Keywords:

Panel utter

change in utter characteristics as a function of the uid to structure mass ratio and the

structural aspect ratio. The paper also presents an exploration of the non-monotonic

ileveren thatl veloc

) andnabe

f thedynamics of this system. The initial models looked at the limiting cases where either the span or the length of the elastic

equation with the appropriate boundary conditions (Guo and Padoussis, 2000; Huang, 1995; Kornecki et al., 1976;

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Journal of Fluids and Structures

Journal of Fluids and Structures 34 (2012) 68830889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.juidstructs.2012.06.009n Corresponding author. Tel.: 1 6507933366; fax: 1 9196608963.E-mail address: scg8@duke.edu (S. Chad Gibbs).member is assumed to be innite. For the rst case, the problem approaches a two-dimensional limit. In the two-dimensional limit the potential ow equations can be solved to determine the aerodynamic forces using the continuous(2011) have explored the potential of using utter for propulsion. Furthermore, Balint and Lucey (2005), Huang (1995Howell et al. (2009) have shown that cantilevered plate utter in the human soft palate can explain snoring and Wataet al. (2002b) has explored this type of utter in the printing industry.

Many structural and aerodynamic models have been developed or applied to improve the understanding oSince the experimental observations of the apping ag by Taneda (1968), many scholars have explored the stability ofthis system experimentally and theoretically. Although extensively explored in the literature, a full understanding of thedynamics of this relatively simple uidstructure interaction remains elusive. In addition to the problems inherentphysical signicance, Doare and Michelin (2011), Dunnmon et al. (2011) and Giacomello and Porri (2011) have recentlyproposed using the phenomena for energy harvesting applications and Eloy and Schouveiler (2010) and Hellum et al.1. Introduction

The interaction between a cantinteraction problem. It is well knowvelocity is increased above a criticaconditions which is modeled using a leading edge torsional spring. The theoretical

results are compared to vibration and aeroelastic test results collected in the Duke

University wind tunnel as well as previous theoretical and experimental results for the

leading edge clamped conguration. The aeroelastic experiments conrmed the validity

of the three-dimensional vortex lattice aerodynamic model over a subset of mass ratios.

& 2012 Elsevier Ltd. All rights reserved.

d exible elastic plate in a uniform axial ow is a canonical uidstructurethis system exhibits a utter instability in low subsonic ow as the free streamity. The structure then enters a large and violent limit cycle oscillation (LCO).Vortex lattice aerodynamics

Flutter experimentstransition in utter velocity between the pinned-free and clamped-free boundaryTheory and experiment for utter of a rectangular plate with a xedleading edge in three-dimensional axial ow

S. Chad Gibbs n, Ivan Wang, Earl Dowell

Duke University, Durham, NC 27708, USA

a r t i c l e i n f o

Article history:

Received 12 July 2011

Accepted 11 June 2012Available online 23 July 2012

a b s t r a c t

This paper explores cantilevered beam utter for both clamped and pinned leading edge

boundary conditions. Specically, a three-dimensional vortex lattice panel method is

coupled with a classical Lagrangian one-dimensional beam structural model to predict

the linear utter boundary for nite size rectangular plates. The paper explores the

journal homepage: www.elsevier.com/locate/jfs

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 69Watanabe et al., 2002a) or discrete approximations. The discrete approximations can be split into the discrete vortexmodels (Alben and Shelley, 2008; Howell et al., 2009; Michelin et al., 2008; Tang and Dowell, 2002; Tang and Padoussis,2007, 2008) or numerical simulations solving the NavierStokes (Balint and Lucey, 2005; Watanabe et al., 2002a). For thesecond case, where the chord length is much larger than the span, a slender body approximation has been used byLemaitre et al. (2005) to explore the dynamics. For the two-dimensional case, Howell et al. (2009) explored the inuence ofspatial connement and Michelin and Llewellyn Smith (2009) and Tang and Padoussis (2009) have modeled the inuenceof cascades on the response of the system.

In addition to these two-dimensional aerodynamic models, researchers have used different structural models toexplore the response of the system. The structural models have largely consisted of linear and non-linear models of beamswith out-of-plane displacements. In general linear structural models are used to explore the aeroelastic stability boundaryas parameters are varied. Non-linear models have been used by Michelin et al. (2008), Tang and Padoussis (2008), Tanget al. (2003), Tang and Padoussis (2007) and Dunnmon et al. (2011) to explore post critical behaviors such as LCOamplitude and hysteresis loops which are observed experimentally. Recently interest in piezoelectric energy harvestinghas motivated the detailed exploration of the non-linear post critical behavior because predicting the amplitude andfrequency of the limit cycle is vital to optimizing the energy harvested from the system (Doare and Michelin, 2011;Dunnmon et al., 2011; Giacomello and Porri, 2011).

