18 April 2005

Dynamics of Non-viscouslyDamped Distributed Parameter

SystemsS 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

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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

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Introduction (1)

Modelling and analysis of damping properties arenot as advanced as mass and stiffness properties.The reasons:

by contrast with inertia and stiffness forces, it isnot in general clear which variables are relevantto determine the damping forces

the spatial location of the damping sources aregenerally unclear - often the structural joints aremore responsible for the energy dissipationthan the (solid) material

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Introduction (2)

the functional form of the damping model isdifficult to establish experimentally, and finally

even if one manages to address the previousissues, what parameters should be used in achosen model is still very much an openproblem

The ‘solution’ over the past 100 years:

Use viscous damping model

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Viscous Damping Model

Introduced by Lord Rayleigh in 1877

instantaneous generalized velocities are theonly relevant variables that determine damping

However,

viscous damping is not the only damping modelwithin the scope of linear analysis.

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Non-viscous Damping Model

Any causal model which makes the energydissipation functional non-negative is a possiblecandidate for a damping model

non-viscous damping models in general havemore parameters and therefore are more likelyto have a better match with experimentalmeasurements

Question:

What non-viscous damping model should beused?

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Equation of Motion

ρ(r)u(r, t) + L1u(r, t) + L2u(r, t) = p(r, t) (1)

specified in some domain D with homogeneouslinear boundary condition of the form

Mu(r, t) = 0; r ∈ Γ

specified on some boundary surface Γ.u(r, t): displacement variableρ(r): mass distribution of the systemp(r, t): distributed time-varying forcing functionL2: spatial self-adjoint stiffness operatorM: linear operator acting on the boundary

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The Damping Operator

The damping operator L1 can be written in the form

L1u(r, t) =

∫

D

∫ t

−∞

C1(r, ξ, t − τ)u(ξ, τ) dτ dξ (2)

where C1(r, ξ, t) is the kernel function.

The velocities u(ξ, τ) at different time instantsand spatial locations are coupled through thekernel function

Eq. (1) together with the damping operator (2)represents a partial integro-differential equation

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The Damping Operator

Any function that makes the energy dissipationfunction

F(t) =1

2

∫

D

{∫

D

∫ t

−∞

C1(r, ξ, t − τ)

u(ξ, τ)dτ dξ} u(r, t) dr (3)

non-negative can be used as a kernel function.The main assumption:

the damping kernel function C1(r, ξ, t) isseparable in space and time

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Viscous Damping

The kernel function is a delta function in both spaceand time:

C1(r, ξ, t − τ) = C(r)δ(r − ξ)δ(t − τ) (4)

the spatial delta function means that thedamping force is ‘locally reacting’ and the timedelta function implies that the force dependsonly on the instantaneous value of the motion

in general this represents the non-proportionalviscous damping model

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Viscoelastic Damping

The kernel function is a delta function in space butdepends on the past time histories:

C1(r, ξ, t − τ) = C(r)g(t − τ)δ(r − ξ) (5)

Represents a locally reacting viscoelasticdamping model where the damping forcedepends on the past velocity time historiesthrough a convolution integral over the kernelfunction g(t)

g(t) is known as retardation function, heredityfunction or relaxation function

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Non-local Viscous Damping

The kernel function is a delta function in time butdepends on the spatial distribution of the velocities:

C1(r, ξ, t − τ) = C(r)c(r − ξ)δ(t − τ) (6)

velocities at different points can affect thedamping force at a given point via a convolutionintegral

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Non-local Viscoelastic Damping

This is the most general form of damping model

the only assumption is that the kernel functionis separable in space and time:

C1(r, ξ, t − τ) = C(r)c(r − ξ)g(t − τ) (7)

all the previous three damping models can beidentified as special cases of this model

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Parametrization of Models (1)

Plausible functional form of the kernel functions inspace and time is requiredRequirement:For a physically realistic model of damping

ℜ

[

G(ω)

∫

D

∫

D

C(r)c(r − ξ)U ∗(ξ, ω)U(r, ω) dξ dr

]

≥ 0

for all ω

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Non-viscous Damping Functions

Damping functions (in Laplace domain) Author, Year

G(s) =

Pnk=1

aks

s + bk

Biot (1955, 1958)

G(s) =E1sα − E0bsβ

1 + bsβBagley and Torvik (1983)

0 < α < 1, 0 < β < 1

sG(s) = G∞

"

1 +

Pk αk

s2 + 2ζkωks

s2 + 2ζkωks + ω2

k

#Golla and Hughes (1985)

and McTavish and Hughes (1993)

G(s) = 1 +

Pnk=1

∆ks

s + βk

Lesieutre and Mingori (1990)

G(s) = c1 − e−st0

st0Adhikari (1998)

