Unsteady thin airfoil theory revisited for a general deforming airfoil

Transcription

1 Journal of Mehanial Siene and Tehnology 4 () () 45~46 DOI.7/s Unsteady thin airfoil theory revisited for a general deforming airfoil Christopher O. Johnston, William H. Mason and Cheolheui Han,* Department of Aerospae and Oean Engineering, Virginia Polytehni Institute and State University, laksburg, VA 46, USA Department of Aeronautial and Mehanial Design Engineering, Chungju National University, 5 Daehak-Ro, Choongbuk, 38-7, Korea (Manusript Reeived January 9, ; Revised July, ; Aepted Otober, ) Abstrat The unsteady thin airfoil theory of von Karman and Sears is extended to analyze the aerodynami harateristis of a deforming airfoil. The von Karman and Sears approah is employed along with Neumark s method for the unsteady load distribution. The wake-effet terms are alulated using either the Wagner or Theodorsen funtion, depending on the desired amberline deformation. The onept of separating the steady and damping terms in the amberline boundary ondition is introdued. The influene of transient and sinusoidal airfoil deformations on the airfoil load distribution is examined. The general equations developed for the unsteady lift and load distribution are evaluated analytially for a morphing airfoil, whih is defined here by two quadrati urves with arbitrary oeffiients. This general amberline is apable of modeling a wide range of pratial amberline shapes, inluding leading and trailing edge flaps. Results of the present model are shown for a variable amber, or morphing, airfoil onfiguration. The influene of transient and sinusoidal motion on the fore oeffiients and load distribution is addressed. Keywords: Morphing airfoil; Thin airfoil theory; Von Karman and Sears approah; Unsteady aerodynamis Introdution A fundamental problem in applied aerodynamis onerns the predition of the time-history of the aerodynami fores ating on an airfoil following a hange in the airfoil geometry []. Reent interest in morphing ontrol devies, whih produe a hingless hange to the airfoil amber, has led to the possibility of amberline shape hanges other than just a onventional hinged flap [, 3]. The aerodynami fores ating on the airfoil during and after the hange in shape have an influene on both the atuator design and airraft dynamis [4, 5]. eause of the large variety of possible shape hanges for a morphing airfoil, it is desirable to formulate the problem in a theoretial framework that provides insight into the unsteady response of the aerodynami fores to a general hange in airfoil shape. A valid approah for this task is the lassial unsteady thin airfoil theory developed by Theodorsen [6] and von Karman and Sears [7]. Although suh a lassial tehnique ontains many approximations (invisid, inompressible, small disturbanes) and one may obtain more aurate results from widely available numerial approahes, the benefit of being able to separate various omponents of the unsteady load distribution or fore and moment oeffiients provides This paper was reommended for publiation in revised form by Assoiate Editor Do Hyung Lee * Corresponding author. Tel.: , Fax.: address: hhan@jnu.a.kr KSME & Springer insight into the problem, whih may then be onfirmed through more advaned methods. Thus, the unsteady thin airfoil theory warrants onsideration. There have been many appliations of unsteady thin airfoil theory applied to trailing edge flaps reported in the literature [6-6], and sine the trailing edge flap is a speial ase of the general deforming amberline disussed later in this paper, these studies have partiular relevane to the present problem. One of the earliest of these studies, the well known appliation to a harmonially osillating trailing edge flap by Theodorsen s [6] provided the lift and pithing moment harateristis of the airfoil. Although Theodosen also presented results for the flap hinge moments, no results were given for the airfoil load distributions. Postel and Leppert [8], along with a number of German researhers (Dietze [9-], Jaekel [], Shwarz [3, 4], and Sohngen [5, 6]) determined the load distribution on a harmonially osillating airfoil. Reently, Mateesu and Abdo [7] obtained both the unsteady fores and the load distribution using the method of veloity singularities. Narkiewiz et al. [8] solved a similar problem whih allowed for arbitrary flap and airfoil osillations. Lieshman [9] and Hariharan and Lieshman [] onsidered arbitrary flap defletions using indiial onepts, although both foused mainly on the ompressible flow ase. Although there has been onsiderable researh on the theoretial unsteady aerodynamis of a trailing edge flap, there has been little researh onerning other deforming amberline

