Just to put you in the context: during an internal turning operation, the overhang (ratio Length/Diameter of the tool [L/D]) is really important to guarantee the stability of the process (minimal vibration as possible). Having said that, it is desirable to increases the overhang to do deep holes, because of this the ratio L/D varies depending on the necessity and consequently the natural frequency of the tool will change.
As you can see in the attached Picture_A and B, I am trying to find the Eigenvalues when the overhang (ratio L/D) changes. Is it possible like in Figure 2 in the attached paper (link below)?
L1 = ratio L/D (changeable);
L2 = Fixation of the tool (content)
L3 = the remaining part of the tool out of the fixation (changeable)
Tool length is content = L1 + L2 + L3