25TH INTERNATIONAL CONGRE SS OF THE AERONAUTIC AL SCIENCES
AN ANALYTICAL METHOD IN COMPUTATIONAL AEROELASTICITY BASED ON WAGNE R FUNCTION
Sh. Shams*, H. Haddadpour**, M.H. Sadr Lahidjani*, M. Kheiri ** *Amirkabir University of Technology, Dep . Aerospace Engineering , Tehran, Iran ** Sharif University of Technology, Dep . Aerospace Engineering , Tehran, Iran
Keywords : Aeroelasticit y, Flutter, Sharp-edged gust, Integro-differential system of equations
1 Abstract circular frequency and plunging (k m ) 2 ω, ω , ω w w θ 1 This paper presents an analytical method in and pitching frequencies (k I ) 2 computational aeroelasticity for an airfoil θ e. a considering two degrees of freedom (heaving ρ density of air and pitching) based on the Wagner integral U f flutter speed of airfoil function. In the obtained aeroelastic equations free stream velocity and its nondimensional U, V of motion, there are some integral parts that value U b ωθ give an integro -differential system of equations. t,σ time Using appropriate approximation for the Wagner function, a new form of equations can ϕ(t ) the Wagner function be obtained by derivation from mentioned ψ(t ) the Kussner function equations. These equations are in the form of ordinary differential equations. Using the obtained equations, the flutter speed is Introduction predicted for a given airfoil and the results are The indicial function was introduced by Wagner compared with the results of other investigators. to describe the lift response of a two Also, the dynamic responses of the airfoil to a dimensional flat plate in incompressible sharp -edged gust are shown in p re -flutter flow [1]. Some years latter , Theodorsen [2] regime for different cases. introduced the frequency response of a two dimensional flat plate airfoil in incompressible flow. The use of Laplace transformation was Nomenclature suggested by Jones [3], and Sears [4] applied c, b airfoil chord and semi chord length this method to solve some problems . Garrick [5] mass per unit length and reduced mass ratio used a convenient approximation according to m, µ mπρ b 2 the Fourier integral transform for the Wagner function. Garrick [6] and Miles [7] used I moment of inertia ab out elastic axis per unit e.a. length Duhamel superposition formula on a simple nondimensional radius of gyration about harmonic motion of an airfoil that leads to 1 rθ 2 2 elastic axis (Ie. a mb ) arbitrary equatio ns of motion. Marzocca et.al nondimensional distance between elastic [8] used a two dimensional rigid/elastic lifting a axis and midchord surface in unsteady incompressible flow. The x nondimensio nal distance between elastic Wagner's function was used to describe the time θ axis and center of mass domain unsteady aerodynamic lift and moment. w,θ airfoil motion in heave and pitch directions Also, the Ku ssner's function was applied for kw , kθ heave and pitch stiffness coefficients gust loads modeling. In recent years (Over the past two decades) two different approaches are developed for unsteady
