Rayleighs Classical Damping Revisited S. Adhikari and A. Srikantha - - PowerPoint PPT Presentation

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Rayleighs Classical Damping Revisited S. Adhikari and A. Srikantha Phani Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL:

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B E College, India, January 2007

Rayleigh’s Classical Damping Revisited

  • S. Adhikari and A. Srikantha Phani

Department of Aerospace Engineering, 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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Bristol Aerospace

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Outline of the presentation

Introduction Background of proportionally damped systems Generalized proportional damping Damping identification method Examples Summary and conclusions

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Introduction

Equation of motion of viscously damped systems: M¨ y(t) + C˙ y(t) + Ky(t) = f(t) Proportional damping (Rayleigh 1877) C = α1M + α2K Classical normal modes Simplifies analysis methods Identification of damping becomes easier

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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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Damped Beam Example

Damped free-free beam: L = 1m, width = 39.0 mm thickness = 5.93 mm

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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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Our Objective

Can we improve the Classical Damping proposed by Lord Rayleigh in 1877 so that we can take account of the frequency variation of the damping factors?

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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

−

M

−1K

2

/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

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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Experimental Example 1

Natural frequencies, Hz Damping factors Natural frequencies, Hz (experimental) (in % of critical damping) (from FE) 33.00 0.6250 30.81 (-6.64 %) 85.00 0.2000 85.24 (0.29 %) 166.00 0.0833 167.61 (0.97 %) 276.00 0.0313 277.73 (0.63 %) 409.00 0.0625 415.67 (1.63 %) 569.00 0.1250 581.42 (2.18 %) 758.00 0.1163 774.94 (2.24 %) 976.00 0.1786 996.20 (2.07 %) 1217.00 0.8621 1245.15 (2.31 %) 1498.00 0.7143 1521.77 (1.59 %) 1750.00 0.3571 1826.06 (4.35 %) Measured data for the beam example

b

Cd = 2MT

  • p1I + p2T + p3T2
  • = 2p2K + 2(p1M + p3K)

√ M−1K.

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Experimental Example 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Natural frequencies (ωj), rad/sec Modal damping factors (ζj)

  • riginal

inverse modal transformation Rayleigh′s proportional damping polymonial fit generalized proportional damping

Fitted and measured damping factors

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Summary

  • 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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Conclusions(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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Conclusions(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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