Viscoelastic and Structural Damping Analysis with …highorder.berkeley.edu/proceedings/aiaa-annual-2012/paper1237.pdf · the help of numerical examples. The material considered is

50 th AIAA Aerospace Sciences Meeting, Multidisciplinary Design Optimization (MDO)January 9 – 12, 2012, Nashville, TN.

AIAA Paper ID 1133940VISCOELASTIC AND STRUCTURAL DAMPING ANALYSIS

WITH DESIGNER MATERIALS

Harry H. Hiltona,1 and Germain Sossoua,b2

aAerospace Engineering Department, College of Engineering andPrivate Sector Program Division, National Center for Supercomputing Applications

University of Illinois at Urbana-Champaign104 S. Wright Street, MC-236, Urbana, IL 61801–2935 USAbSUPMÉCA - Institut Supérieur de Mécanique de Paris

3 rue Fernand Hainaut, 93400 Saint-Ouen, France

ABSTRACT: The interrelationships between viscoelastic, Newtonian vis-cous and structural (Coulomb friction) damping are analyzed in termsof Fourier transforms and complex moduli in the frequency domain andare also interpreted in terms of behavioral responses associated with realmaterial compliances or moduli in the real time space. It is shown thatthe correspondence between viscous and elastic structural damping isspurious, severely limited to only harmonic motion and that it does notextend to more complicated viscoelastic materials beyond Newtonian vis-cous ow dissipation. The dissipation energy generated by viscoelasticand structural damping is also examined. The eects of structural damp-ing on elastic and viscoelastic bending-torsion utter are evaluated withthe help of numerical examples. The material considered is aluminum,but the analysis is general and can be applied to any viscoelastic materialsuch as high polymer composites. It is shown that the presence of in-creased structural damping does not necessarily have stabilizing eects bydecreasing the viscoelastic or elastic utter speed nor are the viscoelastic utter speeds necessarily lower than the corresponding elastic ones.An analysis is developed for analytically designing material properties andgeometric sizing to tailor elastic and viscoelastic damping responses in or-der to control prescribed events subject to pre-selected constrains, suchas cost, weight, geometry, maximum de ections, minimum utter ve-locity, life times, dissipation energy, composite ber orientation, volumefractions, etc. A simple illustrative example and future research directionsare presented.

1Professor Emeritus of Aerospace Engineering and Senior Academic Lead for Computational Structural/SolidMechanics at NCSA / PSP. Corresponding author. AIAA Fellow. Voice: 217-333-2653 FAX: 217-244-0720Email: [email protected]

2Student Intern at UIUC, MS in Mechanical Engineering Candidate 2013 at SUPMÉCA, and BS Engineeringin Aeronautics 2013, ENSMA, Poitiers. Email: [email protected]

1American Institute of Aeronautics and Astronautics

50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition09 - 12 January 2012, Nashville, Tennessee

AIAA 2012-1256

Copyright © 2012 by Harry H. Hilton. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Keywords: designer materials, dissipation energy, flutter, structural damping, viscousdamping, viscoelasticity

LIST OF SYMBOLS

aij, bijkl = relaxation modulus coefficientsAk = aerodynamic coefficientscV D = viscous mechanical dampping coefficientC∗ijkl Cijkl = compliancesC∗ijkln Cijkln = compliance coefficientsD∗ D = bending, torsional, etc., stiffnessesD∗ijkln Dijkl = bending stiffness coefficientsE∗ijkl Eijkl = relaxation moduliEijkl0 E0 = elastic Young’s modulusE∗ijkln Eijkln = relaxation moduli coefficientsESD = structural damping coefficientFA = aerodynamic force (lift)FASV = aero-servo-viscoelastic forceFC = control force (MR, piezo, SM)FIP = cabin pressurizationFV = external vibratory forceg, gm = structural damping coefficientN = summation limit in modulus, compliance Prony seriesNijkl = in-plane forceP Q = viscoelastic differential operatorsS, Sm = parameters to be optimizedt = timeu, um = generalized displacementsV = volumeV∞ = free stream air velocityVF , ωF = flutter velocity and frequencyw = deflectionW = weightx = Cartesian coordinatesα = rigid body angle of attackεkl = strain componentsρair = air densityρ = density of load carrying structureσkl = stress componentsτklmn = relaxation timesθ = angular deflectionω = Fourier transform variableΩ = forced vibration frequency


INTRODUCTION

In flutter and vibration analysis, it is standard practice to augment elastic effects bythe introduction of structural damping coefficients gi [1 – 11], where the latter are essentiallymeasures of losses due to material hysteresis and/or friction in structural joints. In bothinstances, the fundamental dissipation phenomenon is “dry” solid friction and as such, theassociated force and displacement constitutive relations are explicitly independent of frequencyand of displacement velocities, accelerations or their higher time derivatives. (See [12] and[13]). Analytically, the algebraic Hooke’s law is maintained, but the actual, real elastic moduliare replaced by complex values, i.e., E = Eo(1 + ı gE), where Eo is Young’s modulus in theabsence of structural damping. A similar expression is used for the elastic shear modulus,G = Go(1 + ı gG) and for the elastic bulk modulus, K = Ko(1 + ı gK). The three gs displayedhere may or may not be equal depending upon the specific structural damping encounteredin a given structure or component joint, although as a matter of convenience they are veryfrequently assumed or prescribed equal.

Viscoelastic materials, on the other hand, obey differential and/or integral stresses-strainlaws, which relate stresses, strains and their time derivatives of various orders [15 – 19]. Theviscoelastic dissipation process is primarily an involved, highly frequency and temperaturesensitive, material dependent viscous phenomenon with one or more coefficients of viscosity [15– 19] and, as will be shown, totally unrelated to the structural damping mechanism. Historically,and in this paper as well, the term viscous damping refers to Newtonian flow, where the stressesare proportional to the strain velocities through at most only one coefficient of viscosity forshape changes and no more than one other for volume changes. While the structural dampingphenomenon is well understood and experimental values for these damping coefficients arereadily available [1 – 11], its interpretation vis-à-vis viscous damping appears confused [20,21]. The analysis in [3], on the other hand, is correctly based on the correspondence betweenviscous and structural damping on harmonic motion, but has restricted his analysis to onlymotion at the system’s natural frequency. Under these conditions he shows that the structuraldamping coefficient is frequency independent. More recently, in [22] solid friction damping inmechanical oscillators was modeled by using both linear and nonlinear formulations. Thesemodels are of interest, since they simulate decay behavioral patterns which approximate (a)Coulomb friction [12] at high amplitudes and low frequencies, (b) viscous damping at midamplitudes and mid frequencies and, finally, (c) structural damping at small amplitudes andhigh frequencies. However, these approximate similarities do not imply any relations betweenfundamental behavioral responses of structural and viscous damping phenomena. Structuraldamping arises from Coulomb friction [12] in structural joints and as such is not a material


property in the sense of moduli and the like. In [23] a hysteretic damping analysis for compositelaminates is presented, which includes an extensive bibliography on damping.

Since viscoelasticity includes among other mechanisms both elasticity and viscous damp-ing, i.e., velocity dependent Newtonian viscous dissipation, it can readily serve as a vehiclefor the comparison of viscous and structural damping. In this paper, general linear viscoelas-tic stress-strain relations (including structural and viscous damping) are used to interpret thevarious damping processes by a critical examination of complex moduli and of compliances inthe frequency domain and of compliances in the real time space. Such an approach makes itpossible to treat generalized many degree of freedom systems and is not limited to the singlemass, spring and damper combinations of References [20] and [21].

As an additional and separate issue, the concept of designer materials [52] is applied toa 2–D flutter problem (bending and torsion). This protocol consists of an inverse calculus ofvariation analytical determination of elastic and/or viscoelastic properties and geometric sizingsubject to constraints of minimum weight, maximum stress/deformation, flutter velocities, etc.

ANALYSIS

Flutter and Complex Moduli

In a Cartesian system, x = x1, x2, x3, the generalized isothermal anisotropic viscoelasticstress-strain relations are

σij(x, t) =

8>>>>>>>><>>>>>>>>:

tZ

−∞

Eijkl(x, t− t′)∂εkl(x, t′)

∂t′dt′

ortZ

−∞

E∗ijkl(x, t− t′) εkl(x, t′) dt′

(1)

where the relaxation moduli are represented by Prony series [14]8<:

Eijkl(x, t)

E∗ijkl(x, t)

9=; =

NX

n=1

8<:

Eijkln

E∗ijkln

9=; exp

− t

τn

(2)

with E∗ijkln = Eijkln/τn. The Fourier transforms (FT) of a function f(x, t) is defined by

f(x, ω) = Ff(x, t) =∞Z

−∞

f(x, t) exp (−ı ω t) dt (3)


which leads to the complex nonhomogeneous3 moduli designated by Eijkl(x, ω) and defined by

Eijkl(x, ω) = FEijkl(x, t) = ERijkl(x, ω)| z

= storage modulus

+ı EIijkl(x, ω)| z

= loss modulus

(4)

where ERijkl(x, ω) and EI

ijkl(x, ω) are real functions (see Fig. 1).

