Damping Modelling and Identification Using Generalized Proportional Damping S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk IMAC XXIII Generalized Proportional Damping – p.1/29
Outline of the presentation Introduction Methods of damping modelling Background of proportionally damped systems Generalized proportional damping Damping identification method Examples Summary and conclusions IMAC XXIII Generalized Proportional Damping – p.2/29
Introduction Equation of motion of viscously damped systems: y ( t ) + C˙ y ( t ) + Ky ( t ) = f ( t ) M¨ Proportional damping (Rayleigh 1877) C = α 1 M + α 2 K Classical normal modes Simplifies analysis methods Identification of damping becomes easier IMAC XXIII Generalized Proportional Damping – p.3/29
Models of damping Non-proportional viscous damping Non-viscous damping models: fractional derivative model, GHM model Non-linear damping models In general, the use of these damping models will re- sult in complex modes IMAC XXIII Generalized Proportional Damping – p.4/29
Complex modes and damping If natural frequencies ( Ω ∈ R n × n ), damping ratios ( ζ ∈ R n × n ) and complex modes ( Z ∈ R m × n ) are known, then the damping matrix can be identified a : U = ℜ ( Z ) , V = ℑ ( Z ) B = U + V � Ω 2 B − BΩ 2 � C ′ = Ω − 1 + ζ C = U + T C ′ U + a Adhikari and Woodhouse, J.of Sound & Vibration, 243[1] (2001) 43-61 IMAC XXIII Generalized Proportional Damping – p.5/29
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 IMAC XXIII Generalized Proportional Damping – p.6/29
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 IMAC XXIII Generalized Proportional Damping – p.7/29
Difficulties with complex modes Damped free-free beam: L = 1 m, width = 39 . 0 mm thickness = 5 . 93 mm IMAC XXIII Generalized Proportional Damping – p.8/29
Difficulties with complex modes ℑ (u 1 ) ℑ (u 2 ) ℑ (u 3 ) ℑ (u 4 ) 0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005 0 0 0 0 −0.005 −0.005 −0.005 −0.005 −0.01 −0.01 −0.01 −0.01 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 ℑ (u 5 ) ℑ (u 6 ) ℑ (u 7 ) ℑ (u 8 ) 0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005 0 0 0 0 −0.005 −0.005 −0.005 −0.005 −0.01 −0.01 −0.01 −0.01 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 ℑ (u 9 ) ℑ (u 10 ) ℑ (u 11 ) 0.01 0.01 0.02 0.005 0.005 set1 0.01 set2 0 0 0 set3 −0.01 −0.005 −0.005 −0.02 −0.01 −0.01 0 0.5 1 0 0.5 1 0 0.5 1 Imaginary parts of the identified complex modes IMAC XXIII Generalized Proportional Damping – p.9/29
Proportional damping Avoids most of the problems associated with complex modes Can accurately reproduce transfer functions for systems with light damping IMAC XXIII Generalized Proportional Damping – p.10/29
Transfer function −80 −90 Log amplitude of transfer function (dB) −100 −110 −120 −130 −140 original fitted using viscous −150 fitted using proportional −160 0 20 40 60 80 100 120 140 160 180 Frequency (Hz) IMAC XXIII Generalized Proportional Damping – p.11/29
Limitations of proportional damping The modal damping factors: � α 1 � ζ j = 1 + α 2 ω j 2 ω j Not all forms of variation can be captured IMAC XXIII Generalized Proportional Damping – p.12/29
Damping factors −1 10 experiment fitted Pproportional damping Modal damping factor −2 10 −3 10 0 200 400 600 800 1000 1200 1400 1600 1800 Frequency (Hz) IMAC XXIII Generalized Proportional Damping – p.13/29
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. IMAC XXIII Generalized Proportional Damping – p.14/29
