Transfer Function of Coupled Diff Eq

[dimensionless]

I have two coupled harmonic oscillators:

$$\ddot{x}_{1} = -2kx_{1} + kx_{2} + f(t)$$
$$\ddot{x}_{2} = kx_{1} - kx_{2}$$

Mass 1 is at position $$x_{1}$$ and subject to force $$f(t)$$.

I take the Laplacian of the first equation and solve for $$X_{1}$$ to get

$$X_{1} = \frac{ F(p) + k X_{2} }{ p^{2} + 2k }$$

I then do the same for the second to get

$$X_{2} = \frac{ k X_{1} }{ p^{2} + 2k }$$

I then substitute $$X_{1}$$ into $$X_{2}$$, divide out $$F(p)$$, and then wind up with the transfer function

$$T(e^{jwt}) = \frac{ z^{-2} + 2kz^{-4} }{ 1 + 4kz^{-2} + 3k^{2}z^{-4} }$$

My question:

Does this method work for finding the transfer function of coupled differential equations?

 

[quantumdude]

Sure, why not? All you're shooting for is $T(p)=X_1(p)/F(p)$, right? So if you can get there by this method (which you clearly can), then it must work!