1 SH. SHAMS , H . HADDADPOUR , M.H . SADR LAHIDJANI , M . KHEIRI
aerodynamic modeling for aeroelastic 1 Structural Modeling application which are known as Peters ’ The airfoil structure is modeled as a rigid flat aerodynamics and reduced order modeling plate, with two degrees of freedom in heave and (ROM). Peters et.al [ 9] offers a new type of pitch directions (Fig 1). The structural stiffness finite state aerodynamic model. This model is provided by translational and torsional offers finite state equations for the induced flow springs. field which are derived directly from the potential flow. The resultant equations can be exercised in the frequency -domain, Laplace - domain or time -domain and have capability to apply the two or three dimensional problems. Also ROM was introduced, developed and used for aeroelastic problems by many authors [10-12]. Both of the Peters ’ finite state aerodynamic model and ROM describes unsteady aerodynamics in a state space form. But the use of the Wagner function seems to be Figure 1. Schematic of the two -dimensional airfoil most appealing for the researchers to develop simple and exact model for unsteady flow model. analysis. The linear structural equations for this model In this regard, transformation of the aeroela stic by neglecting structural damping can be stated equations in differentials form provides a good as follows [13]: physical interpretation of the different terms in mw + mbxθθ + kww =− Lt( ) (1) these equations. Transforming the integral terms into differentials with the addition of two new second order differential equations and mbxθ w+ Iea.θ + k θ θ = M ea . ( t ) (2) corresponding au gmented states were presented in Poirel and Price study [13]. Details of this where the positive direction of ( w,θ ) is process are given in Dinyavari and Friedmann shown in Fig . 1. work [14]. This study, presents an analytical approach for 2 Aerodynamic Loading calculating the aeroelastic response of a two - Assuming the subsonic incompressible, dimensional airfoil (typica l section) in time - irrotational unsteady potential flow, the domain. In this method, the resulted integral aerodynamic lift and moment about elastic axis parts from the Duhamel integral part of the can be modeled as [15]: Wagner's function in aeroelastic equations will be omitted by using an appropriate Lt( ) =πρ bw2 − ab θ + U θ approximation of the Wagner's function and by - 1 part integral meth od and therefore a set of two +2(0πρUbw ) +− b (2 a )( θ0) + U θϕ ( 0) ( t ) (3) t fourth -order differential equations with 1 +2πρUb∫ ϕσ() t −+−+ wb (2 a ) θθσ Ud corresponding initial conditions will be 0 obtained. The present formulation will be 31 2 1 Mtea. . ()=πρ bawb −+−− (8 a )() θ U 2 a θ examined i n the time and frequency domain and +2πρUb2(1 + aw ) ( 0) +−+ b ( 1 a )( θ 0) U θ ( 0) ϕ ( t ) the obtained results will be compared with those 2 2 (4) t of oth er investigations. +2πρUb2(1 + a )() ϕσ t −+−+ wb ( 1 a ) θθσ Ud 2∫0 2 The unsteady aerodynamic loads can be computed using appropriate approximation of
