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 , , w w 1 This paper presents an analytical method in and pitching frequencies (k I ) 2 computational aeroelasticity for an airfoil e. a considering two degrees of freedom (heaving 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 give an integro -differential system of equations. t, time Using appropriate approximation for the Wagner function, a new form of equations can (t ) the Wagner function be obtained by derivation from mentioned (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, m 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 r 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 axis and center of mass domain unsteady aerodynamic lift and moment. w, airfoil motion in heave and pitch directions Also, the Ku ssner's function was applied for kw , k 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 + mbx + kww =− Lt( ) (1) these equations. Transforming the integral terms into differentials with the addition of two new second order differential equations and mbx w+ Iea. + k = 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, ) 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( ) = bw2 − ab + U approximation of the Wagner's function and by - 1 part integral meth od and therefore a set of two +2(0Ubw ) +− b (2 a )( 0) + U ( 0) ( t ) (3) t fourth -order differential equations with 1 +2Ub∫ () t −+−+ wb (2 a ) Ud corresponding initial conditions will be 0 obtained. The present formulation will be 31 2 1 Mtea. ()= bawb −+−− (8 a )() U 2 a examined i n the time and frequency domain and +2Ub2(1 + aw ) ( 0) +−+ b ( 1 a )( 0) U ( 0) ( t ) the obtained results will be compared with those 2 2 (4) t of oth er investigations. +2Ub2(1 + a )() t −+−+ wb ( 1 a ) 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 (mbx − b a )( w ++ Ie. a . b (8 + a )) function is approximated by [16]: 2 1 −2Ub(2 + a ) ( 0) w 31 3 1 2 − 1t − 2 t +(b U ( −− a ) 2 Ub( − a ) ( 0)) (t ) =1 − ce1 − ce 2 (5) 2 4 2 1 −2Ub(2 + a ) ( 0) w c = 0.165 U Where 1 , c2 = 0.335 , 1 = 0.0455 b and 2 2 1 +(2k − Ub(2 + a ) ( 0) (9) = 0.3 U . 2 b 31 2 −2Ub(4 − a ) ( 0)) 2 1 1 3 Aeroelastic Modeling +2Ub(2 + aw )( ( 0) +− b ( 2 a )( 0)) ( t ) =2( Ub 2 1 + aeI)( − 1t+ eI − 2 t Combining the structural and aerodynamic 2 11 2 2 equations (Eqs. 1-4), the aeroelastic equations − eI− 1t − eI − 2 t ) of motion can be obtained as follows: w11 ww 2 2 w 2 3 Where =cU − b(1 − a ) and (m+ b )( w + mbx − ab ) i i i i 2 t 2 +bU + kww =c2 and I = exd i () . wi ii ix ∫ (6) 0 +2Ubtw ( ) ( 0) +−+ b (1 a )( 0) U ( 0) 2 Using the coefficients defined in table (1): t =−2Ub () t − wb +−+ (1 a ) U ∫0 2 Table (1) : Define Coeffi cients for equations (10,11 ) (mbx− abw3 )( ++ I b 4 (1 + a 2 )) e. a . 8 A= m + b 2 +bU3 (1 − a ) + k 3 2 A′ = mbx − b a 2 1 1 (7) 3 −2Ub(2 + atw )()( 0) +−+ b ( 2 a )( 0) U ( 0) B= mbx − ab t ′ 41 2 2 1 1 BI=e. a . + b(8 + a ) =2Ub(2 + a )()∫ t −+−+ wb ( 2 aU ) 0 C= 2 Ub (0) C′ = −2 Ub2(1 + a ) ( 0) In order to eliminate the integral parts, using by - 2 2 2 1 part integral method and some simplification D= Ub +2 Ub(2 − a ) ( 0) ′ 31 3 1 2 along with using Eq (5) instead of the Wagner D= bU(2 −− a ) 2 bU( 4 − a ) ( 0) function, equations (6) and (7) will lead to E= k + 2 Ub (0) equations (8, 9). Equations ( 8) and (9) which are w ′ 2 1 ordinary differential equations of the aeroelastic E= −2 bU(2 + a ) ( 0) 2 2 1 system. F=2(0)2 Ub + Ub(2 − a ) ( 0) ′ 2 2 1 Fk= −2 bU(2 + a ) ( 0) (m+ bw2 )( + mbx − ab 3 ) + 2(0 Ub) w 31 2 −2b U(4 − a ) ( 0) +(2bU2 + Ub 2 (1 − a ) ( 0)) 2 1 G=−2 Ubw( ( 0) + b (2 − a )( 0)) +(2kw + Ub (0)) w G′ =2 bU2 (1 + aw )(( 0) +− b ( 1 a )( 0)) 2 2 1 2 2 +(2Ub (0) + 2 Ub(2 − a ) ( 0)) (8) 1 −2Ubtw( )( ( 0) + b (2 − a )( 0)) A simple form of the aeroelastic equa tions − 1t − 2 t =2(Ub − 1 eI1 − 2 eI 2 can be written as follows: − 1t − 2 t + w1eI1 ww + 2 eI 2 w ) 3 SH. SHAMS , H . HADDADPOUR , M.H . SADR LAHIDJANI , M . KHEIRI Aw+ B + Cw + D the following equations we arrive at following equations: +Ew + F + G (4) ( 4) Aw+ B +[] A( 1 ++ 2 ) Cw − t − t 510) = − eI1 − eI 2 11 2 2 +[]B( 1 + 2 ) + D − t − t 1 2 +[]A 12 + C( 1 ++ 2 ) Ew + w1eI1 ww + 2 eI 2 w +[]B 12 + D( 1 ++ 2 ) F +[]C + E( +−− ) w (14) Aw′+ B ′ + Cw ′ + D ′ 12 1 2 w1 w 2 +[]D + F ( +++ ) +Ew′ + F ′ + G ′ 12 1 2 1 2 +E − − w (11) []12w1 2 w 2 1 ′− 1t ′ − 2 t = 1eI1 + 2 eI 2 +[]F 12 + 1 2 + 2 1 ′− 1t ′ − 2 t +G [ ++( ) + ] = 0 − w1eI1 ww − 2 eI 2 w 1 2 12 Where = 2 Ub , ′ =2 Ub2(1 + a ) xi x i xi2 x i AwB′′′(4)+ ( 4) +[ A( ++ ) Cw ′ ] and x denote andw .