1. Modelling and identification of damping in aero-structures ECERTA Project Liverpool 26 November 2007
Marco Prandina

2. Classification of damping
Damping, the dissipation of energy by a vibrating structure, can be classified into three main categories :
Material damping : energy dissipation due to microstructural mechanisms as irreversible intercrystal heat flux, grain boundary viscosity, etc.
Boundary damping : dissipation associated with junctions or interfaces between parts of the structure (joints) and contacting surfaces (friction).
Fluid contact damping : energy radiation into surrounding medium and dissipation associated with local viscous effects.

3. Structural damping identification
Damping is sometimes neglected or over-simplified in dynamic design of structures. However, a good identification of structural damping is important for a good modeling of structures, especially when the model is to be used in:
Predicting transient responses.
Predicting decay times.
Other characteristics dominated by energy dissipation or instability (e.g. flutter).

4. Problems in damping identification
Damping is one of the least well-understood aspect of vibration analysis for many reasons :
Absence of a universal mathematical model to represent all damping forces.
It’s not clear which state variables the damping forces will depend on.
Mechanisms that dissipate energy in a system are different in nature and amplitude, engineering choices must be made before starting to develop a model.
Damping can’t be measured by static tests like stiffness and inertia. Dynamic tests are needed and they are normally more affected by noise.

5. Inverse problem
Identification of damping is a typical inverse problem: the aim is to find the model parameters of a dynamic system using measurements such as Frequency Response Functions, modal data, known system parameters (mass and stiffness matrix of a Finite Element Model, for example) and any other useful information.
FRF
Modal data
Known parameters
Model
Unknown
model
parameters

6. Main issues in inverse problem
There are some practical issues in solving the inverse problem of damping identification for a real structure:
Spatial and modal incompleteness of data
Ill-conditioning of matrices
Non-uniqueness of solutions
Computational time for real big structures
Noise and errors in measurement

7. Incompleteness of data in modal analysis
Modal matrix
Modal
Incompleteness
Spatial Incompleteness
Spatial Incompleteness
Spatial Incompleteness

8. Testing existing models
Numerical simulation of two existing damping identification models was performed to test how much they are affected by frequency range incompleteness of data on a clamped cantilever beam modeled as a simple ten-elements FEM with a dashpot on its free end. x2 x4 x6
x20 x1 x3 x5
x19 c

9. Damping matrix

10. Tested models : Adhikari - Woodhouse
Assumptions :
Damping is small, so eigenvalues and eigenvectors of the damped system will be close to the undamped ones. First order perturbation method is applicable.
Data needed :
Complex frequencies (eigenvalues) and mode shapes (eigenvectors) measured by experiments.

11. Tested models : Adhikari - Woodhouse
xi = mode shapes of the undamped system (real)
zi = mode shapes of the damped system (complex)
wi = natural frequencies of the undamped system (real)
i = mode shapes of the damped system (complex)
Under the assumption of “small damping”, the following approximation can be used :

12. Tested models : Adhikari - Woodhouse
Substituting zj into the quadratic pencil egeinvalues equation obtained by the separation of variables
Premultiplying by xkT and using the orthogonality properties of the undamped mode shapes we obtain
Where

13. Tested models : Adhikari - Woodhouse
For the case k = l (neglecting the second order terms involving l(j) and C’kl,  k K l ), we obtain
For the case k K l , again retaining only the first order terms, we obtain
From the imaginary part of experimental complex eigenvalues and eigenvectors it’s now possible to extract the C’ matrix in modal coordinates and then convert it to the original coordinate system by the relation

14. Tested models : Pilkey – Park - Inman
Assumptions :
M, C and K are symmetric.
Damping is subcritical, so eigenvalues and eigenvectors arise in complex conjugate pairs.
Data needed :
Complex frequencies (eigenvalues) and mode shapes (eigenvectors) from experiments.
M matrix for the iterative method
M and K matrices for the direct method

15. Tested models : Pilkey – Park - Inman
Based on Lancaster’s works on the second order pencil eigenvalues equation, this method proves that if the eigenvectors are normalized so that
(for the iterative and for the direct Pilkey-Park-Inman method, respectively), the damping matrix can be calculated by the equation
Where the overbar indicates the complex conjugate

