Dynamics of Non-viscously Damped Distributed Parameter Systems S Adhikari , Y Lei and M I Friswell Department of Aerospace Engineering, University of Bristol, Bristol, U.K. URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html 18 April 2005 Non-viscously Damped Systems – p.1/35
Outline of the Presentation Introduction Models of damping Equation of motion Outline of the solution method Incorporation of boundary conditions Numerical examples & results Conclusions & future works 18 April 2005 Non-viscously Damped Systems – p.2/35
Introduction (1) Modelling and analysis of damping properties are not as advanced as mass and stiffness properties. The reasons: by contrast with inertia and stiffness forces, it is not in general clear which variables are relevant to determine the damping forces the spatial location of the damping sources are generally unclear - often the structural joints are more responsible for the energy dissipation than the (solid) material 18 April 2005 Non-viscously Damped Systems – p.3/35
Introduction (2) the functional form of the damping model is difficult to establish experimentally, and finally even if one manages to address the previous issues, what parameters should be used in a chosen model is still very much an open problem The ‘solution’ over the past 100 years: Use viscous damping model 18 April 2005 Non-viscously Damped Systems – p.4/35
Viscous Damping Model Introduced by Lord Rayleigh in 1877 instantaneous generalized velocities are the only relevant variables that determine damping However, viscous damping is not the only damping model within the scope of linear analysis. 18 April 2005 Non-viscously Damped Systems – p.5/35
Non-viscous Damping Model Any causal model which makes the energy dissipation functional non-negative is a possible candidate for a damping model non-viscous damping models in general have more parameters and therefore are more likely to have a better match with experimental measurements Question: What non-viscous damping model should be used? 18 April 2005 Non-viscously Damped Systems – p.7/35
Equation of Motion ρ ( r )¨ u ( r , t ) + L 1 ˙ u ( r , t ) + L 2 u ( r , t ) = p ( r , t ) (1) specified in some domain D with homogeneous linear boundary condition of the form M u ( r , t ) = 0; r ∈ Γ specified on some boundary surface Γ . u ( r , t ) : displacement variable ρ ( r ) : mass distribution of the system p ( r , t ) : distributed time-varying forcing function L 2 : spatial self-adjoint stiffness operator M : linear operator acting on the boundary 18 April 2005 Non-viscously Damped Systems – p.8/35
The Damping Operator The damping operator L 1 can be written in the form � t � L 1 ˙ u ( r , t ) = C 1 ( r , ξ , t − τ ) ˙ u ( ξ , τ ) dτ d ξ (2) D −∞ where C 1 ( r , ξ , t ) is the kernel function. The velocities ˙ u ( ξ , τ ) at different time instants and spatial locations are coupled through the kernel function Eq. (1) together with the damping operator (2) represents a partial integro-differential equation 18 April 2005 Non-viscously Damped Systems – p.9/35
The Damping Operator Any function that makes the energy dissipation function � t F ( t ) = 1 �� � C 1 ( r , ξ , t − τ ) 2 D D −∞ u ( ξ , τ ) dτ d ξ } ˙ ˙ u ( r , t ) d r (3) non-negative can be used as a kernel function. The main assumption: the damping kernel function C 1 ( r , ξ , t ) is separable in space and time 18 April 2005 Non-viscously Damped Systems – p.10/35
Viscous Damping The kernel function is a delta function in both space and time: C 1 ( r , ξ , t − τ ) = C ( r ) δ ( r − ξ ) δ ( t − τ ) (4) the spatial delta function means that the damping force is ‘locally reacting’ and the time delta function implies that the force depends only on the instantaneous value of the motion in general this represents the non-proportional viscous damping model 18 April 2005 Non-viscously Damped Systems – p.11/35
Viscoelastic Damping The kernel function is a delta function in space but depends on the past time histories: C 1 ( r , ξ , t − τ ) = C ( r ) g ( t − τ ) δ ( r − ξ ) (5) Represents a locally reacting viscoelastic damping model where the damping force depends on the past velocity time histories through a convolution integral over the kernel function g ( t ) g ( t ) is known as retardation function, heredity function or relaxation function 18 April 2005 Non-viscously Damped Systems – p.12/35