2 45 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~46 Table. Geometri oeffiients for speifi amberlines represented by Eq. (4). (a) Trailing-edge (TE) flap (b) Leading-edge (LE) flap () Conformal TE flap (d) Conformal LE flap (e) NACA 4-digit (Config. A) (f) NACA 4-digit (Config. A) A C A C - x -x x ( x ) ( x ) ( x ) x x x x x x ( x ) ( x ) ( x ) ( x ) x - ( ) x x x x Table. Camberlines Represented by Eq. (4) and Table. TE Flap LE Flap Conformal TE Flap Conformal LE Flap x x ( ) [5] determined the lift and pithing moment for ubi polynomial amberlines. No mention was made of the load distribution. Singh [6] and Malean [7] presented a numerial model for the unsteady aerodynamis of deforming airfoils, although they did not disuss the agreement of their methods with theory. Other soures of information regarding the modeling of deforming amberlines are the theoretial studies of membrane airfoils. For example, Llewelyn [8] and oyd [9, 3] disuss the appliation of steady thin airfoil theory to various single-segment polynomial amberlines, whih are similar to those of interest for morphing appliations. The aim of this paper is to outline a systemati appliation of unsteady thin airfoil to a general deforming amberline, whih will be omposed o two paraboli urves. This shape has the ability to model a wide variety of amberline shapes, and it an be redued to a onventional trailing edge flap. This feature is onvenient for validation and for illustrating the differenes of the trailing edge flap with other more ompliated onfigurations. For the quasi-steady fore oeffiient and load distribution alulation, the present method ombines the approahes of Glauret [3] and Allen [33]. The von Karman and Sears approah [7] for the unsteady fore oeffiient alulation is adopted along with Neumark s method [34] for the unsteady load distribution. The wake-effet terms are alulated using either the Wagner or Theodorsen funtion. The influene of transient and sinusoidal airfoil deformations on the airfoil load distribution are examined. It should be noted that a feature of the present study is the analyti determination of the unsteady load distribution for the general deforming amberline. This has not been presented previously in the literature.. Unsteady thin airfoil method NACA 4-digit (Config. A) NACA 4-digit (Config. ) shapes. Shwarz [3, 4] studied an osillating paraboli ontrol surfae. This shape was meant to aount for the visous effets of a onventional flap by smoothing out the hinge line. Reent interest in morphing airraft has reated interest in this paraboli, or onformal flap, for use as a hingless ontrol surfae [, ]. Unfortunately, the previous work of Shwarz [3, 4] has gone unnotied in the English-speaking literature exept for the brief mention by Garrik [3] and the disussion in the translated artile by Shwarz [4]. Spielburg [4] studied the unsteady aerodynamis of an airfoil with an osillating paraboli amberline. This study was restrited to a irular ar paraboli amberline, whih was meant to represent hordwise aeroelasti deformations. Mesari and Kosel Considering the inompressible small disturbane flow around a two dimensional thin airfoil, von Karman and Sears [7] represented the lift and the pithing moment in terms of the following three omponents: quasi-steady, apparent mass and wake indued. Their results are based on the oordinate system with its origin fixed to the mid-hord and with its x- axis set parallel to the hord. The origin of the oordinate system in the present method is set to the leading edge with its x- axis parallel to the hord (See Table ). Applying the transformation x = ( os ) / to von Karman and Sears results, the three (quasi-steady, apparent mass, wake-indued) omponents of the lift are written as follows: L = ρu Γ (a) (a) ρ L = γ ( ) ossind 4 t (b) (b) L ρ = γ ξ U dξ () () ξ ξ where Γ and γ are the quasi-steady irulation and vortiity

