188434519 Aeroelastic Flutter

8/16/2019 188434519 Aeroelastic Flutter

1/15

EAS 3406

AEROELASTIC

BY

DR. MOHAMMAD YAZDI HARMIN

ASSIGNMENT 2:

FLUTTER

NAME : ONG THIAM CHUN

NO. MATRIC : 158347


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DISCUSSION

a) This aeroelastic calculation assumes that the unsteady aerodynamic damping is constant. In

order to take account of structural damping in flutter analysis, one needs to obtain Rayleigh

coefficients. Due to complexity in finding Rayleigh coefficient, the structural damping is alsoneglected. Based on “Introduction to Aircraft Aeroelasticity and Loads” by Jan R. Wright,

the inclusion of structural damping actually has small effect on delaying flutter speed.

Figure 1: Vf and Vg plots against Air Speed (m/s)

Critical flutter speed takes place where one of the modes has negative damping ratio. As the

airspeed increases, the frequencies begin to converge. One of the modes has increasing damping

ratio, as a contrast to this phenomena, another mode shows decreasing damping ratio which

eventually become negative at around 111.6m/s. The frequency does not coalesce when flutter

happens. This damping ratio decrement happens at a shallow gradient, so we called the flutter as

a soft flutter.


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b) Figure 2: Effect of changes in flexural axis on flutter speed

The mass axis is fixed while flexural axis is varying. Notice that even the flexural axis and mass

axis are coincide at mid-chord, flutter is still taking place. When flexural axis is located at mid-

chord, there is no inertial coupling. However, the two mode shapes are coupled when flexural

axis is no longer located at mid-chord. From Figure 2, the flutter speed decreases as the flexural

axis is moving further away from leading edge. An increment of flexural axis also indicates the

increases in eccentricity between flexural axis and aerodynamic centre. This phenomenon can be

explained based on moment of the wing. As the eccentricity increases, lift forces at aerodynamic

centre will create a greater moment effect as the moment arm (eccentricity) of lift force to

flexural axis is longer. This moment may push the leading edge upwards, lead to higher effective

angle of attack, and this is where the degradation on flutter speeds performance started with the

increasing distance of flexural axis and leading charge.


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c)

Figure 3: Effect of flap stiffness on flutter speed

Figure 4: Effect of pitch stiffness on flutter speed


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Generally, Figure 3 shows that flutter speed decreases along with the increases in flap stiffness

and have a sudden increase again at 4.6 x 107 Nm/rad of flap stiffness. On the other hand, the

increment in pitch stiffness improves its flutter speed except in the lower pitch stiffness (rapid

increase with a sharp drop), which is opposite to that in increment of flap stiffness. The result

may indicate that the changes in flap and pitch stiffness does not necessary improve dynamic

aeroelasticity performance in the same trend (example: increases in pitch stiffness or decreases in

flap stiffness improve dynamic aeroelasticity performance), there is uncertainty shown in Figure

3 and Figure 4. One may conclude that this theoretical analysis serves as reference for suitable

flap and pitch stiffness to improve dynamic aeroelasticity performance in this particular wing

configuration and altitude. The limits of increment in pitch stiffness and decrement in flap

stiffness should be noticed to avoid degradation in performance. More studies should be done to

investigate the factors for such inconsistency in the result shown in Figure 3 and Figure 4.

d)

Figure 5: Vf plot


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Figure 6: Vg plot

Figure 7: Output of the calculation

In frequency matching k-method, the stable region is where the damping coefficient, g are

negative and the system becomes unstable when g become positive. With the same configuration

in a), the k-method analysis yields a slightly higher flutter speed than that in a). This is due to the

inclusion of unsteady reduced frequency effects. The unsteady aerodynamic damping term, ̇

is assumed as constant in the analysis in part a) at different air speed. However, the aerodynamic

stiffness and damping matrices are reduced frequency-dependent. The frequency dependency of

̇

is given by the equation below:

̇

Frequency Matching k-method seems to show a more realistic result than the method using in

part a) because it includes the unsteady aerodynamic condition and reduced frequency. If an

aircraft is analysed based on steady condition or neglecting some unsteady terms, the aircraftmay perform well below the actual capable performance. Steady state analysis is much simpler

and quicker within safety margin of aircraft performance but come with a cost of unrealistic and

degrading the actual performance of aircraft. K-method or p-k method is the better analysis

method for such dynamic aeroelastic problem which in real life is unsteady and frequency

dependent.


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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_1950.

pdf on 21 November 2013.

