Thread: second order problem

1. second order problem

I have a dynamic model like below:

xddot = u1;
yddot=u1 tan (theta) + (xdot tan (theta) - ydot)

I want to convert it into the second order chained form as:
y1ddot = v1;
y2ddot = v2;
y3ddot = y2v1 + (additional terms)

hope someone can give an idea how to start to get the derivation in second order chained form pattern.
Thank you very much.

2. So, as I understand it, your current problem is this:

$\ddot{x}=u_{1}$

$\ddot{\theta}=u_{2}$

$\ddot{y}=u_{1}\tan(\theta)+\dot{x}\tan(\theta)-\dot{y},$

and you want to convert to

$\ddot{y}_{1}=v_{1}$

$\ddot{y}_{2}=v_{2}$

$\ddot{y}_{3}=y_{2}v_{1}+\dots$

Is that correct? If so, can I ask if you have the freedom to assign the $y_{k}$ and $v_{k}?$ If so, I would go with the following assignments:

$y_{1}=x$

$v_{1}=u_{1}$

$y_{2}=\tan(\theta).$

Differentiating $y_{2}$ yields the following:

$\dot{y}_{2}=\sec^{2}(\theta)\,\dot{\theta},$ and

$\ddot{y}_{2}=\sec^{2}(\theta)(\ddot{\theta}+2 \dot{\theta}^{2}\tan(\theta))=\sec^{2}(\theta)(u_{ 2}+2\dot{\theta}^{2}\tan(\theta)).$

So, let

$v_{2}=\sec^{2}(\theta)(u_{2}+2\dot{\theta}^{2}\tan (\theta)),$ and

$y_{3}=y.$

We would thus have the following system:

$\ddot{y}_{1}=v_{1}$

$\ddot{y}_{2}=v_{2}$

$\ddot{y}_{3}=v_{1}y_{2}+\dot{y}_{1}y_{2}-\dot{y}_{3}.$

Is that what you were seeking?

3. Dear Sir Ackbeet,

You are correct. That's what i'm seeking. I still need help to convert another system to that form, but I will post that in another thread.

Thank you very much Sir.

Originally Posted by Ackbeet
So, as I understand it, your current problem is this:

$\ddot{x}=u_{1}$

$\ddot{\theta}=u_{2}$

$\ddot{y}=u_{1}\tan(\theta)+\dot{x}\tan(\theta)-\dot{y},$

and you want to convert to

$\ddot{y}_{1}=v_{1}$

$\ddot{y}_{2}=v_{2}$

$\ddot{y}_{3}=y_{2}v_{1}+\dots$

Is that correct? If so, can I ask if you have the freedom to assign the $y_{k}$ and $v_{k}?$ If so, I would go with the following assignments:

$y_{1}=x$

$v_{1}=u_{1}$

$y_{2}=\tan(\theta).$

Differentiating $y_{2}$ yields the following:

$\dot{y}_{2}=\sec^{2}(\theta)\,\dot{\theta},$ and

$\ddot{y}_{2}=\sec^{2}(\theta)(\ddot{\theta}+2 \dot{\theta}^{2}\tan(\theta))=\sec^{2}(\theta)(u_{ 2}+2\dot{\theta}^{2}\tan(\theta)).$

So, let

$v_{2}=\sec^{2}(\theta)(u_{2}+2\dot{\theta}^{2}\tan (\theta)),$ and

$y_{3}=y.$

We would thus have the following system:

$\ddot{y}_{1}=v_{1}$

$\ddot{y}_{2}=v_{2}$

$\ddot{y}_{3}=v_{1}y_{2}+\dot{y}_{1}y_{2}-\dot{y}_{3}.$

Is that what you were seeking?

4. You're welcome!