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my question is very easy: I started study dynamic systems and I have this system:

$Mz_1''(t) +\beta z_1'(t) = F(t)$

$Mz_2''(t) - \beta z_2'(t)+kz_2(t)=\beta z_1'(t)$

where $ M, \beta, k $ are constants and ' is the order of diff. equation.

Now I have to choose the state variables, that they are as the order of the system. Which is the order of this system? 4 or 2? So how much state variables can I choose?

Regards

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  • $\begingroup$ solve the first equation and substitute it in the second equation and then solve the second equation $\endgroup$
    – E.H.E
    Oct 17, 2015 at 20:20
  • $\begingroup$ I tryed to find $ z_1' $ and substitute it in the second equation, but I still have $z_1''$ . $\endgroup$
    – linofex
    Oct 17, 2015 at 20:36
  • $\begingroup$ Do you know how to get the complementary solution and the particular solution? $\endgroup$
    – E.H.E
    Oct 17, 2015 at 20:39
  • $\begingroup$ I don't solve the system I have to choose the order of the system and then choose the state variables and write i / s / o rappresentation of the mechanical system $\endgroup$
    – linofex
    Oct 17, 2015 at 20:41

1 Answer 1

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Generally each differentiation can be selected as a state variable. You have two second order differential equations, which should give 4 state variables. There are of course infinitely many choices of state variables, but the easiest one is

$$ x_1 = z_1, x_2 = z_1', x_3 = z_2, x_4 = z_2' $$

Now, can you write this in matrix form as follows:

$$\begin{bmatrix}x_1' \\ x_2' \\ x_3' \\ x_4'\end{bmatrix} = \begin{bmatrix}* & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * \end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4\end{bmatrix} + \begin{bmatrix}* \\ * \\ * \\ *\end{bmatrix}$$

Can you find what each * should be?

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  • $\begingroup$ Yes! This is what I asked! Yesterday I solved the system in the same mode, but I had got a doubt about the number of state variables! Thank you very much! $\endgroup$
    – linofex
    Oct 18, 2015 at 21:02

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