ANovelAerodynamicForceandFlutteroftheHigh-Aspect-Ratio ...

Research ArticleA Novel Aerodynamic Force and Flutter of the High-Aspect-RatioCantilever Plate in Subsonic Flow

Li Ma1 Minghui Yao 1 Wei Zhang 1 Kai Lou2 Dongxing Cao 1 and Yan Niu 1

1College of Mechanical Engineering Beijing University of TechnologyBeijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures Beijing 100124 China2Water Transport Planning and Design Co Ltd Communications Construction Company Limited Beijing 100007 China

Correspondence should be addressed to Minghui Yao merry_mingming163com

Received 11 April 2020 Revised 13 May 2020 Accepted 27 May 2020 Published 19 June 2020

Academic Editor Konstantin V Avramov

Copyright copy 2020 Li Ma et al )is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)is paper focuses on the derivation of the aerodynamic force for the cantilever plate in subsonic flow For the first time a newanalytical expression of the quasi-steady aerodynamic force related to the velocity and the deformation for the high-aspect-ratiocantilever plate in subsonic flow is derived by utilizing the subsonic thin airfoil theory and Kutta-Joukowski theory Results showthat aerodynamic force distribution obtained theoretically is consistent with that calculated by ANSYS FLUENT Based on thefirst-order shear deformation and von Karman nonlinear geometric relationship nonlinear partial differential dynamicalequations of the high-aspect-ratio plate subjected to the aerodynamic force are established by using Hamiltonrsquos principle Galerkinapproach is applied to discretize the governing equations to ordinary differential equations Numerical simulation is utilized toinvestigate the relation between the critical flutter velocity and some parameters of the system Results show that when the inflowvelocity reaches the critical value limit cycle oscillation occurs )e aspect ratio the thickness and the air damping havesignificant impact on the critical flutter velocity of the thin plate

1 Introduction

With the development and popularization of unmanned aerialvehicles in fields of detection disaster prevention and disasterreduction researchers paid more attention on aeroelasticproblems of high-aspect ratio airfoils in subsonic airflowFlutter is self-excited oscillation of a flight vehicle under thecoupling effect of the aerodynamic pressure the elastic forceand the inertia force Cantilever plates in the axial flowmay losestability at sufficiently high flow velocity Analysis of the lineartheory indicates that there is a critical dynamic pressure )emotion of the panel becomes unstable when the dynamicpressure is higher than the critical value Once the instabilitythreshold is exceeded flutter will take place Under the con-dition of flutter in the system the energy of the surroundingairflow will be continuously pumped into the plate to sustainthe flutter motion )e lightweight and high performances ofthe modern aircraft make the aeroelastic problems of theaircraft more prominent

One of the key issues of aeroelasticity is flutter whichusually leads to the disaster of the aircraft [1] )us theaeroelasticity problem of the aircraft comes into our sightZhang et al [2] applied the composite multilayer plate tosupersonic aircraft under the aerodynamic pressure to in-vestigate excessive nonlinear vibration suppression of theplate Flutter is a fluid-structure coupling problem Cher-nobryvko et al [3] discussed nonlinear dynamic stabilityconical shells in a supersonic gas stream Amabili andPellicano [4] found that flutter is very sensible to small initialimperfections of the structure Vedeneev [5] studied flutterof an elastic thin plate and obtained the exact solution bysolving the structural kinetic equation coupling with thehydrodynamic equation When solving dynamic problemsof the fluid-structure coupling system researchers preferadopting aerodynamic models to simulate the external flowfield Based on these aerodynamic models aerodynamiccharacteristics related to motion state of the system such asthe displacement velocity and acceleration are obtained

HindawiShock and VibrationVolume 2020 Article ID 8841590 17 pageshttpsdoiorg10115520208841590

)en nonlinear dynamic equations of airfoils are derivedand aerodynamic characteristics of the system areinvestigated

Aerodynamic force is a key factor in the analysis of theflutter phenomena which directly determines the buckingform of the plate structure So it is necessary to select aproper aerodynamic force model in the study of flutterproblem Up to now scholars have proposed abundanttheories of the aerodynamic force According to the de-pendence of aerodynamic forces on time and spatialaerodynamic theories can be roughly divided into threetypes such as the steady quasi-steady and unsteady aero-dynamic theories )e first type is the steady aerodynamicmodel namely assuming the force acting on the liftingsurface of the wing do not change with time )e steadyaerodynamic model is mainly used for static aeroelasticanalysis such as the thin airfoil theory [6] )e second kindis the quasi-steady aerodynamic theory in which we assumethe aerodynamic force at any time is only related to themotion state of the wing at that time By using this theorythe reduced frequency of motion is smaller and vibrationcharacteristics analysis of the airfoil subjected to the aero-dynamic force can be simpler Lin et al [7] proposed a quasi-steady piston theory to calculate the flutter problem of acantilever plate in supersonic flow Hu et al [8ndash10] inves-tigated aeroelastic vibration of the plate subjected to theaerodynamic force obtained from the first order pistontheory Brouwer and McNamara [11] optimized the pistontheory and verified it Dowell and Ganji [12] extended thepiston theory to higher order terms in several expansionsand analyzed the flutter of single degree of freedom panelOwing to the ignoring of the propagation effect of smallperturbations in the subsonic airflow the piston theory isnot applicable in the subsonic airflow )us scholars gavesome new methods to compute aerodynamic Grossmanquasi-steady theory [13] can also be applied to calculate thetotal lift force and the total torque of high-aspect ratio wingswhich can be simplified as an elastic beam without pretwistand the axial extension Obviously this theory is not suitablefor plates with the small aspect ratio )e third one is theunsteady aerodynamic theory which considers the influenceof change of circulation and wake flow on the aerodynamicforce of a moving airfoil Xu et al [14] deduced the solutionof the unsteady aerodynamic force for a slender airfoilwhich is modeled as a beam structure However this methodcan only be used to solve problems of 2D plate structures inthe incompressible airflow Based on the model constructedby applying )eodorsen unsteady aerodynamic theory re-searchers investigated stability of wings However Cordeset al [15] pointed out )eodorsen function cannot capturethe experimental transfer functions in frequency depen-dence when investigating the unsteady lift force of an airfoil