2 AN ANALYTICAL METHOD IN COMPUTATIONAL AE ROELASTICITY BASED O N WAGNER FUNCTION
the Wagner function. Therefore, the Wagner 3 41 2 (mbxθ −πρ b a )( w ++ Ie. a . πρ b (8 + a )) θ function is approximated by [16]: 2 1 −2πρUb(2 + a ) ϕ ( 0) w 31 3 1 2 −ε1t − ε 2 t +(πρb U ( −− a ) 2 πρ Ub( − a ) ϕ ( 0)) θ ϕ(t ) =1 − ce1 − ce 2 (5) 2 4 2 1 −2πρUb(2 + a ) ϕ ( 0) w c = 0.165 U Where 1 , c2 = 0.335 , ε1 = 0.0455 b and 2 2 1 +(2kθ −πρ Ub(2 + a ) ϕ ( 0) (9) ε = 0.3 U . 2 b 31 2 −2πρUb(4 − a ) ϕ ( 0)) θ 2 1 1 3 Aeroelastic Modeling +2πρUb(2 + aw )( ( 0) +− b ( 2 a )( θ0)) ϕ ( t ) =2(πρ Ub 2 1 + aeI)( λ−ε1t+ λ eI − ε 2 t Combining the structural and aerodynamic 2 θ11 θθ 2 2 θ equations (Eqs. 1-4), the aeroelastic equations −λeI−ε1t − λ eI − ε 2 t ) of motion can be obtained as follows: w11 ww 2 2 w
2 3 Where λ=cU ε − ε b(1 − a ) and (m+πρ b )( w + mbxθ − πρ ab ) θ θi i i i 2 t 2 +πρbU θ + kww λε=c2 and I = exdεi σ () σσ . wi ii ix ∫ (6) 0 +2πρUbtw ϕ( ) ( 0) +−+ b (1 a )( θ 0) U θ ( 0) 2 Using the coefficients defined in table (1): t =−2πρUb ϕ() t − σ wb +−+ (1 a ) θ U θ ∫0 2 Table (1) : Define Coeffi cients for equations (10,11 ) (mbx−πρ abw3 )( ++ I πρ b 4 (1 + a 2 )) θ θ e. a . 8 A= m + πρ b 2 +πρbU3 (1 − a ) θ + k θ 3 2 θ A′ = mbxθ − πρ b a 2 1 1 (7) 3 −2πρUb(2 + atw )()( ϕ 0) +−+ b ( 2 a )( θ0) U θ ( 0) B= mbxθ − πρ ab t ′ 41 2 2 1 1 BI=e. a . +πρ b(8 + a ) =2πρUb(2 + a )()∫ ϕ t −+−+ σ wb ( 2 aU ) θ θ 0 C= 2πρ Ub ϕ (0) C′ = −2πρ Ub2(1 + a ) ϕ ( 0) In order to eliminate the integral parts, using by - 2 2 2 1 part integral method and some simplification D=πρ Ub +2 πρ Ub(2 − a ) ϕ ( 0) ′ 31 3 1 2 along with using Eq (5) instead of the Wagner D=πρ bU(2 −− a ) 2 πρ bU( 4 − a ) ϕ ( 0) function, equations (6) and (7) will lead to E= k + 2πρ Ub ϕ (0) equations (8, 9). Equations ( 8) and (9) which are w ′ 2 1 ordinary differential equations of the aeroelastic E= −2πρ bU(2 + a ) ϕ ( 0) 2 2 1 system. F=2(0)2πρ Ub ϕ + πρ Ub(2 − a ) ϕ ( 0) ′ 2 2 1 Fk=θ −2πρ bU(2 + a ) ϕ ( 0) (m+πρ bw2 )( + mbx − πρ ab 3 ) θ + 2(0 πρϕ Ub) w θ 31 2 −2πρb U(4 − a ) ϕ ( 0) +(2πρbU2 + πρ Ub 2 (1 − a ) ϕθ ( 0)) 2 1 G=−2πρ Ubw( ( 0) + b (2 − a )( θ 0)) +(2kw + πρ Ub ϕ (0)) w G′ =2πρ bU2 (1 + aw )(( 0) +− b ( 1 a )( θ 0)) 2 2 1 2 2 +(2πρϕUb (0) + 2 πρ Ub(2 − a ) ϕθ ( 0)) (8) 1 −2πρϕUbtw( )( ( 0) + b (2 − a )( θ 0)) A simple form of the aeroelastic equa tions
−ε1t − ε 2 t =2(πρUb − λθ1 eI1 θθ − λ 2 eI 2 θ can be written as follows:
−ε1t − ε 2 t +λw1eI1 ww + λ 2 eI 2 w )
3 SH. SHAMS , H . HADDADPOUR , M.H . SADR LAHIDJANI , M . KHEIRI