Non-local Viscous Damping The kernel function is a delta function in time but depends on the spatial distribution of the velocities: C 1 ( r , ξ , t − τ ) = C ( r ) c ( r − ξ ) δ ( t − τ ) (6) velocities at different points can affect the damping force at a given point via a convolution integral 18 April 2005 Non-viscously Damped Systems – p.13/35
Non-local Viscoelastic Damping This is the most general form of damping model the only assumption is that the kernel function is separable in space and time: C 1 ( r , ξ , t − τ ) = C ( r ) c ( r − ξ ) g ( t − τ ) (7) all the previous three damping models can be identified as special cases of this model 18 April 2005 Non-viscously Damped Systems – p.14/35
Parametrization of Models (1) Plausible functional form of the kernel functions in space and time is required Requirement: For a physically realistic model of damping � � � � C ( r ) c ( r − ξ ) U ∗ ( ξ , ω ) U ( r , ω ) d ξ d r ℜ G ( ω ) ≥ 0 D D for all ω 18 April 2005 Non-viscously Damped Systems – p.15/35
P n " # Non-viscous Damping Functions P P n Damping functions (in Laplace domain) 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 ζ k ω k s sG ( s ) = G ∞ 1 + k α k Golla and Hughes (1985) s 2 + 2 ζ 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 18 April 2005 Non-viscously Damped Systems – p.16/35
Parametrization of Models (2) g ( t ) = g ∞ µ exp( − µt ) so that G ( ω ) = g ∞ µ i ω + µ c ( r − ξ ) = α 2 exp( − α | r − ξ | ) and C ( r ) , g ∞ , µ and α are all positive The damping force: � t � C ( r ) g ∞ µ exp( − µ { t − τ } ) D −∞ α 2 exp( − α | r − ξ | ) ˙ u ( ξ , τ ) d ξ dτ 18 April 2005 Non-viscously Damped Systems – p.17/35
Special Cases if α → ∞ , µ → ∞ one obtains the standard viscous model in (4) if α → ∞ and µ is finite one obtains the local non-viscous model in (5) if α is finite but µ → ∞ one obtains the non-local viscous damping model in (6) if both α and µ are finite one obtains the non-local viscoelastic damping model in (7) 18 April 2005 Non-viscously Damped Systems – p.18/35
��� ��� Damped Euler-Bernoulli Beam x� x� 1� 2� x� x� L� R� x� 0� Homogeneous Euler-Bernoulli beam with non-viscous damping Objectives: To obtain eigenvalues and eigenvectors of the system 18 April 2005 Non-viscously Damped Systems – p.19/35
Equation of Motion (1) Part within the damping patch: � x 2 � t EI ∂ 4 w ( x, t ) + ρA∂ 2 w ( x, t ) α + 2 exp ( − α | x − ξ | ) ∂x 4 ∂t 2 −∞ x 1 � g ∞ µ exp ( − µ ( t − τ )) ∂w ( ξ, t ) � dξdτ = f ( x, t ) � ∂t � t = τ (8) when x ∈ [ x 1 , x 2 ] 18 April 2005 Non-viscously Damped Systems – p.20/35
Equation of Motion (2) Part outside the non-viscous damping patch: EI ∂ 4 w ( x, t ) + ρA∂ 2 w ( x, t ) ∂w ( x, t ) + C 0 = f ( x, t ) (9) ∂x 4 ∂t 2 ∂t when x ∈ ( x L , x 1 ) ∪ ( x 2 , x R ) . Appropriate boundary conditions must be satisfied at x = x L and at x = x R relevant continuity conditions at the internal points x 1 and x 2 must be satisfied 18 April 2005 Non-viscously Damped Systems – p.21/35
Outline of the Solution Method Transform the equations into Laplace domain differentiate with respect to the spatial variable to eliminate the spatial correlation terms (possible due to the exponential assumption) express the BCs corresponding to the higher order derivatives in terms of the known BCs repeat the process for all three segments merge the solutions from the three segments by matching the displacements and their derivatives at the interfaces 18 April 2005 Non-viscously Damped Systems – p.22/35
Eigensolutions of the Beam The eigenvalues λ j are the roots of � ¯ � � det M ( s ) exp Φ ( s )( x L − x 1 ) T ( x 1 , s ) � ¯ � � + N ( s ) exp Φ ( s )( x R − x 2 ) T ( x 2 , s ) = 0 The corresponding mode shapes are � ¯ � exp Φ ( λ j )( x − x 1 ) T ( x 1 , λ j ) u 0 ( λ j ) , x L ≤ x ≤ x 1 ψ j ( x ) = T ( x, λ j ) u 0 ( λ j ) , x 1 ≤ x ≤ x 2 � ¯ � exp Φ ( λ j )( x − x 2 ) T ( x 2 , λ j ) u 0 ( λ j ) , x 2 ≤ x ≤ x R 18 April 2005 Non-viscously Damped Systems – p.23/35
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