Since exact solutions of unsteady potential equations arefew we can only acquire approximate analytical solutionsthrough other methods Peters [16] obtained a semiem-pirical nonlinear and unsteady ONERAmodel according todata of wind tunnel airfoils Dunnp and Dugundjij [17]analyzed linear flutter of the composite wing using theharmonic balance method by adopting ONERA dynamic

stall model Sadr et al [18 19] developed the ONERAmodelHowever the)eodorsen model and the ONERAmodel arenot suitable for the aerodynamic analysis of plate structuresZhang and Ye [20] established the integral aerodynamicreduced order model [21] based on the Volterra seriestheory However introduction of the integral form leads tothe computational complexity Zhang and Ye [22] developedthe Volterra series theory and expressed the aerodynamicforce as the sum of multiple convolutions However theaerodynamic force obtained is implicit so it is hard to couplethe implicit expression with kinetic equations of the plateGuo et al [23 24] put forward the nonlinear harmonicbalance method to analyze the unsteady flow in the tur-bomachinery and the airfoil Based on proper orthogonaldecomposition Luo et al [25] presented a hybrid modelingmethod for reconstructions of flow field and aerodynamicoptimization Nonlinear regression methods instead of thelinear regression widely used are adopted to establish PODbasis modes which behave with good description perfor-mance in system space Wang et al [26] made a compre-hensive review on the latest studies about the aeroelasticmodeling )e CFD method is adopted in Refs [27ndash29]however this method is not conducive to perform pneu-matic elastic servo analysis Munk et al [30] studied the limitcycle of a two-dimensional cantilever plate under subsonicflow based on the vortex lattice method Castells Marin andPoetsch [31] used the doublet lattice method to model thelifting surface which is more accurate than the NLRImethod Xie et al [32] obtained the aeroelasticity defor-mation of the geometrically nonlinear high-aspect ratiowing which is in great agreement with the experimentalresult Pashaei et al [33] modeled the airflow by using thevortex lattice method and studied the effect of energyharvesting properties of the metal composite on the fluttermargin and limit cycle oscillation amplitudes Ramezaniet al [34 35] established the aerodynamic model byadopting CFDCSD coupling numerical computationalmethod Chen et al [36ndash38] constructed reduced-ordermodels of high speed vehicles based on CFD simulationsResults show that the ROM approach can significantlyspeedup unsteady aerodynamic calculations of a system

Most of the results obtained from vibration of platesunder subsonic airflow are related to flutter of panels Flutterof the panel is similar to that of the wing)emain differencelies in that there is only one surface subjected to theaerodynamic force for the panel However for the plateboth surfaces are exposed to the airflow Study of panelflutter began in the early 1970s Tang et al [39 40] theo-retically and experimentally researched the aeroelastic re-sponse of a wing and a cantilever plate under subsonicairflow Dowell et al [41 42] proposed the linear potentialflow theory which can calculate the pressure distribution ofany point on the surface of the panel However it is notsuitable for plate or shell structures with airflow acting onboth sides

Generally achievements have been made in the study of2D infinite plate panels and beam structures in subsonicairflow And aerodynamic models have already been appliedin investigation of flutter However research works on

2 Shock and Vibration

explicit expressions of aerodynamic forces of cantileverplates and shells under subsonic airflow are still few In thispaper for the first time an analytical expression of the quasi-steady aerodynamic force for the high-aspect ratio cantileverplate in subsonic flow is induced based on the subsonic thin-airfoil theory and KuttandashJoukowski lift theorem Overallaerodynamic force theoretically calculated by using theexplicit expression we derived has a good agreement withthat obtained by ANSYS FLUENT In addition the aero-dynamic model constructed based on it could be applied toflutter analysis of cantilever plates and shells with the high-aspect ratio Considering lateral vibration and deformationof the mean camber line of the cross section nonlineardynamic equations of transverse vibration of the high-aspectratio cantilever plate are derived by utilizing the quasi-steadyaerodynamic model Influences of parameters of the systemon the critical flutter velocity are investigated