Aw+ Bθ + Cw + D θ the following equations we arrive at following equations: +Ew + Fθ + G ϕ (4) ( 4) Aw+ Bθ +[] A( ε1 ++ ε 2 ) Cw −εt − ε t (10) = −βeI1 − β eI 2 θ11 θθ 2 2 θ +[]B(ε1 + ε 2 ) + D θ −εt − ε t 1 2 +[]Aεε12 + C( ε 1 ++ ε 2 ) Ew +βw1eI1 ww + β 2 eI 2 w +[]Bεε12 + D( εε 1 ++ 2 ) F θ +[]Cεε + E( εε +−− ) β β w (14) Aw′+ B ′θ + Cw ′ + D ′ θ 12 1 2 w1 w 2 +[]Dεε + F ( εε +++ ) β βθ +Ew′ + F ′θ + G ′ ϕ 12 1 2 θ1 θ 2 +Eεε − βε − βε w (11) []12w1 2 w 2 1 ′−ε1t ′ − ε 2 t =βθ1eI1 θθ + β 2 eI 2 θ +[]Fεε12 + βεθ1 2 + βε θ 2 1 θ ′−ε1t ′ − ε 2 t +G [ϕ ++ ( ε εϕ ) + εε ϕ ] = 0 −βw1eI1 ww − β 2 eI 2 w 1 2 12
Where β= 2 πρUb λ , β′ =2 πρUb2(1 + a ) λ xi x i xi2 x i AwB′′′(4)+θ ( 4) +[ A( ε ++ ε ) Cw ′ ] and x denote θ andw . 1 2 i i i +[]B′(ε1 + ε 2 ) + D ′ θ Omitting the integral parts (Iiw , Iiθ ) , eq uations +[]A′εε + C ′( ε ++ ε ) Ew ′ will transform the Eqs(10), (11) into ordinary 12 1 2 ′ ′ ′ differential equations, those are so friendly to +[]Bεε12 + D( εε 1 ++ 2 ) F θ +C′εε + E ′( εε +++ ) β ′′ β w solve. So , multiplying equations (10, 11) by []12 1 2 w1 w 2 (15) ε1t ′ ′ ′′ e and differentiation with respect to t (time) +[]Dεε12 + F ( εε 1 +−− 2 ) βθ1 βθ θ 2 will yield to following equations: ′ ′ ′ +[]Eεε12 + βεw1 2 + βε w 2 1 w ′ ′ ′ +[]F εε1 2 − βεθ1 2 − βε θ 2 1 θ ε1t ε 1 t Ae(ε1 w+ w ) + Be ( εθθ 1 + ) +G ′ [ϕ ++ ( ε1 εϕ 2 ) + εεϕ 12 ] = 0 ε1t ε 1 t +Ce(ε1 w ++ w ) De ( εθθ 1 + ) The required initial conditions for the new ε1t ε 1 t eq uations of motion can be obtained from the +Ee(ε1 w ++ w ) Fe ( εθθ 1 + ) (12 ) old I .Cs and by putting them into Eqs.(10-13), ε1t +Ge (ε1 ϕ + ϕ ) the following additional I.Cs will be obtained.
ε1t( εε 12− ) t ε 2 t =−βθβθ1ee − θ 2 []( εε1 − 2 ) Ie 2 θ + θ ++βewε1t β e( εε 12− ) t[( εε −+ ) Iwe ε 2 t ] 1 w1 w 2 1 2 2 w w(0)= [( BCBCw′ − ′ )( 0) AB′− AB ′ Ae′ε1t(ε w+ w ) + Be ′ ε 1 t ( εθθ + ) 1 1 +−(BD′ BD ′ )θ ( 0)( + BE ′ − BEw ′ )( 0) (16) ′ε1t ′ ε 1 t +Ce(ε1 w ++ w ) De ( εθθ 1 + ) +−(BF′ BF ′ )θ ( 0)( + BG ′ − BG ′ )( ϕ 0)]
ε1t ε 1 t +Ee′(ε1 w ++ w ) Fe ′ ( εθθ 1 + ) (13) +G′ e ε1t (ε ϕ + ϕ ) 1 1 θ (0)= [(AC′ − AC ′ )( w 0) ′ ′ ε1t( εε 12− ) t ε 2 t AB− AB =+βθβθ′e θ ′ e[]( εε1 −+ 2 ) Ie 2 θ θ 1 2 +−(AD′ AD ′ )θ ( 0)( + AE ′ − AEw ′ )( 0) (17) −−β′ewε1t β ′ e( εε 12− ) t[( εε −+ ) Iwe ε 2 t ] w1 w 2 1 2 2 w +−(AF′ AF ′ )θ ( 0)( + AG ′ − AG ′ )( ϕ 0)] Similarly, multiplying left and right hand sides of equations (12, 13) by e(ε 2 −ε1 )t factor and differentiating with respect to t will o mit
I 2w and I 2θ . After simplifying and rearranging,
4 AN ANALYTICAL METHOD IN COMPUTATIONAL AE ROELASTICITY BASED O N WAGNER FUNCTION