3 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~ values, respetively, and γ (ξ) is the vortiity in the wake at the loation ξ. It is onvenient to represent γ using Glauert s Fourier series [3] whih is written as + os γ (, t) = U A( t) An () t sinn sin + () n= where A = w(, t) d/ and A, os / n = w t n d. In these equations, w is the downwash that is used to impose the instantaneous boundary ondition for no flow through the amberline. For a time-varying amberline defined by z, the downwash, w, is written as z z w = +. U t x Substituting Eq. (3) into Eqs. () and () leads to the following equations for the quasi-steady and apparent mass ontributions to the lift L = ρu ( A + A/) (4a) U L = ρ ( A + A ). 8 (4b) Note that the dot above the A and A in Eq. (4b) indiates the time-derivative of the oeffiient, whih atually indiates a time derivative of w only sine the rest of the equation is independent of time. The unsteady pressure distribution is formulated in terms of the equations presented by Neumark [34], whih onveniently orrespond to the three lift omponents presented in Eq. (). Applying the transformation x = ( os ) / to p (quasi-steady omponent) and p (apparent mass omponent) and p (wake indued omponent) in [34] results in + os p = ρuγ( ) = ρu A An sinn sin + (5a) n= p = ρ ( d/ dt) γ sin d (5b) n ρu x γ ξ p = dξ. x (5) ξ ξ These equations require that the irulatory ( γ ) and nonirulatory ( γ n ) quasi-steady vortiity distributions be defined. From Neumark [34], γ and γ n may be written as γ ( ) (3) Γ = (6a) sin U w(, t)sin γ (6b) n = d sin os os where Γ = Ub w(, t) ( os ) /( + os ) sind. The quasi-steady irulation strength ( Γ ) in Eq. (6a) is obtained by integrating Eq. () aross the hord, whih results in the following A Γ = U A +. Substituting this equation for Γ into Eq. (6a) results in γ ( ) = U( A + A)/sin. Similarly, the nonirulatory omponent of the vortiity distribution ( γ n ) may be written as ( t) os U w, sin γ. (8) n = UA UA + d sin sin os os Substituting Eq. (8) into p in Eq. (5b) and performing the integration results in p = ρ ( d / dt)[uasin UA γb ( ) sin d ] + (9) where the basi load distribution ( ( t) U w, sin. γ b ( ) = d os os γ b (7) ) is defined as For many pratial appliations, the funtion z, whih defines the amberline shape, an be represented in the following form: zx (, τ ) = ψ( x) β( τ) () where ψ defines the form of the amberline (for example a flapped or a paraboli amberline) and β defines the time varying magnitude of the amberline (for example the flap defletion angle or magnitude of maximum amber) as a funtion of the nondimensional time ( τ = U t/ ). This form for z will be used throughout the remainder of this paper. To identify the aerodynami fores resulting from the steady amberline slope and the unsteady amberline shape hange, w will be written in general as z z w= + = ws + w x τ d () ψ where w s is the steady omponent ( ws = β ) whih and x is proportional to β and w d is the damping omponent ψ β ( wd = ) whih is proportional to / τ dβ dτ. oth w s and w d result in their own quasi-steady irulation omponents ( Γ,s and Γ,d ), whih onsequently result in their own apparent-mass and wake-effet omponents through Eq. (5b) and (5). Therefore, the terminology of steady and damping, denoted by a subsript s and d, will be used throughout this paper to distinguish between lift or pressure omponents resulting from w s and w d, respetively. Using this onept, the quasi-steady lift (L ) and apparent mass lift (L ) an be written

4 454 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~46 in terms of the lift omponents due to the amber (β ), the amberline shape-hange with time ( β ), and amberline shape-hange aeleration ( β ) as follows: L dl dl = CL, = β + β q q dβ q dβ = K β + K β = ( A + A ) β + ( A + A ) β, s, d, s, s, d, d (a) L dl dl = CL, = β + β q q dβ q dβ (b) = K, sβ + K, dβ = ( A, s + A, s) β + ( A, d + A, d) β 4 4 where q is the dynami pressure, and K,s and K,d are the apparent mass lift omponents due to the time-rate-of-hange of γ,s and γ,d, respetively. The Fourier oeffiients in Eq. () are defined as follows: A, s ψ = d (3a) x ψ Ans, = os( n ) d (3b) x A, d ψ = d (3) ψ And, = os( n ) d. (3d) The quasi-steady and apparent-mass load distribution an also be separated into steady and damping omponents as follows: Δp( ) dδp dδp Δ C p, = β + β q q dβ q dβ = A, sχ( ) + T, s( ) β + A, dχ( ) + T, d( ) β (4a) Δp( ) dδp dδp " =Δ C p, = β + β " q dβ dβ " = T ( ) β + T ( ) β (4b), s, d where the load distribution funtions are defined as T ψ (, t) sin 4 ( ) = x d (5a), s os os T ( ) = ( t) 4 ψ, sin d (5b), d os os dδp, s, s, s, s q dβ dδp, d ", d, d, d qdβ T = A sin A + T sind (5) T = A sin A + T sind (5d) with χ( ) = 4( + os ) /sin. In these equations, T,s and T,d are steady and damping omponents due to the steady and damping basi load