Rayleigh s Classical Damping Revisited

Transcription

1 Rayleigh s Classical Damping Revisited S. Adhikari and A. Srikantha Phani Department of Aerospace Engineering, University of Bristol, Bristol, U.K. S.Adhikari@bristol.ac.uk URL: Classical Damping Revisited p.1/26

2 Bristol Aerospace Classical Damping Revisited p.2/26

3 Outline of the presentation Introduction Background of proportionally damped systems Generalized proportional damping Damping identification method Examples Summary and conclusions Classical Damping Revisited p.3/26

4 Introduction Equation of motion of viscously damped systems: Mÿt) + Cẏt) + Kyt) = ft) Proportional damping Rayleigh 1877) C = α 1 M + α 2 K Classical normal modes Simplifies analysis methods Identification of damping becomes easier Classical Damping Revisited p.4/26

5 Limitations of proportional damping The modal damping factors: ζ j = 1 ) α1 + α 2 ω j 2 ω j Not all forms of variation can be captured Classical Damping Revisited p.5/26

6 Damped Beam Example Damped free-free beam: L = 1m, width = 39.0 mm thickness = 5.93 mm Classical Damping Revisited p.6/26

7 Damping factors 10 1 experiment fitted Pproportional damping Modal damping factor Frequency Hz) Classical Damping Revisited p.7/26

8 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? Classical Damping Revisited p.8/26

9 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 1 C = CM 1 K, b) MK 1 C = CK 1 M, c) MC 1 K = KC 1 M. This can be easily proved by following Caughey and O Kelly s 1965) approach and interchanging M, K and C successively. Classical Damping Revisited p.9/26

10 Caughey series Caughey series: C = M N 1 j=0 α j M 1 K ) j The modal damping factors: ζ j = 1 ) α1 + α 2 ω j + α 3 ωj ω j More general than Rayleigh s version of proportional damping Classical Damping Revisited p.10/26

11 Generalized proportional damping Premultiply condition a) of the theorem by M 1 : M 1 K ) M 1 C ) = M 1 C ) M 1 K ) Since M 1 K and M 1 C are commutative matrices M 1 C = f 1 M 1 K) Therefore, we can express the damping matrix as C = Mf 1 M 1 K) Classical Damping Revisited p.11/26

12 Generalized proportional damping Premultiply condition b) of the theorem by K 1 : K 1 M ) K 1 C ) = K 1 C ) K 1 M ) Since K 1 M and K 1 C are commutative matrices K 1 C = f 2 K 1 M) Therefore, we can express the damping matrix as C = Kf 1 K 1 M) Classical Damping Revisited p.12/26

13 Generalized proportional damping Combining the previous two cases C = M β 1 M 1 K ) + K β 2 K 1 M ) Similarly, postmultiplying condition a) of Theorem 1 by M 1 and b) by K 1 we have C = β 3 KM 1 ) M + β 4 MK 1 ) K Special case: β i ) = α i I Rayleigh damping. Classical Damping Revisited p.13/26

14 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 β 1 M 1 K ) + K β 2 K 1 M ), or b) C = β 3 KM 1 ) M + β 4 MK 1 ) K for any β i ), i = 1,, 4. Classical Damping Revisited p.14/26

15 Example 1 Equation of motion: M q+ [Me M 1 2 K /2 sinhk 1 M lnm 1 K) 2/3 ) + K cos 2 K 1 M) 4 ] M 1 K 1 M tan 1 K π q + Kq = 0 It can be shown that the system has real modes and 2ξ j ω j = e ω4 j 1 /2 sinh ω 2 j ln 4 3 ω j ) + ω 2 j cos 2 1 ω 2 j ) 1 ωj tan 1 ω j π. Classical Damping Revisited p.15/26

16 Damping identification method To simplify the identification procedure, express the damping matrix by C = Mf M 1 K ) Using this simplified expression, the modal damping factors can be obtained as 2ζ j ω j = f ) ωj 2 or ζ j = 1 f ) ωj 2 = fωj ) say) 2ω j Classical Damping Revisited p.16/26

17 Damping identification method The function f ) can be obtained by fitting a continuous function representing the variation of the measured modal damping factors with respect to the frequency With the fitted function f ), the damping matrix can be identified as or 2ζ j ω j = 2ω j fωj ) Ĉ = 2M M 1 K f ) M 1 K Classical Damping Revisited p.17/26

18 Example 2 Consider a 3DOF system with mass and stiffness matrices M = , K = Classical Damping Revisited p.18/26

19 Example 2 Modal damping factor Frequency ω), rad/sec Damping factors Classical Damping Revisited p.19/26

20 Example 2 Here this continuous) curve was simulated using the equation fω) = 1 15 e 2.0ω e 3.5ω) sin ω 7π) ω 3 ) From the above equation, the modal damping factors in terms of the discrete natural frequencies, can be obtained by 2ξ j ω j = 2ω j 15 e 2.0ω j e ) 3.5ω j sin ω ) j ω 3 7π j). Classical Damping Revisited p.20/26

21 Example 2 To obtain the damping matrix, consider the preceding equation as a function of ωj 2 and replace ωj 2 by M 1 K and any constant terms by that constant times I. Therefore: C =M 2 15 M 1 K [ I sin [ e M K e 3.5 M 1K ] 1 )] [ ] M 1 K I M 1 K) 3/2 7π Classical Damping Revisited p.21/26

22 Experimental Example 1 Natural frequencies, Hz Damping factors Natural frequencies, Hz experimental) in % of critical damping) from FE) %) %) %) %) %) %) %) %) %) %) %) Measured data for the beam example C d = 2MT p 1 I + p 2 T + p 3 T 2 = 2p 2 K + 2p 1 M + p 3 K) M 1 K. Classical Damping Revisited p.22/26

23 Experimental Example 1 Modal damping factors ζ j ) original inverse modal transformation Rayleigh s proportional damping polymonial fit generalized proportional damping Natural frequencies ω ), rad/sec x 10 4 j Fitted and measured damping factors Classical Damping Revisited p.23/26

24 Summary 1. Measure a suitable transfer function H ij ω) 2. Obtain the undamped natural frequencies ω j and modal damping factors ζ j 3. Fit a function ζ = fω) which represents the variation of ζ j with respect to ω j for the range of frequency considered in the study 4. Calculate the matrix T = M 1 K 5. Obtain the damping matrix using Ĉ = 2 M T f T) Classical Damping Revisited p.24/26

25 Conclusions1) 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 of variations in the modal damping factors with respect to the frequency. Classical Damping Revisited p.25/26

26 Conclusions2) 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. Classical Damping Revisited p.26/26