Solving an ODE-45 from Euler-Lagrange Diff. Eqn.

[Axecutioner]

I need to find the equation of motion of a double pendulum, as shown here:

I've gotten as far as the two euler-lagrange differential equations, simplified to this:

K₁$\ddot{θ}$₁ + K₂$\ddot{θ}$₂cos(θ₁ - θ₂) + K₃$\dot{θ}$₂²sin(θ₁ - θ₂) + K₄sin(θ₁) = 0
K₅$\ddot{θ}$₂ + K₆$\ddot{θ}$₁cos(θ₁ - θ₂) + K₇$\dot{θ}$₁²sin(θ₁ - θ₂) + K₈sin(θ₂) = 0

Assuming initial conditions $\ddot{θ}$_1o, $\dot{θ}$_1o, θ_1o, $\ddot{θ}$_2o, $\dot{θ}$_2o, θ_2o

What would these equations of motion be?
θ₁(t) =
θ₂(t) =

I was told it could be done in MATLAB but I don't have the software or know how to use it yet so any help would be appreciated. Step-by-step solution would be even better. Thanks in advance.

 

[dikmikkel]

Have you considered the rotational energy aswell? You have to have a moment of inertia dependent term.
The procedure for solving this type of problems is as follows:
1. Define your generalized coordinates for each of the masses: e.g. (x1,y1) = (l1sin(theta(t),l1cos(theta(t)) ...
2. Find the velocity and simplify it. Remember the chain rule for differentiation.
3. Find the Potential energy V and kinetic T($T = 1/2m_1v_1^2+1/2m_2v_2^2+1/2I_1 \omega_1^2+1/2I_2\omega_2^2)$ where omega = dtheta/dt
and the lagrangian is L = T-V
4. Write down your dynamical equations, 2 masses gives 2 equations. and there you go.
$\dfrac{\partial L}{\partial \theta_1} - \dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{\theta_1}}=0$
Summary: Be strict defining coordinate system. Remember the chain rule and product rule for differentiation. Concentrate it is hard algebra.
The equations of motion comes from solving this differential equation system. Sometimes it is assumed that sin(theta) = theta for theta<<1 and that simplifys a lot.
But in general this problem is chaotic in nature, and you would probably solve nummericaly in matlab.

 

[Axecutioner]

Here's what I've done so far, before I posted the OP:

http://i.imgur.com/c1Sdb.jpg

Some of the K values are the same but just keep each one different, since this entire double pendulum case is a simplification of part of a project I'm working on.

And Rotational Kinetic energy and Kinetic energy are the same for a point mass ( 1/2mv^2 = 1/2mL^2w^2 = 1/2Iw^2) so adding both would be wrong.

And I do not want to use the small angle approximation because the actual pendulum in my project is being torqued with an angle dependent force, and the angles involved cover a wide range.

 

[dikmikkel]

Iv'e tried solving the system in maple nummerically without luck. I don't think i am able to get an analytical solution, sorry. And I've never seen a analytical solution.

 

[asteriatic]

I understand your frustration in trying to solve these equations of motion for the double pendulum. However, I can assure you that this is a common challenge in the field of mathematical physics and there are various methods to solve it.

Firstly, I would recommend using MATLAB or any other software that is capable of solving differential equations numerically. This will save you a lot of time and effort in solving the equations manually.

To use MATLAB, you will need to input the equations in the form of a function with the dependent variables and their derivatives. You can then use the built-in ODE solver, such as ODE45, to obtain the solutions for θ1(t) and θ2(t).

If you prefer to solve the equations manually, you can use the method of small oscillations to simplify the equations and solve them using standard techniques. However, this method may not give an accurate solution for large oscillations.

Another approach is to use the Lagrangian formalism to derive the equations of motion and then solve them using numerical methods or analytical techniques. This method is more rigorous and can handle large oscillations, but it requires a good understanding of Lagrangian mechanics.

In conclusion, there are various ways to solve the equations of motion for a double pendulum, and the best approach will depend on your level of expertise and the accuracy required for your solution. I would recommend starting with MATLAB or seeking help from a colleague or a mentor who is familiar with these types of problems. I hope this helps and wish you success in your endeavor.