The critical velocities predicted by the two-dimensional models are similar to each other regardless of the solution

Nomenclature

E Youngs modulush plate thicknessI area moment of inertia rsSh3=12K , M stiffness and mass matrixKa torsional spring stiffnessL plate chord length in the ow directionm mass per unit length rshSN number of structural modes includedpx,t aerodynamic pressure difference per

unit lengthPn1=2i aerodynamic force on the ith panel at a time

step between n and n1Qn generalized aerodynamic forceqn~t nth non-dimensional generalized coordinateS plate span length in the normal to the ow

direction~S aspect ratio S=L

Sc, Ss structure elements in chordwise ( x!) and

spanwise ( y!) direction

U1, ~U free stream and utter uid velocityVd,i Z component of the velocity of the panel at

the ith colocation pointWc, Ws wake elements in chordwise ( x

!) and span-

wise ( y!) direction

wx,t displacement at beam location (x) at time (t)x, y streamwise and spanwise coordinatesGni ith circulation strength at time step nDx, Dy chord (x) and span (y) and dimension of

vortex lattice panelm mass ratio raL=rshra, rs uid and structure material densityfn,i nth non-dimensional mode shape at the

ith panelon, o natural and utter frequencies in radians~ non-dimensionaltechnique used. Unfortunately they do not match published experimental results which have been reported by Taneda(1968), Kornecki et al. (1976), Watanabe et al. (2002b), Yamaguchi et al. (2000), Tang et al. (2003), Eloy et al. (2008) andDunnmon et al. (2011). In fact, across the range of parameters tested the utter boundaries predicted by the two-dimensional models are consistently below the experimentally observed values. Even when Huang (1995) attempted tocreate a two-dimensional experimental model by having test pieces span the wind tunnel, the experimentally observedcritical velocities were still higher than the theoretical predictions.

This discrepancy has motivated the application of three-dimensional aerodynamic models. Many of the initial three-dimensional aerodynamic models were used to explore the utter characteristics of a single conguration. For exampleTang et al. (2003) used an unsteady three-dimensional vortex lattice model (VLM) and a non-linear structural model toexplore the utter boundary and post critical behavior of a single aluminum plate. The success of initial three-dimensionalsimulations to match the utter boundary between theory and experiment has prompted the most recent explorations ofthe stability boundary in parameter space with three-dimensional aerodynamic models by Eloy et al. (2007, 2008). Ingeneral these simulations have shown much better agreement with the experimental results. Furthermore an explorationof the three-dimensional effects of spanwise connement by Doare et al. (2011) demonstrates that the small distancebetween wind tunnel walls and experimental specimen required to produce the two-dimensional limit experimentallywould be prohibitively difcult to achieve. Three-dimensional effects are believed to explain the systemic discrepanciesbetween two-dimensional aerodynamic theoretical predictions and experimental observations for the critical uttervelocity.

With the new understanding of the importance of three-dimensional effects on the quantitative behavior of this uidstructure system there is a need to analyze the impact of different inuences such as structural boundary conditions, windtunnel wall connement and experimental support structure with a three-dimensional aerodynamic model. The three-dimensional unsteady vortex lattice model remains a versatile means to explore the aforementioned inuences. Numerical

simulations have the benet of being able to model the effect of different congurations without changing the frameworkof the analysis. The work presented here is a continuation of the work done by Tang et al. (2003). In this paper the VLMaerodynamic model is generalized and used to explore the stability boundary in parameter space. Specically the criticalow velocity as a function of mass ratio m and aspect ratio ~S is explored and compared with new experimental results aswell as experimental and theoretical results found in the literature. In general the qualitative trends and quantitativevalues match the experimental results.

In addition, the paper explores the effect of the leading edge boundary condition on the critical utter velocity using aleading edge torsional spring. The transition between the two limiting cases includes a surprising, non-monotonictransition in critical utter velocity.

@x x xb ,t @x x xb ,t

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 688370Fig. 1. Schematic of panel geometry.w9x xb ,t 0,@2w

@x2

x xb ,t

0, 3

w9x xb ,t 0,@w

@x

x xb ,t

0, 4

where xb is the geometric location of the given boundary.2. Theory

Energy methods are used to derive the equations of motion for the aeroelastic system. All the structural modelings areclassical but a brief overview is included for completeness. First the equations of motion and resulting natural frequenciesand mode shapes for a beam with both clamped-free and pinned-free boundary conditions are presented. Next a torsionalspring at the xed leading edge is included to model the transition between the pinned-free and clamped-freecongurations. Then a brief description of the vortex lattice method is given. Finally the form of the coupled aeroelasticequations is given and a solution technique using the eigenvalues of the linear system is outlined.

2.1. Structural model with torsional spring

The derivation of the structural model begins with dening the energies of a beam. Assumptions for this applicationinclude: (1) small displacements and (2) one-dimensional chordwise bending motion. The rst assumption of smalldisplacements is required for the linear analysis which is conducted. The assumption of primarily bending motion is anassumption which has been conrmed by previous experimental observations (Dunnmon et al., 2011; Tang and Padoussis,2008) and one that is used by most other theoretical models studying this system. Fig. 1 shows a diagram of the elasticmodel of the beam that was analyzed. The goal is to solve for wx,t.

