Effect of gravity on body fixed to a single joint - 1 DOF

I've been creating some random structures (beams, arms, segments, whatever you prefer), which i can move relative amongst each other by using actuators mounted across the joints. All by a whole lot of trigonometry. Now I wanted to add the gravitational effect to the model, which is the source of my question.

I want to examine the displacement of the center of gravity induced by the gravitational force, when the CoG is not balanced over the joint fixed to the ground. I found a small model of a stick falling from an initial angle on Wolframalpha.com which suggested using the following 2nd order ODE:

(1/12*m*L^2+1/4*m*L^2*cos(theta)^2)theta'' - 1/4*m*L^2*cos(theta)*sin(theta)*(theta')^2 = 1/2*m*g*L*cos(theta)

This, I've tried to implement in my model, since it's the exact same case (just a more complex stick). But I cannot figure out how to get this fitted to the ode45 function!

I have initial position of the CoG and no intial angular velocity. Since its a 2nd order ode I guess it'll have to be transformed into two first order ode's, but unfortunately my skill is lacking in this whole field. I would very much appreciate any help :-) Thanks in advance!

Answer 1

[Andrew Newell]

See this example of how to convert a second order ODE to a linear system. Reformulate the above equation, and then we'll do the next step.