Help with motions of a double pendulum using RK4

[Rezalt]

Hello everyone, :)I hope you consider this the right place for this question.
I have been working way to long on making a model of a double pendulum using the RK4 method with excel.

I have these four equations that I should be able plug into RK4
View attachment 3722

I know that the first two equations for example shows the angular velocity of each pendulum, but I just can't figure out how to write this into RK4/excel. I have tried in many ways... with the last attempt seemingly beeing the worst.I will try and shorten down my excel sheet and attach it to this post.
https://drive.google.com/file/d/0B15dVNHE5_8IaEtOcFNNdVM4SGM/view?usp=sharing

Hopefully some of you can help, because I'm getting tired of feeling like an idiot. :PBest regards

- Rezalt

 

[jvicens]

Dear Rezalt,

Thank you for reaching out to us with your question. It is definitely the right place to ask for help with your double pendulum model.

To start, let's review the RK4 method. RK4 stands for "Runge-Kutta fourth order" and it is a numerical method used to solve differential equations. It is commonly used in modeling and simulation because it is relatively easy to implement and provides accurate results.

In order to use RK4, we need to first understand the differential equations that describe the system we are trying to model. In your case, you have four equations that describe the angular velocity and acceleration of each pendulum. These equations are:

dθ1/dt = ω1
dω1/dt = -g(2m1+m2)sin(θ1) - m2gsin(θ1-2θ2) - 2sin(θ1-θ2)m2(ω2^2+ω1^2cos(θ1-θ2))
dθ2/dt = ω2
dω2/dt = 2sin(θ1-θ2)(ω1^2m1+g(m1+m2)cos(θ1)+ω2^2m2cos(θ1-θ2))

Where:
θ1 = angle of first pendulum
θ2 = angle of second pendulum
ω1 = angular velocity of first pendulum
ω2 = angular velocity of second pendulum
g = acceleration due to gravity
m1 = mass of first pendulum
m2 = mass of second pendulum

Now, let's take a look at your excel sheet. It looks like you have already set up the initial conditions, constants, and time step (dt). The next step is to set up the RK4 method for each of the four equations.

The general form of the RK4 method is:

y(t+dt) = y(t) + (1/6)(k1 + 2k2 + 2k3 + k4)

Where:
y(t+dt) = the value of y at the next time step
y(t) = the current value of y
dt = time step
k1 = slope at the current time step
k2 = slope at the midpoint of the current time step
k3 = slope at the midpoint of the current time step using k2
k4 =