2 Aerodynamic Force Derivation for the High-Aspect-Ratio Cantilever Plate inSubsonic Flow

21 Analysis of the Aerodynamic Force for the High-Aspect-Ratio Cantilever Plate in Subsonic Flow Field )e schematicdiagram of the cantilever plate considered is shown inFigure 1 )e wing is simplified as a high-aspect ratiocantilever plate)e span length chord length and thicknessof the plate are a b and h respectively )e velocity of thesubsonic airflow along the chordwise direction is denated asUinfin Cross section of the cantilever plate is marked as A(X Y Z) is the inertial coordinate system and the origin ofit is in point O e

0x is the spanwise direction e

0y is the

chordwise direction and e0

z is the thickness direction re-spectively Based on the strip assumption KuttandashJoukowskilift theorem and linearized small perturbation theory weinduce an aerodynamic force model of a high-aspect ratiocantilever plate in subsonic airflow

)e strip assumption of the plate with high-aspect ratiocan be briefly introduced as follows Actually the airflow onwings is a three-dimensional fluid However if then thegeometric dimension of the cantilever plate does not changealong the chordwise direction the aspect ratio is high and theinflow velocity is not change along the spanwise )e velocityof the fluid can be considered as a component in 2D plane(YOZ plane) and the component in the spanwise axis (X-axis)is zero in the most part of the spanwise region )us everychord section can be analyzed as a 2D airfoil with an infinitespanwise length Diagrams of each section along the chordwisedirection such as section A can be shown in Figure 1

At the beginning of the 20th century Joukowski pro-posed KuttandashJoukowsi lift theorem which established therelation of the lift and circular rector of moving objects in theair as shown in equation (1) In the incompressible low-velocity inviscid and straight uniform flow field the forcedistribution on unit length of the spanwise of the closed 2Dwing is perpendicular to the direction of the airflow Its valuecan be expressed by product of the density of the fluid ρ flow

velocity Uinfin and the vortex strength c(Y) of unit arc lengthof the wing

dL ρUinfinc(Y)dY (1)

)e vortex strength c(Y) on unit arc length is positive ina clockwise direction When using KuttandashJoukowski lifttheorem (inviscid potential flow theory) to solve the lift ofthe airfoil the chief problem among the issues is how toobtain the local vortex strength c(Y)

)e airfoil whose ratio of the maximum thickness andthe chord length is less than 12 is defined as a thin airfoilFor a thin airfoil we can use linearized small perturbationtheory of the low speed flow around a thin airfoil inaerodynamics to calculate the local vortex strength c(Y))e problem about flow around a thin airfoil means thesmall attack angle and the small bending In addition a thinairfoil problem means the thin thickness )us boundaryconditions and pressure coefficient of the airfoil can belinearized )erefore based on principle of superpositionthe attack angle the bending and the thickness can beconsidered separately and then superposed )e potentialflow around a thin airfoil can be decomposed into threesimple linear potential flows which include the flow arounda curved plate without an attack angle a symmetrical airfoilwithout an attack angle and a flat plate with a small attackangle )e flow around a symmetrical airfoil without anattack angle cannot generate the lift force so we only need toconsider the lift forces generated by the small attack angeland the bending of the airfoil When the straight uniformairflow flows across the mean camber line of 2D airfoil with asmall attack angle we can use the surface vortex on the meancamber line to simulate the distribution of the local vortexstrength c(Y) which has a trigonometric series solutionLateral displacement of the mean camber line of thechordwise cross section is set up as WY )e chordwiseposition of an arbitrary point in the middle arc line can bewritten as

Y b

2(1 minus cos θ) (0le θle π) (2)

When the plate vibrates slightly a very small displace-ment dWY

appears According to the mathematical definitionof limit the tangent value dWY

dY is equal to the corre-sponding angle value )us we express the attack angle asdWY

dY that is the tangent value of the attack angle

ZY

Xa

b

A

O h

Uinfin

Figure 1 Model of the high-aspect-ratio cantilever plate

Shock and Vibration 3

Substitute equation (2) into dWYdY then the expression of

the attack angle is changed into function Kθ which is relatedto WY and θ )erefore expression of WY is derived as aftermentioned If the attack angle Kθis integrable as given inequation (4) the local vortex strength c(Y) can be expressedas c(θ)

c(θ) 2Uinfin A0 cotθ2

+ 1113944infin

n1An sin(nθ)⎛⎝ ⎞⎠ (3)

where

A0 α minus1π

1113946π

0Kθdθ (4a)

An 2π

1113946π

0Kθ middot cos(nθ)dθ (n 1 2 3 4 ) (4b)

According to equation (3) when the analytic expressionsof the attack angle and the mean camber line are given thereis a unique trigonometric series solution of the local vortexstrength c(θ) Coefficients of the solution can be confirmedby equations (4a) and (4b)

Linearized small perturbation theory of the flow aroundthe thin airfoil is a steady theory so it can only solve thesteady local vortex strength of the thin airfoil without de-formation and vibration However when a plate vibratesadditional attack angle caused by vibration velocity andtrailing vortex caused by changing circular rector will impactlocal vortex strength as shown in Figure 2 Lateral vibrationvelocity Vw and inflow velocity Uinfin will generate a newadditional attack angle θ2 which is approximately VwUinfinas shown in Figure 3

