By vibration signature, do you mean the frequency responses from the two inputs to the output? Are F1 and F2 constants? Is the system of equations complex (you show exp(i*2*omega*t) which is a complex number)?
Given frequency response measurements, there are various functions that can produce a model that fit the measured response - n4sid, oe, pem, and in R2012a, also tfest, procest, ssest. For structured estimation, you can also use grey box modeling techniques (idgrey/pem in R2011b or earlier, greyest in R2012a), but for simple systems like yours (unless I missed something; see questions above), functions like OE and PEM can deliver the results, provided you do have the right data.