Before examining the aeroelastic system it is useful to look at the unforced system to determine the in-vacuum naturalfrequencies and mode shapes. The results are later used to solve the aeroelastic equations of motion. The familiar equationof motion for a beam in bending is

m@2w

@t2

@

2

@x2EI

@2w

@x2

0, 1

and the natural boundary condition combinations are given in Eq. (2) for a free boundary, Eq. (3) for a pinned boundaryand Eq. (4) for a clamped boundary

@2w2

0, @3w3 0, 2

At this point the equations are non-dimensionalized and normalized using a characteristic length equal to the length ofthe beam (L) and a characteristic time T L2

m=EI

p. Using these factors the non-dimensional equations of motion become

@2 ~w

@~t2 @

4 ~w

@ ~x4 0, 5

with the boundary conditions applied at ~xb 0 and ~xb 1.The non-dimensional natural mode shapes are solved using separation of variables, where it is assumed that

~w ~x, ~t q~tf ~x. The familiar forms of the solution are

q~t A expfl~tgB expfl~tg,

f ~x C sinhl

p~xD cosh

l

p~xE sin

l

p~xF cos

l

p~x: 6

Using this form for the spatial mode shapes f ~x and applying the boundary conditions, the natural frequencies andmode shapes up to an arbitrary constant can be determined. A summary of the approximate non-dimensional (radians/non-dimensional time) natural frequencies for the pinned-free and clamped-free beams is given in Table 1. The massnormalized mode shapes for both boundary conditions can be seen in Fig. 2.

The leading edge spring can either be modeled by incorporating the potential energy due to the spring into theequations of motion or modifying the boundary conditions to include the restoring moment due to the torsional spring. Forthis paper the boundary condition method is used because the resulting mode shapes are the natural modes of the springsystem and therefore the elastic portion of the aeroelastic equations remain uncoupled. This minimizes the number ofstructural modes required to capture the dynamics in the aeroelastic simulations. The boundary conditions at the pinnededge with the torsional spring can be determined by applying a force balance at ~x 0. Here the torsional force applied bythe spring modeled by Hooks law must be identically equal to bending moment. Mathematically this can be written as

@2 ~w

@ ~x20 ~K a

@ ~w

@ ~x, 7

where ~K a KaL=EI. In order to ensure the mode shapes satisfy this boundary condition as well as the three other natural

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 71boundary conditions, the assumed solution is substituted into the boundary condition equations. This process yields the

Table 1Approximate non-dimensional natural frequencies.

Mode

number

Pinned-free

frequency

Clamped-free

frequency

1 0 :5172p22 234

2p2 1:492p23 334

2p2 312 2p2^ ^ ^n n34

2p2 n12 2p2

Fig. 2. The solid line corresponds to the clamped-free mode shapes, dashed line to the pinned-free mode shapes, and the dotted line to the ~K a 1000mode shapes. All mode shapes normalized to a generalized mass of one.

following matrix equation:

1 1 1 1

~K a 1 ~K a 1cosh

l

psinh

l

pcos

l

psin

l

p

sinhl

pcosh

l

psin

l

pcos

l

p

26664

37775

C

D

E

F

26664

37775

0

0

0

0

26664

37775: 8

The set of four coupled equations captured in Eq. (8) can be used to solve for the natural frequencies l by determiningthe values of l which make the determinate of the matrix equal to zero. There are an innite number of frequencies thatwill satisfy this requirement. Depending on the number of mode shapes desired, the nullspace of the matrix can be used todetermine the values for C, D, E, F up to an arbitrary constant for each of the ls which satisfy the determinant equation. Acommon choice for the constant is one that normalizes the generalized mass to one.

Before moving on to the aeroelastic analysis, it is useful to explore the transition between pinned-free and clamped-freenatural modes. An observation from the natural frequency analysis which is of interest for the current aeroelastic analysisis the separation between the rst two natural frequencies. Fig. 3 clearly shows that these two frequencies initially movecloser than the pinned-free limit, before moving apart. This may be signicant for the aeroelastic analysis because theseparation of frequencies for coalescence type utter is known to impact the utter velocity.

2.2. Vortex lattice aerodynamic method

For this system the forcing function for the elastic model comes from the aerodynamics. Specically a generalized forceis a force can be expressed as

Qn Ffnx dA: 9

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 688372For this application the aerodynamic forces are calculated using a vortex lattice method which has been developed andutilized in previous works, for example see Hall (1994) and Tang et al. (2003). This method is a lattice method ofaccounting for the discrete vortex laments as they progress through time and space and allows for the modeling of athree-dimensional ow and includes the effect of the wake created by the unsteady structural motion. The wake of thesystem is prescribed to be in the plane of the plate system as seen in Fig. 4, allowing the aerodynamics to remain linear. Forthis specic application, a certain type of vortex lament called a horseshoe vortex is used. The reason to track the vortexlaments is that the strength of the circulation inside the lament directly correlates to the forces applied. A signicantmodication to previous implementations (Hall, 1994; Tang et al., 2003) is the non-dimensionalization of the aerodynamicrelationships yielding equations which are only dependent on two parameters, the mass ratio m and aspect ratio ~S andwhich yield a non-dimensional utter velocity ~U UL

EI=m

p.