16. Results of simulation Adhikari – Using 20/20 modes
Pilkey – Using 20/20 modes

17. Results of simulation Adhikari– Using 15/20 modes
Pilkey – Using 15/20 modes

18. Results of simulation Adhikari – Using 10/20 modes
Pilkey – Using 10/20 modes

19. New approach – Orthogonality equations
At the present time, a new model updating approach is under study based on the orthogonality equations derived from the quadratic pencil. It can be proven that the following equations yields diagonal matrices
These three orthogonality equations and the eigenvalues quadratic equation
can be used to write model updating equations to find the updating parameters.

20. New approach – Orthogonality equations
In Datta, Elhay and Ram paper’s on orthogonality of the quadratic pencil, is also proven that
And it also can be proven that the modal parameters from the Duncan state space analysis a and b defined as
Can be related to the diagonal Di matrices as

21. New approach – Orthogonality equations
The four main equations can be rearranged as

22. New approach – Model updating
The three matrix can be written (for the simple example of the clamped beam) as
For i = 1 to ns number of substructures where
MA, KA, CA are the initial analytical mass, stiffness and damping matrices
Mi, Ki, MRi MTi are respectively the mass, stiffness, rotational diagonal mass and translational diagonal mass matrices of each substructure
i , i , i , i , i are the updating parameters

24. New approach
And put together in a single matrix equation as

25. New approach
Finally, to obtain real system matrices, the updating parameters must be real
Totally, if m modes are correctly measured, we are able to write 8m2 equations in the form

26. New approach
The updating parameters can be now calculated using the linear last squares mathematical optimization technique by the pseudo-inverse
This new approach, tested on the simple clamped beam example, gives very good results in the identification off all three matrices together even using just 2 measured modes.

27. Results – Damping matrix
Original Damping Matrix
Identified Damping Matrix (4/20 modes)

28. New approach
The new method seems to work very well, but it must be said that all the information about the structure under consideration were known and it was easy to choose the right substructure and analytical matrices for this example.
Future work includes a deeper investigation on the new approach and in particular:
- Test the new approach with more complex structures in order to validate it.
- Test the new approach with real structures.
- Comparing the three different models with real structures.

29. Future work
Future work will not only include the validation of the new method proposed but also new interesting topics.
A new important issue is to separate the different sources of damping in the identification process in order to model them more accurately.
The final target will be to define a procedure which makes possible the extraction of separated coefficients for viscous damping, Coulomb friction and any other source present in a big structure (e.g. wing) using Ground Resonance Tests data only.
A collaboration with Politecnico di Torino will start on this topic with a modal analysis experiment on a beam with different sources of damping.

30. Future work
Until now, the research was focused only on viscous linear damping with a constant C matrix.
This new topic will include the problem of non-linearity in the identification of the damping parameters.
Coulomb friction is a clear example of local non-linearity (material and geometric distributed non-linearities will not be considered at this time) and it will add many issue in modelling and in increasing computational time for large order Finite Element Models.

31. Future work experiment - beamz y x
“wings”
cantilever beam y z x

32. Experiment – Coulomb damping
Friction material 2
Friction material 1 z x y

33. Experiment – Viscous magnetic damping
Conductive material M
Gap
Rare-earth magnet

34. Conclusions
Modal incompleteness on two main damping identification techniques was numerically tested and a new approach was proposed. Results obtained by the new method were encouraging. However, the example is simple and further investigations are needed.
In the simulated tests, only modal incompleteness was tested, model reduction or modal expansion techniques are under study at the present time to simulate spatial incompleteness.
The new experiment (without applying Coulomb friction) will be able to provide some real data to test the new method.
New topic will include more and different sources of damping to make a more accurate model and evaluate which sources are the most important and which can be neglected.

35. Acknowledgements Prof. John E. Mottershead
Prof. Ken Badcock and all the ECERTA team members
Dr. Simon James
Prof. Cristiana Delprete, Dr. Elvio Bonisoli and Dr. Carlo Rosso from Politecnico di Torino
Marie Curie Actions

36. Thank you Grazie