1. Progress in identification of damping: Energy-based method with incomplete and noisy data Marco Prandina University of Liverpool

2. Practical issues in damping identification Absence of a universal mathematical model to represent all damping sources. Spatial and modal incompleteness of data. Computational time for real big structures. Compared to mass and stiffness identification, damping identification needs dynamic tests (normally more affected by noise and erros compared to static tests).

3. Data available from FEM and experiments Mass matrix M (FEM) Stiffness matrix K (FEM) Mode shape matrix  = [  1,  2, …,  m ] (FEM) Natural frequencies  i (FEM) Complex mode shape matrix  = [  1,  2, …,  m ] (Experiments) Complex frequencies i (Experiments) Time histories (Experiments) Frequency Response Function (FRF) matrix H(  ) (Experiments)

4. Desired output Since it is practically impossible to model damping in all its features, a common approach is to consider viscous damping (dissipative forces which depend only on instantaneous generalized velocities) for its simplicity and generality. Even if it is well recognized that it is not physically realistic, in most cases it is sufficient to model the main dynamic characteristics of a system. (Output: matrix of damping coefficient C ) Coulomb friction is one of the non-linear sources of damping present in real structures (dissipative forces which depend on the sign of instantaneous generalized velocities) that contributes to the total dissipation of energy. (Output: matrix of Coulomb friction coefficient C F )

5. Energy dissipation method d denotes the damping forces and, for this method to be applicable, is required to have a form of Where  represent damping coefficients. Both viscous damping and Coulomb friction fall in that category:

6. Energy dissipation method Pre-multiplying by and integrating along a finite time interval T 1, the energy equation is obtained

7. Energy dissipation method In the case of periodic excitation (so that f(t) and x(t) are periodic) the contribution of conservative forces to the total energy over a full cycle of periodic motion is zero. So, if T 1 = T (period of the excitation):

8. Example : diagonal viscous damping matrix One of the simplest cases is a system with only diagonal viscous damping matrix. In this case, the energy equation is reduced to or

9. Example : diagonal viscous damping matrix In matrix form it is possible to write the equation In this case (diagonal) the number of unknowns is n (size of the model) and only one equation is available for each different excitation f(t). Assuming to excite the structure m times, m different equations are obtained and put together in one matrix equation.

10. Example : diagonal viscous damping matrix

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

12. Example : diagonal viscous damping matrix

13. Example : diagonal viscous damping matrix Spatial incompleteness and numerical integration of accelerations NoisePhase distortion Incompleteness

14. Example : diagonal viscous damping matrix Spatial incompleteness and numerical integration of accelerations Damping coefficients are very sensitive to phase distortion in integrated signals. Special care should be taken in estimating velocities to use this method. In order to reduce the problem of phase distortion and noise in numerically integrated velocity signals, a single frequency harmonic excitation close to k th natural frequency is used for each test. The response can be written in the form Where  (k) is the k th mode shape (obtained from FEM model) and q(t) is a modal coordinate. The response will be harmonic too and for this linear case will have the same frequency of the excitation (  k ).

15. Example : diagonal viscous damping matrix Spatial incompleteness and numerical integration of accelerations The modal coordinate may be represented using In this example there are no nonlinearities, so higher harmonics are not needed ( n =1) The two coefficients A and B can be identified using a least-square procedure using this equation and the measured values of accelerations from experiment in each time instant.

16. Example : diagonal viscous damping matrix Spatial incompleteness and numerical integration of accelerations For each time instant

17. Example : diagonal viscous damping matrix Spatial incompleteness and numerical integration of accelerations Once the two coefficients A k and B k are identified (for each of the m excitations), the analytical integration of the modal coordinate Can be used to reconstruct the full vector of velocities time histories by using (the integration constant C 1 is zero because the mean values of the velocity signals have to be zero)

18. Example : diagonal viscous damping matrix Simulated acceleration from experiment (10% noise) - DOF 19 – 1 st mode

19. Example : diagonal viscous damping matrix Velocity obtained by analytical integration - DOF 19 – 1 st mode

20. Example : diagonal viscous damping matrix It is now possible to fill the matrix A of velocities integrals and the vector e of excitation energies for each of the m excitations.

