Hi Amit Sharma You may have underdamped system.

Hope this helps!

Hi Amit,

If I am right , in the case of an underdamped system, as Dr. Martin says, the poles are complex numbers and the natural frequencies are the absolute values or magnitudes of the poles.

When characteristic roots of a system are complex conjugates, the system is under damped and the imaginary part gives damped natural frequency.

When the characteristic roots are complex conjugates with zero real part, the system is undamped and the imaginary part gives natural frequency (ωn). The ωn can be used to determine stability of the system as follows.

ωn2 < 0, Unstable equilibrium (ωn is complex)

ωn2 = 0, Neutral equilibrium (ωn is zero)

ωn2 > 0, Stable equilibrium (ωn is real positive).

Thanks for your answer.

In that case I have to take absolute value or not

No! because the unstable system has no meaning for natural frequency.

Since you are simulating a plate, I think there should not be any issue of stability. First you check your model and boundary conditions, it may introduce instability.

Also check that your software is giving output in which form- directly natural frequencies or characteristic roots?

Again thanks for the information.

I have to calculate 6 natural frequency mathematically. If I take four terms in my deflection function as a solution it is fine. It gives me four frequencues. But when I increase my terms from 4 to 6 it will give some real root and some complex roots

Thanks in advance

Hi Amit Sharma,

What are the BC and which deflection functions do you consider?

plate is clamped along all the edges i.e., clamped boundary condition. Deflection function is taken as series which satisfy the clamped edge condition.

Thanks in Advance

Maybe you'll find something helpful in the article attached.

Than you very much for your help. But paper is not in English languag. Thanks once again

Using http://www.etranslator.ro/german-english-online-translator.php something can be read.

Thank you very much sir.

Dear Amit:

the presence of damping in your system may produce complex roots. Regarding the vibration of rectangular plates, perhaps this article could be useful for you:

Article On the vibration analysis of rectangular clamped plates usin...

Dear all

I agree with the above authors - negative damping is not expected for your case.

Amit - on a more practical note - try using a solver that solves for real modes instead of one that solves for complex modes. The latter does not provide as useful data unless your physical damping modelling is accurate and meaningful. If you have not input any damping or, very little damping, then I am not surprised of your complex modal solver produces all sorts of data as you then likely are within the numerical solver tolerances.

As a side note, a while back, I was using a complex mode solver and it gave me different results using the exact same input data. I asked the supplier who simply replied that "the calculation of complex modes is a hobby for gentlemen". They have since cleaned up the solver performance to now produce repeatable results but I suspect that their feedback comment still stand.

For what it is worth, I would just like to add the reflection that negative damping does make physical sense in some situations, e.g. negative damping is what you get for brake squeal and it is then just another way of stating the power input from the rotation. Similar situations can be found in aeroacoustics, flow-strutcure interaction, rotor dynamics and so on, where gross motion interact with vibration to produce an input that can be stable or unstable.

On top of the above is the more esoteric so called Lock-In phenomenon where resonance may sustain also under conditions where it usually would not, simply because amplitude is too high for response to come down in amplitude to utilize other (e.g. linear) loss mechanisms that would be available at lower amplitude levels. Known examples are found for so called stepping frequencies found in corrugated tubes.

Not exactly a Lock-In but, at least to me, quite an impressive analysis is so called modes from shock nonlinearities. https://www.code-aster.org/spip.php?article750 . I would expect these two modes to exhibit quite different damping values.

The link shown between heat and damping here also is quite cool.

Article Damping heat coefficient – Theoretical and experimental rese...

Please find also some more ramblings of mine on damping here

https://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/

It is a big world out there and our understanding of damping is poor.

Sincerely

Claes

Thanks all for your help.

Yes