Identification of Damping S Adhikari University of Bristol, - - PowerPoint PPT Presentation

June 2005, University of Catania

Identification of Damping

S Adhikari

University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html

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Outline of the Presentation

Introduction Models of damping Complex frequencies and modes Viscous and Non-viscous damping identification Generalized proportional damping Identification of Generalized proportional damping Simulation and Experimental Results Conclusions

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Introduction

There are two ways to understand the dynamics of complex structures: The first is the experimental approach. A carefully conducted experiment can provide crucial information regarding the system

• dynamics. However, the experimental process

is time consuming, expensive and it may be not be possible to dynamically test a complex structure under desired loading conditions. The alternative is to ‘replace’ the actual structure by a mathematical model and perform numerical experiments in a computer.

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Quality of a model

The quality of a model depends on: Fidelity to (experimental) data: The results obtained from a numerical or mathematical model undergoing a given excitation force should be close to the results

• btained from the vibration testing of the same structure

undergoing the same excitation. Robustness with respect to (random) errors: Errors in estimating the system parameters, boundary conditions and dynamic loads are unavoidable in practice. The

• utput of the model should not be very sensitive to such

errors.

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Quality of a model

Predictive capability In general it is not possible to experimentally validate a model over the entire domain of its scope of application. The model should predict the response well beyond its validation domain.

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Viscously damped systems

Equation of motion: M¨ q(t) + C ˙ q(t) + Kq(t) = f(t) (1) Proportional damping (Rayleigh 1877) C = α1M + α2K Classical normal modes Simplifies analysis methods Identification of damping becomes easier

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Models of damping

Non-proportional viscous damping Non-viscous damping models: fractional derivative model, GHM model, convolution integral model Non-linear damping models In general, the use of these damping models will re- sult in complex modes

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Non-proportional damping

Modes becomes complex if damping is non-proportional Approximate natural frequencies and modes: λj ≈ ±ωj + iC′

jj/2,

zj ≈ xj + i

N

• k=1

k=j

ωjC′

kj

(ω2

j − ω2 k)xk.

ωj: undamped natural frequencies; xk: undamped modes; C′ = XTCX: modal damping matrix.

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Non-viscous damping models

Fractional derivative model: Fd = l

j=1 gjDνj[q(t)], where

Dνj[q(t)] = dνjq(t) dtνj = 1 Γ(1 − νj) d dt t q(t) (t − τ)νj dτ Special case: νj = 1 : ⇛ viscous damping Convolution integral model: Fd = t

0 G(t − τ) ˙

q(τ)dτ G(t) is a matrix of the damping kernel functions. Special case: G(t − τ) = Cδ(t − τ) : ⇛ viscous damping

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Non-viscously damped systems

Equation of motion: M¨ y(t) + t

−∞

G(t − τ) ˙ y(τ) dτ + Ky(t) = 0 (2) Approximate natural frequencies and modes: λj ≈ ±ωj+iG′

jj(±ωj)/2,

zj ≈ xj+i

N

• k=1

k=j

ωjG′

kj(ωj)

(ω2

j − ω2 k) xk.

G(ω): Fourier transform of G(t); G′(ωj) = XTG(ωj)X

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Damping functions in the Laplace domain

Damping functions Author, Year G(s) =

k=1

aks s + bk Biot (1955, 1958) G(s) = E1sα − E0bsβ 1 + bsβ Bagley and Torvik (1983) 0 < α < 1, 0 < β < 1 sG(s) = G∞

1 +

k αk

s2 + 2ξkωks s2 + 2ξkωks + ω2

k

Golla and Hughes (1985) and McTavish and Hughes (1993) G(s) = 1 +

k=1

∆ks s + βk Lesieutre and Mingori (1990) G(s) = c1 − e−st0 st0 Adhikari (1998) G(s) = c1 + 2(st0/π)2 − e−st0 1 + 2(st0/π)2 Adhikari (1998)

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Basic questions of interest

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? Is it possible to establish experimentally the spatial distribution of damping?

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Basic questions of interest

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? 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?

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Viscous damping identification

If natural frequencies (Ω ∈ Rn×n), damping ratios (ζ ∈ Rn×n) and complex modes (Z ∈ Rm×n) are known from measurments, then the damping matrix can be identified

a:

U = ℜ (Z) , V = ℑ (Z) B = U+V C′ =

• Ω2B − BΩ2

Ω−1 + 2ζΩ C = U+TC′U+

aAdhikari and Woodhouse, J.of Sound & Vibration, 243[1] (2001) 43-61

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Non-viscous damping identification

Damping model used for fitting: G(t) = µe−µt C µ = ω1vT

1 Mv1

vT

1 Mu1

X = U − 1 µ [VΩ] B = X+V C′ =

• Ω2B − BΩ2

Ω−1 + 2ζΩ C = X+TC′U+

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Simulation example

m . . .

u

k

u

k

u

m

u

k

u

m

u

k

u

m

u

k g(t) g(t) N− th

u

Linear array of N spring-mass oscillators, N = 30, mu = 1 Kg, ku = 4 × ×103N/m. The kernel functions have the form G(t) = C g(t) (3) Here g(t) is some damping function and C is a posi- tive definite constant matrix.