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

10 8 6 4 2 0 -2

ER

EI

MO

DU

LUS

LOG (!)

Fig. 1 Complex relaxation modulus

The governing elastic equilibrium equations for flexible isotropic isothermal elastic liftingsurfaces, fuselages, etc., with time independent masses, subjected to aerodynamic and inertialforces with generalized displacements um(x, t), m = 1, 2, · · · , M and u(x, t) = u1, u2, · · · , uMcan be expressed in the generalized form

LEmuE(x, t)

= m(x) ∂

2uE(x, t)∂t2| z

inertia (T1E)

+ D0(x)uE(x, t)| z internal elastic

restoring forces (T2E)

+ Lm

x, t, ρ, V∞,u

E(x, t), ∂uE(x, t)∂t

,∂2uE(x, t)

∂t2

| z lift, aerodynamic torque etc, (self−excited closed loop, T3E)

− Fvibm(x, t,Ω)| z vibratory orstatic forces

(open loop, T4E)

= 0

m = 1, 2, · · · ,M x = x1, x2, x3 (5)3Nonhomogeneous materials include functionally graded materials (FGMs) as a subset.


where Lm are operators describing unsteady aerodynamic contributions, ρ is the atmosphericdensity, V the vehicle flight velocity. The symbols D0 represent spatial differential operatorsassociated with bending, torsion, etc., and depend on moduli and geometry but are not func-tions of Poisson’s ratios [27 – 35]. An optional term may be added to this governing relation

to represent external mechanical viscous damping consisting of c(x) ∂uE(x, t)∂t

. While this ac-tion alters the flutter velocity and frequency values it is not part of the material viscoelasticresponses.

The forces Fvibm are for open loop vibration contributions and may include internal cabinpressurizations, but no generalized displacements uE(x, t). The elastic-viscoelastic correspon-dence principle (analogy) [15, 19, 25, 26] consists of the application of FTs to Eq. (5) and of thesubsequent substitution of complex viscoelastic moduli E(x, ω) and G(x, ω) for elastic moduliEo(x) and Go(x) respectively, or essentially replacing the real and frequency independent elasticstiffnesses by complex viscoelastic stiffness functions and differential/integral operators. This,then leads to governing viscoelastic relations in the FT space

Lm

u(x, ω)

=

− ω2 m(x) u(x, ω) + D(x, ω) u(x, ω) + Lρ, V∞, x, ω,u(x, ω)

− F vib (x, ω,Ω) = 0

m = 1, 2, · · · ,M (6)

that invert into the real time space t as

Lm u(x, t) = m(x) ∂2u(x, t)∂t2| z

inertia (T1VE)

+tZ

−∞

D(x, t− t′)∂u(x, t′)∂t′

dt′

| z internal viscoelastic resisting forces(T2VE)

+ Lm

x, t, ρ, V∞,u(x, t), ∂u(x, t)

∂t,∂2u(x, t)∂t2

| z lift, aerodynamic torque,etc. (self−excited closed loop, T3VE)

− Fvib(x, t,Ω)| z vibratory orstatic forces

(open loop, T4VE)

= 0

m = 1, 2, · · · ,M x = x1, x2, x3 (7)

Additionally, of course, aero-servo control forces and sensors in the form which may dependon one or more of the indicated variables

FASV = FASV

0@x, t,u(x, t), ∂u(x, t)

∂t,∂2u(x, t)∂t2

,

tZ

0

u(x, t′) dt′1A

| z aero−servo−elastic/viscoelastic forces (closed loop)

(8)


may be added to these viscoelastic systems by placing −FASV or −FASV in the middle part of(5) to (7) as has been demonstrated in [24].

It can be readily shown [15, 19] that for simple harmonic motion the FT variable ω is theoscillatory frequency and that in the case of flutter [25, 26] it becomes the flutter frequency,while V∞ = Vf plays the role of the flutter speed. The latter two are, of course, pairs ofeigenvalues at which a given flight structure can experience harmonic motion with frequencyωf . The flutter velocity Vf can readily be replaced by the flutter Mach numberMf . Viscoelasticresponses may also be characterized on an energy axis involving all potential energy at one endand all dissipation at the other which is shown schematically in Fig. 2 . Elasticity and viscousdamping represent the two degenerate viscoelastic extremes at opposite ends of the energyscale, i.e., elasticity is 100% potential energy and zero damping, while Newtonian viscous flowis all dissipation and no potential energy storage.

ENERGY REPRESENTATION OF

MATERIAL PROPERTIES

PE = 100% PE = 0% DE = 0% DE = 100%

VISCOELASTIC

•_________________________________________________•

ELASTIC VISCOUS FLOW

PE = Potential energy DE = Dissipative energy

Fig. 2 Viscoelastic Energy Distributions

In general, linear viscoelastic material behavior (including elasticity and viscous damp-ing), i.e. the stress-strain or constitutive relations, can also be expressed by relations betweengeneralized displacements um and forces Fm. For isotropic materials they are given by [15, 19]

Fm(x, t) =tZ

−∞

Em(t− t′) ∂um(x, t′)∂t′

dt′ m = 1, 2, · · · ,M (9)

In particular, for each m these reduce to

F em = Emo(1 + ı gm) uem m = 1, 2, · · · ,M (10)

for elastic structural damping and to


F vsdm = Emo(1 + ı gm) uvsdm + η uvsdm (11)

for combined viscous and structural damping. The structural damping coefficient g should notbe confused with the gravitational constant to which it bears no relation. (When g = 0 inEq. (11), then only viscous damping takes place coupled with an elastic response.) Similar, butmore involved, expressions may also be written for anisotropic materials [13], but will not beintroduced here for the sake of simplicity. They are, however, treated briefly at the end of thenext section.

The application of Fourier transforms to Eqs. (3), leads to [15, 19, 25, 26]

Fm = Em um (12)

where the Em are frequency dependent viscoelastic complex moduli (4). Note that Eqs. (14)are symbolically equivalent to the F.T. of Eqs. (4) and that the F.T. of the elastic Eqs. (5)gives a complex modulus for structural damping, which is frequency insensitive.

Such an omega dependence is due to the intrinsic nature of the time differential or integralviscoelastic stress-strain laws of Eqs. (1) and (55). It can be readily seen from Eqs. (4), that inthe general viscoelastic case, the complex moduli with structural damping are

Em(ω) = (1 + ı gm)ERm(ω) + ı EI

m(ω) = ERm(ω) + ı E

′Im(ω) (13)

whereE′Im(ω) = gmE

Rm(ω) + EI

m(ω) (14)

Furthermore, elastic structural damping is also included in Eqs. (6), by virtue of the com-plex moduli defined by Eqs. (4), except that then the ER

m and EIm are frequency independent.

In any event, the expressions on the right hand sides of Eqs. (6) (i.e., the generalized forces)are unaffected by the nature of the elastic or viscoelastic materials. Therefore, the funda-mental difference is that in the elastic case with or without structural damping, are frequencyindependent, while for viscoelastic materials the stiffness parameters are always frequency func-tions. For nonhomogeneous viscoelastic materials with structural damping, one needs only toreplace the elastic moduli in Eqs. (6) with E ′(x, ω) in the FT space, where E(ω) is given byEq. (13) . Again note, that for elastic structural damping, the stiffness parameters in Eqs.(6) have a form identical to frequency independent viscoelastic ones. Table 2 illustrates thecomplex moduli representations in the four combinatorial cases considered.

The structural damping terms ı g um may be thought of as out of phase components of thedisplacements um, and, as such, bear some resemblance to velocity effects, i.e., viscous damp-ing (Fig. 3). However, examination of the FT of the viscous damping term η um, in Eqs. (11),


Table 2Complex Moduli E(!) = ER(!) + E0

I(!) and Compliances CE(t)

Complex Modulus E(ω)Material Compliance CE(t)

Real Part Imaginary Part

Elastic E0 0 CE0 = 1E0

Elastic with E0 g E0CE0

1 + ı gstructural damping

Viscous damping 0 η ωt

η

Viscous damping with E0 g E0 + η ω1η

exp−(1 + g)t

τ

structural damping

Viscoelastic with ER(ω) g ER(ω) + EI(ω) Eq. (21)structural damping

um

1

um

1

ı g um

1

÷ um

1

Fig. 3 Viscoelastic and Structural Damping Phase Relations

clearly shows that it is equal to ı ω η u for a time independent viscosity coefficient η. Conse-quently, as long as structural damping coefficients g are frequency independent, they cannotphenomenologically relate to viscous damping, unless one postulates a η inversely proportionalto ω – not the ordinary coefficient of viscosity to be sure. Also note that the FT of Eq. (11) for


g = 0 is E = E0(1 + ı ω∗/√E0 M∗), where ω∗ = ω/ωN and the natural frequency ω2

N = E0/M∗

with M∗ the system mass. Therefore, the complex modulus for viscous damping is frequencydependent, and only at the natural frequency can a frequency independent correspondence beestablished between structural and viscous damping when g = η

√E0 M∗. This relation be-

tween the complex moduli applies to any motion, and such a correspondence between g andη is not limited to harmonic motion as has been discussed earlier by Fung [3]. At all otherfrequencies, of course, the frequency dependent relationship g = η ω∗

√E0 M∗ is valid for any

elastic structural or viscous damping complex modulus, but is not physically realistic as it doesnot describe dry friction.