Wright, J. R. (2007). Introduction to Aircraft Aeroelasticity and Loads. Dynamic Aeroelasticity –

Flutter (pp167-pp199). West Sussex, England: John Wiley & Sons, Ltd.

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

Webster College, 2013.

Mohammad Yazdi Harmin. (2013). Lecture on Dynamic Aeroelasticity – Flutter. Personal

Collection of Mohammad Yazdi Harmin, University Putra Malaysia, Selangor.


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APPENDIX

Aeroelastic Equation

where

Frequency Matching – k method

=


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MATLAB coding

clear all; ; clc %Parameters m = 100; % unit mass / area of wing c = 1.7; % chord in m

s = 7; % semi span in m xcm = 0.5*c; % position of centre of mass from nose xf = 0.5*c; % position of flexural axis from nose e = xf/c-0.25;% eccentricity between flexural axis and aero centre (1/4chord) K_k = 1.2277e7; % Flap Stiffness in Nm/rad K_t = 7.2758e5; % Pitch Stiffness in Nm/rad Mthetadot = -1.2; % unsteady aero damping term roll = 1.225; % air density in kg/m^3 aw = 2*pi; % 2D lift curve slope M = (m*c^2/(2*xcm))-m*c; % leading edge mass term b=c/2;

%Inertial, Damping and Stiffness Matrix a11=(m*s^3*c)/3 + M*s^3/3; % I kappa based on pg 179 a12 = ((m*s^2)/2)*(c^2/2 - c*xf) - M*s^2*xf/2; %I kappa theta a21 = a12; a22= m*s*(c^3/3 - c^2*xf + xf^2*c) + M*(s*xf^2); % I theta A=[a11,a12;a21,a22];

b11=c*s^3*aw/6; b12=0; b21=-e*c^2*s^2*aw/4; b22=-c^3*s/8*Mthetadot; B=[b11 b12; b21 b22]

c11=0; c12=c*s^2*aw/4; c21=c11; c22=-e*c^2*s*aw/2; C=[c11 c12; c21 c22]

D=zeros(2)

e11=K_k; e12=0; e21=e12; e22=K_t; E=[e11 e12; e21 e22]

% %Find Eigenvalue of Lambda, Frequency and Damping Ratio ii=1; for V=1:0.1:180

Q=[zeros(2) eye(2); -A\(roll*V^2*C+E) -A\(roll*V*B+D)]; lambda=eig(Q);

for j = 1:4 im(j) = imag(lambda(j)); re(j) = real(lambda(j)); freq(j,ii) = sqrt(re(j)^2+im(j)^2); damp(j,ii) = -100*re(j)/freq(j,ii);


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freq(j,ii) = freq(j,ii)/(2*pi);end Vel(ii) = V; ii=ii+1; end

%Find Flutter speed for dd=1:1:4 for l=1:1:numel(damp(1,:))

if damp(dd,l)1 vflut_high(dd)=Vel(l); vflut_low(dd)=Vel(l-1); dd_high(dd)=damp(dd,l); dd_low(dd)=damp(dd,l-1); coef(dd)=dd_high(dd)/-dd_low(dd); vflut(dd)=(vflut_high(dd)+coef(dd)*vflut_low(dd))/(coef(dd)+1); break

else vflut(dd)=1000;

end end

end vflutter=min(vflut);

%Display result figure(1) subplot(2,1,1); plot(Vel,freq,'k'); title('Vf plot','fontsize',15); xlabel ('Air Speed (m/s) '); ylabel ('Frequency (Hz)'); grid

subplot(2,1,2); plot(Vel,damp,'k'); hold on title('Vg plot','fontsize',15);

xlabel ('Air Speed (m/s) '); ylabel ('Damping Ratio (%)'); grid on; hold on plot(vflutter,0,'ro','markerfacecolor','r'); text(vflutter,0,['\leftarrowFlutter Speed:',...

num2str(roundn(vflutter,-2)),'m/s'],'fontsize',12,'fontweight','bold')

fprintf('The flutter speed is %5.2f. m/s.',vflutter)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %b) effect of flexural axis on flutter speed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

iii=1; for xf=0.1*c:0.1*c:0.8*c e=xf/c-0.25;

%Inertial, Damping and Stiffness Matrix a111=(m*s^3*c)/3 + M*s^3/3;a122= m*s*(c^3/3 - c^2*xf + xf^2*c) + M*(s*xf^2); a112 = ((m*s^2)/2)*(c^2/2 - c*xf) - M*s^2*xf/2;a121 = a112; A1=[a111,a112;a121,a122];


