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I'm dealing with system identification for the first time in my life and am in desperate need of help :) The system is spring-mounted and I'm analyzing the vertical and torsional displacements. However, it seems like the vertical and torsional oscillations are coupled (shouldn't be in theory..). So I want to create a model that takes this coupling into account. However this turned out to be much harder than I first thought...

The classical damped harmonic oscillator

x" + 2*gamma*x' + omega_0^2*x = 0 (1)

can be rewritten as:

x(t) = A*exp(-gamma*t)*cos(omega*t + phi) (2)

if omega_0^2 > gamma^2, (omega=sqrt(omega_0^2-gamma^2))

I'm supposing a coupled system between vertical and torsional oscillations would look something like this:

h" + 2*gamma*h' + omega_0^2*h + k1*a' + k2*a = 0 (3)

a" + 2*gamma*a' + omega_0^2*a + k3*h' + k4*h = 0 (4)

where k1, k2, k3 and k4 are coupling coeff.

But from here I have no idea how to continue... The main goal is of course to determine the coupling coeff and validating the model with experiment data. Does anyone have any suggestions of how to determine the coupling coeff? Or know how to express (3) and (4) as h(t) and a(t)? Furthermore I'm not sure that eqs (3) and (4) are the best to describe this coupled system... Any suggestions at this point are highly appreciated!

Thanks,

Maria