In the case of vertical string of which top end is fixed and bottom end is free, tension in the string is present due to gravitational force. While analyzing the dynamics of such system, practically, neither the displacement nor the slope at the free end will be zero. Then what boundary condition should be used at the bottom end?
A good question. Please see the answer, which I wrote in the attached sheet.
What motions do you assume? Longitudinal and/or transversal?
If the end is really free, and there is not any force, then the BCs should reflect it.
That is, u'(l, t)=0 (no tension), w"(l, t)=Mb/IE=0 (no bending moment), and w'''=Q/IE=0 (no shear).
Zsolt Sir, I am considering the transverse vibrations. When string is vertically suspended, I am assuming it is under tension due to gravity. Though at the bottom end, we can assume the tension to be zero, physically we don't see the slope at the end is zero during the vibration. So, how do we really incorporate real physical boundary conditions in to the model?
For transversal vibration you must express the strain with y'.
For the case of free end, the strain must be zero. I am a bit sceptic that this would menni y'(l, t) =0...
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