EAS 3406

AEROELASTIC

BY

DR. MOHAMMAD YAZDI HARMIN

ASSIGNMENT 1:

EFFECT OF WING SWEEP

ON

STATIC AEROELASTIC BEHAVIOUR

NAME : ONG THIAM CHUN

NO. MATRIC : 158347

Assumption for static aeroelastic equation:

i. Rigid wing with 2 roots, one for bending control (flapping) and one for twisting

control (pitch).

ii. Strip theory is assumed with strip aligned in the streamwise direction.

iii. Span and streamwise chord are kept constant regardless of sweep angle.

iv. Trim is not considered in this case.

v. Sweep angle > 0, sweepback; Sweep angle < 0, forward sweep.

Figure 1: Graph of Divergence Speed, Vdiv (m/s) vs Sweep Angle, 𝜦 (deg)

Figure 2: Lift per span (N/m) vs Sweep Angle, 𝜦 (deg)

Lift per span ,dLdy

=q α w c [ (θ0+θ ) cosΛ+κsinΛ ] Equation for Figure 2

Figure 3: Graph of Divergence Speed, Vdiv (m/s) vs Flap Stiffness, Kκ (Nm/rad) at sweep angle = 10º (sweepback)

Figure 4: Graph of Divergence Speed, Vdiv (m/s) vs Pitch Stiffness, Kθ (Nm/rad) sweep angle = 10º (sweepback)

Figure 3: Graph of Divergence Speed, Vdiv (m/s) vs Flap Stiffness, Kκ (Nm/rad) at sweep angle = -10º (forward sweep)

Figure 6: Graph of Divergence Speed, Vdiv (m/s) vs Pitch Stiffness, Kθ (Nm/rad)sweep angle = -10º (forward sweep)

Figure 7: Graph of Divergence Speed, Vdiv (m/s) vs Flap Stiffness, Kκ (Nm/rad) at sweep angle = 0º (unsweep)

Figure 8: Graph of Divergence Speed, Vdiv (m/s) vs Pitch Stiffness, Kθ (Nm/rad)sweep angle = 0º (unsweep)

V ¿=√ 2K κ Kθ

ρ α w [Kκ( c2 scos2 Λ4 )−K θ( cs2 tanΛ

2+ c2 ssin2 Λ

4 )]Equation for Figure 3, 4, 5, 6, 7, and 8, by altering values of Kκ and Kθ

DISCUSSION

b) Effect of sweep angle on divergence speed and lift distribution

Based on Figure 1, the graph shows the divergence speed increases with sweepback

wing while decreases with forward sweep wing. This may due to the contribution of flap

angle of forward sweep wing on the effective angle of attack of wing.

Figure 9: Effect of sweep angle to leading edge incidence angle.

In forward sweep (or sweepforward) wing, twisting at point B is lower than that at

point A as predicted by linear twist theory: θ= ys

θT . The bending moment during flight will

“push” point A higher than B based on ∆ θFlap=cκsinΛ

c (κ is negative for flap upward during

steady level flight, 𝜦 is negative for forward sweep), thus the angle of incidence is higher so

do the angle of attack. The wing in forward sweep will then suffer the bending effect

continuously and then, lead to divergence earlier than unsweep and sweep back wing. In

some studies, they showed that divergence speed of forward sweep wing is lower than flutter

speed, which makes the divergence non-detectable as flutter is often the indication for pilot to

slow down the aircraft or lower the aircraft nose up angle to prevent divergence. For

sweepback wing, the divergence speed is very high and most probably greater than flutter

speed.

Velocity is fixed at 60m/s for analysis in variation of lift distribution at different

sweep angle. The angle of incidence could also be the explanation for lift per span graph in

Figure 2. It shows a trend of lift per span decreases exponentially with increases in sweep

angle. This could be due to the higher angle of incidence of forward sweep wing, hence the

lift per span is more than that in sweepback wing as lift is proportional with the effective

angle of attack.

c) Effect of flap and pitch stiffness on divergence speed

In the study of effect of flap and pitch stiffness on divergence speed, three distinct

results are found especially the case of increasing flap stiffness. For every wing, the

divergence speed is proportional to the pitch stiffness. However, increasing flap stiffness only

improve divergence speed of forward sweep wing but degrade that in sweepback wing, and

unsweep wing is independent of flap stiffness increment. This shows that the flap angle and

flap stiffness are the major factors that affect divergence speed of wing. Once again, the angle

of incidence will be highlighted in the next explanation. Bending on wing is restricted by flap

stiffness, implies that higher flap stiffness will have a smaller bending effect on wings. From

Figure 9, the angle of incidence in unsweep wing is not affected by bending which is why the

flap stiffness of wing does no effect on the angle of incidence so do the divergence speed. In

sweepback wing, the divergence speed is superior than unsweep and forward sweep wings

due to the reduced angle of incidence by bending effect. The increases in flap stiffness will

then limit the bending at point D and reduce the effect of reduction in angle of incidence as

compared in a lower flap stiffness wing, thus divergence speed is lower as shown in Figure 3.