Caughey series Caughey series: N − 1 � � � j M − 1 K C = M α j j =0 The modal damping factors: � α 1 � ζ j = 1 + α 2 ω j + α 3 ω 3 j + · · · 2 ω j More general than Rayleigh’s version of proportional damping IMAC XXIII Generalized Proportional Damping – p.15/29
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 = M f 1 ( M − 1 K ) IMAC XXIII Generalized Proportional Damping – p.16/29
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 = K f 1 ( K − 1 M ) IMAC XXIII Generalized Proportional Damping – p.17/29
Generalized proportional damping Combining the previous two cases � � � � M − 1 K K − 1 M C = M β 1 + K β 2 Similarly, postmultiplying condition (a) of Theorem 1 by M − 1 and (b) by K − 1 we have � KM − 1 � � MK − 1 � C = β 3 M + β 4 K Special case: β i ( • ) = α i I → Rayleigh damping. IMAC XXIII Generalized Proportional Damping – p.18/29
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 � � � � M − 1 K K − 1 M (a) C = M β 1 + K β 2 , or � KM − 1 � � MK − 1 � (b) C = β 3 M + β 4 K for any β i ( • ) , i = 1 , · · · , 4 . IMAC XXIII Generalized Proportional Damping – p.19/29
� M � 2 Example 1 Equation of motion: � − 1 K / 2 sinh( K − 1 M ln( M − 1 K ) 2 / 3 ) M ¨ q + − M e � √ � M − 1 K 4 + K cos 2 ( K − 1 M ) K − 1 M tan − 1 ˙ q + Kq = 0 π It can be shown that the system has real modes and � 1 � 1 � � ln 4 1 tan − 1 ω j j / 2 sinh 2 ξ j ω j = e − ω 4 + ω 2 j cos 2 3 ω j π . √ ω j ω 2 ω 2 j j IMAC XXIII Generalized Proportional Damping – p.20/29
Damping identification method To simplify the identification procedure, express the damping matrix by � � M − 1 K C = M f Using this simplified expression, the modal damping factors can be obtained as � � ω 2 2 ζ j ω j = f j � � 1 = � ω 2 or ζ j = f ( ω j ) (say) f j 2 ω j IMAC XXIII Generalized Proportional Damping – p.21/29
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 2 ζ j ω j = 2 ω j � f ( ω j ) �� � � � M − 1 K � M − 1 K or C = 2 M f IMAC XXIII Generalized Proportional Damping – p.22/29
Example 2 Consider a 3DOF system with mass and stiffness matrices 1 . 0 1 . 0 1 . 0 2 − 1 0 . 5 , M = K = 1 . 0 2 . 0 2 . 0 − 1 1 . 2 0 . 4 1 . 0 2 . 0 3 . 0 0 . 5 0 . 4 1 . 8 IMAC XXIII Generalized Proportional Damping – p.23/29
Example 2 0.02 original recalculated 0.018 0.016 0.014 Modal damping factor 0.012 0.01 0.008 0.006 0.004 0.002 0 0 1 2 3 4 5 Frequency ( ω ), rad/sec Damping factors IMAC XXIII Generalized Proportional Damping – p.24/29
Example 2 Here this (continuous) curve was simulated using the equation e − 2 . 0 ω − e − 3 . 5 ω � � � � � f ( ω ) = 1 1 + 1 . 25 sin ω � 1 + 0 . 75 ω 15 7 π From the above equation, the modal damping factors in terms of the discrete natural frequencies, can be obtained by e − 2 . 0 ω j − e − 3 . 5 ω j � � � � � � 2 ξ j ω j = 2 ω j 1 + 1 . 25 sin ω j 1 + 0 . 75 ω 3 . j 15 7 π IMAC XXIII Generalized Proportional Damping – p.25/29
Example 2 To obtain the damping matrix, consider the preceding equation as a function of ω 2 j and replace j by M − 1 K and any constant terms by that ω 2 constant times I . Therefore: � � √ √ � C = M 2 − 1 K − e − 3 . 5 − 1 K M M e − 2 . 0 M − 1 K 15 � 1 � �� � � M − 1 K I + 0 . 75( M − 1 K ) 3 / I + 1 . 25 sin × 7 π IMAC XXIII Generalized Proportional Damping – p.26/29
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 = � f ( ω ) which represents the variation of ζ j with respect to ω j for the range of frequency considered in the study √ M − 1 K 4. Calculate the matrix T = 5. Obtain the damping matrix using C = 2 M T � � f ( T ) IMAC XXIII Generalized Proportional Damping – p.27/29
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