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Models of non-viscous damping

Model 1 (exponential): g(1)(t) = µ1e−µ1t Model 2 (Gaussian): g(2)(t) = 2 µ2 π e−µ2t2 The damping models are normalized such that the damping functions have unit area when integrated to infinity, i.e., ∞ g(j)(t) dt = 1.

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Characteristic time constant

For each damping function the characteristic time constant is defined via the first moment of g(t) as θ = ∞ t g(t) dt. Express θ as: θ = γ Tmin. γ is the non-dimensional characteristic time constant and Tmin is the minimum time period. We expect: γ ≪ 1 : near viscous γ → O(1) : strongly non-viscous

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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: γ = 0.02, damping model 2

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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: γ = 0.5, damping model 1

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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: γ = 0.5, damping model 1; γfit = 0.49

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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: γ = 0.5, damping model 2; γfit = 0.63

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Experimental setup

Damped free-free beam, L = 1m, width = 39.0 mm, thickness = 5.93 mm Clamped damping mechanism Instrumented hammer for impulse input

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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

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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 (damping between 4-5 nodes)

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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 (damping between 4-5 nodes); γfit = 1.31

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Summary so far

A method is proposed to identify a non-proportional non-viscous damping model in vibrating systems from complex modes and natural frequencies. 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.

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Difficulties with complex modes

the expected ‘shapes’ of complex modes are not clear (complex) scaling of complex modes can change their geometric appearances the imaginary parts of the complex modes are usually very small compared to the real parts – makes it difficult to reliably extract complex modes

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Difficulties with complex modes

the phases of complex modes are highly sensitive to experimental errors, ambient conditions and measurement noise and often not repeatable in a satisfactory manner

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Difficulties with 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

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Proportional damping

Avoids most of the problems associated with complex modes Can accurately reproduce transfer functions for systems with light damping

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Transfer function

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 proportional

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Some observations

It is possible that more than one damping model with corresponding correct sets of parameters may represent the system response equally well. 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. As a consequence proportional viscous damping can be used as a valid model.

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Limitations of proportional damping

The modal damping factors: ζj = 1 2 α1 ωj + α2ωj

• Not all forms of variation can be captured

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Damping factors

200 400 600 800 1000 1200 1400 1600 1800 10

−3

10

−2

10

−1

Frequency (Hz) Modal damping factor

experiment fitted Pproportional damping

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Conditions for proportional damping

Theorem 1 A viscously damped linear system can possess classical normal modes if and only if at least one of the following conditions is satisfied: (a) KM−1C = CM−1K, (b) MK−1C = CK−1M, (c) MC−1K = KC−1M. This can be easily proved by following Caughey and O’Kelly’s (1965) approach and interchanging M, K and C successively.

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Caughey series

Caughey series: C = M

N−1

• j=0

αj

• M−1K

j The modal damping factors: ζj = 1 2 α1 ωj + α2ωj + α3ω3

j + · · ·

• More general than Rayleigh’s version of

proportional damping

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Generalized proportional damping

Premultiply condition (a) of the theorem by M−1:

• M−1K

M−1C

• =
• M−1C

M−1K

• Since M−1K and M−1C are commutative

matrices M−1C = f1(M−1K) Therefore, we can express the damping matrix as C = Mf1(M−1K)

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Generalized proportional damping

Premultiply condition (b) of the theorem by K−1:

• K−1M

K−1C

• =
• K−1C

K−1M

• Since K−1M and K−1C are commutative

matrices K−1C = f2(K−1M) Therefore, we can express the damping matrix as C = Kf1(K−1M)

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Generalized proportional damping

Combining the previous two cases C = M β1

• M−1K
• + K β2
• K−1M
• Similarly, postmultiplying condition (a) of

Theorem 1 by M−1 and (b) by K−1 we have C = β3

• KM−1

M + β4

• MK−1

K Special case: βi(•) = αiI → Rayleigh damping.

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Generalized proportional damping

Theorem 2 A viscously damped positive definite linear system possesses classical normal modes if and only if C can be represented by (a) C = M β1

• M−1K
• + K β2
• K−1M
• , or

(b) C = β3

• KM−1

M + β4

• MK−1

K for any βi(•), i = 1, · · · , 4.

