MDOFs damped system: transfer function estimantion

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I need to calculate the transfer function of a 4 DOFs system, in particular I need to calculate the system natural frequencies. Do you know to figure out these? On books I have found how to get it in the undamped case. Thanks
 

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I need to know how to find a transfer function for 2DOF system with damping. If anyone answers this question will be helpful to two persons....thanks, Milind.
 
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AlephZero
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I need to calculate the transfer function of a 4 DOFs system, in particular I need to calculate the system natural frequencies. Do you know to figure out these? On books I have found how to get it in the undamped case. Thanks
The answer depends on how you want to model the damping.

In the general case where you have known physical sources of damping (e.g. dashpots) in the model, you have a 4x4 quadratic eigenproblem and both the eigenvalues and vectors (mode shapes) will be complex. In other words, the motion of the different DOFs in a mode are not in phase with each other.

In practice, for small levels of damping where the physical cause of the damping is not known explicitly, you would use a modal damping model based on the undamped modes and frequencies.
 
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The answer depends on how you want to model the damping.

In the general case where you have known physical sources of damping (e.g. dashpots) in the model, you have a 4x4 quadratic eigenproblem and both the eigenvalues and vectors (mode shapes) will be complex. In other words, the motion of the different DOFs in a mode are not in phase with each other.

In practice, for small levels of damping where the physical cause of the damping is not known explicitly, you would use a modal damping model based on the undamped modes and frequencies.
for a viscous damping on this book (pag815):
http://books.google.com/books?id=AKjnOnhGDXYC&dq=vibration+fundamental+and+practice+silva&hl=it&ei=DTkhTtmAKcOr-gbDq7mVAw&sa=X&oi=book_result&ct=result&resnum=1&sqi=2&ved=0CCkQ6AEwAA

I found that from an undamped transfer function, it is necessary to substitute the stiffness coefficients (k) with k-jc, where c is the damping coefficients.
 

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