The set of governing equations for the vortex lattice method can be segmented into four types of equations that governthe circulation on a given element. Furthermore, for the square lattice that was constructed for this problem the equationsfor a given column are the same for all the elements in that column. The four types of equations are:

A11- Over the plate structure (downwash inuence).

Fig. 3. This solid line is the separation between the rst two natural frequencies as ~Ka is varied. The dotted line is the separation between the rst twopinned-free frequencies.dded to the homogeneous structural equation. The generalizedZ

points on the elastic structure.

ral model through a set of downwash relationships. The

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 73dothecon

2.4

genBe

whthe

equThe vortex lattice aerodynamic model interacts with the structu2.3. Non-dimensional downwash state relations W11- First column of the wake (shed wake). W12- Second to last column in the wake (wake convection). W13- Last column of the wake (wake relaxation).

For a detailed description of the form of the aerodynamic equations see the existing literature (Hall, 1994; Tang et al.,2003). The relationships in each of the four sectors combine to give a set of equations that are equal in number to thenumber of elements in the aerodynamic mesh. In general the relationships capture induced downwash and convectionrelationships and are discretized in time to be relationships between the state at time n and time n1. The modicationsto the strictly aerodynamic equations given by A11, W11, W12, and W13 were limited to normalizing the wash andcirculation using the characteristic length and time scales derived from the elastic equation of motion. These relationshipscombine to create the following governing aerodynamic relationship:

sCn1zCn Vd, 10where s and z are aerodynamic matrices that are constructed to satisfy the relationships prescribed in the differentsections of the aerodynamic mesh A11, W11, W12, and W13 and Vd is the vertical velocity of the plate at the collocation

Fig. 4. Expanded schematic of vortex lattice mesh.wnwash equation is governed by the movement of the elastic plate. The downwash created by the horseshoe vortices invortex lattice mesh must equal to the time rate of change of the panel displacement (including the effect of owvection) ~w at the collocation points. This can be written as

~Vd d ~w

d~t

fluid

@ ~w@ ~x

plate

~U1@~w

@~t

plate

: 11

. Non-dimensional generalized force

The nal coupling equation is the generalized force caused by the aerodynamic ow. In order to calculate theeralized force a transformation from the tracked circulations to the induced force must be dened. An application ofrnoullis equation yields the following non-dimensionalized aerodynamic force:

~Pn1=2i

m~S

~U1 ~Gn

i ~Gn1i

Xk

~Gn1k ~Gn

k " #

, 12

ere ~Gn

i indicates the strength of the ith circulation element at time step n, and the sum over k indicates a sum over allelements upstream of and including the ith circulation element.Before moving on it is important to discuss the non-dimensionalization of Eq. (12). From the normalizing of the elastications of motion previously discussed in the torsional spring section, the non-dimensional aerodynamic force is given

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 688374by ~P PL2=EI. This process produces two non-dimensional parameters. The rst one is the aspect ratio of the system ~Swhich is equal to the beam span, S, divided by the beam chord, L. The second is the mass ratio m which is dened as a ratioof the mass of the air to the mass of the beam, specically, m raL=rsh.

Finally the pressure dened in Eq. (12) is used in the governing aeroelastic equations through the generalized forceterms Qn which are of the form

Qn1=2n qntXSti 1

~Pn1=2i fn,i: 13

Creating a vector Q with the nth term equal to Qn allows the governing matrix equation to be written as

M qKqQ : 14

2.5. Vortex lattice based aeroelastic simulations

The coupled governing aeroelastic equations outlined in the previous section are solved using an eigenanalysis of thesystem, although time marching is also possible. First the elastic equations of motion are placed into state space and timediscretized. The best way to illustrate this process is to start by looking at the ith equation for the relationship dened inEq. (14). This relationship can be expressed as

Mi qiKi qi Qi: 15As is common for transforming an equation into state space, it is necessary to dene two state variables y1 qi and y2 _qiand discretized the variables as follows:

_y2n1=2 y

n12 yn2Dt

, 16a

yn1=21 yn11 yn1

2, 16b

yn1=22 yn12 yn2

2 y

n11 yn1Dt

_y1 n1=2: 16c

The last equation is just a discrete relationship between y1 and y2. Moving both the discrete representations of the half-time step to one side and setting equal to zero one can obtain the following relationship:

yn11 yn1Dt

yn12 yn2

2 0: 17

Furthermore, the denitions in Eq. (16) can be used to re-write Eq. (15) as

Miyn12 yn2Dt

!K i

yn11 yn12

!Q n1=2i : 18

Eq. (18) is then combined with the time discretized equations for the aerodynamics to yield a matrix equation of theform given in Eq. (19)

k Hn1OHn 0: 19In this case k and O are large sparse matrices containing sectors that are described in terms of previously dened sets ofequations

k s bC1 D1

" #, O x 0

C2 D2

" #, H

~Cy

( ): 20

s and x contain the aerodynamic terms, b contains the downwash relationships, D1 and D2 contain the elastic terms andC1 and C2 contain the generalized force relationships.

