1. LOCATION AND IDENTIFICATION OF DAMPING PARAMETERS

2. Contents Objectives of the research
Introduction on damping identification techniques
New energy-based method
Numerical simulation
Experimental results
Conclusions
Future works

3. Objectives of the research
- Better understanding of damping in structures from an engineering point of view
Defining a practical identification method
Validate the method with numerical simulations
Test the method on real structures

4. Damping in structures
Damping in structures can be caused by several factors:
Material damping
Damping in joints
Dissipation in surrounding medium

5. Issues in damping identification
Absence of a mathematical model for all damping forces
Computational time
Incompleteness of data
Generally small effect on dynamics

6. Identification techniques
Techniques for identifying the viscous damping matrix
Perturbation method
Inversion of receptance matrix
Lancaster’s formula
Energy-dissipation method
Prandina, M., Mottershead, J. E., and Bonisoli, E., An assessment of damping identification methods, Journal of Sound and Vibration (in press), 2009.

7. Theory
The new method is based on the energy-dissipation method, starting from the equations of motion of a MDOF system
The energy equation can be derived

8. Theory
In the case of periodic response, the contribution of conservative forces to the total energy over a full cycle of periodic motion is zero. So if T1 = T (period of the response)
And the energy equation can be reduced to

9. Diagonal viscous damping matrix
The simplest case is a system with diagonal viscous damping matrix. In this case the energy equation becomes

10. Diagonal viscous damping matrix

11. Underdetermined system
The energy system of equations is usually underdetermined since the number of DOF can be greater than the number of tests. To solve the problem there are different options:
Change the parameterization of the damping matrix
Increase the number of different excitations
Define a criterion to select the “best” columns of matrix A

12. Smallest angle criterion
Angle between a column ai of matrix A and the vector e
Similarly, an angle between a set of columns B and the vector e can be calculated using SVD an QR algorithm

13. Numerical example Accelerometers (dof 7, 11 and 19)
Dashpots (dof 3, 5, 13 and 17) 2 4 6 8 10 12 14 16 18 20 1 3 5 7 9 11 13 15 17 19

14. Procedure
Accelerations are measured on DOF 7, 11 and 19 for a set of 8 different excitations at frequencies close to first 8 modes, random noise is added.
Velocities in all DOF are obtained by expanding these 3 measurements using the undamped mode shapes
Best columns of A are selected using smallest angle criterion
The energy equation is solved using least squares non-negative algorithm (to assure the identified matrix is non-negative definite)

15. Results Case 1 N DOF of dashpots Damping coefficients (Ns/m) Angle
Exact 3 5 13 17
0.01
0.5
0.1 1 N
DOF of identified dashpots
Identified damping coefficients (Ns/m)
Angle 1 - 17
1.084
12.557 2 5
0.581
1.042
1.029 3 13
0.506
0.124
0.989
0.263 4
0.01
0.501
0.099
1.002
0.001

16. Results Case 2 N DOF of dashpots Damping coefficients (Ns/m) Angle
Exact 3 5 13 17
0.1 N
DOF of identified dashpots
Identified damping coefficients (Ns/m)
Angle 1 - 19
0.107
6.505 2 13
0.151
0.059
0.404 3 5 15 17
0.212
0.127
0.055
0.124 4
0.101
0.098
0.099
0.1
0.001

17. Results
0.1
N=1
0.107
N=2
0.151
0.059
N=3
0.212
0.127
0.055

18. Results Case 2 – Damping factors Mode Correct N=1 Error % N=2 N=3 1
3.96%
0.03%
0.00% 2
44.33%
0.11% 3
24.65%
12.72%
1.07% 4
16.81%
9.74%
0.94% 5
73.36%
7.95%
2.88% 6
14.71%
1.86%
4.69% 7
2.82%
4.12%
11.59% 8
54.20%
15.05%
15.93% 9
35.40%
31.19%
0.61% 10
47.89%
27.98%
2.50%

19. Nonlinear identification
The method can be applied to identify any damping in the form
In case of viscous damping and Coulomb friction together, for example, the energy equation can be written as

20. Nonlinear identification
New matrix A
Viscous
Coulomb Friction

21. Experiment setup

22. Magnetic dashpot

23. Experiment procedure
The structure without magnetic dashpot is excited with a set of 16 different excitations with frequencies close to those of the first 8 modes
The complete set of accelerations is measured and an energy-equivalent viscous damping matrix is identified as the offset structural damping
The measurement is repeated with the magnetic dashpot attached with the purpose of locating and identifying it

24. Experiment procedure Velocities are derived from accelerometer signals
Matrix A and vector e are calculated, the energy dissipated by the offset damping is subtracted from the total energy
The energy equation (In this case overdetermined, since there are 16 excitations and 10 DOFs) is solved using least square technique

25. Experimental results Magnetic viscous dashpot on DOF 9
Damping coefficients
Expected (Ns/m)
Identified (Ns/m) C1 C2 C3 C4 C5 C6 C7 C8 C9
1.515
1.320
C10
0.032

26. Further experiments
Further experiments currently running will include more magnetic dashpots in different DOFs.
They will also include nonlinear sources of damping such as Coulomb friction devices.

27. Coulomb friction device

28. Advantages of the new method
Estimation of mass and stiffness matrices is not required if a complete set of measurements is available
Can identify non-viscous damping in the form
Robustness against noise and modal incompleteness
Spatial incompleteness can be overcome using expansion techniques

29. Conclusions New energy-based method has been proposed
Numerical simulation has validated the theory
Initial experiments on real structure give reasonably good results, further experiments are currently running

30. Future works Coulomb friction experiment
Extend the method to include material damping
Try different parameterizations of the damping matrices

31. Acknowledgements Prof John E Mottershead Prof Ken Badcock
Dr Simon James
Marie Curie Actions