If you have only geometric nonlinearities: A beam with axial load (N). If you increase the tensile load from zero the frequency will rise since the overall stiffness is increased (think guitar strings!). If you have compression the overall stiffness decreases and the frequency goes down untill buckling. Also there are purely geometric effects - if you have a cable under gravity load (think suspension bridge!) the in-plane frequency of the rope will be a function of the deflected shape and the axial tension etc.

If you have also material nonlinearities with increase of the load, thus crack opening and increase in inelastic response the frequency will decrease and also the eigenshape may change.

Basically, eigen frequently analysis is based on an assumption that the vibration amplitude is very small. For an example, a spring--mass system, the differential of the loading by the deflection is the spring rate. So, you can calculate the frequency. If the spring was highly loaded, the shape of the spring is changed and the loading--deflation relationship is no more linear---you have different differential---different spring rate---different frequency. This is the one---load dependent eigen frequency caused by non-linear effect. 

Did this help your understanding?

Yes, the eigen frequency is dependent from the load in case of nonlinear behavior. The eigen frequency depends on the stiffness of the system, which is going to be changed during the loading. It case of RC structures, the more load, the more damage, the less the stiffness, and the less the eigen frequency or the more the natural period.

For a confirmation, I say “highly loaded” means static load, not the vibration.

I’m wondering whether the example of guitar string is appropriate. In my understanding, the restoring force of strings is purely non-linear effect; hence linear analyses cannot determine the eigen value. Actually, I asked some experts; they are also not sure (automobiles do not have “string” at all). I think the model of guitar strings is rather complex.

So far, I think the spring--mass model is easier to help your intuitive understanding.

@ Yoshiki Fukada :    I am getting a better understanding.

@ Stoyan Andreev :  Thanks for your valuable input

@ Tammam Bakeer : You confirmed my understanding. Thanks a lot.. :)

Now one question, when we have non-linearity (like varying stiffness) can we even call the frequency of maximum response as Eigen frequency ? or is it a term just associated with linear systems?

I don’t know exactly what do you mean “maximum response”. Remember that the concept of eigen frequency is valid only for small amplitude vibrations.

@ Yoshiki Fukada : I meant the frequency at which response in maximum? Or in a simpler way, is the concept of Eigen Frequency only for linear systems or is it also the same for small amplitude vibration of cracked beams?

Basically, the concept is only for linear system. However, an approximation is available if the amplitude is sufficiently small. The approximation regards the system as linear one. This is why we use differential. Let's establish a mathematical model by yourself; you can obtain better understandings. 

Good luck. 

If there is a discontinuity, the situation is different. The approximation is no more valid. Because it is not differentiable. If you consider a vibration of cracked body and the crack open--close with the vibration. You can not use eigen frequency analysis. 

I understand the point of views. But I am yet to get a clear idea. Maybe I put it like this.

• Say, I have a specimen with a crack of length 'a' mm. From experiments, I have different Eigen frequencies for 1g and 2g base acceleration loads. Same sample is tested and crack size is always 'a' mm. Except the load magnitude, everything remains same. How do I explain this deviation in Eigen Frequency?

Cheers

I have no idea what is your model. Please explain.

Dear all, please find the image I have added.

@ Mr Fukada : In the above image, the Eigen frequency of the model is say 180 Hz. I apply 1g global acceleration load (sweeping frequencies from 150 Hz - 200 Hz) and see that maximum response is at 180 Hz which is as expected.

For the same model, if I instead apply 2g acceleration load (over a frequency range) should I expect the Eigen frequency to be shifted from 180 Hz? If yes, why should it shift?

The problem is in experiments, I see a shift in the Eigen frequency but no such effect is observed in simulation. I am trying to understand if it is physical to expect a shift or it is more an experimental artifact.

Thanks

I don't understand what do you mean "acceleration at 180 Hz" and how did you measure the eigen frequency. Even I don't know what did you conduct---experiments or analyses (simulation). 1g or 2g accelerations are steady load input; there is no frequency. If they are vibrations, sey "amplitude". 

@ Yoshiki Fukada: I apply a load of amplitude 1g (9.81m/s^2) with a frequency sweep ranging from 150 Hz - 200 Hz. That is what I have mentioned in my last post. 

I hope you understand now.

You still haven't explained properly yet. I asked you three questions but you answered for only one. And, this is more important, we already gave you sufficient knowledge about eigen frequency; but you ignore them. Without applying these knowledge, you asking instant solution. It is not right for your learning.

Physical understanding is not written by words, but it is an imagination. We cannot give you the imagination, we only can give you some assistances. Only yourself can establish your own imagination. 

Moreover, you'd better to check what is eigen value. It is basically purely mathematical. I think you use this word slightly different meanings. 

Sincerely,

@ Yoshiki Fakuda: Please ignore it. It was just a small conceptual question and I believe I have my question clearly posed. Moreover, I just wanted expert opinion and not an instant solution.

Nevertheless, thanks for the input

Shree

Only now, I understood what do you mean "maximum response"---it is resonance. In a linear system, resonances occur near the eigen frequencies, therefore observation of resonances is a mean to "estimate" the eigen frequencies. But a resonance frequency it self is not eigen frequency. "A resonance occurred" does not mean an eigen frequency exists, because... I already told. 

In your system, the crack is only a mean to introduce strong non-linearity. The fundamental fact is the non-linearity, but not the crack. It is very important to identify what is the fundamental factors in the system. This is also important for your good communication with others. 

Have a good day,