On the other hand, increment of flap stiffness is favoured in forward sweep wing to reduce

the angle of incidence in order to have a higher divergence speed. Hence, to increase wing

divergence speed for different wings:

i) Forward sweep : Increase the flap stiffness and pitch stiffness

ii) Unsweep : Increase the pitch stiffness

iii) Sweepback : Decrease the flap stiffness and pitch stiffness

All in all, sweepback wing is still having the highest divergence speed in this fixed

range of flap and pitch stiffness despite the increment in flap stiffness did pull down the

divergence speed of the wing.

Common Advantage for Both Forward Sweep And Swept Back Wing

It delays formation of shock wave by decreasing Critical Mach number, MCR, Drag

Divergence Mach number, MDD. In sweep wing configuration, the aerodynamic properties of

local section of wing are governed by normal component of airflow to leading edge.

Figure 10: Sweepback wing

The condition above shows that the MDD and MCR are delayed (in other word,

increased) as it would need a higher freestream Mach number to reach MCR and MDD. The

effective t/c ratio is thinner as effective chord length is also increased by 1/cos(𝜦) for the

same thickness of wing. Hence, MDD is increased.

Common Disadvantage For Both Forward Sweep And Sweepback Wing

Without high lift device, maximum lift coefficient, CLmax decreases due to loss of

control in pitching situation. Sweep wings will contribute pitching moment (nose down for

forward sweep with aerodynamic centre (ac) ahead of centre of gravity (cg), nose up for

swept back with ac behind cg) when loss of lift.

Advantage of Sweepback Wing

This configuration increases pilot view (especially for fighter). Besides that, it will

reduce shockwave drag in supersonic flight if the sweep angle is bigger than Mach angle, µ

as shown in Figure 11. As this wing flutters before diverge (to give warning to pilot about

divergence), it is more preferable than forward sweep wing to be implemented in airliner with

M eff=Mcos ( Λ)

M DD ,eff =M DD ∙√cos Λ

Λ is sweep angle

tc=( 1−M DD,eff

K )32

high speed. As shown above, the sweepback wing has higher divergence speed compare to

other configuration due to effect of bending in reduction of wing incidence as in Figure 9.

Disadvantage for Sweepback Wing

There is a spanwise flow from wing root to wing tip. Hence, the boundary layer will

thicken toward the wingtips, increases drag, and thickening of boundary towards wing tip.

This increases the tendency for early separation and tip stalls before root. This tip stall

reduces the effectiveness of aileron roll control as aileron is located outboard section of wing.

Manoeuvrability of aircraft with this wing also is lower than that with forward sweep wing.

Figure 11: Mach cone and wing Figure 12: Spanwise flow at wing

Advantage of Forward Sweep Wing

The spanwise flow along the wing is inwards from wing tip to wing root in this

configuration. The roll control effectiveness of aileron increases as the boundary layer

thickening is towards wing root. In other words, root stalls before tip and rolling can be

recovered by aileron despite loss of lift. Therefore, forward sweep wing aircraft has better

manoeuvrability and controllability even in higher angle of attack than straight or swept

back wing aircraft.

Disadvantage of Forward Sweep Wing

Forward sweep wing tends to diverge before flutter actually happen, makes the

divergence undetectable. Lift force tends to bend wing tip upwards due to aeroelasticity in

cruising. This increases angle of incidence at wing tip, and further increasing lift will causes

more bending of wing tip, end up to wing divergence. This also happen when an airplane

turns and applies high G-loads on forward-swept wings, wing tips are twisted with leading

edge bending upward (wash-in effect).

REFERENCE

Babister, A. W. (June 1950). Flutter and Divergence of Sweptback and Sweptforward

Wings. Retrieved from

https://dspace.lib.cranfield.ac.uk/bitstream/1826/7209/3/COA_Report_No_39_JUN_19

50.pdf on 21 November 2013.