distributions, respetively. The first term in Eq. (4b) is the apparent mass term (Δp,s ) due to the timerate-of-hange of γ,s, and the seond term in Eq. (4b) is the damping term (Δp,d ) due to the time-rate-of-hange of γ,d. Colleting the terms defined in the previous paragraphs for the quasi-steady and apparent-mass ontributions to the lift and load distribution, the following equations an be written: C = K β + ( K + K ) β + K β + C (6a) L, s, d, s, d L, Δ Cp = A, sχ( ) + T, s( ) β + (6b) A, dχ + T, d( ) + T, s( ) β + T, d( ) β +ΔCp, where the wake-effet omponents (C L, and ΔC p, ) depend on the time variation of β. For transient and sinusoidal variations of β, the wake-effet omponent of the lift oeffiient, C L,, is written as follows: C τ = ( K Δ βτ + K Δβ( τ)) φτ (transient) (7a) + K β σ + K β σ φ τ σ dσ L,, s, d τ (, s, d ) τ ( β β ) C = [ C( k) ] K + K (sinusoidal) (7b) L,, s, d where φ is the Wagner funtion and C (k) is the Theodorsen funtion, k = ω/u is the redued frequeny. The Wagner funtion may be modeled using the approximation introdued.9τ.6τ by Jones [35], whih is written as φτ =.65e.335e. The Theodorsen funtion is a omplex number written as Ck = Fk + igk. For both the transient and sinusoidal ases, the load distribution is written as follows: + os Δ C = C sin p, L, (8) where the time dependene is ontained in C L,. The appliation of Theodorsen s funtion to a sinusoidal osillation of L an be generalized by applying the separation of steady and damping terms defined previously to a sinusoidal osillation of β. Consider a time-varying β defined as follows: ikτ β = βe = β os( kτ) + isin ( kτ) (9) where β is the magnitude of the osillation and k = ω/ U and τ = Ut /. Note that k is defined differently than the lassially defined k, where k =k. From Eqs. (6a) and (7b), the total unsteady lift oeffiient may be written for sinusoidal motion as L( τ ), sβ (, d, s) β, dβ [ Ck ]( K, sβ K, dβ ) C = K + K + K + K () + +

5 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~ Substituting Eq. (9) into () and keeping only the real part results in L C ( τ, k) = Z k β os kτ + Z k β sin kτ () where Z( k) = K, dk K, sf + K, dkg and Z k = K, sk + K, sg+ K, dkf. The lift an also be presented in terms of a magnitude and phase angle as follows: ( τ, ) β os( τ φ ) C k = C k + () L L L where CL Z Z φ L = tan Z/ Z. Likewise, the load distribution an be written as p = +, C ( τ,, k) =Π k, β os kτ +Π k, β sin kτ (3) where ( k ) ( k, ) A, sχ + T, s( ) T, d( ) k Π, = χ ( K, sf K, s kk, dg) + (, dχ +, d( ) +, s( ) ) A T T k χ + + Π = ( K, sg kk, df kk, d). and 3. Appliation to a general deforming amberline The equation for a general amberline onsisting of two quadrati segments may be written as zn, = βτ anx + bnx+ n ; n = (4) for x x and n = for x x where x is the loation at whih the two segments onnet. The oeffiients in Table represent six basi amberline shapes of possible interest and Table illustrates these shapes. Applying these definitions to Eq. (4), while also applying the x = os /, results in the following: transformation of ( ) ws, n = βτ Hnos + In (5a) β ( τ) wdn, = ( En os + Fn os + Gn ) (5b) where Hn = an, In = an + bn, En = an/4, Fn =an/ bn/, and G = a /4 + b /+. n n n n 3. Quasi-steady load distribution Having defined the steady and damping omponents of downwash w in Eq. (), the quasi-steady load distribution and fore oeffiients may be alulated. The main hallenge in obtaining the load distribution is evaluating the integral for the basi load distribution (Eq. (4)). Substituting Eq. (5) into Eq. (5) results in the following two equations for the steady and damping omponents of the nondimensional basi load distribution T, s T, d [ ( I I) + ( HH)os ] ( ) + ( ) + + [ H + ( ) H ] sin sin tan / os ( ) = ln sin tan / os ( G G) + ( FF)os + ( E E)(+ os ) sin tan ( / ) os + ( ) = ln. sin tan ( / ) os + ( F + Eos ) + + ( )( F + Eos ) sin + ( EE)sin (6a) (6b) Eq. (6a) may be validated for a onventional TE flap by omparing it with Spene s result [36] and for a onformal TE flap by omparing with Johnston s result [37]. For an NACA 4-digit airfoil ase, Eq. (6a) an be shown to ompare well with Pinkerton s approximate result [38]. As shown in Eq. (4a), the omponent of the quasi-steady load distribution is not dependent on T,s or T,d, but dependent only on A,s and A,d. This omponent of the load distribution, alled the additional load distribution, has the shape of an angle of attak produed load distribution (represented by ψ). Eq. () shows that the quasi-steady lift depends on the first three Fourier oeffiients (the fourth oeffiient, A 3,s, is alulated as well to determine A,d ). The steady oeffiients are evaluated as follows, A, s = I ( ) I ( H H ) sin + + H( ) + H + A, s = ( I I ) sin ( H H) os + (7) 3( I I) os + A, s = sin 3 ( H H)( + os ) 3 A3, s = 6 ( H H ) os sin + ( I I ) sin3. 