For the linear system that was analyzed, all the information that is gleaned from the time history analysis can bedetermined from an eigenanalysis without time stepping the solution. The eigenanalysis for this system was done on Eq.(19) assuming the solution for the circulation and the state variables has the form

HH expfl~tg: 21Substituting this relationship into Eq. (19) yields

expflD~tgkOH 0: 22It is clear that Eq. (22) is in the form of an eigenvalue problem once one denes L expflD~tg. As before, the real and

imaginary parts of the eigenvalues are the values that will determine the stability and frequency of the system. The

eigenvalue of the system with the largest magnitude corresponds to the structural motion found from time marching.After determining l lnfLg=D~t the values are sorted by their proximity to the eigenvalues of the previous velocity. Theaeroelastic damping is real part of l and the aeroelastic frequency, o, is the imaginary part of l. The utter velocity isdetermined by running a series of different velocities and tracking the modal damping and frequency. The velocity atwhich the damping becomes positive represents the utter velocity and the corresponding frequency from the eigenvalueis the utter frequency.

The eigenanalysis for the linear system has two advantages over a time history analysis. First, the frequency anddamping values found through the eigenanalysis can be recovered for each of the individual modes. This allows a cleardenition of the mode which drives the system unstable and the creation of a root locus plot to analyze how frequenciesevolve with damping. Second, because only the largest eigenvalues are important for the analysis, Matlabs eigs functioncan be used to solve efciently and quickly for only the largest eigenvalues.

3. Theoretical aeroelastic simulations

Using the vortex lattice aeroelastic theory, the utter characteristics of clamped-free and pinned-free beams withvarying mass ratios and aspect ratios is conducted and compared to the results produced by previous researchers studyingthe same system but using different methods. The simulation conguration is given in Fig. 5. As shown in the gure therigid airfoil which is present in the experimental conguration shown in Fig. 10 is not included in the vortex lattice meshand therefore not included in the aeroelastic simulations. The inuence of the airfoil can be included in the aeroelasticmodel by including bound circulation on the xed airfoil which is governed by Eq. (11), where the downwash on the rigidstructure is equal to zero. The airfoil is not included to allow for theoretical results that can be compared to previous

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 75theoretical simulations. However, initial simulations done comparing the utter velocity with and without the rigid airfoilshow that the airfoil can act to destabilize the system and lower the theoretical utter velocity by up to 20% for small massratio. A further exploration of the inuence of the leading edge airfoil, which is present in experiments, is not be exploredin this paper, but could be the subject of future research.

The rst question studied in detail is the change in utter characteristics from the pinned-free to clamped-freeboundary conditions. Using m 0:277, ~S 0:5, and N10 appropriate in-vacuum beam modes, a set of simulations is runwith a varying magnitude pinned edge spring. The simulation is run using 150 panel elements and 300 wake elements inthe streamwise direction and 10 elements in the spanwise direction.

Fig. 6(b) shows the transition between pinned-free and clamped-free utter frequencies. The analysis suggests amonotonic transition for the utter frequency between the pinned-free and clamped-free congurations. This resultmatches the transition in frequency behavior observed in the structural model, an unsurprising result. Furthermore thecritical values for ~K a are in the same range as they were for the transition in the natural frequency analysis for the uttermode. The result also demonstrates that for this conguration utter occurs between the rst and second frequencies forall torsional spring stiffness values.

However, the utter velocity transition from pinned-free to clamped-free does not monotonically transition from thepinned-free case to the clamped-free case, see Fig. 6(a). This unexpected result shows that a small torsional spring at theleading edge of a pinned-free beam will actually drive the utter velocity below the pinned-free critical velocity. In fact, for

Fig. 5. Aeroelastic simulation model.

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 688376small values of torsional spring stiffness, making the spring stiffer will actually drive down the utter velocity. It ishypothesized that the larger effect on the natural frequencies of the rst mode at small torsional spring stiffness valuesthat initially moves the rst two natural frequencies closer may be one of the causes of the reduction in the utterboundary. Because of the inherent torsional stiffness for many pinned systems, this is a critical result because it suggeststhat using the pinned-free model may not be a conservative estimate, and therefore an effort to quantify the torsionalstiffness of the pinned connection must be explored. This result could also be signicant for applications in energyharvesting where a lower utter velocity is desired. Looking closer at the inection point of the utter velocity curve, itappears to occur at a ~K a where the distance between the rst two spring natural frequencies is moving towards theclamped-free limit. It is at this critical point where the torsional spring begins to become strong enough to force theresponse to behave more like the clamped-free case. Recall Fig. 3.