)e additional attack angle of the thin plate θ2 is definedas VwUinfin so flow theory of the thin airfoil can be used tocalculate the local vortex strength caused by θ2 Substituteequation (2) into VwUinfin expression of the additional attackangle is marked as Qθ which is related to Vw and θ )usexpression of Vw is focused on If the additional attack angleQθ is integrable the local vortex strength caused by vibrationvelocity can be expressed as a function cprime(θ) as follows

cprime(θ) 2Uinfin A0prime cotθ2

1113888 1113889 + 1113944

infin

n1Anprime sin(nθ)⎛⎝ ⎞⎠ (5)

where

A0prime α minus1π

1113946π

0Qθdθ (6a)

Anprime

2π

1113946π

0Qθ middot cos(nθ)dθ (n 1 2 3 4 ) (6b)

)e total circular rector in the nonviscous flow fieldmust be conserved)us the integrable total circular rectorof every point in that field must be zero When a platevibrates equivalent wake vortex is generated by change ofthe circular rector )e wake vortex generated has a stronginfluence on the local vortex strength of airfoils )ereforethe wake vortex will also affect deformation of the plateDeformation will also generate new wake vortex Both the

wake vortex and deformation of the plate have influence onthe local vortex Comparatively speaking wake vortex has asmaller influence on the local vortex strength so we in-troduce the quasi-steady hypothesis in which the influ-ences of the wake vortex on the local vortex strength areneglected To sum up under the quasi-steady hypothesiswe only need to consider the influence of the mean camberlinersquos deformation and the additional angle of attack onlocal vortex strength

22 Interpolation Functions of the Mean Camber Linersquos De-formation and Vibration Velocity In order to use equations(3)ndash(6a) and (6b) to calculate the local vortex strengthcaused by the additional attack angle and deformation of themean camber line it is necessary to know analytic expres-sions of the mean camber linersquos deformation of the chordsection WY and the vibration velocity distribution functionVw Moreover Kθ and Qθ should be integrable on θHowever when cantilever plate and shell structures vibratedeformation and velocity are different at different timewhich may not satisfy the condition mentioned above )usa fitting method of the interpolation function is applied toexpress the deformation function WYand the vibrationvelocity distribution function Vw )erefore the sum ofseveral interpolation functions can satisfy the condition Ifthese interpolation functions are integrable on θ aftersubstituting Y into the deformation function WY and thedistribution functionVw WY and Vw of the vibration ve-locity can be approximately expressed as integrable func-tions about θ Nonequidistant Lagrange interpolationmethod satisfies the condition abovementioned so this

Z

Yθ1

θ2 asymp VwUinfin

Uinfin

Uinfin

Vw

VwUinfin + θ1

Figure 2 Additional attack angle of the plate

Z

Z

Y

Y

Vw gt 0

Vw lt 0

VwUinfin lt 0

VwUinfin gt 0

Uinfin

Figure 3 Equivalent diagram of the additional attack angle of thewing

4 Shock and Vibration

method is applied to fit WY and Vw Interpolation functionsare several polynomial functions of curves that pass throughpoints given on the 2D plane

Since the first mode of the cantilever plate vibration isthe bending deformation the first interpolation functionWX1 is set up as a constant which is the average value ofdeformation of the mean camber line at Y 0 and Y b asshown in Figure 4 WX1is expressed as

WX1 12

(w(x b t) + w(x 0 t)) (7)

After the first fitting of the mean camber linersquos defor-mation difference between the deformation curve CC0 ofthe mean camber line and the first interpolation functionWX1 is CC1 as shown in Figure 5

Since the second order mode of the cantilever platevibration is the torsion deformation the second interpola-tion function WX2 is set up as a linear function which isshown in Figure 6

Let values of the interpolation function at Y 0 and Y

b be equal to values of residual CC1 at Y 0 and Y brespectively )us slope of the first interpolation functioncan be calculated as follows

1b

w(x b t) minus WX1( 1113857 minus w(x 0 t) minus WX1( 1113857( 1113857 (8)

)e second interpolation function WX2 can be writtenas follows

WX2 1b

(w(x b t) minus w(x 0 t)) middot Y + w(x 0 t) minus WX1

(9)

Difference between residual CC1 after the first fitting andthe second interpolation function WX2 is marked as CC2which can be expressed as CC2 W minus WX1 minus WX2

)e third interpolation function WX3 is set up as aquadratic function as shown in Figure 7 Values of CC2 atY 0 and Y b are 0 In order not to increase the residualat Y 0 and Y b in the third fitting the third inter-polation function is taken as K2 middot Y(b minus Y) )e extremepoint of the interpolation function appears at Y b2thus it is necessary to makeK2 times Y (b minus Y)|Yb2 CC2|Yb2 in order to get a minimaldifference between the third interpolation function andthe residual CC2 According to the equation unknowncoefficient K2 and the third interpolation function WX3can be calculated WX3 is written as follows

WX3 w xb

2 t1113888 1113889 minus Wx11113888 1113889

4b2

middot Y(b minus Y) (10)

Difference between the residual CC2 in the second fittingand the third interpolation function WX3 is marked as CC3)e difference can be expressed asCC3 W minus WX1 minus WX2 minus WX3

)e fourth interpolation function WX4 is chosen as acubic function which is shown in Figure 8 Values of CC3 at

W

Y

CC0

Figure 4 Deformation of the mean camber line

W

WX1

w (x 0 t) w (x b t)CC1

Y

Figure 5 )e first interpolation function on the plane (Y Z)