3 For a TE flap, these oeffiients an be shown to agree with Glauert s [3] or Allen s [33] result. For the damping quasisteady terms, only A,d must be determined beause of the relationships for A,d and A,d given in Eq. (3). 3. Apparent mass load distribution Examining Eq. (4b) for the apparent mass load distribution ( C p, ), it is lear that the only omponent not yet known is the integral term required for T,d (the Fourier oeffiients were required for the quasi-steady terms and T,s does not need

6 456 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~46 to be determined beause it is equal to T,d ). Thus, the only required integration for T, ( ) is the following T, d sin d d + sin (3E+ G)os + 3( F)os ln Eos3 + sin = 3 + sin (3E G)os 3( F)os + + ln Eos3 + sin (4E+ G+ 6Fos ) sin + Eossin 6 sin sin 4 os sin sin sin sin + F + E + E + + 3( E( ) + E)sin 3 + 6( F( ) + F + Esin )( ossin ) ( E( ) + E)sin3 (8) where the terms E, F, and G represent E -E, F -F, and G -G, respetively. Note that this term is proportional to the aeleration of the amberline deformation (β ). The apparent mass load distribution for the onventional TE flap ase an be shown to agree with the result obtained by Postel and Leppert [8]. Table 3. Lift and pithing moment parameters for a pithing and heaving airfoil. PITCH (AOUT X A ) HEAVE K,s K,d ( x ) 3/4 a - K,s / K,d ( x ) / / a - / J,s J,d - /8 J,s - /8 J,d ( )( x ) / 5/8 a /8 Table 4. Load distribution parameters for a pithing and heaving airfoil. PITCH (AOUT X A ) HEAVE A,s A,d / xa - T,s T,d sin (/ ) 4x os sin sin T,d a Fig.. Steady omponent of the quasi-steady load distribution for the Morphing Configuration A amberline with various maximum amber loations. 4. Results and disussion Two simple and pratial amberline motions, pith and heave, provide a good example of applying the general deforming amberline results derived in the previous setion. For an airfoil pithing around an axis x a (measured from the leading edge), the shape funtion ψ is ψ = xa x = xa ( /)( os ). For a heaving airfoil, ψ is written simply as ψ =. Tables 3 and 4 present the parameters defined in the present method for the pith and heave ases. The omplete lift and load distribution equations (Eqs. () and (4)) resulting from this derivation agree well with the known result (pp. 6-7 of isplinghoff, et al. [39]). To illustrate the apability of the general deforming amberline derivation disussed in the previous setion, the results of the Morphing Configuration A amberline are presented. This ase is of interest to morphing airraft design where a similar amberline deformation ould be applied. The unsteady load distribution and fore oeffiients for this ase have not been presented previously in the literature. Figs. and show the quasi-steady load distribution due to the steady (T,s ) and damping (T,d ) omponents, whih are obtained from Eqs. (6a) and (6b). The influene of the maximum amber loation on these load distribution omponents is shown by omparing the results for three maximum amber loation (x ) values. Similarly, Fig. 3 shows the damping omponent of the apparent mass load distribution (T,d ) due to the time-rate-of hange of γ,d, whih is obtained from Eqs. (5d) and (8). Figs., and 3 represent the only onfiguration-speifi load distribution omponents required for the omplete unsteady load distribution defined in Eq. (6b), sine T,s is equal to T,d and the wake-indued omponent is represented by χ. Therefore, the unsteady load distribution for transient or sinusoidal motion may be determined by superimposing these load distributions. As shown in Fig., the position of peak values of the steady omponents aords with the position of the maximum amber. oth Fig. and Fig. 3 show that the position of the maximum amber does not affet the positions of the peak values of the damping oeffiients. Thus, it an be said that the positions of the peak values of the damping omponents of both

