1dof resonant frequency between modal and harmonic

filjan2

Hello,

I would like to ask for guidance about issue of the resonant frequency discrepancy obtained with different methods.

Let me describe a problem step-by-step:
1DOF problem was considered to check numerical results with analytical approach:

As you can see below, either frequency for damped or undamped systems are aligned with Code_aster output of modal analyses:

table-limit.txt

4kB

Next step was to check output for the same 1DOF model for harmonic analysis (physical and modal bases). Both harmonic approaches gave the same curves output with extremum (i.e. resonant frequency) a bit shifted left - read extremum from both curves equals 3.73Hz (see attachments below) but I expected 3.82 Hz obtained based on analytical and numerical modal analysis!

I would like to ask you about the possible reason of this difference. I guess it is rather some misconception in my understanding than solver problem because I observed the same differences in the other well-known commercial solver.
Bigger damping ratio leads to an increase of this difference. And the question at the end - which resonant frequency is more reliable?
Resultant damping ratio (0.2) is not typical in engineering, but I have not encountered any limitation of any analytical approach usage to only some small values - of course as far it is less than 1.

Best regards,
Filip

jeanpierreaubry

filjan2
maybe this has to do with the load function?
it would be interesting if you shared the files

and just a typo the pulsation omega is in rad/s, not Hz!

filjan2

@jeanpierreaubry thank you for your response. Please find input files in the attachment. I mainly followed the guide you wrote, so some names might seem to be familiar for you.

single-dof-mesh.txt

206B

single-dof-modal-qep.txt

2kB

single-dof-harmo-direct.txt

3kB

I am not attaching harmonic modal based as its gives the same output as direct one.

What regards units - indeed you are right but as far radian is dimensionless this spreadsheet (Smath) recognizes 1/s as Hz.

Best regards and I count on your guidance
Filip

jeanpierreaubry

filjan2
try and plot the REEL and IMAG component of DZ, it is interesting
and tell me what you think
then i think i have to dive again in,the text books

filjan2

jeanpierreaubry

I found the explanation in one of yours bibliography book - "Dynamics of structures" by Cloug & Penzien-please see p.42. chapter 3-3 Resonant Response and p.27 2-6 Damped Vibration.*

The reason of this difference is of the way of getting the resonance frequency. Modal analysis is similar to finding roots of equation but harmonic is like the interpretation of extreme values/zeros from phase-freq and magnitude-freq characteristics. To obtain these curves one needs an assumption about exerted load (force here) and this assumption after transformation creates extra "2" beside zeta² for 1 DOF system (See below). Now analytical results align with FEA. I guess that assumption of base motion instead of force creates the same resonant frequency to modal-based approach, but it will be next step to check. Now we see that assumed harmonic force slightly shifts resonance.

Best regards
Filip

_* the same harmonic approach is described here http://pioneer.netserv.chula.ac.th/~pphongsa/teaching/vibration/Ch3.pdf

jeanpierreaubry

filjan2
so here is the answer!
but i guess there is a typo in your answer, or we do not have same edition of Clough, or i am dumb blind
as i cannot see what you state "p.27 2-6 Damped Vibration"

i let you carry out the next step "base motion" and tell me your finding

anyhow as stated bu Clough the practical result of this shift are somewhat in the width of uncertainty

filjan2

jeanpierreaubry

Hello again,

I used the third edition of Clough&Penzien.

Harmonic analysis loaded by 'base motion' gives the same resonance frequency - the peaking frequency is independent of the load type - it was not right question to take a closer look.

I came to another conclusion but trivial one: Damped modal eigenfrequency is just not close to the resonant frequency when damping ration is not close to 0.
All non-zero initial conditions provides decaying response of the system and aligned with damped modal frequency but if any driven force (as base motion or force) exists and its frequency equals modal frequency the system does not ensure its maximum response.

Here you see response of the system with a usage of DYNA_NON_LINE with non-zero initial state. The system is tensioned in z direction by 1mm and then released:

Circles mark each time-point crossing the zero displacement position - time between each consecutive circle is a real half-period of the system. By simple transformation it equals freq=3.8168 Hz (very close to the theoretical modal analysis with damping) (FFT of this signal does not provide good resolution because of short time of amplitude with relevant amplitude - that is why I count halfperiods).

Then I made two models with a sinusoidal loads as the resonance and the modal-damped frequency and indeed the first one gives a bit higher response:

I also found a similar discusion and distinction of frequency in electrical eletrical engineering forum:
[https://electronics.stackexchange.com/questions/692613/what-is-the-difference-between-natural-frequencies-and-resonance-frequency-in-se].

To conclude:
If one is looking for the precise resonance frequency of a damped system then relying on a modal analysis (with or without damping) does not provide valid solution.
Best regards
Filip