w (0) = present method predict the same value for flutter 1 speed for the given airfoil. [((Aε+ CB )′ − ( A ′ ε + CBw ′ ))( 0) AB′− AB ′ 1 1 [17] +((Bε1 + DB )′ − ( B ′ ε 1 + DB ′ ))( θ 0) Table (2) : Given airfoil specifications 3 +((Cε1 + EB )′ − ( C ′ ε 1 + EBw ′ ))( 0) ρ 0.002378 Sluges/ft 2 ′ ′ ′ Ie. a . 1.606 Sluges .ft +((Dε1 + FB ) − ( D ε 1 + FB ))( θ 0) (18) ′ kθ 1003.75 ft .lb /rad +((Eε1 − βw1 − β w 2 ) B c= 2 b 5.18 ft −(E′ε + β ′ + β ′ ))( B w 0) 1 w1 w 2 m 1 Sluges +((Fε1 + βθ + β θ ) B ′ 1 2 kw 100 lb /ft −(F ′ε1 − β θ ′ − βθ ′ )B ) θ ( 0) 1 2 xθ 0.1 +(BG′ − BG ′ )((ε1 ϕ0) + ϕ ( 0))] a -0.2
U σ ω θ (0) = In following figures, , and are bωθ ωθ ωθ 1 [((Aε1+ CA )′ − ( A ′ ε 1 + CAw ′ ))( 0) nondimensional speed, damping factor, and AB′− AB ′ nondimensional frequency, respectively. +((Bε1 + DA )′ − ( B ′ ε 1 + DA ′ ))( θ 0) ′ ′ ′ +((Cε1 + EA ) − ( C ε 1 + EAw ))( 0) (19) 0.1 Nondimensional speed ((Dε1+ FA )′ − ( D ′ ε 1 + FA ′ ))( θ 0) 00 0.5 1 1.5 2 2.5 +((Eεββ1 −−ww )( AE′ − ′ εββ 1 ++ ww ′ ′ ))( Aw 0) 12 12 -0.1 +((Fεββ ++ )( A′ − F ′ εββ −− ′ ′ )A )θ ( 0)
1θ1 θ 2 1 θ 1 θ2 r -0.2 o t c a ′ ′ F -0.3 +(AG − AG )((ε1 ϕ0) + ϕ ( 0))] g n i p m
a -0.4 D
-0.5 4 Case study -0.6 present study p-k method (Theodorsen) [17] Verification of introduced formulation is -0.7 p method (Peters) [17]
carried out by considering a two dimensional -0.8 linear airfoil with specifications as shown in table (2). First the damping and frequency parts (a) of the eigenvalues of the aeroelastic system are
co mpared with the results that obtained based 1 on the Theodorsen and Peters theories. The
results of present formulation which are 0.8 y c n
obtained using the P method are compared with e u q e r
the results of the Theodorsen using P-k method F l
a 0.6 n o i
and Peters using P method, respecti vely. Figures s n e m i
(2a, 2b) show the good correspondence between d n o 0.4 the different theories. The difference between N present study the results of the present theory and Peters ’ p-k method (Theodorsen) [17] theory with those of the Theodorsen theory is 0.2 p method (Peters) [17]
due to using P and P -k methods. Also from 0 0.5 1 1.5 2 2.5 these figures, we can determine the dynamic Nondimensional speed (b) instability speed of aeroelastic system (flutter Figure 2. damping(a); and frequency (b).of the aeroelastic speed). All of the mentioned theories and system versus nondimensional airspeed
5 SH. SHAMS , H . HADDADPOUR , M.H . SADR LAHIDJANI , M . KHEIRI
In order to show the applicability of the present So, the second order initial conditions would formulation, the response of the aeroelastic be found as following system to a sharp -edged gust is determined. The sharp -edged gust is the most critical case for 1 w(0)= [( BCBCw′ − ′ )( 0) gust modeling. It can change the angle of attack AB′− AB ′ of the system largely and rapidly and can induce +−(BD′ BD ′ )θ ( 0)( + BE ′ − BEw ′ )( 0) (22) large lift and load factor beyond the structural strength. In order to account the gust response, +−(BF′ BF ′ )θ ( 0)( + BG ′ − BG ′ )( ϕ 0) first it is necessary to develop the formulation +(BH′ − BH ′ )ψ ( 0)] and to find the higher order initial