21. Example : diagonal viscous damping matrix Underdetermined system of equations A m x n c n x 1 = e m x 1 m < n Example: n = 20 DOF m = 8 (excite the structure at first 8 freq.) 4 dashpot in DOF 3,5,13,17 with c = 0.1

22. Example : diagonal viscous damping matrix Underdetermined system of equations 0.11132.191.58453.637.20865.6320.441289.5044.581668.0882.231966.30135.022170.23203.542284.72287.612329.79386.722336.54 1.191245.8112.582391.0438.431833.3664.85452.8670.7190.9048.241805.2613.965030.350.688138.2938.089818.65138.8310118.0 0.71613.364.96420.597.765.453.75713.000.011333.423.04621.785.865.502.12970.000.712329.5913.572664.82 2.441626.639.3719.303.102217.381.642577.448.240.111.762467.552.601946.106.82175.190.044472.8316.486355.31 6.352845.669.581722.610.986479.3310.68165.770.007011.6610.79177.821.136205.187.931194.271.916742.7421.9914033.3 1.56363.110.411416.411.60167.310.041654.231.790.000.041660.581.63156.220.331284.690.84860.273.563406.29 2.03116.570.042464.761.22842.251.48513.440.002465.241.49499.751.22868.510.062342.291.38384.083.704968.14 3.6325.711.562812.150.005496.431.612718.463.180.381.562809.030.005509.331.672620.152.7714.196.2211243.2 A = c =c = = 43.02 10.32 1.92 1.52 1.36 0.45 0.38 0.43 = e unknown

23. Example : diagonal viscous damping matrix Underdetermined system of equations = 43.02 10.32 1.92 1.52 1.36 0.45 0.38 0.43 1.67.2135.0287.6 12.638.414.038.1 5.07.85.90.7 9.43.12.60.0 9.61.01.11.9 0.41.6 0.8 0.01.2 1.4 1.60.0 2.8 If it is possible to select the right column of matrix A the problem will become over determined and several techniques (e.g. Least Square Method with positive constraints) could be applied to solve the identification problem. c unknowns < number of equations

24. Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors 0.11132.191.58453.637.20865.6320.441289.5044.581668.0882.231966.30135.022170.23203.542284.72287.612329.79386.722336.54 1.191245.8112.582391.0438.431833.3664.85452.8670.7190.9048.241805.2613.965030.350.688138.2938.089818.65138.8310118.0 0.71613.364.96420.597.765.453.75713.000.011333.423.04621.785.865.502.12970.000.712329.5913.572664.82 2.441626.639.3719.303.102217.381.642577.448.240.111.762467.552.601946.106.82175.190.044472.8316.486355.31 6.352845.669.581722.610.986479.3310.68165.770.007011.6610.79177.821.136205.187.931194.271.916742.7421.9914033.3 1.56363.110.411416.411.60167.310.041654.231.790.000.041660.581.63156.220.331284.690.84860.273.563406.29 2.03116.570.042464.761.22842.251.48513.440.002465.241.49499.751.22868.510.062342.291.38384.083.704968.14 3.6325.711.562812.150.005496.431.612718.463.180.381.562809.030.005509.331.672620.152.7714.196.2211243.2 A = 43.02 10.32 1.92 1.52 1.36 0.45 0.38 0.43 e = aiai The aim is to select between the column of A a certain number of vectors that linearly combined (by multiplying them to positive damping coefficients to be identified later) are able to represent vector e. Considering all the possible combinations and comparing the error is practically impossible because the number of combinations is too big even for a small example like the 10-elements beam.

25. Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors One way of selecting the best subset of vectors is to calculate the angle between one (or more) column and the vector e. The angle between two vectors a and b can be calculated by The column with the smallest angle between itself and vector e is the vector that is best able to represent vector e by a one dimensional optimization. However, it is not guarantee that it is one of the right ones where the dashpots actually are. The first step is to select a finite number of columns (in this case, three) with a relatively small angle to start with.

26. Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors Once the first three vectors are selected with this criteria, the concept of angle between a subspace and a vector is introduced. Using QR algorithm and SVD decomposition, the angle between a subset of vectors F and a vector G can be calculated by Where  i is the i th singular value (SVD), Q F and Q G are orthogonal matrices from the QR algorithm applied to F and G. Again, it is now possible to select the best three combinations of the previous three selected vectors with the smallest angles and to proceed with more vectors until a suitable solution is reached.

27. Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors DOFAngle 196.51 176.81* 137.73* DOFAngle 19130.4044 1971.0223 19111.4423 17191.7041 1752.1994* 17112.5832 13190.4044 1370.8801 13111.1061 DOFAngle 1913170.2558 191350.2754 191380.3432 197130.3942 19750.7085 197140.923 1911130.3571 191190.746 191170.9911 1719130.2558 171971.022 1719111.1318 175150.1247 17530.8312* 17521.1307 1711131.0561 1711191.1318 171131.8333 1319170.2558 131950.2754 131980.3432 137190.3942 137120.6408 13780.643 1311190.3571 131190.6353 1311120.803

28. Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors 191317100.1705 19131790.1895 191317180.1936 19135150.1802 1913580.1874 19135120.195 1913850.1874 19138170.2366 19138150.2585 1971350.1953 19713170.2554 19713110.265 1975130.1953 1975110.258 197590.4115 19714130.392 1971450.7003 19714200.818 191113170.1998 19111350.2256 19111370.265 19119130.3541 19119150.3679 1911950.5407 1911750.258 19117130.265 1911790.6516 171913100.1705 17191390.1895 171913180.1936 17197130.2554 1719750.6492 17197140.8994 171911130.1998 17191190.7374 171911100.9144 17515160.0221 1751540.0256 17515200.0285 1753130.0005 1753150.1031 1753190.253 1752150.1041 175270.3408 1752110.4017 171113190.1998 17111390.5124 171113100.7381 171119130.1998 17111990.7374 171119100.9144 1711350.4818 17113131.0178 17113191.0464 131917100.1705 13191790.1895 131917180.1936 13195150.1802 1319580.1874 13195120.195 1319850.1874 13198170.2366 13198150.2585 1371950.1953 13719170.2554 13719110.265 13712190.283 13712160.4375 13712150.5185 1378190.277 1378160.4611 137840.5247 131119170.1998 13111950.2256 13111970.265 1311930.2104 1311920.3196 13119180.322 131112190.3356 13111240.416 13111290.4257

29. Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors After each step it is possible to calculate the damping coefficients that solve the energy equations for dashpots located in the selected DOFs. Least square method with positive constraints has been used. IF the selected DOFs are the right ones, the values of damping coefficients are close to the exact ones. If DOFs are selected wrongly, an equivalent (from an energetic point of view) damping matrix is obtained.

30. Example : diagonal viscous damping matrix “Wrong” case Exact solution

31. Example : diagonal viscous damping matrix “Correct” case Exact solution

32. Conclusions - A method to identify damping forces in the form has been presented and numerically tested. - A method to reconstruct full velocities time histories from noisy accelerometer signals has been presented and applied to the identification method. - A subspace selection criteria to solve underdetermined systems of equations has been introduced and it is currently under study to optimize the method avoiding wrong selection.

33. Future work - One important task is to find a way to pick the right columns avoiding the wrong selection. Choosing the right excitations (at present time, the method has been tested only using excitations at the first m natural frequencies without any criteria) in a more efficient way could be the first step. - Testing the same example with Coulomb friction too. - In the case of wrong selection of DOFs, the identification of the damping matrix is still good from an energy point of view. It could be useful to check if from a dynamic point of view (in the frequency range of interest) it is a good approximation of the real behaviour. - Test the method with real data (experimental test almost ready, waiting for magnets)

34. Thank you Ok, let’s consider damping for the next flight test so…