Another natural parameter to explore with this model is the aspect ratio ~S. Fig. 7 shows the current theoreticalprediction for a m 0:6 plate as the aspect ratio is varied. Again, for this set of simulations, 150 panel elements and 300wake elements in the streamwise direction, 10 spanwise elements and 10 clamped-free modes were used. The resultshown in Fig. 7 matches previous theoretical results published by Eloy et al. (2008). Also shown in the gure are theexperimental data points collected by Eloy et al. (2008). For the linear analysis presented here the only data that the modelshould be compared to are the unlled squares because the gap in the utter velocity down to the lled in squaresrepresents a hysteretic effect which is not captured by the current linear model. A recent publication suggests that thehysteresis arises due to spanwise deformations in the structure (Eloy et al., 2012; Zhao et al., 2011) before the onset ofutter which become less important once the structure begins to utter. This may explain why the current theoreticalpredictions match the lower utter velocities as they are observed after these deformations have been eliminated by aviolent utter motion.

Fig. 6. Flutter velocity (a) and frequency (b) vs. ~Ka . Small values of ~Ka correspond to a pinned-free case (dotted) and large values correspond to aclamped-free case (dashed). The rst and second natural frequency evolution results are also included as the thin lines in (b). The thick lines correspond

to the aeroelastic results.

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 77The next set of simulations which is conducted provides a comparison between the pinned-free and clamped-free utters as afunction of the mass ratio. This is explored both from a utter frequency and a utter velocity perspective. Aspect ratios of 0.5, 1.0and 1.5 are simulated. This set of simulations is conducted with the same lattice properties as the previous analyses.

Fig. 8(a) shows the comparison between the utter velocities of the pinned-free and clamped-free beams. It is clear fromthe results that for mass ratios between 0.1 and 1, the pinned-free and clamped-free utter boundaries are very similar. This isalso the case for the frequency comparison shown in Fig. 8(b). For both cases the utter mode, as identied by the frequency issecond mode utter (second mode for pinned-free is often called its rst bending mode, because the rst mode is a rigid bodymotion). Also for both cases the frequency of the utter falls below the respective in-vacuum second mode frequency. Anothercommon trend which is observed is that the frequency of oscillation begins to decrease as the mass ratio increases.

A phenomenon observed both using the lattice method discussed here and in alternate analysis done with differentaerodynamic theories, see Eloy et al. (2007) and Guo and Padoussis (2000), is a transition in utter mode to third modeutter at a higher frequency and velocity as the mass ratio increases above a critical value. At the mass ratios where thereis utter in both modes, there is an interesting behavior in the modal damping evolution. At the lower velocity the secondmode goes unstable in its normal manner. However, instead of having a damping value whose magnitude continues togrow, the damping levels out. Simultaneously the third mode begins to become less negatively damped and thefrequencies of the second and third modes begin to come together. At the velocity corresponding to the third mode utter,the third mode becomes unstable and the second mode becomes stable again. This transition is shown in Fig. 9(a) and (b).If a time marching analysis is done, all that is observed would be the jump in frequency and utter shape at the upperutter velocity, while the eigenanalysis allows the tracking of the stability of the individual modes. As with previousworks, this transition occurs at a lower mass ratio for the pinned-free case. Unfortunately the current experimental modelwould not allow for testing of mass ratios where higher mode utter is predicted.

Fig. 7. Flutter velocity as a function of the aspect ratio at a mass ratio m of 0.6 and clamped-free boundary conditions. The thick line corresponds to thecurrent authors theoretical predictions, the dashed line is taken from Eloy et al. (2008). The squares are previously published experimental data points

(Eloy et al., 2008). The empty squares correspond to the velocity at which the system becomes unstable as the velocity increases and the lled in squares

correspond to the velocity where the response returns from unstable oscillations to stable as the ow velocity is decreased.It is clear from Figs. 8(a) and (b) that the difference between the pinned-free and clamped-free cases would be morenoticeable in the utter frequency than in the utter velocity. In fact the difference between the clamped-free and pinned-free utter velocity values is so small, it may not be observable during experiments.

Overall, this implementation of the vortex lattice panel method for modeling the aerodynamics produced resultssimilar to the theoretical results of previous researchers. Although the vortex lattice method may take longer to run asingle simulation, it has value in that it can be modied to capture aerodynamic nonlinearities and other real-worldconsiderations such as wind tunnel walls and experimental support structures (Attar, 2003; Preidikman and Mook, 2000).

4. Experiments

Vibration and aeroelastic experiments are conducted on samples of varying sized 3003 aluminum plates. For the0.381 mm thick aluminum the length is varied from 200 mm to 300 mm in increments of 25 mm. For the 0.25 mm thickaluminum the length is varied between 225 mm and 275 mm in 25 mm increments. All congurations share an aspectratio ~S 0:5. The properties for this material are assumed to be the common values for the alloy given in Table 2.

To record the frequency content of the plate movements two methods are used. First, a small piezoelectric patch isattached at the root of the plate. The properties of the piezoelectric patch are given in Table 2. The piezoelectric patch ischosen to be small so that it will not affect the motion of the system. This is veried by the vibration experiments. Thesecond method includes an accelerometer placed at the root of the plate. Results are not sensitive to the measurementdevice and they are interchanged throughout the experimental process. For both methods, the sensor signal is collectedand analyzed in real time by the Spectral Dynamics SD380 spectrum analyzer for frequency content.