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b111=c*s^3*aw/6; b112=0; b121=-e*c^2*s^2*aw/4; b122=-c^3*s/8*Mthetadot; B1=[b111 b112; b121 b122];

c111=0; c112=c*s^2*aw/4; c121=c111; c122=-e*c^2*s*aw/2; C1=[c111 c112; c121 c122];

%Find Eigenvalue of lambda kk=1; for V=1:1:200

Q1=[zeros(2) eye(2); -A1\(roll*V^2*C1+E) -A1\(roll*V*B1+D)]; lambda1=eig(Q1); Vel1(kk)=V;

%Find frequency and damping ratio for jj = 1:4

im1(jj) = imag(lambda1(jj)); re1(jj) = real(lambda1(jj)); freq1(jj,kk) = sqrt(re1(jj)^2+im1(jj)^2); damp1(jj,kk) = -100*re1(jj)/freq1(jj,kk); freq1(jj,kk) = freq1(jj,kk)/(2*pi);end

%Find Flutter Speed kk=kk+1; for ddd=1:1:4 for ll=1:1:numel(damp1(1,:))

if damp1(ddd,ll)1 vflut_high1(ddd)=Vel1(ll); vflut_low1(ddd)=Vel1(ll-1); dd_high1(ddd)=damp1(ddd,ll); dd_low1(ddd)=damp1(ddd,ll-1); coef1(ddd)=dd_high1(ddd)/-dd_low1(ddd);

vflut1(ddd)=(vflut_high1(ddd)+coef1(ddd)*vflut_low1(ddd))/(coef1(ddd)+1); break

else vflut1(ddd)=1000;

end end

end

end xff(iii)=xf/c; vflutter1(iii)=min(vflut1); iii=iii+1; end

%Display result figure(2) plot(xff,vflutter1) title(['Flutter speed, V_f_l_u_t_t_e_r (m/s) vs Flexural axis,'...


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'x_f at x_c_m = 0.5c']) grid on xlabel('Flexural axis, x_f per chord (m/m)') ylabel('Flutter speed, V_f_l_u_t_t_e_r (m/s)')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% c)effect of flap and pitch stiffness on flutter speed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %Reset value xf=0.5*c; e = xf/c-0.25; %Find Eigenvalue of Lambda and Flutter Speed ii=1; for K_k=0.5e7:0.1e7:5e7 %Structural Stiffness e211=K_k; e212=0; e221=e212; e222=K_t; E2=[e211 e212; e221 e222];

k=1; for V=1:1:200

Q2=[zeros(2) eye(2); -A\(roll*V^2*C+E2) -A\(roll*V*B+D)]; lambda2=eig(Q2); Vel2(k)=V;

for j = 1:4 im2(j) = imag(lambda2(j)); re2(j) = real(lambda2(j)); freq2(j,k) = sqrt(re2(j)^2+im2(j)^2); damp2(j,k) = -100*re2(j)/freq2(j,k); freq2(j,k) = freq2(j,k)/(2*pi); % convert frequency to hertz

end

k=k+1;

for dd=1:1:4 for l=1:1:numel(damp2(1,:))

if damp2(dd,l)1 vflut_high2(dd)=Vel2(l); vflut_low2(dd)=Vel2(l-1); dd_high2(dd)=damp2(dd,l); dd_low2(dd)=damp2(dd,l-1); coef2(dd)=dd_high2(dd)/-dd_low2(dd); vflut2(dd)=(vflut_high2(dd)+coef2(dd)*vflut_low2(dd))/(coef2(dd)+1); break

else vflut2(dd)=1000;

end end

end end K_kappa(ii)=K_k; vflutter2(ii)=min(vflut2);


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ii=ii+1; end

%Display Result figure(3) plot(K_kappa,vflutter2)

title(... 'Flutter Speed, V_f_l_u_t_t_e_r(m/s) vs Flap Stiffness, K_\kappa(Nm/rad)') grid on xlabel('Flap Stiffness, K_\kappa (Nm/rad)') ylabel('Flutter Speed, V_f_l_u_t_t_e_r (m/s)')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Reset Value K_k=1.2277e7;