However, while such a proposition satisfies the consistency of expressions in the ω FT space,Eqs. (10) and (11) demonstrate that even an inversely frequency dependent viscosity coefficientη or a constant one at the natural frequencies cannot restore correspondence in the real timespace between the elastic and viscous damping cases for general displacement functions u(x, t)encountered in creep, relaxation and other non-oscillatory motions. As a matter of fact, evenin relatively simple motion where u is proportional to a single exponential function exp(ı ω t),the correspondence between viscous and structural damping is lost in those non mechanicalvibration problems, such as for instance flutter, which have highly nonlinear sensitivities tofrequency eigenvalues. For convenience and completeness, one usually represents viscoelasticstress and strain behavior in terms of mechanical models, such as, for instance, the generalizedKelvin model (GKM) [5] shown in Fig. 4. Consequently, it follows from Eq. (11) and froman examination of the GKM that viscoelastic damping represents a much more complicatedphenomenon than either elastic or viscous structural damping, since the complex compliancesCE = 1/E, CG = 1/G, CK = 1/K, etc. which are of the form

CG = C0G

1 + ı gG+ 1

ı ω ηN+1+

NX

n=1

1Gn [1 + ı (ω τn + gG)] (15)

with similar relations for the others and where the relaxation times τn and Gn are all materialproperty, temperature sensitive parameters [3, 15, 19] (Fig. 4). Viscoelastic compliances in theabsence of structural damping are given by Eq. (15) with g = 0. Similarly, the expressions (15)also include viscous damping as a degenerate case of the form N = 0, C0G = 0 and with allGn =∞. The elastic case can be obtained from ηN+1 = Gn =∞.

These two distinct phenomena, i.e., structural and viscoelastic (including viscous) damp-ing, may be interpreted in yet another fashion by examining their complex representations. Foreach generalized displacement qm, the corresponding elastic modulus with structural dampingcan be represented by E0(1 + ıgE) = Re exp(ı ∆e) and the expressions Re = E0

p1 + g2

E and∆e = tan−1(gE) are both frequency independent. (For the sake of simplicity of representation,


Elastic Action

Viscoelastic Action

Newtonian Flow

2G1 2h 1

2G

2h

2h

2h

2G2 2

N N

N+1

Skl

Skl

Fig. 1. THE GENERALIZED KELVIN MODEL

Fig. 4 The Generalized Kelvin Model (GKM)

the subscripts m are not included here.) Complex viscoelastic moduli may be written in asimilar fashion as seen in Eq. (13) with

E(ω) = Rve(ω) exp [ı∆ve(ω)] (16)

where

Rve(ω) = ER(ω)

1 + g2E + 2 gE eE(ω) +

heE(ω)

i21/2

(17)

and∆ve(ω) = arctan

hgE + eE(ω)

i(18)

are both frequency dependent with eE(ω) = E′I(ω)/ER(ω). These values are shown in Figs. 13

and 14 for 2024 aluminum [10, 11]. Aluminum was selected in order to demonstrate thatviscoelastic flutter phenomena are not limited to the usually suspect high polymer composites.


The Rvemin and Rvemax values correspond to ω = 0 and ∞ (i.e. t = ∞ and 0) respectivelyand the eE(ω) peak in the neighborhood of 15 Hz, which is of the order of magnitude of theflutter frequencies for the examples considered in Reference [26]. This reference has an incorrectformulation and solution based on the then prevailing erroneous belief bordering on an urbanlegend that viscoelastic Poisson’s ratios (PRs) are time independent. The rectified solution forthe here applicable viscoelastic Timoshenko beams appears in [27] and the correct formulationsbased on proper viscoelastic PRs may be found in [28 – 35].

The Re vectors for the elastic structural damping at the temperatures of Figs. 13 and 14are equal to Rvemax and the angles ∆e are equal to ∆vemin at all temperatures. These resultsare also displayed in Table 3. Similar values for an extensive number of distinct polymers maybe found in [43] and [44].

Table 3Viscoelastic Damping Properties of 2024 Al [39]

Tempe- Structural E′I/ER Rvemax Rvemin ∆vemax ∆vemin

rature Damping psi x 10−7 psi x 10−7 degrees degreesF Coefficient

80 0 .00283 1.070 1.060 .162 080 .05 .0528 1.071 1.061 3.02 2.86200 0 .00585 1.038 1.020 .335 0200 .05 .0559 1.039 1.021 3.20 2.86340 0 .0144 .990 .966 .826 0340 .05 .0644 .991 .967 3.69 2.86450 0 .0258 .954 .900 1.48 0450 .05 .0758 .955 .901 4.33 2.86

Dissipation Energy

A comparison of the dissipation energies generated by viscoelastic, viscous and structuraldamping processes is next in order. They can be considered together by referring to the mechan-ical models of Fig. 4. For an isotropic, isothermal linear viscoelastic material the stress-strainrelations for change in shape and in volume are, respectively [15, 19]

2Ekl(x, t) =tZ

−∞

CG(x, t− t′) ∂Skl(x, t′)

∂t′dt′ or Skl(x, t) = 2

tZ

−∞

G(x, t− t′) ∂Ekl(x, t′)

∂t′dt′

(19)


ε(x, t) =tZ

−∞

CK(x, t− t′) ∂σ(x, t′)∂t′

dt′ or σ(x, t) =tZ

−∞

K(x, t− t′) ∂ε(x, t′)

∂t′dt′ (20)

where Skl and Ekl are the stress and strain deviators, ε and σ the mean strains and stressesand where the compliances are of the form

CG(t) = CG0

1 + ı g+ t

ηN+1| z = CGN+1

+NX

n=1

1ηn|z

= CGn

1− exp

−(1 + g) t

τn

(21)

and with a similar expression for the volumetric compliance CK(t), both of which are obtainedby the FT inversion of Eq. (15). Note that Eq. (21) defines the general viscoelastic compliancesin the presence of structural damping. These compliances may also be written in the mannerof Eq. (13), i.e.

CG(ω) = 1(ı ω)2 G(ω)

= CGR(ω) − ı C ′GI(ω) = GR(ω)− ı G′I(ω)[GR(ω)]2 +G′I(ω)]2 (22)

As can be seen from Eqs. (11) and (15), the introduction of structural damping effectsinto viscoelasticity results in changes in the denominators of the sums of Eq. (15) and in firstterm multiplication by 1/(1 + ı g). Effectively, upon FT inversion, this serves to shift the timeto t+ g t in the exponential terms of Eq. (21).

The dissipation energy per unit volume at any point x and at any time t > 0 is

DE(x, t) =tZ

−∞

NX

n=1

S(n)kl (x, t′) E(n)

kl (x, t′) dt′ +tZ

−∞

MX

m=1

σ(m)(x, t′) ε(m)(x, t′) dt′ (23)

where superscripts (n) and (m) denote quantities associated with each dashpot n or m in theFig. 4 GKM. The stress-strain relations for each dashpot are given by [15, 19]

S(n)kl = 2 ηn E(n)

kl (24)

and

2E(n)(x, t) =tZ

−∞

Cn(x, t− t′) ∂ [Skl(x, t′)]∂t′

dt′ (25)

where Skl is the total stress deviator in the GKM and is the sum of of any dashpot and thestress deviator of its corresponding elastic paired spring. Equations similar to (24) and (25)can also be written for volumetric changes.