G(s) = c1 + 2(st0/π)2 − e−st0

1 + 2(st0/π)2Adhikari (1998)

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Parametrization of Models (2)

g(t) = g∞ µ exp(−µt) so that G(ω) = g∞ µ

iω+µ

c(r − ξ) = α2exp(−α|r − ξ|) and

C(r), g∞, µ and α are all positive

The damping force:

∫

D

∫ t

−∞

C(r) g∞ µ exp(−µ{t − τ})

α

2exp(−α|r − ξ|)u(ξ, τ) dξ dτ

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Special Cases

if α → ∞, µ → ∞ one obtains the standardviscous model in (4)

if α → ∞ and µ is finite one obtains the localnon-viscous model in (5)

if α is finite but µ → ∞ one obtains thenon-local viscous damping model in (6)

if both α and µ are finite one obtains thenon-local viscoelastic damping model in (7)

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Damped Euler-Bernoulli Beam������x L x R

x 1 x 2

x 0

Homogeneous Euler-Bernoulli beam withnon-viscous damping

Objectives:

To obtain eigenvalues and eigenvectors of thesystem

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Equation of Motion (1)

Part within the damping patch:

EI∂4w(x, t)

∂x4+ ρA

∂2w(x, t)

∂t2+

∫ x2

x1

∫ t

−∞

α

2exp (−α |x − ξ|)

g∞µ exp (−µ(t − τ))∂w(ξ, t)

∂t

∣

∣

∣

∣

t=τ

dξdτ = f(x, t)

(8)

when x ∈ [x1, x2]

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Equation of Motion (2)

Part outside the non-viscous damping patch:

EI∂4w(x, t)

∂x4+ρA

∂2w(x, t)

∂t2+C0

∂w(x, t)

∂t= f(x, t) (9)

when x ∈ (xL, x1) ∪ (x2, xR).

Appropriate boundary conditions must besatisfied at x = xL and at x = xR

relevant continuity conditions at the internalpoints x1 and x2 must be satisfied

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Outline of the Solution Method

Transform the equations into Laplace domain

differentiate with respect to the spatial variableto eliminate the spatial correlation terms(possible due to the exponential assumption)

express the BCs corresponding to the higherorder derivatives in terms of the known BCs

repeat the process for all three segments

merge the solutions from the three segments bymatching the displacements and theirderivatives at the interfaces

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Eigensolutions of the Beam

The eigenvalues λj are the roots of

det[

M(s) exp(

Φ(s)(xL − x1))

T (x1, s)

+N(s) exp(

Φ(s)(xR − x2))

T (x2, s)]

= 0

The corresponding mode shapes are

ψj(x) =

exp(

Φ(λj)(x − x1))

T(x1, λj)u0(λj), xL ≤ x ≤ x1

T(x, λj)u0(λj), x1 ≤ x ≤ x2

exp(

Φ(λj)(x − x2))

T(x2, λj)u0(λj), x2 ≤ x ≤ xR

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Boundary Conditions

The matrices M(s) and N(s) depend on the boundaryconditions:

Clamped-clamped (C-C):

M(s) =

I2×2 O2×2

O2×2 O2×2

, N(s) =

O2×2 O2×2

I2×2 O2×2

Free-Free (F-F):

M(s) =

O2×2 I2×2

O2×2 O2×2

, N(s) =

O2×2 O2×2

O2×2 I2×2

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Example 1: The System

Part 1 Part 2

Damped beam with step variation in the system propertiesand pinned boundary conditions (Friswell and Lees, 2001)

Parameters Part 1 Part 2

Li 1m 2m

ρAi 10 kg/m 20 kg/m

ci 0 Ns/m2 10 Ns/m2

EIi 100 Nm2 100 Nm2

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Example 1: Results

Proposed method Friswell and Lees (2001)-2.2552 ± 1.2711i -2.2552 ± 1.2711i-1.7936 ± 10.903i -1.7936 ± 10.903i-1.5741 ± 24.863i -1.5741 ± 24.863i-1.7876 ± 43.165i -1.7876 ± 43.165i-1.8781 ± 68.118i -1.8781 ± 68.118i-1.6984 ± 99.327i -1.6984 ± 99.327i-1.6775 ± 133.66i -1.6775 ± 133.66i-1.8549 ± 174.00i -1.8549 ± 174.00i

The first eight eigenvalues of the beam

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Example 2: The System (1)

g ( t ) , c ( x )

C wR

K θ R

M wR

C θ R

K wR

x L x 1

x R x 2

C w2 g ( t ) C w1 g ( t )

Euler-Bernoulli beam with complex boundary conditions andmiddle supports

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Example 2: The System (2)

The numerical values (in SI units) of the systemparameters are as follows:

Case 1: Local viscous damping:L=1, EI=1, m=16, MwR=4, KwR=8, CwR=4,g∞=1.6,g(t)=δ(t), c(x) = δ(x),CθR = KθR = Cw1 = Cw2=0

Case 2: Non-local non-viscous damping:L=1, EI=1, m=16, MwR=4, KwR=8, CwR=4, g∞=16,g(t) = µ exp (−µt),c(x) = α exp (−α |x|),KθR = 8, CθR = Cw1 = Cw2=4

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Example 2: Results

jλj

Proposed method Yang and Wu (1997)

1 -0.2705 ± 1.1451i -0.2705 ± 1.1451i

2 -0.1357 ± 4.4930i -0.1357 ± 4.4930i

3 -0.0896 ± 13.2586i -0.0896 ± 13.2586i

4 -0.0727 ± 26.8877i -0.0727 ± 26.8877i

5 -0.0647 ± 45.4297i -0.0647 ± 45.4297i

6 -0.0602 ± 68.8941i -0.0602 ± 68.8941i

7 -0.0575 ± 97.2862i -0.0575 ± 97.2862i

8 -0.0557 ± 130.6088i -0.0557 ± 130.6088i

9 -0.0546 ± 168.8632i -0.0546 ± 168.8632i

10 -0.0537 ± 212.0505i -0.0537 ± 212.0505i

First ten eigenvalues of the beam for Case 1

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Example 2: Results

jλj

µ = ∞, α = ∞ µ = 100, α = 10 µ = 100, α = 0.1 µ = 1, α = 10

1 -0.67921 ± 1.1780i -0.65439 ± 1.1966i -0.56874 ± 1.2444i -0.52895 ± 1.4404i

2 -0.98559 ± 6.4492i -0.92155 ± 6.4835i -0.61797 ± 6.4677i -0.49096 ± 6.5182i

3 -1.3833 ± 16.611i -1.2245 ± 16.723i -1.0592 ± 16.703i -0.61363 ± 16.656i

4 -1.4263 ± 31.584i -1.2455 ± 31.754i -1.1706 ± 31.727i -0.73244 ± 31.620i

5 -1.0714 ± 51.442i -0.90384 ± 51.521i -0.83720 ± 51.483i -0.74997 ± 51.456i

6 -1.0353 ± 76.208i -0.79867 ± 76.276i -0.74770 ± 76.237i -0.71290 ± 76.207i

7 -1.3874 ± 105.87i -0.92919 ± 106.11i -0.90512 ± 106.08i -0.71677 ± 105.88i

8 -1.4200 ± 140.46i -0.91144 ± 140.68i -0.89997 ± 140.67i -0.75157 ± 140.47i

9 -1.0745 ± 179.97i -0.78527 ± 180.03i -0.77680 ± 180.01i -0.75400 ± 179.98i

10 -1.0541 ± 224.42i -0.74574 ± 224.45i -0.74022 ± 224.44i -0.73084 ± 224.42i

First ten eigenvalues of the beam for Case 2

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Example 2: Results

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

x (m)

ℜ(ψ j(x

)) Mode 1

Mode 2

Mode 3

Mode 4

µ = ∞, α= ∞

µ = 100, α= 10

µ = 100, α= 0.0

µ = 1, α= 10

Real parts of the first four modes for Case 2

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Example 2: Results

−0.5 0 0.5−5

0

5

10

15x 10

−3

x (m)

ℑ(ψ 1(x)

)

Mode 1

−0.5 0 0.5−2

0

2

4

6x 10

−3

x (m)

ℑ(ψ 2(x)

)

Mode 2

−0.5 0 0.5−4

−2

0

2

4x 10

−4

x (m)

ℑ(ψ 3(x)

)

Mode 3

−0.5 0 0.5−2

−1

0

1

2x 10

−3

x (m)

ℑ(ψ 4(x)

)

Mode 4

µ = ∞, α= ∞µ = 100, α= 10

µ = 100, α= 0.0

µ = 1, α= 10

Imaginary parts of the first four modes for Case 2

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Conclusions (1)

A method to obtain the natural frequencies andmode-shapes of Euler-Bernoulli beams withgeneral linear damping models has beenproposed

it is assumed that the damping force at a givenpoint in the beam depends on the past historyof velocities at different points via convolutionintegrals over exponentially decaying kernelfunctions

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Conclusions (2)

conventional viscous and viscoelastic dampingmodels can be obtained as special cases of thisgeneral linear damping model

the choice of damping models effects theimaginary parts of the complex modes

future work will discuss computational issues,forced vibration problems and experimentalidentification of non-viscous damping models

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Open Problems

To what extent different damping models with‘correct’ sets of parameters influence thedynamics?

which aspects of dynamic behavior are wronglypredicted by an incorrect damping model?

how to choose a damping model (not theparameters!) for a given system?

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