# 2nd order non linear ODE

a.mlw.walker
So I have never solved a second order NON LINEAR ode before, but my current project has thrown one at me.

I am going to use matlab to solve it when I have it in the correct state, but at the mo, I cant get it to that...
Its three coulped equations, in terms of x,theta and x2. They are in a matrix, ma + cv + kx = force

for each one.

The way they are coupled though is a trigonometric relationship between x and theta and x2.

I.E
theta = f(x)
x2 = f(x)
and x changes with time.

What should I do to include this relationship?

gato_
The question is too vague. Post the equations

a.mlw.walker
OK.
Please see attached PDF. It was easier to do it in word than using latex.

Doing all those differentials correctly is very difficult, and it quicly becomes quite ugly.

#### Attachments

• physicsforums.pdf
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gato_
Sorry, but I think I don't understand where the constraint between $$(\theta,x_{2},x)$$ is comming. Could you post the original problem? Anyway, instead of explicitly using the constraints in the equations of motion, you might try using lagrange multipliers, which work better in this cases. Add this to your lagrangian:

$$\Lambda_{1}[x_{2}-rSin\theta+\sqrt{l^{2}-r^{2}Cos\theta}]+\Lambda_{2}[2rxCos\theta-x^{2}+l^{2}-r^{2}]$$

And derive the equations of motion without substituing the constraints. When simulating, the equations for the evolution of the unknowns $$\Lambda_{1,2}$$ are $$\dot{\Lambda_{1,2}}=0$$, as they are constants

a.mlw.walker
The original problem is two pistons attached to one flywheel, where one of them has an applied force on it, and both are experiencing dampening, x is the displacement of the piston experiencing the external force, and x2 is the displacement on the other piston (attached image).

then the equations I posted before have come from this diagram.

I havent heard of Lagrange multipliers, i shall look them up

#### Attachments

• two pistons.jpg
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