Do internal / external stresses affect modal frequency?

Question

Imagine a cantilevered beam, fixed on one end and with a tension line connecting the other end to ground (e.g. perpendicular to the length of the beam).

If I increase the tension, does the modal frequencies of the beam change?

• On the one hand, tension affects the frequency of a string. So, would pulling on the beam (in tension) cause it to have a higher frequency?
• On the other hand, the bending stiffness of a beam is not affected by a lateral force. So, if the stiffness is not affected, then neither should the frequency.

What can help me reconcile these two comments?

Bonus aspects of the question:

1. If the stress is high enough to affect the slope of the stress-strain curve, then the stiffness changes because the modulus changes.
2. In Nastran, you cannot apply a load while performing an eigenmode / eigenvalue normal modes analysis. Does that mean it is not a linear phenomena?

Answer 1

Consider a small piece cut out of a structure that has non-zero internal stress.

To maintain equilibrium, there must be some forces applied to the boundary of the piece. (Of course when it was part of the complete structure, those forces came from the stress in the adjacent parts of the structure.)

When you deform the piece further, those external forces can do work, which is simply "force $\times$ distance" applied to the geometry of the deformation.

You can describe this work for all possible deformations as being equivalent to an additional stiffness matrix, called the stress stiffness. (Historically it was sometimes called the "geometric stiffness", but that name is not very self-explanatory!)

So, when a structure with internal stresses vibrates, the relevant stiffness matrix is not the elastic stiffness $K_e$, but the sum of the elastic and stress stiffness $K_e + K_\sigma$.

$K_\sigma$ can either increase or decrease the vibration frequencies. In fact $K_\sigma$ is the same stress stiffness that is used in finite element elastic buckling models, and the buckling load is the magnitude of the stress when the lowest natural frequency of the structure is zero.

This is not a "linear" phenomenon in the sense that $K_\sigma$ depends on the particular stress distribution in the whole structure, and that (obviously) depends on the applied loads, including inertia loads if the structure is rotating, etc).

You can model this in Nastran, for example using SOL 106. You first do a static analysis to find the internal stresses (that step can be either a linear or nonlinear stress analysis) and then a vibration analysis of the stressed structure. If the stress analysis is nonlinear, you may want to do several vibration analyses with different levels of applied load.

The natural frequencies of a stretched string is a simple special case of the general situation, and the usual "Dynamics 101" derivation of the frequencies just ignores all the terms in the general equations which are 0 in that special case, without really explaining what is going on.

Answer 2

You can look at it as springs in series and find K effective of the two.

$$ \frac{1}{K_e}= \frac{1}{K_1}+\frac{1}{K_2}$$ $$ \frac{1}{K_e}= \frac{L^3}{3EI}+ \frac{1}{K_{wire}} $$

Then it gets a bit complicated. you need to find the mode shapes of the beam and calculate the beam's participatory mass for each mode and apply that to the wire as a spring mass system.

Answer 3

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. 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 Hook'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.