7 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~ Fig.. Damping omponent of the quasi-steady load distribution for the Morphing Configuration A ase with various maximum amber loations. Fig. 5. Steady and damping apparent mass lift oeffiients for the Morphing Configuration A ase with various maximum amber loations. Fig. 3. Damping omponent of the apparent mass load distribution for the Morphing Configuration A ase with various maximum amber loations. Fig. 4. Steady and damping quasi-steady lift oeffiients for the Morphing Configuration A ase with various maximum amber loations. Fig. 6. Influene of the redued frequeny on the lift-oeffiient magnitude for Morphing Configuration A and. quasi-steady and apparent mass load distributions are not muh affeted by the loation of the maximum amber. The load distribution due to the damping aerodynami fore omponent is shown to be equivalent for the morphing amberline with the maximum amber loated at equal distanes from the airfoil enter. In ontrast, the load distribution due to the steady aerodynami fore term hanges sign for these equivalent onfigurations. Figs. 4 and 5 show the influene of the maximum amber loation on the quasi-steady and apparent-mass lift omponents, respetively. The K,s term represents the quasi-steady lift per-unit β due to the amberline shape and K,d represents the quasi-steady lift per-unit dβ/dτ generated by the rate of the amberline shape-hange. Fig. 4 shows that when x is shifted aft the K,s value inreases whereas the K,d term beomes more negative. For the lift omponents shown in Fig. 5, the K,s term represents the apparent mass lift per-unit dβ/dτ while K,d represents the apparent mass lift per-unit d β/dτ. It is seen that K,s is negative for x / values less than ½ while K,d is negative for all x values. Thus, it an be said that the steady

8 458 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~46 Fig. 7. Influene of the redued frequeny on the lift-oeffiient phaseangle for Morphing Configuration A and. lift effetiveness inreases for a morphing amberline as the position of maximum amber is shifted aft, while the damping lift beomes more negative. To illustrate the appliation of the present approah to a sinusoidal ase, the differenes between the lift oeffiient for the Morphing Configuration A and Configuration ases are presented. Fig. 6 shows the values of CL / K, s for these two morphing onfigurations. Note that K,s is different for eah ase as shown in Fig. 4, although it is used here to normalize eah ase to the same osillation in the steady-state C L. The only differene between Configurations A and is w d. In fat, w d for Configuration is a ombination of the w d for Configuration A along with a heaving motion. The effet of maximum amber loation (x ) is seen to have a similar effet for both ases. Fig. 6 shows that the magnitude of lift is signifiantly larger for Configuration A. Fig. 7 presents the phase angle of the lift oeffiient for the two onfigurations. For Configuration A, only a negative phase angle is shown while for the Configuration the phase angle beomes positive as the redued frequeny is inreased. As shown in Figs. 6 and 7, the study of two different variable amber onfigurations, identified here as Configuration A and, indiates that in the absene of amberline aeleration, the only differene between the two onfigurations is the magnitude of the damping additional load distribution. For a positive hange in lift, the time rate of the magnitude of the amberline terms produe positive lift for Configuration and negative lift for Configuration A. This omponent of the lift an have a signifiant effet on the maneuverability of a flight vehile, and it is therefore a notable result. The effet of maximum amber loation is seen to have a similar effet for both ases. For osillatory motions, the magnitude of lift is signifiantly larger for Configuration A for all values of redued frequeny. The phase angle for lift is positive and inreases with inreased redued frequeny for Configuration, while it inreases slightly, but remains negative, for Configuration A. 5. Conlusions The unsteady thin airfoil theory for the aerodynami analysis of deforming airfoils is derived based on the theory of von Karman and Sears for both transient and osillatory shape hanges. It is found that the steady lift effetiveness inreases for a morphing amberline as the position of maximum amber is shifted aft, while the damping lift beomes more negative. In ase of the study of two different variable amber onfigurations, the time rate of the magnitude of the amberline terms produe positive lift for Configuration and negative lift for Configuration A for a positive hange in lift. Thus, the maneuverability