Differentiation of Eqs. (25) and substitution into (24) gives

S(n)kl (x, t) = ηn

8<:

tZ

0

∂Cn(x, t− t′)∂t′

Skl(x, t′) dt′ + Cn(x, 0) Skl(x, t)

9=; (26)

with a similar expression for σ(m). Finally, the introduction of Eqs. (24), (25) and (26) into(23) results in

DE(x, t) =tZ

0

8<:N+1X

n=1

12 ηn

24

t′Z

0

∂Cn(x, t′ − ξ)∂ξ

Skl(x, ξ) dξ + Cn(x, 0) Skl(x, t′)

35

24

t′Z

0

∂Cn(x, t′ − ξ)∂ξ

Skl(x, ξ) dξ + Cn(x, 0) Skl(x, t′)

359=; dt′

+tZ

0

8<:

MX

m=1

1ηvm

24

t′Z

0

∂CKm(x, t′ − ξ)∂ξ

σ(x, ξ) dξ + CKm(x, 0) σ(x, t′)

359=; dt′ (27)

The total dissipative energy DET (t) is the volume integral

DET (t) =Z

V

DE(x, t) dx1 dx2 dx3| z = dV

(28)

where V is the total volume of the flexible body. It is readily seen that Eqs. (27) and (28) dependboth on material properties (Cn, CKm) and the loading process. Consequently, Eq. (28) is anextremely useful expression for comparing the dissipation properties of various materials at oneor more desired loading or strain histories.

In the viscous no structural damping case, Eqs. (19) through (27) simplify to only oneterm in each C, i.e.,

CG(t) = 1η

exp−(1 + gG) t

τ

(29)

and with a similar K subscripted expression for CK and where τ = η/Go and τv = ηv/Ko.For elastic structural damping alone, only the first term of Eq. (21) remains and due to thenature of the Dirac delta functions the stress-strain relations (19) and (20) reduce to the usualalgebraic elastic ones. The results are summarized in Table 2.

For anisotropic materials, similar but more complicated flutter and dissipation energyexpressions can readily be derived. However, they may require as many as 21 complex moduli


or compliances (instead of the two isotropic ones used in the foregoing development) to fullydescribe anisotropic material behavior [41]. The anisotropic relations between stress σkl andstrains εkl now become in the FT space

σkl(x, ω) = Bklmn(x, ω) εmn(x, ω) (30)

where Bklmn are complex anisotropic moduli. For the sake of economy of length, the anisotropicflutter and dissipation energy analysis and results are not included in this paper, however, theprevious isotropic analysis can easily be extended to anisotropic materials by rewriting Eq. (9)as [26]

Fm =tZ

−∞

Eml(x, t− t′)∂uml(x, t′)

∂t′dt′ (31)

and subsequently redefining each material property in the relations following Eqs. (9) withl ranging from 1 to 6. This serves to expand the discussion from each isotropic Em to sixanisotropic Eml, but does not change any of the fundamental principles and interactions con-sidered above.

Extensive listings of damping properties of real materials may be found in References [42]to [44].

Designer Materials with Optimum Damping and Other Properties

Although the viscoelastic loss factor defined by4

LFt(t) = EI(t)ER(t) or LF ω(ω) = EI(ω)

ER(ω)(32)

is a popular material property descriptor [43], it is not a representative measure of the vis-coelastic damping energy. It must also be noted that these two definitions are not equivalentas their FTs are not equal, i.e.

LF t(t) =∞Z

−∞

EI(t)ER(t) exp (−ı ω t) dt 6= LF ω(ω) =

∞Z

−∞

EI(t) exp (−ı ω t) dt

∞Z

−∞

ER(t) exp (−ı ω t) dt

(33)

Since in this Section one seeks comparisons of issues and topics related among other pa-rameters to dissipative energy, one can select, within reason, some convenient form related to

4See also Eqs. (16) to (18).


dissipation as a study candidate, although by no means the only possible representation. Forisotropic materials one such form is

gDE(ω) =Z

V

Skl(x, ω) Ekl(x, ω) dV| z

deviatoric contribution

+Z

V

σ(x, ω) ε(x, ω) dV| z

volumetric contribution

(34)

which yieldsgDE(ω) = gDE

R

(ω) + ıgDEI

(ω) =

Z

V

SR

kl(x, ω)ER

kl(x, ω)− SI

kl(x, ω)EI

kl(x, ω) + σR(x, ω) εR(x, ω)− σI(x, ω) εI(x, ω)

dV

| z measure of storage energy

+ ı

Z

V

SR

kl(x, ω)EI

kl(x, ω) + SI

kl(x, ω)ER

kl(x, ω) + σR(x, ω) εI(x, ω) + σ

I(x, ω) εR(x, ω)dV

| z measure of dissipative energy

(35)where V is the entire structural volume and dV = dx1dx2dx3. The dissipative measure can alsobe evaluated for a specific time duration −∞ ≤ t ≤ tA from

gDEI(tA) = <

8<:

tAZ

−∞

∞Z

−∞

gDEI

(ω) exp (ı ω t) dω dt

9=; (36)

It has been proven in Refs. [50] and [51] that isotropic as as well as anisotropic viscoelasticmaterial damping is most sensitive to the slope in Region C and to the ratio E∞

E0≥ 0 of Fig. 5. It

should be understood that this ratio bears no affinity to LFt of Eq. (32). Furthermore, it canbe readily seen from Fig. 1 that if one desires to have substantial dissipation in the shorttime operational range then the material properties must be such that the ER and EI curvesshift to the left toward t = 0. These types of short time duration conditions prevail duringimpact loads. Other load histories with dissipation as a prescribed operating requirementwould require material properties where the EI(ω) curves peak in different frequency ω (time)locations/domains. Material properties related to ER(ω) and/or EI(ω) can also be formulatedto cover specific time intervals, relaxation modulus curve shapes, etc., (See Fig. 1).

Indeed, in [52] through an analytical formulation of inverse problems it was proven thatelastic and viscoelastic material properties can be designed/tailored to render desired perfor-mances according to prior specifications and constraints. For instance, these protocols can be


0.0

0.2

0.4

0.6

0.8

1.0

1.2

-10 -8 -6 -4 -2 0 2

MASTER RELAXATION MODULUS

NO

RM

AL

IZE

D

RE

LA

XA

TIO

N

MO

DU

LU

S

LOG (TIME)

A

B

C

DE

E0

E!

t0

tR

Fig. 5 Master relaxation curve [50]

used to analytically design optimum relaxation moduli that guarantee say maximum dissipationin a given time range or a maximum flutter speed or other constraints.

A brief outline of the protocol to be followed is shown in Fig. 6 with Sm where m =1, 2, · · · ,M or

S = S1, S2, · · · , SM (37)

is the numberM of unknown parameters Sm to be optimized. This inverse procedure consistsof the following:

1. Derive governing relations for the problem, which in generic form are5

L (x, t,u(x, t), En, τn) = L (x, t,u(x, t),S) = 0 (38)

where the u = u1, u2, · · · , uM are generalized displacements.

2. Formulate desired constraints on structural weight, cost, maximum stresses and/or de-formations, life/survival times, minimum flutter velocity or Mach number (bVF or cMF ),6

composite fiber configurations, maximum dissipative energy, etc. as

C`(x, t, En, τn, V ∗F ,F(x), · · · ) = 0 ` = 1, 2, · · · , L (39)5Eqs. (7) are typical such relations.6The constraint being specified as V∞ = Vf < bVF or M∞ =Mf < cMF .


where F(x) is a function describing composite fiber orientation, volume fractions, etc.

3. Eliminate the spatial dependence by applying Galerkin’s procedure or in the case ofenergies by integrating over the structural volume V .

4. Solve the governing relations (38) for the remaining temporal functions

bu(S, t) =bZ

a

u(x, t) um(x, t) dx with m = 1, 2, · · · ,M (40)

5. Eliminate the temporal dependence by least square fits or at prescribed life times

eu(S) = bu(S, tLF ) (41)

or at other specified times such as, for instance, where either the dissipative energy orEI(t) peak (Fig. 1), etc. Alternately, another specification could involve a time averagingprocess, such that

eu(S) = 1tLF

tLFZ

0

bu(t′) dt′ (42)

6. FormulateM simultaneous equations of theM unknown parameters through the appli-cation of Lagrangian multipliers λ` [59 – 60], such that

∂

∂Smeu(S) + λ`C(S) = 0 m = 1, 2, · · · ,M (43a)

or∂

∂Sm

ngDEI

(tA,S) + λ`C(S)o

= 0 m = 1, 2, · · · ,M (43b)

or any other proper expression(s) that one wishes to optimize. All the unknown materialproperty, geometry and constraint parameters are denoted by S = S1, S2, · · · , SM.

7. Solve the simultaneous algebraic equations for the Sm, which will establish the elastic E0

or viscoelastic material parameters En and τn and any others dictated by the specifiedconstraints and required (selected) to perform specific a priori prescribed tasks. See Fig. 6.

8. Experience has shown that when flutter is involved as one of the constraints, an analyticalsolution of the flutter determinant in terms of the unknown parameters presents severechallenges. Consequently, from a purely operational point of view, it is best to first solvefor all desired parameters without involving any aeroelastic constraints. Subsequent to


University of Illinois at Urbana-Champaign (UIUC)

AA

E

AE UIUC

EOT_RT_Sub_Template.ppt | 9 Copyright © 2009 Boeing. All rights reserved.

