LOCATION AND IDENTIFICATION OF DAMPING PARAMETERS
Contents
2
- 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
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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
4
Issues in damping identification
- Absence of a mathematical model for all damping forces
- Computational time
- Incompleteness of data
- Generally small effect on dynamics
5
Identification techniques
Techniques for identifying the viscous damping matrix
- Perturbation method- Inversion of receptance matrix- Lancaster’s formula- Energy-dissipation method
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Prandina, M., Mottershead, J. E., and Bonisoli, E., An assessment of damping identification methods, Journal of Sound and Vibration (in press), 2009.
Theory
The energy equation can be derived
7
tg fKxxxxDxM ,,
tttgTt
t
Tt
t
d d ,, 11
TT
fxKxxxxDxMx
The new method is based on the energy-dissipation method, starting from the equations of motion of a MDOF system
Theory
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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)
0d T
tTt
t
KxxMx
tttgTt
t
Tt
t
d d ,, TT
fxxxxDx
And the energy equation can be reduced to
Diagonal viscous damping matrix
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The simplest case is a system with diagonal viscous damping matrix. In this case the energy equation becomes
tttTt
t
Tt
t
d d TT
fxxCx
tttxctxctxcTt
t
Tt
t
nnn
Tt
t
Tt
t
d d...dd T22222
2111
fx
Diagonal viscous damping matrix
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tt
tt
tt
c
c
c
txtxtx
txtxtx
txtxtx
mmmm Tt
t
m
Tt
t
Tt
t
nnTt
tmn
Tt
tm
Tt
tm
Tt
t
n
Tt
t
Tt
t
Tt
t
n
Tt
t
Tt
t
d
...
d
d
...
d...dd
............
d...dd
d...dd
T
2T
1T
22
11
222
21
2
2
2
222
21
2
1
2
122
11
2
1
222
111
fx
fx
fx
ecA
Underdetermined system
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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
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Angle between a column ai of matrix A and the vector e
eeaa
eaTT
T
arccosii
ii
Similarly, an angle between a set of columns B and the
vector e can be calculated using SVD an QR algorithm
eBii QQTarccos
Numerical example
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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)
Procedure
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- 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
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N DOF of identified dashpots Identified damping coefficients (Ns/m) Angle1 - - - 17 - - - 1.084 12.5572 - 5 - 17 - 0.581 - 1.042 1.0293 - 5 13 17 - 0.506 0.124 0.989 0.2634 3 5 13 17 0.01 0.501 0.099 1.002 0.001
Case 1
N DOF of dashpots Damping coefficients (Ns/m) AngleExact 3 5 13 17 0.01 0.5 0.1 1 0
Results
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N DOF of identified dashpots Identified damping coefficients (Ns/m) Angle1 - - - 19 - - - 0.107 6.5052 - - 13 19 - - 0.151 0.059 0.4043 - 5 15 17 - 0.212 0.127 0.055 0.1244 3 5 13 17 0.101 0.098 0.099 0.1 0.001
Case 2
N DOF of dashpots Damping coefficients (Ns/m) AngleExact 3 5 13 17 0.1 0.1 0.1 0.1 0
Results
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0.10.1 0.10.1
N=1
0.107
N=2
0.151 0.059
N=3
0.212 0.127 0.055
Results
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Case 2 – Damping factors
Mode Correct N=1 Error % N=2 Error % N=3 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%
Nonlinear identification
The method can be applied to identify any damping in the form
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xxxD ,,gIn case of viscous damping and Coulomb friction together, for example, the energy equation can be written as
tttsignTt
t
Tt
t
d d TT
fxxCxCx F
Nonlinear identification
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txxtxxtxtx
txxtxxtxtx
mmmm Tt
tmnmn
Tt
tmm
Tt
tm
Tt
tm
Tt
t
nn
Tt
t
Tt
t
n
Tt
t
dsign...dsignd...d
.................
dsign...dsignd...d
112
22
1
1111112
1
2
11
1111
Viscous Coulomb Friction
New matrix A
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
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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
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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xxxD ,, g
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
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