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## #1 2017-08-10 09:24:54

edhage
Member
Registered: 2016-08-23
Posts: 13

### How is reduced damping calculated?

I need some help in finding the documentation that describes how AMOR_REDUIT (reduced damping) is calculated in a modal analysis.

When a modal analysis is performed with CALC_MODES of a hysteretic material with AMOR_HYST = 0.06 for example, the resuling reduced damping is calculated as 0.03 for every mode. Which is correct (loss factor/2 as mentioned in r5.05.04).
This value is found in modes.LIST_PARA()['AMOR_REDUIT'] if modes is the result of CALC_MODES.

When I use a combination of materials with different hysteretic values also AMOR_REDUIT is calculated in CALC_MODES for every mode. This is valueable information for me and I want to know how this is calculated.

Cannot find it in r5.01.03 or r5.05.04 and in docu of CALC_MODES.

Help is appreciated

Greetings, Edward

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## #2 2017-08-16 10:42:24

Johannes_ACKVA
Member
From: Ingenieurbüro für Mechanik, DE
Registered: 2009-11-04
Posts: 595
Website

### Re: How is reduced damping calculated?

When I use a combination of materials with different hysteretic values also AMOR_REDUIT is calculated in CALC_MODES for every mode. This is valueable information for me and I want to know how this is calculated.

Edward

the values of  AMOR_REDUIT are simply the reel parts of the complex eigenvalues. So they are calculated by the eigensolver.

Only in the very special case of a homogeneous loss factor there is this simple equation: reduceddamping=loss factor/2

Be aware: the loss factor is a material property, the reduced damping is a value of the vibrational system

Regards
Johannes_ACKVA

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## #3 2017-08-16 12:54:07

edhage
Member
Registered: 2016-08-23
Posts: 13

### Re: How is reduced damping calculated?

Hello Johannes,

I think what you mean is that the values of AMOR_REDUIT are the imaginary parts of the complex stiffnesmatrix pre- and post-multiplied by the eigenvalues, and not the real part of the eigenvalues.

if you do this than you gain a matrix with the size of the stiffnessmatrix but the terms are not only diagonal.

The terms are diagonal case if you pre- and post multiply the real part of the stiffnessmatrix, the result is omega^2.

But in the case of damping you will have non-diagonal terms (except if loss factor is homogenous, than the imaginary part is a constant factor * real matrix).

Are the non-diagonal terms just disregarded ?

Greetings, Edward

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## #4 2017-08-17 10:29:39

Johannes_ACKVA
Member
From: Ingenieurbüro für Mechanik, DE
Registered: 2009-11-04
Posts: 595
Website

### Re: How is reduced damping calculated?

No. The values of  AMOR_REDUIT are simply the reel parts of the complex eigenvalues.
Look R5.01.03, eq 1.3-2

The eigenvalues lambda are pairs of conjugate complex numbers. The imaginary part is  i*omega, the real part is proportional to the damping.
The greek letter xi is the degree of damping which is in Code-Aster AMOR_REDUIT

This is true for  both cases: wether your stiffn matrix is complex (hysteretic damping) or you have viscous damping (velocity proportional damping matrix)

So, a little correction to my last post: the reel part of the eigenvalue is proportional, but not equal the degree of damping

Regards
Johannes_ACKVA
______________________________________________________________________
CODE-ASTER-courses at Ingenieurbüro für Mechanik, Germany

*** CODE-ASTER INTRO + CONTACT + MATERIAL
09-13 October 2017

*** CODE-ASTER  DYNAMIC ANALYSIS
30 Nov - 01 Dec 2017

Ingenieurbüro für Mechanik
D 91717 Wassertrüdingen / Germany

www.code-aster.de                                                Training & Support for NASTRAN and CODE-ASTER

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## #5 2017-08-17 17:21:23

CLF
Member
From: Sweden
Registered: 2009-01-27
Posts: 90
Website

### Re: How is reduced damping calculated?

Gents

Sorry to but in.

Code Aster (CA) is quite a quirky piece of code with lots of endearing functionality on dynamics.

There are more ways than one to calculate damping.

