I assume you have only real-valued shapes, therefore you might have room only in the poles, or complex-valued eigenfrequencies, with unchanged shapes.

Do not forget to update your FE models with real testing. In Article Competing optical instruments for the estimation of Full Field FRFs

Article Full field optical measurements in experimental modal analys...

Best regards,

Alessandro

Bharatharun Veluchamy It depends on the FEA package used. In PERMAS you might use

$MODDAMP TYPE = RATIO KIND = FUNCTION

0.01

to add 1% viscous modal damping to each mode.

The mode shape is not altered by hysteretic damping distributed homogeneously as a material property, it is simply the amplitude of the response to a dynamic force input which is reduced. This corresponds to a slight out-phasing between the applied force and the response acceleration/displacement, that requires, as rightly said by Alessandro, to use complex (vectorial) formulations.

If the damping model is not hysteretic (i.e. an intrinsic property of the material's young modulus), the physics turn far more complex and the mathematisation non-linear (e.g. viscous or frictional damping). In such cases the mode shape is altered and reveals generally amplitude-dependant...

Bernard Garnier is right and I would add that real-life boundary conditions might introduce other than hysteretical behaviours. As far as I know, the complex-valued representation of shapes is the closest to this deployment, where in some areas of the shape there might be delays or phase differences. To all this, you really might incur in non-linear dependencies, where a linear modelling such as EMA might be valid for a small interval.

Let's consider the equation of a 1 ddl system for free movements : xdd+2.ksi.wn.xd+wn²x=0

ODE easy to solve : the characteristic polynomial roots are +/-i.wn.sqrt(1-ksi²). So, you can see the frequency shift due to your viscous damping : sqrt(1-ksi²). For ksi=0.01, you have a shift with a factor of 0.99995 … I think it's less than the accuracy of your simulation !

We can now write the matricial equation of your diagonalized problem (in the modal basis ie eigenvector basis, considering the mode are mass-normalized) : Xdd+Wn²X=0 (real equation for first / Wn is a diagonal matrix). You are in a harmonic case, and you know that modal solutions respect Xmd=i.Wn.Xm, Xmdd=i.Wn.Xmd=-Wn².Xm.

You want to add a damping term in your equation : Xdd+2.ksi.Wn.Xd+wn².X=0. I've choosen a viscous damping, also diagonal in the modal basis. This is the usual assumption used in the finite element codes while you input a global modal damping. You can see that your eigen vectors Xm are still solution of your new equation, associated to the modified eigenvalues (-Wn² -> -Wn²(1-ksi²)).

Conclusion : with these (common) hypothesis, your mode shapes are not affected by a modal viscous damping, and your eigen frequencies are a little bit down shifted by a sqrt(1-ksi²) factor.

Allessandro and Bernard are right when they write

- that the general case is more complicated to solve (viscous damping matrix not diagonalized in the realeigenmode basis for instance, or other kinds of damping),

- that in an exprimental way, it's more and more complicated...

In your case, you consider 1% damping. This is a small value. You can consider that the effect is very very small on your modal base.

Article A state-space based model updating technique

Equations (36)-(38) might be helpful.

As Jean-Pierre Boudot remarked, the modal damping affects only the now-damped eigenfrequencies, but leaves equal the modeshapes, also in many degrees of freedom systems. Therefore you might add mathematically more damping to each eigenfrequencies without affecting the shapes.

A different thing comes when you are synthesising Frequency Response Functions: here the changed poles have effects on the modal blending, i.e. the superposition of the unchanged modeshapes with damped poles. This indeed might turn to be again complex-valued, therefore the final Operative Deflection Shape, sum of the shapes with right phase, at a specific frequency may have delays/phase shifting in some areas, depending on the added damping to the eigenfrequencies.

It should be also stated here that if the author of the question wants to increase the damping and recalculated eigenvalues it can be done by the solution of the eigenvalue problem using the modal and spectral matrix of the undamped system. If damping was considered Rayleigh, the mode shapes will be the same despite the modal and spectral matrix will be different compared to the undamped system. If general damping was considered the mode shapes would be complex.

So I would proceed in such a way that firstly calculate the alpha, beta coefficients of Rayleigh damping so that I achieved a damping ratio of 1%. It can be easily done if I know the eigenfrequencies of the system. Then synthesize damping matrix as C=alpha*M+beta*K and recalculate spectral and modal matrix for damped system.

Hi

As stated by https://www.researchgate.net/profile/Jean-Pierre-Boudot and https://www.researchgate.net/profile/Nils-Wagner-2 - you do not correct the mode shape, you just shift the natural (eigen) frequency for free vibration a little to get the correct resonance frequency for forced vibration.

This is automatically done in modal response analysis in most FE software and damping is then usually added as modal loss factors.

The above is a good approximation when damping is light and evenly distributed.

/Claes