LOCATION AND IDENTIFICATION OF DAMPING PARAMETERS

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

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

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

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

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.

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

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

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

Diagonal viscous damping matrix

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

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

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

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)

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

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

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

Results Case 2 – Damping factors Mode Correct N=1 Error % N=2 N=3 1 0.014092 0.013534 3.96% 0.014096 0.03% 0.00% 2 0.001496 0.002160 44.33% 0.001495 0.11% 3 0.001024 0.000772 24.65% 0.000894 12.72% 0.001035 1.07% 4 0.000338 0.000395 16.81% 0.000305 9.74% 0.000341 0.94% 5 0.000138 0.000240 73.36% 0.000149 7.95% 0.000134 2.88% 6 0.000190 0.000162 14.71% 0.000193 1.86% 0.000181 4.69% 7 0.000114 0.000117 2.82% 0.000118 4.12% 0.000100 11.59% 8 0.000057 0.000089 54.20% 0.000049 15.05% 0.000048 15.93% 9 0.000106 0.000068 35.40% 0.000073 31.19% 0.000105 0.61% 10 0.000085 0.000044 47.89% 0.000061 27.98% 0.000087 2.50%

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

Nonlinear identification New matrix A Viscous Coulomb Friction

Experiment setup

Magnetic dashpot

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

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

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

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.

Coulomb friction device

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

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

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

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