of a flight vehile of a morphing wing an be affeted by the differene of morphing atuation methods. For osillatory motions, the magnitude of lift of the Configuration A wing is signifiantly larger than the Configuration wing for all the investigated values of redued frequeny. It is expeted that the onveniene of the analyti pressure distributions and fore oeffiients obtained by the present method an be useful for fundamental studies of various morphing airfoil onfigurations. Aknowledgment The researh is partially supported by asi Siene Researh Program through the National Researh Foundation of Korea (NRF) funded by the Ministry of Eduation, Siene and Tehnology (KRF ) and by a grant from the Aademi Researh Program of Chungju National University in 8. Nomenlature A nb, : Fourier oeffiients defined in Eq. (3), n =,,.., and b is the same as defined for T a,b : Chord length C L,n : Lift oeffiient, n =,, and orrespond to the quasi-steady, apparent mass, and wake-effet terms C M,n : Quarter-hord pithing moment oeffiient, n represents the terms defined with C L k : Redued frequeny defined as ω/u L n : Lift fore defined in Eq. (), n =,, and orre spond to the quasi-steady, apparent mass, and wake-effet terms q : Dynami pressure t : Time T a,b : Components of the aerodynami load distribution defined in Eq. (5), a = and orresponding to the quasi-steady and apparent mass terms, and b = s and d orresponding to the omponents resulting from the steady and damping boundary ondition defined in Eq. () K a,b : Components of the lift oeffiient defined in Eq. (), the subsripts are defined for T a,b U : Free-stream veloity

9 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~ w : Indued veloity on the airfoil amberline x : Distane along the airfoil hord aligned with the free-stream veloity x : Loation along the airfoil where the two segments of the general deforming amberline onnet Z n : Components of the osillatory lift oeffiient defined in Eq. () α : Angle of attak β : Defines the time-history of the amberline shape hange χ : Shape of an angle of attak load distribution defined under Eq. (5) C p,n : Nondimensional unsteady pressure loading, n =,, and orrespond to the quasi-steady, apparent mass, and wake-effet terms p n : Unsteady pressure loading defined in Eq. (5), n =,, and orrespond to the quasi-steady, apparent mass, and wake-effet terms γ : Cirulatory quasi-steady vortiity distribution γ n : Nonirulatory quasi-steady vortiity distribution Π n : Components of the osillatory load distribution defined in Eq. (3) : Polar oordinate along the airfoil surfae τ : Nondimensional time ψ : Shape funtion of the airfoil amberline defined in Eq. () ω : Frequeny Referenes [] E. Forster,. Sanders and F. Eastep, Synthesis of a variable geometry trailing edge ontrol surfae, AIAA paper 3-77, April 3. [] E. Forster,. Sanders and F. Eastep, Modeling and sensitivity analysis of a variable geometry trailing edge ontrol surfae, AIAA paper 3-87, April 3. [3] E. Stanewsky, Aerodynami benefits of adaptive wing tehnology, Aerospae Sienes and Tehnology, 4 () [4] C. O. Johnston, et al., Atuator-work onepts applied to unonventional ontrol devies, Journal of Airraft, 44 (7) [5] C. O. Johnston, et al., A model to ompare the flight ontrol energy requirements of morphing and onventionally Atuated wings, AIAA paper 3-76 (3). [6] T. Theodorsen, General theory of aerodynami instability and the mehanism of flutter, NACA Report No. 496 (935). [7] T. von Karman and W. R. Sears, Airfoil theory for nonuniform motion, Journal of the Aeronautial Sienes, 5 () August, (938) [8] E. E. Postel and E. L. Leppert, Theoretial pressure distribution for a thin airfoil osillating in inompressible flow, Journal of the Aeronautial Sienes, 5 (8) August (948) [9] F. Dietze, The air fores on a harmonially osillating selfdeforming plate, Luftfarhtforshung, 6 () (939) 84. [] F. Dietze, Law of aerodynami fore of a jointed plate in harmoni motion, Luftfarhtforshung, 8 (4) (94) 35. [] F. Dietze, Comparative alulations onerning aerodynami balane of ontrol surfaes, in Three Papers from Conferene on Wing and Tail Surfae Interations, (Translated from Lilienthal-Gesellshaft fur Luftfahrtforshung, Marh 94, pp. 6-74, NACA TM 36 (95). [] K. Jaekel, On the alulation of the irulation distribution for a twod-dimensional wing having periodi motion, Luftfarhtforshung, 6 (3) (939) 35. [3] L. Shwarz, Calulation of the pressure distribution of a wing harmonially osillating in two-dimensional