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883784.1. Structural

Vibration testing is done to ensure that the plate frequencies, which are used in the theoretical aeroelastic model, areaccurate representations of the actual natural frequencies of the test specimens. Furthermore the structural testingensures that the test apparatus and frequency measuring piezoelectric patch or accelerometer do not have a large effect onthe test specimens behavior. Fig. 10 shows the experimental apparatus, described in the previous section, which is usedwhen measuring the natural frequencies of the plate. The natural frequencies of the plate are determined by applying animpulse force at the tip of the plate and observing the frequency content of the response.

Overall the natural frequencies measured in experiment matched the expected clamped-free natural frequencies over therange of test specimens (Appendix B). This experiment also helps to validate the time scaling because it is clear that for all themass ratios the non-dimensional frequencies do in fact remain constant. Finally this experiment conrms that the frequencymeasuring device does not signicantly change the natural frequencies and therefore should not affect the response of the system.

4.2. Aeroelastic

The aeroelastic experiments are carried out in the Duke University wind tunnel. The elastic specimen is mounted in thewind tunnel using a rigid airfoil that spans the wind tunnel to provide the leading edge clamp of the elastic plate. As withthe structural experiments, the utter frequency is calculated from the signal of the attached piezoelectric patch oraccelerometer at the moment the structure begins to utter. The ow velocity is measured with a hotwire at the entranceto the wind tunnel test chamber.

Fig. 8. Flutter velocity (a) and frequency (b) vs. m. The thick solid line corresponds to the clamped-free beam with ~S 1:0, the thin dashed line toclamped-free beam with ~S 0:5, and the thick dotted line to a pinned-free beam with ~S 0:5. Also included in the velocity gure is theoreticalpredictions for a clamped-free beam with ~S 1:0 from Eloy et al. (2007).

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 79The utter velocity is measured by slowly incrementing the ow velocity in the wind tunnel up until the specimenentered a limit cycle oscillation. As the velocity of the wind tunnel comes close to the utter velocity, a peak in thefrequency response begins to appear and the velocity increment size is decreased to 0.25 m/s per increment. At each owspeed the velocity is held for 23 s before incrementing again. At a certain velocity, the oscillations grow until the specimenenters a large limit cycle oscillation. The velocity where the beam entered these oscillations is recorded as the uttervelocity and the frequency at this speed is read from the spectrum analyzer. As an aside, similar to experimental results inthe literature, the velocity where the system returned from unstable to stable was different than the recorded linear uttervelocity, producing a hysteresis loop, a non-linear result which is not studied in this work because all the theoretical work islinear. For each specimen the test is repeated three times and the average utter velocity and frequency is recorded.

The goal of the wind tunnel testing is to validate the theoretical model with experimental data points. Specically, astudy of utter as a function of the mass ratio for the clamped-free conguration is conducted. Good agreement betweenthe clamped-free experiment and theory helps validate the aeroelastic model.

The experimental testing for the mass ratio variation is done for the clamped-free conguration because the DukeUniversity wind tunnel has an established experimental setup and test protocol for this conguration. As one can see bylooking at Fig. 11, there is good agreement between theory and current and published experimental values. Quantitativelythis is shown by a small average difference and standard deviation of the difference from the experiments given inTable A2. The averages are calculated by subtracting the theoretical value from the experimental value and then dividingthe difference by the theoretical values. This small difference is consistent with previous comparisons with dimensionalvortex lattice simulations and experiments carried out by Tang et al. (2003) and Dunnmon et al. (2011). For the frequency

Fig. 9. The two gures show the evolution as the mass ratio is increased over the range where the utter characteristics move from second bending tothird bending for the clamped-free conguration. (a) shows the root locus evolution and (b) shows the velocity vs. damping evolution. The triangles

correspond to rst bending, the xs to second bending and the dots to third bending. The solid line is the zero damping and the open circle identies

where a mode becomes unstable.

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 688380Table 2Experimental parameters.

Property Symbol Value

Elastic plate

properties

Alloy 3003

Thickness h 0.381 mm,

0.25 mm

Density rs 2840 kg/m3

Youngs modulus E 72 GPa

Airfoil chord 101 mm

Airfoil span 550 mm

Air density ra 1.2 kg/m3

Piezoelectric patch

properties

Source Measurement

specialtyresults presented in Fig. 11 there is a consistent bias for the experimental values to be under the theoretical values. Thisdiscrepancy may be caused by non-linear effects because the utter frequency measurement is in fact the limit cycleoscillation frequency. An initial exploration of the inclusion of the leading edge airfoil in the theoretical model alsosuggests that the experimental apparatus may also be a cause of the lower utter frequencies and lower utter velocities.This impact would also explain the increasing difference as the mass ratio increases which corresponds to a relativelylarger support structure compared to the size of the elastic specimen. Regardless, the good agreement between theory andexperiment is encouraging and suggests that for the utter velocity and frequency, the vortex lattice aerodynamic methodprovides an accurate model for the linear response of the system.