W

Y = b2WX2

w (x 0 t) ndash WX1CC1

CC2

Y = b

w (x b t) ndash WX1

Y

Figure 6 )e second interpolation function on the plane (Y Z)

cc = 0cc = 0

Y = b2W

Y

CC3

WX3

CC2

Figure 7 )e third interpolation function on the plane (Y Z)

cc = 0 cc = 0 cc = 0

Y = (12 ndash radic36) b Y = b2W

Y

CC3

CC4

Y = b

WX4

Figure 8 )e fourth interpolation function on the plane (Y Z)

Shock and Vibration 5

Y 0 Y b and Y b2 are 0 In order not to increase theresidual at Y 0 Y b and Y b2 in the fourth fittingthe fourth interpolation function is taken asK3 times Y times (b minus Y) times (b2 minus Y) Extreme points of the inter-polation function occur at Y b2 minus

3

radicb6 and

Y b2 +3

radicb6 so it is essential to establish the equation

[K3 times Y times (b minus Y) times (b2 minus Y)]|Y(b2)minus (3

radicb6) CC3|Y(b2)minus

(3

radicb6) in order tomake the fourth interpolation function fit

the residual CC3to the greatest extent According to theequation unknown coefficient K3 and the expression of WX4can be calculated WX4 is shown as follows

WX4 w(x y t) minus WX1 minus WX2 minus WX3( 1113857

1113868111386811138681113868Y(b2)minus (3

radicb6)

((b2) minus (3

radicb6))((b2) +(

3

radicb6))(

3

radicb6)

times Y(b minus Y)b

2minus Y1113888 1113889

(11)

Similarly difference between the residual CC3 in thethird fitting and the fourth interpolation function WX4 ismarked as CC4 which can be expressed asCC4 W minus WX1 minus WX2 minus WX3 minus WX4

)e fifth interpolation function WX5 is set up as theform of a quartic function as shown in Figure 9 Values ofCC4 at Y 0 Y b Y b2 and Y (b2) minus (

3

radicb6) are

zero In order not to increase the residual at Y 0 Y band Y b2 in the fourth fitting the fifth interpolationfunction is taken asK4 times Y times (b minus Y) times ((b2) minus Y) times ((b2) minus (

3

radicb6) minus Y)

Difference at extreme points of the interpolation function

Y (b2) + (3

radicb6) is still exist so it is necessary to let

K4 times Y times (b minus Y) times ((b2) minus Y)times

((b2) minus (3

radicb6) minus Y)|Y(b2)minus (

3

radicb6) CC4|Y(b2)minus (

3

radicb6)

CC4|Yb2+3

radicb6 in order to make the fifth interpolation

function fit the residual CC4 to the greatest extentAccording to the equation unknown coefficient K4 and thefifth interpolation function WX5 can be calculated WX5 isshown as follows

WX5 w(x y t) minus WX1 minus WX2 minus WX3 minus WX4( 1113857

1113868111386811138681113868Y(b2)+(3

radicb6)

((b2) +(3

radicb6))((b2) minus (

3

radicb6))(minus (

3

radicb6))b

times Y(b minus Y)b

2minus Y1113888 1113889

b

2minus

3

radic

6b minus Y1113888 1113889 (12)

Difference between the residual CC4 in the fourth fittingand the fifth interpolation function WX5 is marked as CC5)e difference can be expressed asCC5 W minus WX1 minus WX2 minus WX3 minus WX4 minus WX5 Values ofCC5 at Y 0 Y b Y b2 Y (b2) minus (

3

radicb6) and Y

(b2) + (3

radicb6) are 0

After five times of deformation fitting abovementionedthe total residual CC5 brought by the mean camber linersquosdeformation CC0 of five interpolation functions of the thinplate have the tendency to gradually converge to zero asshown in Figure 10 If the mean camber line is morecomplicated it is necessary to conduct more interpolationfunctions For cantilever plates and shells with high-aspectratio chord deformation is relatively simple so it is enough

to take top four interpolation functions to fit thedeformation

Velocity is the derivative of displacement which iscontinuous while the plate is vibrating so vibration velocitydistribution of the plate is also continuous If we representthe mean camber linersquos vibration velocity of the chordsection as a curve a method that is totally similar to theabove can be used to fit the curve Form of the interpolationfunction of velocity distribution is exactly the same as that ofthe interpolation function of deformation )erefore it onlyneeds to replace items about deformation in equations (7)and (9)ndash(12) with a corresponding item about velocity)enfive interpolation functions for fitting velocity distributionare obtained as follows

cc = 0cc = 0 cc = 0 cc = 0 cc = 0

CC4

CC5

Y = (12 ndash radic36) b Y = (12 + radic36) bW

Y

Y = b2 Y = b

WX5

Figure 9 )e fifth interpolation function on the plane (Y Z)

Y = (12 ndash radic36) b Y = (12 + radic36) bW

cc = 0 cc = 0cc = 0 cc = 0 cc = 0

Y

Y = b2

CC0

CC5

Y = b

Figure 10 Residual error of the deformation in fitting

6 Shock and Vibration

VX1 (zw(x b t)zt) +(zw(x 0 t)zt)

