LOCATION AND IDENTIFICATION OF DAMPING PARAMETERS

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LOCATION AND IDENTIFICATION OF DAMPING PARAMETERS
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Contents

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

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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

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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
• Lancasters 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.
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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

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

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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
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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
3
5
9
13
15
17
19
1
7
11
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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)

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Results
Case 1
N DOF of dashpots DOF of dashpots DOF of dashpots DOF of dashpots Damping coefficients (Ns/m) Damping coefficients (Ns/m) Damping coefficients (Ns/m) Damping coefficients (Ns/m) Angle
Exact 3 5 13 17 0.01 0.5 0.1 1 0
N DOF of identified dashpots DOF of identified dashpots DOF of identified dashpots DOF of identified dashpots Identified damping coefficients (Ns/m) Identified damping coefficients (Ns/m) Identified damping coefficients (Ns/m) Identified damping coefficients (Ns/m) Angle
1 - - - 17 - - - 1.084 12.557
2 - 5 - 17 - 0.581 - 1.042 1.029
3 - 5 13 17 - 0.506 0.124 0.989 0.263
4 3 5 13 17 0.01 0.501 0.099 1.002 0.001
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Results
Case 2
N DOF of dashpots DOF of dashpots DOF of dashpots DOF of dashpots Damping coefficients (Ns/m) Damping coefficients (Ns/m) Damping coefficients (Ns/m) Damping coefficients (Ns/m) Angle
Exact 3 5 13 17 0.1 0.1 0.1 0.1 0
N DOF of identified dashpots DOF of identified dashpots DOF of identified dashpots DOF of identified dashpots Identified damping coefficients (Ns/m) Identified damping coefficients (Ns/m) Identified damping coefficients (Ns/m) Identified damping coefficients (Ns/m) Angle
1 - - - 19 - - - 0.107 6.505
2 - - 13 19 - - 0.151 0.059 0.404
3 - 5 15 17 - 0.212 0.127 0.055 0.124
4 3 5 13 17 0.101 0.098 0.099 0.1 0.001
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Results
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Results
Case 2 Damping factors
Mode Correct N1 Error N2 Error N3 Error
1 0.014092 0.013534 3.96 0.014096 0.03 0.014092 0.00
2 0.001496 0.002160 44.33 0.001495 0.11 0.001496 0.03
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
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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
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Nonlinear identification
New matrix A
Viscous
Coulomb Friction
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Experiment setup
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Magnetic dashpot
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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

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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

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Experimental results

• Magnetic viscous dashpot on DOF 9

Damping coefficients Expected (Ns/m) Identified (Ns/m)
C1 0 0
C2 0 0
C3 0 0
C4 0 0
C5 0 0
C6 0 0
C7 0 0
C8 0 0
C9 1.515 1.320
C10 0 0.032
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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.

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Coulomb friction device
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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

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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

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Future works

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

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Acknowledgements

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