13

Governing Eqs.Ln(S, x, t) = 0

1

ConstraintsCn(S, x, t) = 0

1

Garlerkin eliminates x

LSQ, time averaging, evaluation at design

times eliminates t

Solve for parameters Sm (algebraic eqs.)

Solve forum(t)

1

∂

∂Smun(S) + λn C(S∗) = 0

1Fig. 6 Designer material ow chart [52]

the conclusion of these steps, the flutter velocity can be calculated and compared withits desired (prescribed) value, i.e. VF > Vmax where Vmax is the prescribed largest designflight velocity. (See the Subsection on the “Illustrative Example”.)

An Illustrative Example

A simple example has been constructed with but two parameters in order to graphicallyoffer an illustration of their contributions. It consists of a single elastic I–beam wing sparat room temperature, such as those found UAVs or MAVs (Fig. 7). The height of the sparis fixed and the parameters to be optimized are the flange width (b′) and the thickness (h).Proper elastic moduli (E0) are determined and a minimum structural weight configuration issought. Once these parameters are known the flutter speed is calculated and compared to themaximum prescribed flight velocity of 265 m/s. Fig. 8 indicates that these design conditionsare met for the usual range of elastic G0 and ν0. As the PR values enter the negative range andthe material becomes auxetic with G0 → ∞, indicating that the spar is becoming torsionallyrigid. Consequently, the flutter velocity tends to infinity. The curves marked composite and Alare for a density of ρ = 1575 kg/m3 and 2700 kg/m3 respectively, which closely resemble theactual density values for these two materials.

It is to be noted that the calculated flutter speed for the 2-D optimum parameters isexceedingly high albeit not accurate since incompressible aerodynamics were used to conformwith the vehicle flight regime. Consequently, it leads to a highly over designed flutter condition.However, this should not be unexpected since (a) the C6 of Eq. (51) requires the flight velocity


only to be less than the flutter speed and (b) the flutter velocity VF is not one of the twoselected parameters.

The analysis and design protocol for a spar relative to sizing and material properties isdemonstrated as follows. A single spar UAV wing, Fig. 7, (similar one to an AAI RQ-7 Shadow[58]) is considered by assuming the spar to be a uniform I beam made out of a single elasticmaterial (to be then defined by E0 and G0). A spar box beam could have been readily alsoconsidered for optimization.

Fig. 7 The Shadow Wing Cross Section [58]

0

2000

4000

6000

8000

10000

0

500

1000

1500

2000

2500

3000

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

VF (m/s) COMPOSITE

VF (m/s) Al

G0 (GPa) COMPOSITE

G0 (GPa) Al

FLU

TTER

VEL

OC

ITY

(m/s

) SHEA

R M

OD

ULU

S (GPa)

ELASTIC POISSON'S RATIO

Fig. 8 Shear Modulus and Flutter Speed vs. Elastic PR

Flutter conditions arising from a two-dimensional flexure-torsion mode are investigated.Some properties are shown in Table 4 and the composite has a 70% carbon fiber fraction. Ofcourse in a realistic multi-D, i.e. M-D, environment, composite parameters such as number ofplies, plies angles, stacking sequence, fiber and matrix moduli, etc. are optimization candidates.


Table 4 Elastic Properties

Material Young’s DensityModulus (GPa) kg/m3

Composite 280.9 1575.Aluminum 70. 2700.

Mass inertia and section inertia properties are derived by assuming the entire wing to berepresented only by a single spar with the following requirements:

• The aerodynamic load lift distribution on the wing is modeled as parabolic

q(x2) = q0L2 − x2

2

with q0(N/m3) > 0 (44)

where L is length of the spar (half wing span).

• The unknown parameters to be optimized are

S = b′, h, E0, G0 (45)

and c is kept as a constant.

• The task is to minimize the structural weight of the spar given by

W = 2 ρLh (b′ + c− h) ≤ Wmax (46)

Actually, it is more convenient and efficient to deal with the ratio Wρ

= V when the actualoptimizing process is being carried out. This represents the volume of the spar and itsoptimization is implied by optimizing the geometric parameters b′ and h, which are twoof its primary contributors.

• The constraints are

Geometry:

hmax ≤b′

kor C1(S) = k hmax − b′ ≤ 0 with k ∈

kmin,

b′

hmin

(47a)

b′ ≤ d c or C2(S) = b′ − d c ≤ 0 with d ∈ ]0, 1] (47b)h ≥ hmin or C3(S) = hmin − h ≤ 0 (47c)


where d is a prescribed multiplier and and hmin is the minimum thickness.

Maximum deflection and stress:

w(L) = B

E0 I(b′, c, h) ≤ wmax or C4(S) = B

E0 wmax− I(b′, c, h) ≤ 0 (48)

with

B = 19q0L6

360 and I(b′, c, h) = b′ h3 + 3 b′ h(2 − h)2 + 4h (c− h)3)6 (49)

σ22(0) = M(0)cI(b′, c, h) ≤ σmax or C5(S) = M(0)c

σmax− I(b′, c, h) ≤ 0 (50)

with M(0) = q0 L4/4. Since the constraint B

E0 wmax>M(0)cσmax

was prescribed, C4(S) issatisfied, which implies that the C5(S) condition is also met.

Minimum flutter speed:

VF > Vmax → VF (S) = (SF )Vmax or C6(S) = VF − (SF )Vmax = 0 (51)

with a safety factor (SF ) = 1.2 used in this example.

If L(S, λ) = W (S) +4X

i=1

λi Ci(S), then the stationary points of the structural weight W

satisfy: 8>>>>><>>>>>:

∇(L) = ∇W (S) +4X

i=1

λi∇Ci(S) = 0

λi Ci(S) = 0, ∀ i ∈ 1, · · · , 4

λi > 0, ∀ i ∈ 1, · · · , 4

(52)

It should be noted that when taking in account only C1, · · · , C5, G0 is not included in theensemble S, and then S =b′, h, E0.

Solutions of (52) are

S1 = (k hmin, hmin, E10) and S2 = (d c, d c

k, E2

0) (53)

where Ei0 = B

wmaxI(b′i, ci, h) . The optimum condition is reached at S1.

In this illustrative case, the results can be displayed and physically interpreted through agraphical approach as shown in Figs. 9 and 10 for room temperature Al.7 All the constraints

7Similar graphs can be produced for other materials, such as the composite example.


C1, . . . , C5 of Eqs. (47), (48) and (50) are satisfied within the area defined by the “triangle” inFig. 9 . The curve marked spar moment of inertia is a contour line of 12 I values and is theone which satisfies I = B

wmaxEopt, where Eopt is the value of E required for the contour line

to reach the point (b′, h) = (k hmin, hmin), everywhere above this moment of inertia curve therestraint C4 is met. The minimum weight configuration for a material density of 2700 kg/m3

corresponding to Al alloys is marked by the symbol X on Fig. 9 .

0

0.02

0.04

0.06

0.08

0.1

0 0.002 0.004 0.006 0.008 0.01

MINIMUM FLANGE WIDTH b'

MAXIMUM FLANGE WIDTH b' ~ c

SPAR MOMENT OF INERTIA

FLA

NG

E W

IDTH

b'

(m)

THICKNESS h (m)

X

Fig. 9 The Detailed Graphical Approach for Al( = 2700 kg/m3, see Fig. 7 for dimensions)

In Fig. 10 multiple examples of W/ρ are presented. Note that with the dimensionalconstraint represented by the horizontal straight line indicating b′ = d c, there is an upper limiton the weight value marked as (W/ρ)max.

The flutter determinant is now needed to solve for the flutter velocity and frequency. Since

the included natural torsional frequency ωα = π

2L

rG0J

Idepends on the shear modulus G0,

the latter must be defined.8 A correct choice of that parameter can then be made so that theflutter velocity satisfies the prescribed VF = (SF )Vmax. Fig. 8 details the flutter speeds and theshear moduli for admissible elastic Poisson’s ratio values associated with Al alloys and carbonfiber composites, which together with the E0 Young’s moduli define the G0 shear ones for eachmaterial.

8where J is the polar moment of inertia and I the mass moment of inertia of the spar about the elastic axisper unit length


0

0.05

0.1

0.15

0.2

0.25

0 0.002 0.004 0.006 0.008 0.01

MINIMUM FLANGE WIDTH b'MAXIMUM FLANGE WIDTH b' ~ cSPAR MOMENT OF INERTIA(W/ρ)

min = 7.50E-04

W/ρ = 8.92E-04W/ρ = 1.05E-03(W/ρ)

max = 1.24E-03

FLA

NG

E W

IDTH

b' (

m)

THICKNESS h (m)

Fig. 10 The Graphical Approach for Al( = 2700 kg/m3, see Fig. 7)

DISCUSSION

Damping Considerations

In Fig. 2 depicts a representation of energy exchanges in viscoelastic media, betweenpotential and dissipative energies. The two extreme viscoelastic positions are the elastic oneon the left and the viscous flow one on the right. Elastic action is all stored, i.e. potential,energy while pure viscous damping entails only dissipation. Viscoelastic action when no energyis supplied to the system after the initial input results in decreases in potential energy andincreases in dissipative energy.