2

VX2 1b

zw(x b t)

ztminus

zw(x 0 t)

zt1113888 1113889Y +

zw(x 0 t)

ztminus VX1

VX3 zw(x (b2) t)

ztminus VX11113888 1113889

4b2

(b minus Y)Y

VX4 (zw(x b t)zt) minus VX1 minus VX2 minus VX3( 1113857

1113868111386811138681113868 Y(b2)minus (3

radicb6)

((b2) minus (3

radicb6))((b2) +(

3

radicb6))(

3

radicb6)

1113888 1113889

times Y(b minus Y)b

2minus Y1113888 1113889

VX5 (zwzt) minus VX1 minus VX2 minus VX3 minus VX4( 1113857

1113868111386811138681113868 Y(b2)+(3

radicb6)

((b2) +(3

radicb6))((b2) minus (

3

radicb6))(minus (

3

radicb6))b

1113888 1113889

times Y(b minus Y)b

2minus Y1113888 1113889

b

2minus

3

radic

6b minus Y1113888 1113889

(13)

23 Lift Force Calculation Precondition of calculating liftforce of the high-aspect-ratio cantilever plate is to obtain thelocal vortex strength as shown in equation (1) Local vortexstrength of the cantilever thin plate with high-aspect ratio ismainly caused by two factors deformation of the meancamber line and the vibration velocity Based on linearizedsmall perturbation theory the total local vortex strengthcaused by these two factors can be obtained by using linearsuperposition as shown in equation (14) In the previoussection we give the interpolation functions of the twovariables In order to calculate the local vortex strengthcaused by deformation of the mean camber line local vortexstrength caused by every interpolation function can becalculated respectively and then the linear superpositioncan be carried out Chordwise deformation of the platestructure is relatively simple so top four interpolationfunctions are enough to fit the deformation of the cantileverplate accurately WX1 WX2 WX3 and WX4 are taken tocalculate local vortex First of all attack angle caused by fourinterpolation functions dWX1

dY dWX2dY dWX3

dY anddWX4

dY are calculated respectively Substitute equation (2)into dWX1

dY dWX2dY dWX3

dY and dWX4dY then

functions of the attack angle become related to θ )ese four

functions are set up as K1 K2 K3 and K4 which aresubstituted into equations (4a) and (4b) respectively A0 andA calculated by equations (4a) and (4b) are substituted intoequation (3) to solve the corresponding local vortex of topfour interpolation functions cWX1 cWX2 cWX3 andcWX4

In order to calculate the local vortex caused by the vi-bration velocity local vortex strength caused by each in-terpolation function of the velocity is calculatedrespectively Linear superposition is conducted to obtaintotal local vortex strength caused by the vibration velocityBecause vibration velocity of cantilever plate is relativelysimple it is accurate enough to take four interpolationfunctions to fit velocity distribution of the plate )us topfour interpolation functions VX1 VX2 VX3 and VX4 areselected to calculate the local vortex Firstly angle of attackfunctions VX1Uinfin VX2Uinfin VX3Uinfin and VX4Uinfinoffour interpolation functions are calculated respectively Andsubstitute equation (2) into VX1Uinfin VX2Uinfin VX3Uinfinand VX4Uinfin )en angle of attack functions are translatedinto functions related to θ )ese four functions are set up asQ1 Q2 Q3 and Q4 which are substituted into equations (6a)and (6b) respectively A0prime and An

prime calculated by equations(6a) and (6b) are substituted into equation (5) to solve thecorresponding local vortex cVX1 cVX2 cVX3 and cVX4of top four interpolation functions To sum up the total localvortex strength caused by the mean camber linersquos defor-mation and the lateral vibration velocity is cz which can beexpressed as linear superposition of the local vortex strengthcalculated by abovementioned eight interpolation functionscz is written as follows

cz cWX1 + cWX2 + cWX3 + cWX4 + cVX1 + cVX2

+ cVX3 + cVX4 pw1 middot w(x b t) + pw2 middot w(x 0 t)

+ pw3 middot w xb

2 t1113888 1113889 + pw4 middot w x

b

2minus

3

radicb

6 t1113888 1113889

+ pv1 middotzw(x b t)

zt+ pv2 middot

zw(x 0 t)

zt

+ pv3 middotzw(x (b2) t)

zt+ pv4 middot

zw(x (b2) minus (3

radicb6) t)

zt

(14)

where

pw1 (123

radicminus 36)

Y

b+(10 minus 6

3

radic)1113874 1113875

Uinfinb

times

Y

bminus

Y2

b2

1113971

+

3

radicminus 72

Uinfinb

b

Yminus 1

1113970

(15a)

pw2 minus 26 minus 63

radic+(12

3

radic+ 36)

Y

b1113874 1113875

Uinfinb

Y

bminus

Y2

b2

1113971

+7 +

3

radic

21113888 1113889

Uinfinb

b

Yminus 1

1113970

(15b)

pw3 483

radic Y

b16 minus 24

3

radic Uinfinb

Y

bminus

Y2

b2

1113971

+ 23

radic Uinfinb

b

Yminus 1

1113970

(15c)

Shock and Vibration 7

pw4 363

radicminus 72

3

radic Y

b1113874 1113875

Uinfinb

Y

bminus

Y2

b2

1113971

minus 33

radic Uinfinb

b

Yminus 1

1113970

(15d)

pv1 (12 minus 43

radic)

