Equations of motion of a 2-DoF Free damped vibration system

[mmullan]

Summary:: What are the Equations of motion for a free damped 2-Dof systrem?

Hello,

I am required to calculate the equations of motion for a 2-dof system as shown in the attached file. The system is undergoing free damped vibrations. I have found the equations of motion for no damping but i was wondering what effect damping has on these equations and have not been able to find a book that has the equations for free damped 2 dof motion. The system i am analysing will require the motion to be able to calculate displacement values with changing initial displacements but the initial velocities will always be 0.Would anyone know the damped free vibration equations of motion for a 2 dof system or know how these equations are obtained?

 

[BvU]

Hello @mmullan,

!​

I have found the equations of motion for no damping

There are a few rules in the homework forums that require you to post your attempt at solution before we are allowed to help.

In this case I don't think I'm breaking any rules if I suggest you use the same approach as with the damped harmonic oscillator.

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[Fred Wright]

Please see https://web.mst.edu/~stutts/SupplementalNotes/Lagrange.pdf. An example for a similar system as yours is given using two dash pots, however you can generalize to three dash pots.

 

[mmullan]

Thanks for the help. I need to find an equation for x1(t) and x2(t). From the example you've shown i am unsure how you could take it from the matrix form to the x(t) equations

 

[BvU]

Thanks for the help. I need to find an equation for x1(t) and x2(t). From the example you've shown i am unsure how you could take it from the matrix form to the x(t) equations

Yes, I know. But:
I tried to point out the rules to you: first post what you have, then we can provide help

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[mmullan]

Yes, I know. But:
I tried to point out the rules to you: first post what you have, then we can provide help

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Sorry about that. I am new to the forum and didn't know about these rules. I have attached my workings where i have performed a laplace transform on the equations of motion of the masses according to Newton's second law. I am unsure of the next steps in order to obtain equations for x1(t) and x2(t)

 

[BvU]

Very good. In the undamped case you now proceed to decouple the $\ddot x_i$ and find normal modes. In your work it looks as if you go to a Laplace transform for the coupled $x$.
Would it be sensible to try and find composite coordinates $\ \xi_i\$ (with solutions $\ \xi_i = A_ie^{s_it}\$ where $\ s_1 \in \mathbb {C} \$ ) and see if you can find the matrix for them, so you can decouple ?

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[mmullan]

Sorry but i am unsure how to relate the equations i have found using the free body diagram to the equation for the composite coordinates. could this equation be described as x1(t)=X1e^(s1t)