Fig. 4 depicts a mechanical model, called the generalized Kelvin model or GKM, whichis used to simulate viscoelastic behavior in real materials. None of the components have massand they represent only material responses. The initial spring denotes an instantaneous elasticresponse to input forces or stress. The last damper or dashpot is a pure viscous damping model.All components contribute to viscoelastic responses.

The simplest viscous action is that of a damper or dashpot where the force or stress


is proportional to the velocity and phase relations between stresses and strains vary withtime. Whereas structural damping depends on displacements and exhibits responses wherestresses and strains are ninety degrees out of phase. This is designated by the second term inEqs. (10), where ı =

√−1. (See Fig. 11 ).

0

2

4

6

8

10

0 2 4 6 8 10

RESPONSE TO SIMPLE HARMONIC MOTION

COULOMB FRICTION

VISCOUS DAMPING

σ / E 0

ε

FREQUENCY

Fig. 11 Response to SHM

Simple harmonic motion is used to demonstrate these two dissipative actions and theirresponses. In the simple viscous case the relations between stresses and strains are similar to theprevious representation of Coulomb friction or structural damping. However, note the extremelyimportant difference between the two relations: viscous damping is frequency dependent whilestructural damping is frequency independent. The frequency graph of theses two relationsdemonstrates the striking differences. The symbol “g” is the structural damping coefficient,not to be confused with the gravitational constant. It is generally very small. A value of 0.06indicates an unacceptable very loose “sloppy” joint. Typical values for metal joints are in therange from 0 to 0.05.

Constitutive relations including structural damping are displayed in Eqs. (19) to (21).In both elastic and viscoelastic materials, the presence of structural damping is denoted bythe addition of the complex term ıg and affects the dissipative response of the system. Theviscoelastic relaxation moduli are generally represented by an exponential time series called aProny series [14].

A good understanding of viscoelasticity and structural damping can be gained through the


use of Fourier transforms. The Fourier transform of the relaxation modulus indicates two fre-quency dependent real parts: the storage modulus and the loss modulus. Note that the presenceof structural damping only affects the loss modulus, which is a measure of the damping capacityof the system. Now an extra term appears involving the structural damping coefficient g andthe storage modulus ER. Table 2 displays a summary of the real and imaginary components ofthe complex modulus, which are the Fourier transforms of the relaxation modulus.

An alternate representation of the complex modulus can be formulated in terms of a loca-tion vector #

R and the complex exponential function involving a phase angle ∆ of Eqs. (16) and(17). Note that in the elastic material when the damping coefficient is zero, the vector #

R is theelastic modulus and the phase angle is zero, since no damping is present in this conservativesystem. For elastic materials with structural damping and for viscoelastic materials, this vectorand the phase angle each involve both the storage and loss moduli as well as the structuraldamping coefficient. The viscoelastic quantities are, of course, frequency dependent while theelastic ones with or without structural damping are constants. Both quantities heavily influencethe dynamic motion of a wing and in particular contribute to the presence or absence of flutter.

Table 5 represents the results of a set of elastic and viscoelastic flutter calculations, coveredin detail in the main tutorial on this subject. It is of interest to note that in this example, thepresence of structural damping and/or viscoelastic damping increase the flutter Mach numberover elastic values. Different dimensions, mass distributions and viscoelastic properties canproduce the opposite phenomenon, such that viscoelastic flutter Mach numbers are lower thantheir elastic counter parts. In general, flutter frequencies for flight vehicles are invariably bellow100 Hz and do not present any fatigue issues.

Table 5Some Flutter Results for a 2024 AL Wing

Elastic Viscoelasticg ω Mf ω Mf

0 20.0001 1.3037 26.8586 1.5167

.005 19.9737 1.3056 26.8070 1.5081

.01 19.9465 1.3074 26.7550 1.4994

.05 19.7150 1.3239 29.0102 1.9887


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 100 200 300 400 500

EFFECTIVE LOSS MODULUS / STORAGE MODULUS (2024 AL)

g = 0

g = .05

RA

TIO

L

OS

S T

O S

TO

RA

GE

M

OD

UL

US

TEMPERATURE 0

F

Fig. 12 Maximum loss modulus / storage modulus ratiofor 2024 AL [39]

Viscoelastic materials exhibit strong temperature dependencies. A typical plot of themaximum ratio of the loss to the storage modulus is shown in Fig. 12 . As the temperaturerises, the rate of energy dissipation increases. The coefficient of viscosity roughly increases byone order of magnitude for every 20.

-5

0

5

10

15

20

0 100 200 300 400 500

Rve

CHANGES

Δ DUE TO TEMPERTATURE AT g = 0

Δ DUE TO TEMPERTATURE AT g = .05

Δ AT g = 0 & .05

RE

LA

TIV

E C

HA

NG

ES

TEMPERATURE 0

F

Fig. 13 Rve changes


-1

0

1

2

3

4

5

0 100 200 300 400 500

DAMPING PARAMETERS FOR 2024 AL

Rmax

g=0

Rmin

g=0

Λmax

g=0

Λmin

g=0

Rmax

g=.05

Rmin

g=.05

Λmax

g=.05

Λmax

g=.05

DA

MP

ING

P

AR

AM

ET

ER

S

TEMPERATURE 0

F

Fig. 14 Damping parameters for 2024 AL

Figs. 13 and 14 are for the examples previously displayed in Table 3. They demonstrate theinfluence of temperature and structural damping on the viscoelastic damping parameters, suchas the location vector and phase angle. Minimum and maximum values are shown to establishupper and lower limits. It must be remembered that these parameters are all frequency andtime dependent, thus leading to the displayed envelopes of values.

Fig. 15 depicts the influence of the structural damping coefficient g on flutter velocitiesand frequencies. For the selected wing parameters changing values of g are considerably moreinfluential on viscoelastic flutter than in the equivalent elastic case.

Three important conclusions emerge from the present discussion dealing with dissipa-tion energies, phase relations and the distinctions between structural and viscoelastic damping.Since flutter conditions are represented by highly a nonlinear functions of Vf and ωf , an analyti-cal comparison of viscoelastic and structural damping is not feasible. However, a re-examinationof the bending-torsion supersonic flutter problem for a Timoshenko beam previously analyzedin [26] based on the addition of structural damping effects as exemplified by Eqs. (13 ), leadsto the results displayed in Table 5.

These results are typical for metal wings in supersonic flow and fully account for thematerial property dependence on temperature as the flight and flutter Mach numbers change.It is to be noted that as the structural damping g increases the viscoelastic flutter Machnumber Mf may increase (stabilizing) or decrease (destabilizing). For an elastic aluminumwing with the same mass distribution, geometry and aerodynamics, the corresponding flutter


16

18

20

22

24

26

28

30

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.01 0.02 0.03 0.04 0.05

ELASTIC AND VISCOELASTIC WING FLUTTER

ELASTIC FREQUENCY

VISCOELASTIC FREQUENCY

ELASTIC MACH NO.

VISCOELASTIC MACH NO.

FL

UT

TE

R

FR

EQ

UE

NC

Y

(Hz) F

LU

TT

ER

MA

CH

NU

MB

ER

STRUCTURAL DAMPING COEFFICIENT

Fig. 15 Elastic and viscoelastic wing utter

Mach numbers are smaller than the viscoelastic ones and an increase in structural dampingfor the elastic wing is destabilizing. Even though the viscoelastic action for 2024 aluminum atelevated temperatures is far from being as pronounced as it is in high polymers and composites,the viscoelastic flutter Mach numbers are significantly different from the corresponding elasticones (Table 5). This is due to the highly nonlinear dependence on the flutter frequency and theattendant phase relations which shift in a complicated fashion. In References [25] and [26], ithas been previously noted that viscoelastic flutter Mach numbers may be higher or lower thancorresponding elastic ones for wings of identical geometry, mass distributions and aerodynamicproperties. Dugundji [36] has noted similar behavior due to structural damping in elastic panelflutter.

Designer Materials

The general designer material theory previously developed in [52] is reviewed briefly andapplied among other parameters and conditions to optimizing (maximum or minim values)to dissipative energy. The application of this concept is of particular importance if optimumdamping is sought in combination with other conditions in structures. The importance of theprotocol is its ability to produce simultaneous algebraic relations.

A series of limited 2-D illustrative example serves to detail graphically the evolutionarycontributions of the two selected parameters in the optimization process and is but a small


subset of the realM parameter problem. In a realistic designer material configuration with allthe many parameters and constraints included, the general optimum solution would coalesceto a single point in anM-dimensional space, similar to what is seen in the two parameter plotof Fig. 9 .