conditions to cover the sharp -edged gust effects . In the 1 forwarding section the dynamic responses of the θ (0)= [(AC′ − AC ′ )( w 0) ′ ′ typical section due to a sharp -edged gust will be AB− AB +−(AD′ AD ′ )θ ( 0)( + AE ′ − AEw ′ )( 0) investigated. Those equations can be driven by (23) adding one part to equations (14, 15) as sharp - +−(AF′ AF ′ )θ ( 0)( + AG ′ − AG ′ )( ϕ 0) edged gust effects. These parts are based on Kussner’s function and obtained in a similar +(AH′ − AH ′ )ψ ( 0)] manner which was introduced in present study Also, the third order initial conditions in [18]. sharp -edged gust case are written as
(4) ( 4) Aw+ Bθ +[ A( ε1 ++ ε 2 ) Cw] w (0) = +[]B(ε + ε ) + D θ 1 2 1 [((Aε1+ CB )′ − ( A ′ ε 1 + CBw ′ ))( 0) +[]Aεε12 + C( ε 1 ++ ε 2 ) Ew AB′− AB ′ +((Bε + DB )′ − ( B ′ ε + DB ′ ))( θ 0) +[]Bεε12 + D( εε 1 ++ 2 ) F θ 1 1
+((Cε1 + EB )′ − ( C ′ ε 1 + EBw ′ ))( 0) +[]Cεε12 + E( εε 1 +−− 2 ) βw1 β w 2 w (20) ((Dε1+ FB )′ − ( D ′ ε 1 + FB ′ ))( θ 0) (24) +[]Dεε12 + F ( εε 1 +++ 2 ) βθ1 βθ θ 2 ′ ′ ′ ′ +((Eεββ1 −−ww12 )( BE − εββ 1 ++ ww 12 ))( Bw 0) +[]Eεε12 − βεw1 2 − βε w 2 1 w ′ ′ ′ ′ +((Fεββ1 ++θ1 θ 2 )( B − F εββ 1 −− θ 1 θ2 )B )θ ( 0) +[]Fεε12 + βεθ1 2 + βε θ 2 1 θ +(BG′ − BG ′ )((ε ϕ0) + ϕ ( 0)) +G []ϕ ++ ( ε1 εϕ 2 ) + εε 12 ϕ 1 +(BH′ − BH ′ )((ε1 ψ0) + ψ ( 0))] +H ψ ++ ( ε1 εψ 2 ) + εεψ 12 = 0
(4) ( 4) AwB′′′+θ +[ A( ε1 ++ ε 2 ) Cw ′ ] +[]B′(ε + ε ) + D ′ θ 1 2 θ (0) = ′ ′ ′ +[]Aεε12 + C( ε 1 ++ ε 2 ) Ew 1 ′ ′ ′ [((Aε1+ CA ) − ( A ε 1 + CAw ))( 0) +[]B′εε12 + D ′( εε 1 ++ 2 ) F ′ θ AB′− AB ′ +((Bε1 + DA )′ − ( B ′ ε 1 + DA ′ ))( θ 0) +[]C′εε12 + E ′( εε 1 +++ 2 ) βw ′′ β w w 1 2 (21) ′ ′ ′′ +((Cε1 + EA )′ − ( C ′ ε 1 + EAw ′ ))( 0) +[]Dεε12 + F ( εε 1 +−− 2 ) βθ1 βθ θ 2 (25) ′ ′ ′ ((Dε1+ FA )′ − ( D ′ ε 1 + FA ′ ))( θ 0) +[]Eεε12 + βεw1 2 + βε w 2 1 w
′ ′ ′ +((Eεββ1 −−ww )( AE′ − ′ εββ 1 ++ ww ′ ′ ))( Aw 0) +[]F εε1 2 − βεθ1 2 − βε θ 2 1 θ 12 12 ′ ′ ′ ′ +G ′ []ϕ ++ ( ε1 εϕ 2 ) + εεϕ 12 +((Fεββ1 ++θ1 θ 2 ) A − ( F εββ 1 −− θ 1 θ2 )A )θ ( 0) ′ +(AG′ − AG ′ )((ε ϕ0) + ϕ ( 0)) +H ψ ++ ( ε1 εψ 2 ) + εεψ 12 = 0 1 +(AH′ − AH ′ )((ε1 ψ0) + ψ ( 0))]
6 AN ANALYTICAL METHOD IN COMPUTATIONAL AE ROELASTICITY BASED O N WAGNER FUNCTION
where ψ(t ) stands for the Kussner function 1.3 1.2 and can be stated approximately as following 1.1 [16] 1 Ut 0.9 ψ(s ) =−1 0.5 e−0.130 s − 0.5 es − s , = (26)
o 0.8 i b t a r 0.7 g n i
Also the coefficients H and H ′ are h 0.6 c t i introduced in equation (23 ) as P 0.5 0.4 0.3 H= 2πρ Ubw 0 (27) 0.2 2 1 0.1 H′ =2πρ Ub( + aw )0 2 0 0 0.25 0.5 0.75 1 Where w0 is the gust speed. Time, s (b) For an airfoil with following specifications the Figure 3.Dynamic response t ime histories of the flexible dynamic responses in pre -flutter regimes are airfoil for µ = 14 and A = 0.0375 ; (a) heaving shown. The dynamic responses include the direction; (b) pitching direction. plunging and pitching amplitude variation versus time relative to steady state responses. 1.4 1.3 Table (3) : Given airfoil specifications 1.2 1.1 3 1 ρ 0.002378 Sluges /ft kw 100 lb /ft 0.9 o i t
