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.

 

[astrokreng]

Hello,

Thank you for sharing your problem and equations with me. It seems like you are working on a complex and interesting system. Based on the information you provided, it appears that you are on the right track in terms of finding the natural frequencies of the system.

To answer your question, yes, you will need to solve the system of differential equations to find the natural frequencies of the system. This can be done analytically or numerically, depending on your preference and the complexity of the equations. If you have access to Matlab, you can use the built-in functions for solving systems of differential equations, such as "ode45" or "ode23". These functions use numerical methods to approximate the solution to the equations.

Another approach would be to use the method of undetermined coefficients to solve the equations analytically. This method involves assuming a solution form and then solving for the unknown coefficients. However, this method can be more complex and time-consuming, especially for systems with multiple masses and springs.

In terms of avoiding resonance conditions, it is important to note that resonance occurs when the frequency of the external force matches the natural frequency of the system. To avoid this, you can either adjust the frequency of the external force or adjust the natural frequencies of the system by changing the stiffness or damping coefficients.

I hope this information helps and good luck with your problem. If you need further assistance, please do not hesitate to reach out. I am always happy to help and share my knowledge.