Without knowing actual facts on the inner workings of CA:
1) One solves the undamped problem, i.e. eq. 1.2.2 with C=0, when computing real modes (I believe these are called clean modes in the documentation) and then use the modal damping values in the modal summation procedure. This is computationally effcicient when there are fewer modes than analysis frequencies and is physically motivated for light and evenly distributed damping.  Real valued modes are standing wave pattern of free wave motion.
2) Solving the quadratic problem, i.e. eq 1.3.2 and C > 0, one computes complex modes. As Johannes points out, such analysis produces loss factors. Complex modes are standing wave patterns where there is a net flux of energy across the system. This is seen when animating motion as the node (zero motion) point moves (it stands still for a real valued mode).

Now, looking at U4.52.13 (http://code-aster.org/doc/default/en/man_u/u4/u4.52.13.pdf ), and AMOR_REDUIT, we find mentioned that it uses a GROUP_MA and the potential energy for this group. We find also that AMOR_REDUIT can be used together with MISS3D which is used for a analysis where unbounded regions of soil is connected to the model as well as with RAYLEIGH damping.

The Modal Strain Energy (MSE) method can estimate modal damping to be used for real valued modes and modal summation. The approach is simple, one calculates the % of the total modal strain energy for GROUP_MA, multiplies its local loss factor by this % number and, voila, one has a modal damping value. The procedure can be repeated with varying loss factors and materials across the structure and it works well as long as damping is light.

If memory serves me correctly, this procedure is called composite modal damping in ABAQUS.

Coupling a software like MISS3D to the boundary of a CA FE model produces a full complex valued frequency dependent matrix. Differently stated and using common sense physics - an unbounded region has free waves at any frequency, i.e. solving for modes makes no sense as there is free wave motion, i.e. a mode, at every frequency.

The above mentioned solution scheme is quite heavy. A simplification would be to use a procedure to estimate the modal damping of the bounded domain of the CA FE model, which it seems to me that AMOR_SOL does using a variation of the MSE metod where the portion of energy in the soil part is computed using ENER_SOL  to get a frequency dependent damping value AMOR_SOL, i.e. a loss factor that exists at all frequencies the soil stiffness is defined for.

It appears to me that the AMOR_SOL value can be further simplified and added as modal damping values, AMOR_REDUIT.

This approach is similar to the approach of added modal mass from fluids that load a structure - added mass deals with the real part of the problem while damping deals with the imaginary part of the same problem (or vice versa, depending on how you define matters).

A similar approach is made also for AMOR_RAYLEIGH where the ALPHA and BETA values can be condensed to modal damping values.

The AMOR_REDUIT approach should work well as long as damping is light, say, below 5% of critical or so.

All in all, it seems to me that AMOR_REDUIT exclusively deals with real valued modes and is quite a lovely code for calculating modal damping in various ways. It is by far the most complete implementation of modal damping estimation I have seen in any FE code.

Just my 2 cents
Claes

Last edited by CLF (2017-08-17 17:36:27)

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## #6 2017-08-17 20:56:32

edhage
Member
Registered: 2016-08-23
Posts: 13

### Re: How is reduced damping calculated?

Johannes, Thank you for the explanation and pointing out the paper to me, I will study it.

On a sidenote:
Is the Modal Strain Energy (MSE) method, which is basically a estimation for the modal damping, implemented in Code-Aster ?

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## #7 2017-08-18 09:46:41

CLF
Member
From: Sweden
Registered: 2009-01-27
Posts: 90
Website

### Re: How is reduced damping calculated?

Hi Edward

As I read things in U4.52.13, AMOR_REDUIT with hysteretic damping is MSE, but AMOR_REDUIT has more trix than this up its sleeve.

From your first comment - the structural loss factor ETA (AMOR_HYST) meets the relation ETA = 2*C/Ccrit, where C/Ccrit is the critical damping ratio, a.k.a. modal damping.  So, AMOR_REDUIT is C/Ccrit, while AMOR_HYST is ETA (which in turn is ETA=1/Q - you can look up the Q-factor on Wikipedia). This confusion point is a classic.

/C

Last edited by CLF (2017-08-18 09:53:11)

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