What Frequency Excites Resonance in a Two-Mass, Two-Spring System?

[RussellJ]

Hi, i am currently working on a problem involving a one dimensional, vertical two mass two spring system where the upper mass is free to move with the lower mass adn the lowermass is excited using an oscillating force. The springs are different with a damping force in the form of D = C*x'.


...______ ^x2
...|.M2...|
...|...|
...|_____.|
...\
.../ K2
...\ C2
...___/__ ^x1
...|...|
...|.M1...| <=== Force applied in the form F(t) = Sin(wt)
...|_____.|
...\
.../ K1
...\ C1
_____/_______I am trying to find the frequency w of the force that would excite resonance in M1 or M2 to avoid resonance conditions. so far i have worked out the following differential equations for each mass using a force balance

Mass 1 (M1)

K2*x2 -(M1+M2)*g + Sin(wt) = M1*x1'' +c1*x' + (K1+K2)*x1

Mass 2 (M2)

K1*x1 -(+M2)*g = M2*x2'' +c2*x' + K2*x2
I am having issues going further from here. Do i need to get to a full analytical solution to find the natural frequencies? It has honestly been a while since i have done any work with mechanical vibrations. I am trying to solve this system of equations but have little experience with systems of 2nd order DE's.

A numerical method would be fine, i have access to matlab.

Any suggestions would be appreciated.

 

[RussellJ]

Update: I figure i can find the natural frequency of M2 based on the constants M2, c2 and K2 but i still need the frequency of M1.