flow, Luftfarhtforshung, 7 () (94) 379. [4] L. Shwarz, Aerodynamially equivalent systems for various forms of ontrol surfaes within the sope of the twodimensional wing theory, in Three Papers from Conferene on Wing and Tail Surfae Interations, (Translated from Lilienthal-Gesellshaft fur Luftfahrtforshung, Marh 94, pp. 6-74), NACA TM 36 (95). [5] H. Sohngen, Remarks onerning aerodynamially balaned ontrol surfaes, in Three Papers from Conferene on Wing and Tail Surfae Interations, (Translated from Lilienthal-Gesellshaft fur Luftfahrtforshung, Marh 94, pp. 6-74), NACA TM 36 (95). [6] H. Sohngen, Determination of the lift distribution for optionally non-uniform movement, Luftfarhtforshung, 7 () (94) 4. [7] D. Mateesu and M. Abdo, Unsteady aerodynami solutions for osillating airfoils, AIAA paper 3-7 (3). [8] J. P. Narkiewiz, A. Ling and G. T. S. Done, Unsteady aerodynami loads on an aerofoil with a defleting tab, Aeronautial Journal, August/ September (995) 8-9. [9] J. G. Leishman, Unsteady lift of a flapped airfoil by indiial onepts, Journal of Airraft, 3 () Marh-April (994) [] N. Hariharan and J. G. Leishman, Unsteady aerodynamis of a flapped airfoil in subsoni flow by indiial onepts, Journal of Airraft, 33 (5) Marh-April (996) []. Sanders, F. E. Eastep and E. Forster, Aerodynami and aeroelasti harateristis of wings with onformal ontrol surfaes for morphing airraft, Journal of Airraft, 4 Jan.- Feb. (3) [] E. Forster,. Sanders and F. Eastep, Modeling and sensitivity analysis of a variable geometry trailing edge ontrol surfae, AIAA paper 3-87, April 3. [3] I. E. Garrik, Nonsteady wing harateristis, in Aerodynami Components of Airraft at High Speeds, Vol. VII of High Speed Aerodynamis and Jet Propulsion, edited by Donovan, A. F., and Lawrene, H. R. (957). [4] I. N. Spielburg, The two-dimensional inompressible aerodynami oeffiients for osillatory hanges in airfoil amber, Journal of the Aeronautial Sienes, (6) June (953) [5] M. Mesari and F. Kosel, Unsteady airload of an airfoil

10 46 C. O. Johnston et al. / Journal of Mehanial Siene and Tehnology 4 () () 45~46 with variable amber, Aerospae Siene and Tehnology, 8 (4) [6] J. Singh, Unsteady aerodynamis of deforming airfoils A theoretial and numerial study, AIAA paper 96-6 (996). [7]. J. Malean and R. A. Deker, Lift analysis of a variable amber foil using the disrete vortex blob method, AIAA Journal, 3 (7) July (994) [8] R. P. Llewelyn, The veloity distribution on a polynomial mean line and the onverse, Journal of the Royal Aeronautial Soiety, 68 (964) [9] E. A. oyd, Comment on a onjeture of tanner, Journal of the Royal Aeronautial Soiety, 67 (963) 7-9. [3] E. A. oyd, Generalisation of the ondition for waviness in the pressure distribution on a ambered plate, Journal of the Royal Aeronautial Soiety, 67 (963) [3] E. A. oyd, Comment on the The load distribution on a polynomial mean line and the onverse, Journal of the Royal Aeronautial Soiety, 68 (964) 59. [3] H. Glauert, The Elements of Aerofoil and Airsrew Theory, nd Edition, Cambridge University Press (947). [33] H. J. Allen, General theory of airfoil setions having arbitrary shape or pressure distribution, NACA Report No. 833, 943. [34] S. Neumark, Pressure distribution on an airfoil in nonuniform motion, Journal of the Aeronautial Sienes, 9 (3) Marh (95) 4-5. [35] R. T. Jones, The unsteady lift of a wing of finite aspet ratio, NACA Report No. 68 (94). [36] D. A. Spene, The lift on a thin airfoil with a jet-augmented flap, Aeronautial Quarterly, 9 Aug. (958) [37] C. O. Johnston, Atuator-Work Conepts Applied to Morphing and Conventional Aerodynami Control Devies, M.S. Thesis, Virginia Teh (3). [38] R. M. Pinkerton, Calulated and measured pressure distributions over the midspan setion of the NACA 44 airfoil, NACA Report No. 563 (936). [39] R. L. isplinghoff, H. Ashley and R. L. Halfman, Aeroelastiity, Addison-Wesley (955). Christopher O. Johnston is urrently an Aerospae Engineer in the Aerothermodynamis ranh at NASA Langley Researh Center. He reeived his.s., M.S., and Ph.D. at Virginia Teh, whih is where the present work was performed. Cheolheui Han reeived a.s. degree in Mehanial Engineering from Hanyang University in 993. He reeived his M.S. and Ph.D. degrees from Hanyang University in 998 and 3, respetively. Then, he worked as a visiting post-dotoral researher at the Dept. of Aerospae and Oean Engineering at Virginia Teh, USA. Dr. Han is urrently an Assistant Professor at the Department of Aeronautial and Mehanial Design Engineering.