The statement of the question, using "internal/external stresses" is confusing. I don't understand what "external stress" can be here. You can have external forces but not external stresses. From the question elaboration, it appears to ask whether the internal state of stress affects vibration modes, yet the questioner's example of a "tension line" cannot change the internal state of static stress without altering the boundary conditions on the beam. Altering the boundary conditions will certainly affect vibration modes. In addition, an "internal state of stress" always exists in a material, whether undergoing vibration or not. It seems that the questioner is asking whether the "static" state of stress affects the modes, and static stress are internal stresses of motionless material. I suggest the questioner clear up these issues, and I'll give the answer to the simple question which I believe he really wants to ask: "Does the static internal state of stress affect vibration modes of a beam"?
The answer to that question is No, as long as the elastic limit of stress in the material is not exceeded. A different static state of stress can result in total stresses during vibration that exceed the elastic limit, which may affect the vibration modes. 1018
But the answer to that would be too complicated to discuss here. For purposes here, the simple answer is "No."
Explanation is that, first of all, vibration of a taught string is a different phenomenon, where the restoring (spring) force is the tension itself. With the beam, called a "bar" in vibration science, the restoring force is the elastic force, and even more, it's the change of elastic force, not the total (change + static), caused by the vibration. Static forces don't enter into the equations of motion. If you change the static situation, the dynamic changes of internal stress only occur about a changed mean value, and as long as the response is elastic (obeys <PERSON>'s law), the dynamic solution will not change.
In fact, we know very well that the static state of internal stress normally has no effect on vibration modes Most all hardened metals have some nonzero internal static stresses, and we are not concerned with them when we calculate vibration modes in such materials.
For instance, with 1095 Swedish spring steel, there can be residual internal stresses in high temperature quenching during the hardening process that are not eliminated by sufficient uniform annealing (tempering), and that doesn't affect the vibration of a cantilever made of such material. Such a device is used as sound source in free reed musical instruments, and makers aren't directly concerned with the internal state of stress in the reeds they make. Now, there are complications when you consider other practical effects, such as metal fatigue and breakage. Fatigue can occur over many cycles when stress levels are even below the elastic limit, a static concept. There is also an "endurance limit," which is below the elastic limit, above which fatigue lifetime is greatly reduced. 333