Y

bminus

Y2

b21113888 1113889

32

+

3

radicminus 32

minus4Y

b1113888 1113889 times

Y

bminus

Y2

b2

1113971

minus12

b

Yminus 1

1113970

(15e)

pv2 (minus 43

radicminus 12)

Y

bminus

Y2

b21113888 1113889

32

minus12

b

Yminus 1

1113970

+11 +

3

radic

2minus4Y

b1113888 1113889

Y

bminus

Y2

b2

1113971

(15f)

pv3 minus 163

radic Y

bminus

Y2

b21113888 1113889

32

+8Y

b+ 2

3

radicminus 41113874 1113875 times

Y

bminus

Y2

b2

1113971

minus

b

Yminus 1

1113970

(15g)

pv4 24Y

bminus

Y2

b21113888 1113889

32 3

radicminus 3

Y

bminus

Y2

b2

11139713

radic (15h)

According to equations (1) and (14) the aerodynamicforce Δp can be expressed as

Δp ρUinfincz (16)

Since the aerodynamic force expression is analytic it isconvenient to use analytic and semianalytic method to studythe flutter problem of the cantilever plate

24 Aerodynamic Correction and Error Analysis Value ofitem

(bY) minus 1

1113968in equations (15a) and (15h) at Y 0 is

infinite which leads to an infinite leading edge lift force As amatter of fact leading edge lift force of the wing cannot beinfinite Appearance of such a singularity at the leading edgeof the wing which is attributed to the basic solution of thethin-wall theory gives no consideration to flow around theleading edge namely when air flows past the leading edge ofthe thin plate part of air will pass through the upper panelfrom the lower panel Neglecting thickness of the plate thethin-airfoil theory leads to an infinite streaming velocity andan infinite lift force at the leading edge As a result it isnecessary to correct this problem

According to Ref [43] although there is a singular pointat the leading edge of the plate the pressure distribution on95 chord length range near the trailing edge has a goodconsistency with that of actual measurement )us it isnecessary to add a correlation coefficient in

(bY) minus 1

1113968 )e

infinity value of this function at Y 0 is corrected to beequal to the value of the original curve at Y 095b Aftertrial the item

(bY) minus 1

1113968in equation (15a) is corrected

as(b minus Y)(Y + 005b)

1113968 At the moment the value of

(b minus Y)(Y + 005b)1113968

at Y 0 is equal to the value of(bY) minus 1

1113968at Y 095b )e value of these two functions at

the trailing edge portion of the plate changes little as shownin Figure 11 )e corrected aerodynamic expression isdenoted as Δpprime

If the air on the plate flows at a speed greater than 03times the speed of sound influences of the compressibility ofair on aerodynamic force cannot be neglected )us it isessential to modify the impact of compression Von

KarmanndashChandra Formula is used to estimate the influenceof air compressibility on aerodynamic force and equation(17) is the relationship between the two aerodynamicpressure Δpp on the plate surface in nonsticky steady andsubsonic velocity and 2D compressible flow field and thecorresponding pressure Δpprime in the incompressible flowMainfin is the ratio of the flow velocity Uinfin to the local velocityof sound

To sum up after correction and considering the com-pressibility of air the aerodynamic force expressionΔpp is asfollows

Δpp Δpprime

1 minus Ma2

infin1113968

+(12)Δpprime middot 1 minus1 minus Ma2

infin1113968

( 1113857 (17)

Aerodynamic force Δpp is the linear superposition ofaerodynamic forces calculated by several interpolationfunctions Moreover inflow air must satisfy hypotheses ofirrotational and nonviscous )us the aerodynamic forceΔpp calculated by equation (17) is an approximate result)ere is an error between it and the real aerodynamic forceEffect of this approach is evaluated by estimating magnitudeof the error between the two

Mean camber linersquos deformation and the lateral vibra-tion velocity are mainly considered in theoretical calculationof the aerodynamic force )e essential reason why the liftforce can be generated is to change the attack angle of thewing which indirectly affects the aerodynamic force )uswhat is need is to make a comparison between the lift forcegenerated by deformation of the mean camber line of theplate and that of the corresponding finite element model toillustrate effectiveness of the aerodynamic force theoreticallycalculated

ANSYS FLUNT finite element software is applied tocalculate the aerodynamic force distribution A spline curveof definite shape whose chord length is 1 meter is drew inComputer Aided Design (CAD) Coordinates of controlpoints of the spline are shown in Figure 12 A thin shellmodel of 001 meter thickness is constructed by stretchingthe spline curve we drew to 10 meters along the spanwise

8 Shock and Vibration

direction Assuming the elasticity modulus of the shellstructure in subsonic compressible and nonsticky turbu-lence flow field is infinite )e velocity of the airflow isUinfin 200ms After numerical simulation pressure cloudpictures of the upper and lower surface can be obtained asshown in Figures 13 and 14

)e lift force can be calculated by the pressure differencebetween the upper and lower surface of the airfoil as shownin Figure 15

If we take the spline curve in Figure 12 as a mean camberline of the cross section at a certain moment of the deformedhigh-aspect ratio cantilever plate structure the aerodynamicforce distribution on the section can be calculated accordingto equation (17) As shown in Figure 12 in calculation of theaerodynamic force these items are introduced

w(x 0 t) 0376

w xb

2 t1113888 1113889 0083

w xb

2minus

3

radicb

6 t1113888 1113889 0581

w(x b t) minus 03

w xb

2+

3

radicb

6 t1113888 1113889 minus 045

(18)

