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

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).

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)

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 )

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:

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

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):

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

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.

Example : diagonal viscous damping matrix

Accelerometers (dof 7, 11 and 19) Dashpots (dof 3, 5, 13 and 17)

Example : diagonal viscous damping matrix

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

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 ).

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.

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

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)

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

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

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.

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

Example : diagonal viscous damping matrix Underdetermined system of equations A = c =c = = = e unknown

Example : diagonal viscous damping matrix Underdetermined system of equations = 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

Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors A = 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.

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.

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.

Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors DOFAngle * * DOFAngle * DOFAngle *

Example : diagonal viscous damping matrix How to select “right” columns. Angles between subspaces and vectors

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.

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

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

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.

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)

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