Wright, J. R. (2007). Introduction to Aircraft Aeroelasticity and Loads. Effect of Wing Sweep

on Effective Angle of Incidence (pp 134 – pp 139). West Sussex, England: John Wiley

& Sons, Ltd.

Mohammad Sadraey. “Wing Design”. School of Engineering and Computer Sciences Daniel

Webster College, 2013.

MATLAB coding for calculation

clear allclc %Input---------------------------------------------------------------------K_k=5*10^6 %flap stiffness in Nm/radK_t=5*10^5 %pitch stiffness in Nm/radroll=1.225 %density at sea level in kg/m^3aw=2*pi %lift curve slop in rads=7.5 %semi span in mc=2 %chord size in mtheta_0=2*pi/180; %angle of incidence in rad%-------------------------------------------------------------------------- % Calculation for c)Divergence Speed---------------------------------------i=1for lamb=-25:0.01:25a(i)=K_k*s*c^2*(cosd(lamb))^2/4;b(i)=K_t*((0.5*c*s^2*tand(lamb))+ (c^2*s*(sind(lamb))^2)/4);Vdiv(i)=sqrt((2*K_t*K_k)/(roll*aw*(a(i)-b(i))));angle(i)=lamb;i=i+1;end plot(angle,Vdiv)title(...'Divergence Speed, V_d_i_v(m/s) for different Sweep Angle,\Lambda(deg)')xlabel('Sweep Angle, \Lambda in deg')ylabel('Divergence Speed, V_d_i_v (m/s)')%-------------------------------------------------------------------------- %Calculation for c)Lift Distribution of Wing-------------------------------v=60j=1q=0.5*roll*v^2;for lamb=-25:0.01:25m1_1=(s^2*tand(lamb))/2;m1_2=(c*s*(sind(lamb))^2)/4;m1=q*aw*c*(m1_2+m1_1)+K_k;m2=q*aw*c*((s^2/2)+(c*s*sind(lamb)*cosd(lamb)/4));m3=-q*aw*s*c^2*sind(lamb)*cosd(lamb)/4;m4=K_t-(q*aw*s*c^2*(cosd(lamb))^2/4);M=[m1 m2;m3 m4];M2=[-m2; -m4+K_t];kt=(inv(M))*M2;k=kt(1,1);t=kt(2,1);angle2(j)=lamb;dL(j)=q*aw*c*((theta_0+t)*cosd(lamb)+k*sind(lamb));j=j+1;end figure(2)plot(angle2,dL)title('Lift per span (N/m) for different Sweep Angle, \Lambda(deg)')xlabel('Sweep Angle, \Lambda in deg')ylabel('Lift per span (N/m)')%-------------------------------------------------------------------------- %Calculation for d)effect of flap stiffness on divergence speed------------

lamb=0 %sweep angle in degk=1e=K_t*((0.5*c*s^2*tand(lamb))+ (c^2*s*(sind(lamb))^2)/4);for K_k1=(1*10^6):(5*10^5):(10*10^6)k_k(k)=K_k1;d(k)=k_k(k)*s*c^2*(cosd(lamb))^2/4;Vdiv2(k)=sqrt((2*K_t*k_k(k))/(roll*aw*(d(k)-e)));k=k+1;end figure(3)plot(k_k,Vdiv2)title...('Divergence Speed, V_d_i_v(m/s) against Flap Stiffness,K_\kappa(Nm/rad)')xlabel('Flap Stiffness, K_\kappa (Nm/rad)')ylabel('Divergence Speed, V_d_i_v (m/s)')%-------------------------------------------------------------------------- %Calculation for d)effect of pitch stiffness on divergence speed-----------k=1f=K_k*s*c^2*(cosd(lamb))^2/4;for K_t1=(1*10^5):(5*10^4):(10*10^5)k_t(k)=K_t1;g(k)=k_t(k)*((0.5*c*s^2*tand(lamb))+ (c^2*s*(sind(lamb))^2)/4); Vdiv3(k)=sqrt((2*k_t(k)*K_k)/(roll*aw*(f-g(k))));k=k+1;end figure(4)plot(k_t,Vdiv3)title...('Divergence Speed, V_d_i_v(m/s) against Pitch Stiffness,K_\theta(Nm/rad)')xlabel('Pitch Stiffness, K_\theta (Nm/rad)')ylabel('Divergence Speed, V_d_i_v (m/s)')%--------------------------------------------------------------------------