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 Identification of Damping – p.1/51 June 2005, University of Catania
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 Identification of Damping – p.2/51 June 2005, University of Catania
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. Identification of Damping – p.3/51 June 2005, University of Catania
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 obtained 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 output of the model should not be very sensitive to such errors. Identification of Damping – p.4/51 June 2005, University of Catania
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. Identification of Damping – p.5/51 June 2005, University of Catania
Viscously damped systems Equation of motion: M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) = f ( t ) (1) Proportional damping (Rayleigh 1877) C = α 1 M + α 2 K Classical normal modes Simplifies analysis methods Identification of damping becomes easier Identification of Damping – p.6/51 June 2005, University of Catania
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 Identification of Damping – p.7/51 June 2005, University of Catania
Non-proportional damping Modes becomes complex if damping is non-proportional Approximate natural frequencies and modes: N ω j C ′ � kj λ j ≈ ± ω j + i C ′ jj / 2 , z j ≈ x j + i k ) x k . ( ω 2 j − ω 2 k =1 k � = j ω j : undamped natural frequencies; x k : undamped modes; C ′ = X T CX : modal damping matrix. Identification of Damping – p.8/51 June 2005, University of Catania
Non-viscous damping models Fractional derivative model: F d = � l j =1 g j D ν j [ q ( t )] , where D ν j [ q ( t )] = d ν j q ( t ) � t 1 d q ( t ) = ( t − τ ) ν j d τ 0 d t ν j Γ(1 − ν j ) d t Special case: ν j = 1 : ⇛ viscous damping Convolution integral model: � t F d = 0 G ( t − τ ) ˙ q ( τ )d τ G ( t ) is a matrix of the damping kernel functions. Special case: G ( t − τ ) = C δ ( t − τ ) : ⇛ viscous damping Identification of Damping – p.9/51 June 2005, University of Catania
Non-viscously damped systems Equation of motion: � t y ( t ) + G ( t − τ ) ˙ y ( τ ) d τ + Ky ( t ) = 0 (2) M¨ −∞ Approximate natural frequencies and modes: N ω j G ′ � kj ( ω j ) λ j ≈ ± ω j +i G ′ jj ( ± ω j ) / 2 , z j ≈ x j +i k ) x k . ( ω 2 j − ω 2 k =1 k � = j G ( ω ) : Fourier transform of G ( t ) ; G ′ ( ω j ) = X T G ( ω j ) X Identification of Damping – p.10/51 June 2005, University of Catania
P n " # Damping functions in the P Laplace domain P n Damping functions Author, Year a k s G ( s ) = Biot (1955, 1958) k =1 s + b k G ( s ) = E 1 s α − E 0 bs β Bagley and Torvik (1983) 1 + bs β 0 < α < 1 , 0 < β < 1 s 2 + 2 ξ k ω k s sG ( s ) = G ∞ 1 + k α k Golla and Hughes (1985) s 2 + 2 ξ k ω k s + ω 2 k and McTavish and Hughes (1993) ∆ k s G ( s ) = 1 + Lesieutre and Mingori (1990) k =1 s + β k G ( s ) = c 1 − e − st 0 Adhikari (1998) st 0 G ( s ) = c 1 + 2( st 0 /π ) 2 − e − st 0 Adhikari (1998) 1 + 2( st 0 /π ) 2 Identification of Damping – p.11/51 June 2005, University of Catania
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? Identification of Damping – p.12/51 June 2005, University of Catania
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? Identification of Damping – p.13/51 June 2005, University of Catania
Viscous damping identification If natural frequencies ( Ω ∈ R n × n ), damping ratios ( ζ ∈ R n × n ) and complex modes ( Z ∈ R m × n ) are known from measurments, 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 + 2 ζ Ω C = U + T C ′ U + a Adhikari and Woodhouse, J.of Sound & Vibration, 243[1] (2001) 43-61 Identification of Damping – p.14/51 June 2005, University of Catania
Non-viscous damping identification Damping model used for fitting: G ( t ) = µe − µt C µ = ω 1 v T 1 Mv 1 v T 1 Mu 1 X = U − 1 µ [ VΩ ] B = X + V � Ω 2 B − BΩ 2 � C ′ = Ω − 1 + 2 ζ Ω C = X + T C ′ U + Identification of Damping – p.15/51 June 2005, University of Catania
Simulation example m m k k k m k m k u u u u u u u u u . . . g(t) N− th g(t) Linear array of N spring-mass oscillators, N = 30 , m u = 1 Kg , k u = 4 × × 10 3 N/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. Identification of Damping – p.16/51 June 2005, University of Catania
Models of non-viscous damping Model 1 (exponential): g (1) ( t ) = µ 1 e − µ 1 t � µ 2 π e − µ 2 t 2 Model 2 (Gaussian): g (2) ( t ) = 2 The damping models are normalized such that the damping functions have unit area when integrated to infinity, i.e., � ∞ g ( j ) ( t ) d t = 1 . 0 Identification of Damping – p.17/51 June 2005, University of Catania
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 ) d t. 0 Express θ as: θ = γ T min . γ is the non-dimensional characteristic time constant and T min is the minimum time period. We expect: γ ≪ 1 : near viscous γ → O (1) : strongly non-viscous Identification of Damping – p.18/51 June 2005, University of Catania
Viscous damping identification 25 Fitted viscous damping matrix C kj 20 15 10 5 0 −5 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted viscous damping matrix: γ = 0 . 02 , damping model 2 Identification of Damping – p.19/51 June 2005, University of Catania
Viscous damping identification 8 Fitted viscous damping matrix C kj 7 6 5 4 3 2 1 0 −1 −2 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted viscous damping matrix: γ = 0 . 5 , damping model 1 Identification of Damping – p.20/51 June 2005, University of Catania
Non-viscous Damping Identification 30 25 Fitted coefficient matrix C kj 20 15 10 5 0 −5 −10 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted coefficient matrix: γ = 0 . 5 , damping model 1; γ fit = 0 . 49 Identification of Damping – p.21/51 June 2005, University of Catania
Non-viscous damping identification 12 10 Fitted coefficient matrix C kj 8 6 4 2 0 −2 −4 −6 30 25 20 30 15 25 20 10 15 5 10 5 0 0 j−th DOF k−th DOF Fitted coefficient matrix: γ = 0 . 5 , damping model 2; γ fit = 0 . 63 Identification of Damping – p.22/51 June 2005, University of Catania
Experimental setup Damped free -free beam, L = 1 m, width = 39 . 0 mm, thickness = 5 . 93 mm Clamped damping mechanism Instrumented hammer for impulse input Identification of Damping – p.23/51 June 2005, University of Catania
Measured transfer functions Driving Point Transfer Fucntion Measured 0 Reconstructed −10 −20 Log Amplitude dB −30 −40 −50 −60 −70 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Measured and fitted transfer function of the beam Identification of Damping – p.24/51 June 2005, University of Catania
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