Number of Normal Modes

[sushi75]

Hi everyone!

I start working on structures dynamics, and I ry to get my head out of the equations for a minute and try to make sense of everything. And now I'm getting confused:-(

So I hope someone can bring me some light!!

Basically before any analysis such as frequency reponse, or random vibration analysis, the first step is to identify the normal modes of the structure.

In many books, it says that there are N possible values, N being the number of dof.
From my understanding, there are 6 dof for the structure (3 translations and 3 rotations). So the number of possible frequencies (eigen values), would be N for each translation and each rotation, is that correct?
Thus N would be related on the discretisation chosen for the FE model.

In theory there are a infinite number of natural frequencies if we consider the continuous model (such as a beam), right?

I'm also getting confused as in many books, the example of a spring-mass system is chosen, giving a set of frquencies. Looking at a membrane case, the results give "two-directional freqencies" Fij.
Not easy to understand.

Finally, I wondered what if we have complex eigenvalues for frequencies? is there any phyisical meaning?

Thanks a lot for any help you can provide me, dynamics seems to be a great and intersting topic, but I need to get a full understanding of the fundamental concepts first

Cheers,

T

 

[GregLocock]

"In theory there are a infinite number of natural frequencies if we consider the continuous model (such as a beam), right?"

Yes, in theory.

If you could give an example of the membrane frequencies it would help, it may be that they are saying that the mode with i half waves along one side, and j along the other, has a frequency F[sub]ij[/sub]. In my experience this is defined in the text, and to be honest so are the answers to your other questions. Incidentally each element type of an FEA model can have a different number of DOF, not just 6. Any practical FEA model will have more natural frequencies than are useful.

Cheers

Greg Locock


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[sushi75]

Hi Greg,

Thanks for your quick reply.
To illustrate the statement regarding normal modes of a membrane, please see image attached.

About the general problem, for a FE model with a stiffness matrix size N, there will be N natural frequencies, and 6N mode shapes (some of them may show no displacement though)?

Cheers,

T

 

[GregLocock]

So what doesn't make sense with that table? I thought it was self explanatory.

Cheers

Greg Locock


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[amanuensis]

When doing modal analysis, you change of space.

In modal analysis, the primary space is NOT the space-time.
You have to have a little bit imagination.
If you understand this, modal analysis will have no secret for you.

In modal analysis, the space is NOT the 3d space. A point is NOT given by 6 values (3 coordinates X, Y, Z + 3 rotations).

In modal analysis, the space is composed of stationary waves!!!

A stationnary wave is the equivalent of a point is our ordinary space.

You have to see each stationnary wave (or mode shape) as a point!

So for a beam, the dimension of the space in modal analysis is infinite !

 

[Denial]

About the general problem, for a FE model with a stiffness matrix size N, there will be N natural frequencies, and 6N mode shapes

Change to "...and N mode shapes".

 

[sushi75]

Hi!

Ok I try to rethink things and bring a bit more imagination, so if we consider space in terms of stationary waves, we get N shapes, one for each frequancy.
And the output given by a FE software giving shapes for each "real" dof namely Tx, Ty, Tz, Rx, Ry and Rz are more for analysis purposes, they are just the decomposition of each mode shape on the coordinate system, right?

I know now that dynamic analysis needs more thinking, especially to get a better understandng of data definied on a frequency range.

Hope everyhing will become cristal clear soon!

Cheers

T

 

[amanuensis]

Let's take an example.
Let's suppose a beam, simply supported, length = L, etc..
Let's consider the 3 first modes (N=3)
The modal response is something like this :
P(x) = A1*sin(pi*x/L) + A2*sin(pi*2*x/L) + A3*sin(pi*3*x/L). (Sum of the 3 modes)

You can imagine P(x) (for all x) as just 1 point of coordinates (A1,A2,A3) in a modal space defined by the mode shapes (sin(pi*x/L),sin(pi*2*x/L), sin(pi*3*x/L)).

Let's suppose 1 point of cartesian coordinates (Ax,Ay,Az),
then the analogy is :
A1*sin(pi*x/L) <=> Ax
A2*sin(pi*2*x/L) <=> Ay
A1*sin(pi*3*x/L) <=> Az

 

[sushi75]

Hi,

Thanks a lot, it took me nearly a day but with this new way of thinking, it's much clearer now!

Forgetting the physical space and considering a new "space" defined by the eigen vector as a base ,and thinking in terms of wave definitly makes the picture clear!

cheers
tom

 

[amanuensis]

When complex values appear, it means that damping is taken into account.

Imaginary part of a complex eigenfrequency is related to damping value (damping loss factor).
Imaginary part of a complex eigenvector means that mode is not normal (mode is not a stationary wave).

 

[sushi75]

Hi,

Thanks Amanuensis for the complement of information. The explanation you provided was definitely helpful, and now havinf finally a good understanding of normal modes would be beneficial to understand dynamic analysis.

Before that, going back to modes, having a complex eigenvector (usually we assumed that there is no damping when looking for the normal modes for FRF analysis for instance), means that there is damping, but could you elaborate on the non stationary waves for mode shapes? I understand that this loss of energy could be interpreted as a wave showing decay or attenuation but not sure about it!

Cheers,

T

 

[mpvnaresh]

can any body suggest me books on vibration analysis where i can get the physical significance of the theory (for example what actually mode shapes are?).

 

[IRstuff]

Please start your own thread

TTFN
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[amanuensis]

What is the main difference between normal and complex modes ? The way the phase is taken into account.

With normal modes, the phase exists, but it doesn't matter. Why ? Because the phase is either 0° or 180°. So the amplitude of the mode at each point is given by +something (0°) or -something (180°). Real values are sufficient to display modes. The consequence of this is that movement of the mode reveals nodes. The mode can be seen as a stationnary wave.

With complex modes, the phase can vary from one point to another one. So, in this case, it's more complicated to apprehend the movement of the mode. The phase shift at each point induces a traveling of the nodes during movement. The mode shape is no more constant during movement.