Spring-Mass-Damper System, 2DOF 

i agree

Yes if you have more springs it would change transfer function.

How it will vibrate is going to depend on the type of force input and the values associated with the components. This is demonstrated in some later videos in the vibration and control playlist.

No, because left side is summing all forces, negative and positive, and right side is just m*a, or m*x in this case in positive x direction.

it's correct in the video, it could either be -kx1 -c1x'1 -c2(x'2-x'1) = m1x1" or -kx1 -c1x'1 +c2(x'1-x'2) = m1x1"

Only because the original question asked for the transfer function X_2/F. If you were asked for X_1/F then you would solve for it using Cramer's rule.

@@theryderproject5053 oh . On thank you. This video has been a huge help

I think it is the derivative of this (velocities) because that's the velocity that the damper is moving at - need to take the difference because both ends can move.

@@theryderproject5053 so can we write x2-x1 in the same manner

@@benguyldzzeyrek7388 The direction matters. Try rewatch from around 1.45. I've assumed that m1 is going to move to the right so the force is directed back to the left. This makes the order for the force x1dot-x2dot.

Also watch from 11.15 for a method to double check whether you've got the signs right.

@@theryderproject5053 okay thank you so much