A discussion of the physical significance of the optimization protocol is included in thesub-section above entitled “Designer Materials with Optimum Damping and Other Properties”for the sake of continuity and completeness. It need not be repeated here and the reader isreferred to that sub-section.

Future Work

The designer material concept and protocol can be readily applied in order to establishoptimum moduli, sizing, etc. subject to weight, cost, aeroelastic, failure, maximum deflec-tion/stress, etc., constraints. The cartoon of Fig. 16 represents the hypothetical conditionswhere the various aerospace interest groups are left to their own uncoordinated devices withoutregard to the fact that a flight vehicle is closed loop system [54]. More system areas beyondmaterial properties will be added in the future in the quest to optimize the entire vehicle andthus eventually achieve the desired system of system concept. For instance, the designer aero-dynamics concept has been successfully carried out in [55 – 57], to mention a few instances.

DREAM AIRPLANES? (or to each their own!) Not a system concept!

From C. W. Miller, Vega Aircraft Corp., (Lockheed) 1940.

AERODYNAMICS GROUP

FUSELAGE GROUP

!"#$%"&'(()%"*+(

,-.%/*&0!'(()%"*+(

'$%1''(()%"*+(

'1%20!1(()%"*+(

30#)(()%"*+(

14+1##/)1(()%"*+(

15*0+41#$(()%"*+(

1&1!$%0!/&(()%"*+(

+"31%((+&/#$(()%"*+(

BALSA

310),$(()%"*+(

&"6$(()%"*+(

+%".*!$0"#(()%"*+(

/%4/41#$(()%"*+(

Fig. 16 The Worst Case Scenarios [61]

The number of algebraic relationsM necessary to establish optimum designer material


Table 6 Computational Requirement Estimates forAn Integrated System Optimum Designer Parameter Approach for Entire Vehicles:aerodynamics, control, materials, propulsion, ight performance, structures, etc.

Vehicle UAV / MAV Small General Large TransportAviation Plane (Boeing Dreamliner)

No. of parameters for 90 700 300,000structure & materials only

No. of parameters for 500 70,000 800,000,000entire vehicle (stability,control, aerodynamics,

propulsion, structure, etc.)

Computational hardware Laptop HPC mainframe Blue Waters™(sustained-petascalecomputing system)

on line in 2012

property parameters Sm stemming from Eqs. (43a) or (43b) can be in the millions, if not billions,when considering an entire large transport airplane [53]. (See Table 2.) However, the advent ofthe University of Illinois at Urbana-Champaign (UIUC) National Center for SupercomputingApplications’s (NCSA) Blue Waters® sustained petascale parallel computer system in 2012will usher in an era when such extremely large scale computations become a common every dayreality [62 – 64].

Of course, no attempt is made here to suggest manufacturing techniques that can producematerial properties on command. That is a separate endeavor the beginnings of which areanalyzed and presented in [65].

CONCLUSIONS

It is shown that for any general motion there is no relation between elastic structuraldamping (Coulomb friction) and Newtonian viscous damping except at the natural frequenciesof the equivalent undamped system. The viscoelastic complex and real time relaxation moduliare re-derived to include structural damping.

The results indicate that in the presence of structural damping the real part is unaffected


but the imaginary part includes effects due to the structural damping coefficient and the usualreal and imaginary parts of the complex modulus. The illustrative examples for supersonicflutter of an aluminum wing indicate that an increase in the structural damping coefficientmay increase or decrease the viscoelastic flutter speeds and the elastic flutter speeds are notnecessarily higher than the corresponding viscoelastic ones. In other words neither structuralnor viscoelastic damping necessarily produce stabilizing effects, unless either or both are appliedin appropriate proportions to actual mass, viscoelastic stiffness and lift distributions.

The execution of the designer material concepts will produce, subject to prescribed con-straints, optimum material properties, geometries and sizing on demand, which will improveoverall performances and lighten structural weights.

The integral relations are the Green’s functions corresponding to the differential relations(54) and the two forms are conceptually equivalent. To characterize real materials, values ofRklmn must be in the range of 20 to 30. Such high order DEs when accompanied by spatialfinite element formulations make the differential forms computationally uninviting, practicallyunmanageable and of questionable accuracy. Additionally, extreme care must be exercisedto state proper initial conditions for the additional temporal derivatives of order higher thantwo. Consequently, the integral forms are the constitutive relation characterization protocolsof choice for real viscoelastic materials.

The limited 2-D designer material example graphically illustrates their contribution to theoptimum weight configuration with flutter speed as a prescribed constraint. The example canbe readily generalized to M number of parameters but then the visual effects, graphical andphysical explanations are lost in theM-D space whenM > 2.

APPENDIX A

THE DIFFERENTIAL FORM OF THE CONSTITUTIVE RELATIONS

The integral isotropic and anisotropic relations of Eqs. (9) and (31) may be readily castinto their equivalent differential mode to yield [15, 19, 41] to read

Pkl σkl(x, t) = Qklmn εmn(x, t) (54)

with differential operators

Pkl(x, t) =RklX

j=0

aklj(x, t)∂j

∂tjand Qklmn(x, t) =

RklmnX

j=0

bklmnj(x, t)∂j

∂tj(55)

withbklmn0 =

1 + ı gklmn

Eklmn0(x, t) (56)


The underlined indices indicate no summation, while repeated ones do. Note that the integralforms of the constitutive relations (1) are the Green’s functions of the differential form (54).

While in principle the integral-differential (IE) and differential (DE) forms of the viscoelas-tic constitutive relations are interchangeable, real material require 25 to 30 Prony series terms,which correspond to time derivatives with highest order of 25 to 30. Thus making the DE rep-resentations cumbersome and computationally unattractive. Furthermore, extreme care mustbe exercised to properly state the numerous additional initial conditions corresponding to theincreased order of time derivatives in the governing relations.

ACKNOWLEDGEMENT

Support by grants from the Private Sector Program Division (PSP) of the National Centerfor Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign(UIUC) is gratefully acknowledged.

REFERENCES

1. Scanlan, Robert H. and Robert Rosenbaum (1951) Introduction to the Theory of AircraftVibration and Flutter. The Macmillan Co., New York.

2. Bisplinghoff, Raymond L., Holt Ashley and Robert L. Halfman (1955) Aeroelasticity.Addison-Wesley Publishing Company, Cambridge, MA. (1980) Dover Publications, NewYork.

3. Fung, Y. C. (1955) An Introduction to the Theory of Aeroelasticity. John Wiley & Sons,New York.

4. Bisplinghoff, Raymond L. and Holt Ashley (1962) Principles of Aeroelasticity. John Wiley& Sons, New York.

5. Dowell, Earl H. (1975) Aeroelasticity of Plates and Shells. Noordhoff International Pub-lishing, Leyden.

6. Dowell, Earl H. and Marat Ilganov (1988) Studies in Nonlinear Aeroelasticity. Springer,New York.

7. Hodges, Dewey H. and G. Alvin Pierce (2002) Introduction to Structural Dynamics andAeroelasticity. Cambridge University Press, New York.


8. Dowell, Earl H. and Deman Tang (2003) Dynamics of Very High Dimensional Systems.World Scientific, River Edge, NJ.

9. Dowell, Earl H., Robert Clark, David Cox, Howard C. Curtis Jr., John W. Edwards,Kenneth C. Hall, David A. Peters, Robert H. Scanlan, Emil Simiu, Fernado Sisto andThomas W. Starganac (2004) A Modern Course in Aeroelasticity, 4th ed. Kluwer Aca-demic Publishers, Boston.

10. Donaldson, Bruce K. (2006) Introduction to Structural Dynamics. Cambridge UniversityPress, New York.

11. Wright, Jan R. and Jonathan E. Cooper (2007) Introduction to Aircraft Aeroelasticityand Loads. John Wiley, Hoboken, NJ.

12. Coulomb, Charles-Austin de (1821) Théorie des machines simples, en ayant égard aufrottement de leurs parties et à la roideur des cordages. Bachelier, Paris.

13. Guran, A., Pfeiffer, F. and Popp, K. (Eds.) (1996) Dynamics with Friction. WorldScientific, Singapore.

14. Prony, Gaspard C. F. M. R. Baron de (1795) “Essai experimental et analytique,” Journalde l’École Polytechnique de Paris, 1:24–76.

15. Hilton, Harry H. (1964) “An introduction to viscoelastic analysis,” Engineering Designfor Plastics. 199–276, Reinhold Publ. Corp., New York.

16. Brinson, Hal F. and L. Catherine Brinson (2008) Polymer Engineering Science and Vis-coelasticity: An Introduction. Springer, New York.

