friction.
Unfortunately my equations of motion are spitting out the wrong results when it comes to computation. Something has gone wrong in my manipulations.
I don't expect anyone to go through all of the algebra looking for the mistake(s) (I've spent hours and still can't find it)
I would like to know if there is anyway of setting up my generalised co-ordinates or any other changes I can make at the beginning so that I don't end up with the lengthy and tough to follow equations on page 7-8, and the horrendous equations of motion on page 9. (They hardly fit!)

Note: I need my equations it in terms of the accelerations for the computational aspect, that's why you see the horrible terms at the end. I haven't simplified them because as they stand they are wrong(you really don't need to try make sense of these last few pages unless you have lots of time to waste). Also you may find me substituting numerical specifications in. These just come from the specs of the assignment. I don't introduce them anywhere. There is also a few things I need to fix like calling them "rods" when they are not in fact rods.I also haven't formalised and made clear the last couple of steps in the paper. Bear with me with that.(As I said though, you really don't need to bother looking at these parts!)

Here is my work:
http://myfreefilehosting.com/f/931c205ace_0.41MB

Thanks in advance.
 
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  • #2
You may have a chaotic system which is sensitively dependent on initial conditions so any attempt at computing results from a model will not match reality.
 
  • #3
Yeah it is a chaotic system(computationally I'll still be able to get decent results for very small time intervals using a stiff solver and by setting MATLAB to integrate over much smaller time intervals than it automatically does, although the purpose of the assignment is ). The problem actually arises before I need to solve the DE's numerically. The expressions for the angular accelerations yield the wrong results, if for example I substitute the conditions when the system is in equilibrium, which is clearly just a mistake along the way.

I was just wondering if there was a better way to go about the physics of the system. Hopefully without having to do a lot more work doing things such as deriving Hamilton's equations for the system etc.

Does anyone know to what extent Mathematica can help for the analytical side of things? I have never used it before but apparently its very powerful
 
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Related to Obtaining Lagrangian of complicated pendulum

1. What is the purpose of obtaining the Lagrangian of a complicated pendulum?

The Lagrangian of a complicated pendulum is used to describe the motion of the pendulum in terms of its position and velocity. It allows us to solve for the equations of motion and understand the dynamics of the pendulum.

2. How is the Lagrangian of a complicated pendulum obtained?

The Lagrangian of a complicated pendulum can be obtained by using the Lagrangian formalism, which is a mathematical technique based on the principle of least action. This involves calculating the kinetic and potential energy of the pendulum and then using them to construct the Lagrangian function.

3. What factors affect the Lagrangian of a complicated pendulum?

The Lagrangian of a complicated pendulum is affected by the length of the pendulum, the mass of the pendulum bob, and the angle at which the pendulum is released. Other factors, such as air resistance and friction, may also affect the Lagrangian, but they are often neglected in simplified models.

4. Can the Lagrangian of a complicated pendulum be calculated for any type of pendulum?

Yes, the Lagrangian of a complicated pendulum can be calculated for any type of pendulum as long as the motion is constrained to a specific plane and the pendulum does not experience large deformations or rotations. However, the complexity of the calculations may vary depending on the specific type of pendulum.

5. What are the advantages of using the Lagrangian to describe the motion of a complicated pendulum?

The Lagrangian provides a more generalized and elegant approach to solving for the motion of a complicated pendulum compared to traditional methods. It also allows us to take into account constraints and varying factors, making it a more accurate representation of the pendulum's motion. Additionally, the Lagrangian can be easily extended to more complicated systems, making it a valuable tool for studying complex pendulum systems.

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