5. Conclusions and future work

This paper presents a theoretical framework for implementing a non-dimensional vortex lattice aerodynamic basedmethod for utter analysis of elastic panels in axial ow. The utter characteristics as a function of mass ratio, aspect ratio,and boundary conditions are presented and compare well with previous research done using alternative aerodynamicmodels. Furthermore an exploration using a leading edge torsional spring stiffness ~K a to model the aeroelastic transitionbetween pinned-free and clamped-free utter suggests that neither the clamped-free nor the pinned-free boundary

Fig. 10. Experimental apparatus. Included are (1) the elastic structure being tested, (2) the airfoil support structure and (3) the hotwire ow velocitymeasurement device. Not included are the accelerometer or piezoelectric sensor placed on the opposite side of the elastic structure and the spectrum

analyzer used to measure the frequency.

Series DT series patch

Size 30 mm by 12 mm

Spectrum analyzer

properties

Manufacturer Scientic Atlanta

Name Spectral dynamics

SD380

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 6883 81condition corresponds to the minimum utter velocity. Instead there is a critical ~Ka near unity which produces the lowestutter velocity. This is a new result which has implications in the way pinned-free systems should be analyzed in thefuture. This result also has implications in the design of energy harvesters from ag utter. For example, to lower theutter velocity for a given system, one could lower the torsional stiffness at the root of the plate.

The experiments validated the aeroelastic model and add more data to the growing clamped-free utter boundaryexperimental data set. The accuracy of the model conrms that the vortex lattice method is an accurate model of thethree-dimensional low subsonic aerodynamics to use in utter calculation. Additional experiments for a pinned-freeboundary condition plate as well as a wider range of mass ratios, specically those which utter in different modes thanthe ones explored experimentally in this paper, would be valuable to further validate the applicability of the vortex latticemethod for utter prediction.

The next step in this research is to explore the non-linearities that dominate the motion of the panels at velocities abovethe utter boundary. Specically, the introduction of structural and aerodynamic non-linearities is required to predict thelimit cycle oscillation observed experimentally. Furthermore a study of the inuence of the surrounding walls in the windtunnel on the utter boundary should be included for completeness. This inuence has been recently explored by Doareet al. (2011), and could be integrated into the aerodynamic methodology presented here through the use of images. Finally amore detailed exploration of the experimental support structure on the aeroelastic simulations should be conducted.

Appendix A. Experimental data points

Tables A1 and A2 contain the data collected from the Duke University wind tunnel testing. All velocities are in the unitsof normalized velocity and all the frequencies are in the units of radians/non-dimensional time. The results include the

Fig. 11. Mass ratio variation with experiment. This gure includes new experimental data (x), previous experimental data from Huang (1995) for~S 0:6 to 1:5(n), Eloy et al. (2008) for ~S 1:0 (B), and Eloy et al. (2012) ~S 0:5 (&). Also included in the gure are theoretical results for ~S 0:5 (thickline) and ~S 1:0 (thin line).

S. Chad Gibbs et al. / Journal of Fluids and Structures 34 (2012) 688382Table A1Experimental datapoints for a clamped-free plate.

m Theory Experiment Difference (%)

~U flutter ~o flutter ~U flutter ~o flutter ~U flutter ~oflutter

0.185 15.50 17.45 13.11 17.42 15.40 0.21

0.208 14.69 17.42 13.15 16.47 10.50 5.50

0.222 14.27 17.40 13.54 17.40 5.12 0.04

0.231 14.03 17.39 12.60 17.20 10.22 1.11

0.254 13.47 17.35 12.46 17.73 7.44 2.190.277 12.98 17.31 11.98 17.95 7.69 3.700.277 12.98 17.31 12.90 17.43 0.64 0.680.312 12.36 17.25 8.96 14.58 27.53 15.46

0.333 12.05 17.21 12.03 17.43 0.17 1.280.347 11.85 17.18 8.67 13.56 26.89 21.06

0.381 11.44 17.11 8.62 15.66 24.72 8.48difference standard deviation which is the standard deviation of the set of difference from all the experimental data points.This measurement gives a way to quantify the ability of the experimental trend to match the theoretical trend. A smalldifference standard deviation means the data closely follows the theoretical trend while a small difference demonstratesthat the theory is close to the experiment.

Appendix B. Natural frequencies

Fig. B1 contains the natural frequency data that was collected to validate the structural dynamics portion of theaeroelastic model.

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Table A2Experimental vs. theoretical results summary.

Quantity Velocity

(%)

Frequency

(%)

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Difference

standard deviation

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Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial flowIntroductionTheoryStructural model with torsional springVortex lattice aerodynamic methodNon-dimensional downwash state relationsNon-dimensional generalized forceVortex lattice based aeroelastic simulations

Theoretical aeroelastic simulationsExperimentsStructuralAeroelastic

Conclusions and future workExperimental data pointsNatural frequenciesReferences