17. Lakes, Roderic S. (1999) Viscoelastic Solids. CRC Press, Boca Raton.

18. Lakes, Roderic S. (2009) Viscoelastic Materials. Cambridge University Press.

19. Christensen, Richard M. (1982) Theory of Viscoelasticity - An Introduction, 2nd ed. Aca-demic Press, New York.

20. Soroka, W. W. (1949) “Note on the relation between viscous and structural dampingcoefficients,” Journal of the Aeronautical Sciences 16:409–410, 448.

21. Pinster, W. (1949) “Structural damping,” Journal of the Aeronautical Sciences 16:699.


22. Dahl, P. R. (1976) “Solid friction damping of mechanical vibrations,” AIAA Journal14:675–1682.

23. Saravanos, D. A. and Christos C. Chamis (1990) “Mechanics of damping for fiber com-posite laminates including hygrothermal effects,” AIAA Journal 28:1813–1819.

24. Merrett, Craig G. and Harry H. Hilton (2008) “Generalized linear aero-servo-viscoelastici-ty: Theory and applications,” Proceedings 49 th AIAA/ASME/ASCE/AHS Structures,Structural Dynamics and Materials Conference, AIAA Paper AIAA-2008-1997.

25. Hilton, Harry H. (1957) “Pitching instability of rigid lifting surfaces on viscoelastic sup-ports in subsonic and supersonic potential flow,” Proceedings of Third Midwestern Con-ference on Solid Mechanics 1–19. Edwards Brothers, Ann Arbor, MI.

26. Hilton, Harry H. and Vail, Curtis F. (1993) “Bending-torsion flutter of linear viscoelasticwings in two dimensional flow,” Proceedings AIAA/ASME/ASCE/AHS/ASC 34th Struc-tures, Structural Dynamics and Materials Conference, AIAA Paper 93-1475, 3:1461–1481.

27. Hilton, Harry H. (2009) “Viscoelastic Timoshenko beam theory” Mechanics of Time-Dependent Materials 13:1–10.

28. Hilton, Harry H. and Sung Yi (1998) “The significance of anisotropic viscoelastic Pois-son ratio stress and time dependencies,” International Journal of Solids and Structures35:3081–3095.

29. Tschoegl, Nicholas W. (1997) “Time dependence in material properties: an overview,”Mechanics of Time–Dependent Materials 1:3–31.

30. Hilton, Harry H. (2001) “Implications and constraints of time independent Poisson ratiosin linear isotropic and anisotropic viscoelasticity,” Journal of Elasticity 63:221–251.

31. Tschoegl, Nicholas W., Wolfgang G. Knauss and Igor Emri (2002) “Poisson’s ratio in linearviscoelasticity – a critical review,” Mechanics of Time-Dependent Materials 6:3–51.

32. Lakes, Roderic S. and Alan Wineman (2006) “On Poisson’s ratio in linearly viscoelasticsolids,” Journal of Elasticity 85:45–63.

33. Khan, Kamran A. and Harry H. Hilton (2010) “On inconstant Poisson’s ratios in non-homogeneous elastic media,” Journal of Thermal Stresses 33:1–8.


34. Hilton, Harry H. (2009) “The illusive and fickle viscoelastic Poisson’s ratio and its relationto the elastic–viscoelastic correspondence principle,” Journal of Mechanics of Materialsand Structures 4:1341–1364.

35. Hilton, Harry H. (2011) “Clarifications of certain ambiguities and failings of Poisson’sratios in linear viscoelasticity,” Journal of Elasticity 104:303–318. DOI 10.1007/s10659-010-9296-z

36. Dugundji, John (1966) “Theoretical considerations of panel flutter at high supersonicMach numbers,” AIAA Journal 4:1257–1266.

37. Mathauser, Eldon E. (1956) “Compressive stress-strain properties of 2024-T3 aluminumalloy sheet at elevated temperatures,” NACA TN 3853.

38. Heldenfelds, Richard R. and Eldon E.Mathauser (1956) “A summary of NACA researchon the strength and creep of aircraft structures at elevated temperatures,” NACA RML56D06.

39. Vosteen, Louis F. (1958) “Effect of temperature on dynamic modulus of elasticity onstructural alloys,” NACA TN 4348.

40. Granick, Neil and Jesse E. Stern (1963) “Material damping of aluminum by a resonant-dwell technique,” NASA TN D-2893.

41. Hilton, Harry H. and Stanley B. Dong (1964) “An analysis for anisotropic, nonhomoge-neous, linear viscoelasticity including thermal stresses,” Development in Mechanics 58–73.Pergamon Press, New York.

42. Lazan, Benjamin J. (1968) Damping of Materials and Members in Structural Mechanics.Pergamon Press, New York.

43. Nashif, Ahid D., David I. G. Jones and John P. Henderson (1985) Vibration Damping.John Wiley & Sons, New York.

44. Jones, David I. G. (2001) Handbook of Viscoelastic Vibration Damping. John Wiley &Sons, New York.

45. Soong, T. T. and Gary F. Dargush (1997) Passive Energy Dissipation Systems in Struc-tural Engineering. John Wiley & Sons, New York.

46. Ungar, Eric E. (1988) “Damping of panels,” Noise and Vibration Control (L. L. Beranek,ed.) (revised edition) 434–475. Institute of Noise Control Engineering, Washington.


47. Ungar, Eric E. (1992) “Structural damping,” Noise and Vibration Control Engineering(L. L. Beranek and I. L. Ver, eds.) 451–481. John Wiley & Sons, New York.

48. Cremer, Lothar and Manfred Heckl (1987) Structure-Borne Sound: Structural Vibrationsand Sound Radiation at Audio Frequencies 169–239. Springer Verlag, Berlin.

49. Bert, Charles W. (1973) “Material damping: an introductory review of mathematicalmodels, measures and experimental techniques,” Journal of Sound and Vibrations 29:129–153.

50. Yi, Sung and Harry H. Hilton (1992) “Analytical formulation of optimum material proper-ties for viscoelastic damping,” Journal of Smart Materials and Structures 1:113–122.

51. Beldica, Cristina E. and Harry H. Hilton (1999) “Analytical simulations of optimumanisotropic linear viscoelastic damping properties,” Journal of Reinforced Plastics andComposites 18:1658–1676.

52. Hilton, Harry H., Daniel H. Lee and Abdul Rahman A. El Fouly (2008) “General anal-ysis of viscoelastic designer functionally graded auxetic materials engineered/tailored forspecific task performances,” Mechanics of Time-Dependent Material 12:151–178.

53. Hilton, Harry H. (2009) “A novel approach to structural analysis: Designer/engineeredviscoelastic materials vs. ’off the shelf’ property selections,” Journal of Vacuum Technol-ogy & Coating 10:23–29.

54. Hilton, Harry H. (2012) “A rational integrated approach to designer systems of sys-tems – Tailored aerodynamics, aeroelasticity, stability and control, geometry, materials,structures, propulsion, performance, sizing, weight, cost,” Proceedings AIAA ComplexAerospace System Exchange Event (CASE), Pasadena, CA.

55. Liebeck, Robert H. (1973) “A class of airfoils designed for high lift in incompressible flow,”Journal of Aircraft 10:610–617.

56. Cagle, Christopher M. and Robin W. Schlecht (2007) “Composite elastic skins for shapechanging structures,” NASA Tech Briefs LAR–16599–1.http://www.techbriefs.com/content/view/1113/34/

57. Brinkmeyer, A. W., M. Santer, A. Pirrera and Paul M. Weaver (2012) “Morphing com-posite panel with pseudo-bistable viscoelastic behavior,” SEM XII International Congress& Exposition on Experimental and Applied Mechanics, SEM Paper 404.


58. Ananymous (2011)http://en.wikipedia.org/wiki/AAI_RQ-7_Shadow#Specifications_.28200_Family.29

59. Lagrange, Joseph L. (1788) Mécanique analytique. Gauthier-Villars et fils, Paris.

60. Lagrange, Joseph L. (1762) “Essai d’une nouvelle methode pour déterminer les maxima etles minima des formules integrales indefinies,”Mélanges de philosophie et de mathématiquede la Société royale de Turin.

61. Bruhn, E. F. (1943) Analysis and Design of Airplane Structures. Tri-State Offset Co.,Cincinnati.

62. Anonymous (2010) “About the Blue Waters project,” www.ncsa.illinois.edu/BlueWaters/

63. Anonymous (2010a) “Blue Waters News Search,”www.ncsa.illinois.edu/BlueWaters/news_search.php

64. Ananymous (2011) “Parallel@Illinois,”http://www.parallel.illinois.edu/blue-waters-spc.html

65. Van Krevelen, Dirk W. (1990) Properties of Polymers – Their Correlation with ChemicalStructure; Their Numerical Estimation and Prediction from Additive Group Contribu-tions, 3rd Ed., Elsevier, Amsterdam.


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Viscoelastic and Structural Damping Analysis with …highorder.berkeley.edu/proceedings/aiaa-annual-2012/paper1237.pdf · the help of numerical examples. The material considered is