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Example 1

Equation of motion:

M¨ q+

• Me

−

−1K

/2 sinh(K−1M ln(M−1K)2/3)

+ K cos2(K−1M)

4

• K−1M tan−1

√ M−1K π

• ˙

q + Kq = 0

It can be shown that the system has real modes and

2ξjωj = e−ω4

j /2 sinh

1 ω2

j

ln 4 3ωj

• + ω2

j cos2

1 ω2

j

• 1

√ωj tan−1 ωj π .

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Damping identification method

To simplify the identification procedure, express the damping matrix by C = Mf

• M−1K
• Using this simplified expression, the modal damping

factors can be obtained as 2ζjωj = f

• ω2

j

• r

ζj = 1 2ωj f

• ω2

j

• =

f(ωj) (say)

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Damping identification method

The function f(•) can be obtained by fitting a continuous function representing the variation

• f the measured modal damping factors with

respect to the frequency With the fitted function f(•), the damping matrix can be identified as 2ζjωj = 2ωj f(ωj)

• r
• C = 2M
• M−1K

f

• M−1K
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Example 2

Consider a 3DOF system with mass and stiffness matrices M =   1.0 1.0 1.0 1.0 2.0 2.0 1.0 2.0 3.0   , K =   2 −1 0.5 −1 1.2 0.4 0.5 0.4 1.8  

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Example 2

1 2 3 4 5 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Frequency (ω), rad/sec Modal damping factor

• riginal

recalculated

Damping factors

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Example 2

Here this (continuous) curve was simulated using the equation

• f(ω) = 1

15

• e−2.0ω − e−3.5ω

1 + 1.25 sin ω 7π 1 + 0.75ω3 From the above equation, the modal damping factors in terms of the discrete natural frequencies, can be obtained by

2ξjωj = 2ωj 15

• e−2.0ωj − e−3.5ωj

1 + 1.25 sin ωj 7π 1 + 0.75ω3

j

• .

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Example 2

To obtain the damping matrix, consider the preceding equation as a function of ω2

j and replace

ω2

j by M−1K and any constant terms by that

constant times I. Therefore: C =M 2 15

• M−1K
• e−2.0

√ M

−1K − e−3.5

√ M

−1K

• ×
• I + 1.25 sin

1 7π

• M−1K

I + 0.75(M−1K)3/2

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Steps to follow

• 1. Measure a suitable transfer function Hij(ω)
• 2. Obtain the undamped natural frequencies ωj

and modal damping factors ζj

• 3. Fit a function ζ =

f(ω) which represents the variation of ζj with respect to ωj for the range of frequency considered in the study

• 4. Calculate the matrix T =

√ M−1K

• 5. Obtain the damping matrix using
• C = 2 M T

f (T)

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Summary(1)

Rayleigh s proportional damping is generalized The generalized proportional damping expresses the damping matrix in terms of any non-linear function involving specially arranged mass and stiffness matrices so that the system still posses classical normal modes This enables one to model practically any type

• f variations in the modal damping factors with

respect to the frequency

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Summary(2)

Once a scalar function is fitted to model such variations, the damping matrix can be identified very easily using the proposed method The method is very simple and requires the measurement of damping factors and natural frequencies only (that is, the measurements of the mode shapes are not necessary) The proposed method is applicable to any linear structures as long as one have validated mass and stiffness matrix models which can predict the natural frequencies accurately and modes are not significantly complex

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References

Adhikari, S. (1998), Energy Dissipation in Vibrating Structures, Master’s the- sis, Cambridge University Engineering Department, Cambridge, UK, first Year Report. Bagley, R. L. and Torvik, P. J. (1983), “Fractional calculus–a different ap- proach to the analysis of viscoelastically damped structures”, AIAA Jour- nal, 21 (5), pp. 741–748. Biot, M. A. (1955), “Variational principles in irreversible thermodynamics with application to viscoelasticity”, Physical Review, 97 (6), pp. 1463– 1469. Biot, M. A. (1958), “Linear thermodynamics and the mechanics of solids”, in “Proceedings of the Third U. S. National Congress on Applied Me- chanics”, ASME, New York, (pp. 1–18). Golla, D. F. and Hughes, P. C. (1985), “Dynamics of viscoelastic structures

• a time domain finite element formulation”, Transactions of ASME, Journal
• f Applied Mechanics, 52, pp. 897–906.

Lesieutre, G. A. and Mingori, D. L. (1990), “Finite element modeling

• f frequency-dependent material properties using augmented thermody-

namic fields”, AIAA Journal of Guidance, Control and Dynamics, 13, pp. 1040– 1050. McTavish, D. J. and Hughes, P. C. (1993), “Modeling of linear viscoelastic space structures”, Transactions of ASME, Journal of Vibration and Acoustics, 115, pp. 103–110.

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