x a 0.8 V 1(ft/s ) θ 0.0 r g
n 0.7 i
r v θ 0.5 a 0.0 a 0.6 e H c 1 ft 0.5 0.4 0.3 The results found for sharp -edged gust with unit 0.2 0.1 (w0 = 1 ft s ) value in four different conditions 0 and zero initial conditions. 0 0.25 0.5 0.75 1 Time, s (a) 1.3 1.2
1.2 1.1
1.1 1 1 0.9 0.9 0.8 o 0.8 o i i t t 0.7 a a r 0.7 r g g n n 0.6 i i v 0.6 h a c t e
i 0.5
H 0.5 P 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1
0 0 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Time, s (a) Time, s (b) Figure 4.Dynamic response time histories of the flexible airfoil for µ = 14 and A = 0.0845 ; (a) heaving direction; (b) pitching direction. 7 SH. SHAMS , H . HADDADPOUR , M.H . SADR LAHIDJANI , M . KHEIRI
1.4 1.2
1.3 1.1 1.2 1 1.1 0.9 1 0.8 0.9 o o i i t t 0.7 a a 0.8 r r g g n n 0.7 0.6 i i h v c a t
e 0.6 i 0.5 P H 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Time, s Time, s (a) (b) 1.2 Figure 6.Dynamic response time histories of the flexible 1.1 airfoil for µ = 21 and A = 1.335 ; (a) heaving 1 direction; (b) pitching direction. 0.9
0.8 o i t 0.7 a r 5 Conclusion g
n 0.6 i h c t
i 0.5 It was shown that modeling of the unsteady P 0.4 aerodynamics by a two -state representation of 0.3 the Wagner function and simplifying the 0.2 integro -differential aeroelastic equations of a 0.1 two -dimensional airfoil (typical section) with 0 0 0.25 0.5 0.75 1 the use of some mathematical methods can Time, s reduce these equatio ns to a set of fourth order (b) ordinary differential equations. Good simplicity Figure 5.Dynamic response time histories of the flexible of these equations makes them as a unique tool airfoil for µ = 21 and A = 0.338 ; (a) heaving to obtain dynamic responses to different inputs direction; (b) p itching direction. in time domain solutions. These equations were used to predict flutter speed in comp arison with the Theodorsen and Peters ’ aerodynamics of an 1.6 1.5 airfoil and its dynamic responses due to sharp - 1.4 edged gust with the given specifications. The 1.3 results show that the present method is a 1.2 1.1 powerful simplified analytical method for
o 1
i aeroelastic calculati ons with the least number of t a
r 0.9
g states in comparison with other methods.
n 0.8 i v
a 0.7 e
H 0.6 0.5 References 0.4 1 0.3 Wagner, H.,“Uber die Entstehung des dynamischen 0.2 Auftriebes von Tragflugeln ”, Zeitschrift fur angewandte 0.1 Mathematik und Mechanik , Vol .5, No . 1, February 1925 0 2 0 0.25 0.5 0.75 1 Theodorsen, T ., General Theory of Aerodynamic Time, s Instability and the Mechanism of Flutter, NACA Rept. (a) 496 , May 1934 , pp . 413 -433 .
8 AN ANALYTICAL METHOD IN COMPUTATIONAL AE ROELASTICITY BASED O N WAGNER FUNCTION
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