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

3

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

6

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

8

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

10

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

13

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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