Can the Spatial Distribution of Damping be Measured? S. A DHIKARI , - - PowerPoint PPT Presentation

Can the Spatial Distribution of Damping be Measured?

• S. ADHIKARI, J. WOODHOUSE AND A. SRIKANTH PHANI

Cambridge University Engineering Department Cambridge, U.K.

Can the Spatial Distribution of Damping be Measured? – p.1/22

Outline of the Talk

Introduction Models of damping: Viscous and non-viscous damping Complex frequencies and modes Theory of damping identification Numerical Results Experimental Results Conclusions

Can the Spatial Distribution of Damping be Measured? – p.2/22

Viscously Damped Systems

Equations of motion: M

C

K

(1) Approximate Complex frequency and modes:

: Undamped natural frequency, x

: Undamped modes

C

are the elements of the damping matrix in modal coordinates.

Can the Spatial Distribution of Damping be Measured? – p.3/22

Some General Questions of Interest

• 1. From experimentally determined complex modes can one

identify the underlying damping mechanism? Is it viscous or non-viscous? Can the correct model parameters be found experimentally?

• 2. Is it possible to establish experimentally the spatial

distribution of damping?

• 3. Is it possible that more than one damping model with

corresponding correct sets of parameters may represent the system response equally well, so that the identified model becomes non-unique?

• 4. Does the selection of damping model matter from an

engineering point of view? Which aspects of behaviour are wrongly predicted by an incorrect damping model?

Can the Spatial Distribution of Damping be Measured? – p.4/22

Non-viscously Damped Systems

Equations of motion: M

• ✁

• ✁

(2)

is

matrix of kernel functions. Approximate Complex frequency and modes:

is the Fourier transform of

,

in modal coordinates.

Can the Spatial Distribution of Damping be Measured? – p.5/22

Non-viscous Damping Identification

Damping model used for fitting:

• C

Determine the complex natural frequencies,

, and complex mode shapes,

, from a set of measured transfer functions. Denote

,

. Set

,

and

. Obtain the relaxation parameter

• ✠

M

M

Can the Spatial Distribution of Damping be Measured? – p.6/22

Non-viscous Damping Identification

Obtain undamped modal matrix

• ✘

. Evaluate the matrix

• ✮

• ✙

. From the

matrix, get C

and

C

Use

• ✮

• ✁

to get the coefficient matrix in physical coordinates.

Can the Spatial Distribution of Damping be Measured? – p.7/22

Simulation Example

N− th . . .

u

k

u

k

u

m

u

k

u

m

u

k g(t) g(t)

u

m

u

m

u

k

Linear array of N spring-mass oscillators,

,

,

. Simplest case: the kernel functions have the form

C

(3) Here

is some damping function and C is a positive definite constant matrix.

Can the Spatial Distribution of Damping be Measured? – p.8/22

Models of Non-viscous Damping

MODEL 1 (exponential):

• ✑

• ✑

• MODEL 2 (Gaussian):

• ✫

• ✫

• ✄

The damping models are normalized such that the damping functions have unit area when integrated to infinity, i.e.,

• ✍

Can the Spatial Distribution of Damping be Measured? – p.9/22

Characteristic Time Constant

For each damping function the characteristic time constant is defined via the first moment of

as

• ✠

Express

• as:
• ✠

. The constant

is the non-dimensional characteristic time constant and

is the minimum time period. Expect:

near viscous

strongly non-viscous

Can the Spatial Distribution of Damping be Measured? – p.10/22

Viscous Damping Identification

5 10 15 20 25 30 5 10 15 20 25 30 −5 5 10 15 20 25

k−th DOF j−th DOF Fitted viscous damping matrix Ckj

Fitted viscous damping matrix for

, damping model 2

Can the Spatial Distribution of Damping be Measured? – p.11/22

Viscous Damping Identification

5 10 15 20 25 30 5 10 15 20 25 30 −2 −1 1 2 3 4 5 6 7 8

k−th DOF j−th DOF Fitted viscous damping matrix Ckj

Fitted viscous damping matrix for

• , damping

model 1

Can the Spatial Distribution of Damping be Measured? – p.12/22

Non-viscous Damping Identification

5 10 15 20 25 30 5 10 15 20 25 30 −10 −5 5 10 15 20 25 30

k−th DOF j−th DOF Fitted coefficient matrix Ckj

Fitted coefficient matrix of exponential model for

• , damping model 1;
• fit

Can the Spatial Distribution of Damping be Measured? – p.13/22

Non-viscous Damping Identification

5 10 15 20 25 30 5 10 15 20 25 30 −6 −4 −2 2 4 6 8 10 12

k−th DOF j−th DOF Fitted coefficient matrix Ckj

Fitted coefficient matrix of exponential model for

• , damping model 2;
• fit

• Can the Spatial Distribution of Damping be Measured? – p.14/22

Non-viscous Damping Identification

20 40 60 80 100 120 140 160 180 −160 −150 −140 −130 −120 −110 −100 −90 −80

Frequency (Hz) Log amplitude of transfer function (dB)

• riginal

fitted using viscous fitted using exponential

Original and fitted transfer function

• ✩

(

) for

• , damping model 2

Can the Spatial Distribution of Damping be Measured? – p.15/22

Experimental Setup

Damped free-free beam:

• ✁

m, width =

mm thickness =

mm

Clamped damping mechanism Instrumented hammer for impulse input

Can the Spatial Distribution of Damping be Measured? – p.16/22

Measured Transfer Functions

200 400 600 800 1000 1200 1400 1600 1800 2000 −70 −60 −50 −40 −30 −20 −10

Frequency (Hz) Log Amplitude dB

Measured Reconstructed

Driving Point Transfer Fucntion

Measured and fitted transfer function of the beam

Can the Spatial Distribution of Damping be Measured? – p.17/22

Viscous Damping Fitting

2 4 6 8 10 12 2 4 6 8 10 12 −10 −5 5 x 10

6

k−th DOF j−th DOF Fitted viscous damping matrix, Ckj

Fitted viscous damping matrix for damping between 4-5 nodes

Can the Spatial Distribution of Damping be Measured? – p.18/22

Non-viscous Damping Fitting

2 4 6 8 10 12 2 4 6 8 10 12 −5 5 10 x 10

6

k−th DOF j−th DOF Coefficient of fitted exponential model, Ckj

Fitted coefficient matrix of exponential model for damping between 4-5 nodes;

• fit

• ✁

Can the Spatial Distribution of Damping be Measured? – p.19/22

Measured Complex Modes

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u1)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u2)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u3)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u4)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u5)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u6)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u7)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u8)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u9)

0.5 1 −0.01 −0.005 0.005 0.01

ℑ (u10)

0.5 1 −0.02 −0.01 0.01 0.02

ℑ (u11)

set1 set2 set3

Imaginary parts of the identified complex modes

Can the Spatial Distribution of Damping be Measured? – p.20/22

Summary and Conclusions

A method is proposed to identify a non-proportional non-viscous damping model in vibrating systems from complex modes and natural frequencies. Numerical results show that the method generally predicts the spatial location of the damping with good accuracy. If the fitted damping model is wrong, the procedure yields a non-physical result by fitting a non-symmetric coefficient matrix. That is, the procedure gives an indication that a wrong model is selected for fitting.

Can the Spatial Distribution of Damping be Measured? – p.21/22

Summary and Conclusions

It is possible that more than one damping model with corresponding correct sets of parameters may represent the system response equally well. This means that by measuring transfer functions it is not possible to identify the governing damping mechanism uniquely. Different damping models can be fitted with the identified poles and residues of the transfer functions so that they are approximated accurately by all models. Can the spatial distribution of damping be measured? — Yes! – provided the complex modes are known with sufficient accuracy.

Can the Spatial Distribution of Damping be Measured? – p.22/22