In calculation of the lift force generated only by de-formation of the mean camber line velocities in the aero-dynamic force expression Δppare taken as zero namely

zw(x 0 t)

zt 0

zw(x (b2) t)

zt 0

zw(x (b2) minus (3

radicb6) t)

zt 0

zw(x b t)

zt 0

zw(x (b2) +(3

radicb6) t)

zt 0

(19)

Substituting abovementioned data into equation (17)aerodynamic force distribution calculated is shown in Fig-ure 15 Trend of value theoretical calculated has a goodagreement with that obtained by ANSYS FLUENT Incontrast with value obtained by ANSYS FLUENT the resulttheoretical calculated is relatively small In particular theerror is nearly about 20 at the trailing edge as shown inFigure 15

)e comparison above indicates that the aerodynamicforce fitted by utilizing interpolation functions can predictthe influence of deformation on aerodynamic distributionapproximately Value of the aerodynamic force obtained byANSYS is relatively large Source of the error in the the-oretical value is mainly due to the hypothesis of nonstickyand the neglecting of influence brought by the trailingvortex

3 Establishment and Dispersion ofAerodynamic Equations for theCantilever Plate

)e schematic diagram of the wing is shown in Figure 1which is simplified as a cantilever plate in subsonic flow)erectangular plate is characterized by a times b times h where a is thespan length b is the chord length and h is the thickness A isthe cross section of the wing (X Y Z) is the inertial co-ordinate system and the origin of it is located at the leadingedge of the fixed end of the cantilever plate which is markedby point O

Considering the first-order shear deformation Kirchhoffhypothesis [44] and scale effect the displacement filed of theplate is established Displacement of any point along x yand z direction can be expressed by that of the neutral planeof the plate as follows

u(x y z t) u0(x y t) + zφx(x y t) (20a)

v(x y z t) v0(x y t) + zφy(x y t) (20b)

w(x y z t) w0(x y t) (20c)

6

5

4

3

2

1

00 005 015 025 035 045 055 065 075 085 095 1

Yb

radic(b ndash Y)(Y + 005b) radic(bY) ndash 1

Figure 11 Correction of the coefficient of the aerodynamic force

376cm 5810cm 83cm ndash45cm ndash30cm

W

Y = 0 Y = 12 ndash radic36 Y = 12 + radic36Y = 12 Y = 1

Y

Figure 12 )e coordinate graph of the spline curve

Shock and Vibration 9

where u0 v0 and w0 are displacements along X Y and Z

directions of points on the neutral plane of the bladeand φx and φy are rotation angles around y-axis and x-axis

Strains of the von Karman plate which are computed byusing the nonlinear strain-displacement relation are ob-tained as follows

εxx zu0

zx+12

zw

zx1113888 1113889

2

+ zzφx

zx ε(0)

xx + zε(1)xx (21a)

εyy zv0

zy+12

zw

zy1113888 1113889

2

+ zzφy

zy ε(0)

yy + zε(1)yy (21b)

cxy zu0

zy+

zv0

zx+

zw

zx1113888 1113889

zw

zy1113888 1113889 + z

zφx

zy+

zφy

zx1113888 1113889

c(0)xy + zc

(1)xy

(21c)

cxz φx +zw

zxcyz φy +

zw

zy (21d)

)e constitutive relation of the isotropic material isexpressed as follows

σxx E

1 minus (υ)2εxx + υεyy1113872 1113873 (22a)

σyy E

1 minus (υ)2εyy + υεxx1113872 1113873 (22b)

τxy E

2(1 + υ)cxy (22c)

τxz E

2(1 + υ)cxz (22d)

τyz E

2(1 + υ)cyz (22e)

where E is the elasticity modulus of the material and υ isPoissonrsquos ratio

294e + 04237e + 04181e + 04125e + 04681e + 03117e + 03

ndash448e + 03ndash101e + 04ndash158e + 04ndash214e + 04ndash271e + 04ndash327e + 04ndash383e + 04

ndash496e + 04ndash553e + 04ndash609e + 04ndash666e + 04

ndash440e + 04

ndash722e + 04ndash779e + 04ndash835e + 04

Figure 13 Pressure of the lower surface of the airfoil

294e + 04237e + 04181e + 04125e + 04681e + 03117e + 03

ndash448e + 03ndash101e + 04ndash158e + 04ndash214e + 04ndash271e + 04ndash327e + 04ndash383e + 04

ndash496e + 04ndash553e + 04ndash609e + 04ndash666e + 04

ndash440e + 04

ndash722e + 04ndash779e + 04ndash835e + 04

Figure 14 Pressure of the upper surface of the airfoil

ndash30ndash20ndash10010203040

ANSYSTheory

Figure 15 Comparison between the theoretical result and theANSYS result

10 Shock and Vibration

In order to simplify the calculation relation between theinternal forces and stresses is expressed as follows

Qx

Qy

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

τxz

τyz

⎧⎨

⎩

⎫⎬

⎭dz (23a)

Nx

Ny

Nxy

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ 1113946

h2

minus h2

σxx

σyy