%Find Eigenvalue of Lambda and Flutter Speed ii=1; for K_t=1e5:0.1e5:10e5 e311=K_k;

e312=0; e321=e12; e322=K_t; E3=[e311 e312; e321 e322]; k=1; for V=1:1:200

Q3=[zeros(2) eye(2); -A\(roll*V^2*C+E3) -A\(roll*V*B+D)]; lambda3=eig(Q3); Vel3(k)=V;

for j = 1:4 im3(j) = imag(lambda3(j)); re3(j) = real(lambda3(j)); freq3(j,k) = sqrt(re3(j)^2+im3(j)^2);

damp3(j,k) = -100*re3(j)/freq3(j,k); freq3(j,k) = freq3(j,k)/(2*pi); % convert frequency to hertz end k=k+1; for dd=1:1:4 for l=1:1:numel(damp3(1,:))

if damp3(dd,l)1 vflut_high3(dd)=Vel3(l); vflut_low3(dd)=Vel3(l-1); dd_high3(dd)=damp3(dd,l); dd_low3(dd)=damp3(dd,l-1); coef3(dd)=dd_high3(dd)/-dd_low3(dd); vflut3(dd)=(vflut_high3(dd)+coef3(dd)*vflut_low3(dd))/(coef3(dd)+1); break

else vflut3(dd)=1000;

end end

end end K_theta(ii)=K_t; vflutter3(ii)=min(vflut3); ii=ii+1;


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end

%Display Result figure(4) plot(K_theta,vflutter3) title(...

'Flutter Speed, V_f_l_u_t_t_e_r(m/s) vs Pitch Stiffness, K_\theta(Nm/rad)') grid on xlabel('Pitch Stiffness, K_\theta (Nm/rad)') ylabel('Flutter Speed, V_f_l_u_t_t_e_r (m/s)') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %d) K-method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Reset Value K_theta=7.2758e5; j=1; for k=1:-0.01:0.1

kk(j)=k; Mthetadot=-5/(2+5*k); b11=c*s^3*aw/6; b12=0; b21=-e*c^2*s^2*aw/4; b22=-c^3*s*Mthetadot/8; B=[b11 b12; b21 b22]; F=A-i*roll*(b/k)*B-roll*((b/k)^2)*C; lambda4=eig(F*inv(E)); for l=1:1:2

F_array(l,j)=F(l); lambda_array(l,j)=lambda4(l); omega(l,j)=1/sqrt(real(lambda4(l))); g(l,j)=imag(lambda4(l))/real(lambda4(l)); V4(l,j)=omega(l,j)*c/(2*k); freq4(l,j)=omega(l,j)/(2*pi);

end j=j+1; end

for dd=1:1:2 for l=1:1:numel(g(1,:))

if g(dd,l)>=0 && l>1 vflut_high4(dd)=V4(dd,l); vflut_low4(dd)=V4(dd,l-1); g_high(dd)=g(dd,l); g_low(dd)=g(dd,l-1); coef4(dd)=g_high(dd)/-g_low(dd); vflut4(dd)=(vflut_high4(dd)+coef4(dd)*vflut_low4(dd))/(coef4(dd)+1); red_freq(dd)=(kk(l)+coef4(dd)*kk(l-1))/(coef4(dd)+1); break

else vflut4(dd)=1000; red_freq(dd)=1000;

end end end

vflutter4=min(vflut4)


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red_freq1=red_freq(vflut4==vflutter4) flutter_freq=(vflutter4*red_freq1/b)/(2*pi)

Vel4(1,length(V4))=0; Vel4(2,:)=V4(1,:); Vel4(3,:)=V4(2,:);

Freq(1,length(freq4))=0; Freq(2,:)=freq4(1,:); Freq(3,:)=freq4(2,:);

%Display Result figure (5) title('Vf plot','fontsize',15) xlabel ('Air Speed (m/s) '); ylabel ('Frequency (Hz)'); grid on; hold on plot(V4(1,:),freq4(1,:),'b','linewidth',1.5); hold on plot(V4(2,:),freq4(2,:),'g','linewidth',1.5); hold on plot(Vel4,Freq,'--');hold on legend('Mode 1','Mode 2','Reduced Frequency, k') text(max(V4(1,:)),freq4(1,j-1),['\leftarrownode for V_i at k_i'],...

'fontsize',12,'fontweight','bold')

for l=1:1:length(freq4) for ll=1:1:2 plot(V4(ll,l),freq4(ll,l),'ko','markerfacecolor','k','markersize',5) hold on end

end

figure(6) xlabel ('Air Speed (m/s) '); ylabel ('Damping coefficient, g'); grid on; hold on; title('Vg plot','fontsize',15) plot(V4(1,:),g(1,:),'linewidth',1.5);hold on plot(V4(2,:),g(2,:),'g','linewidth',1.5); hold on plot(vflutter4,0,'or','markerfacecolor','r');hold on legend('Mode 1','Mode 2') text(vflutter4,0,['\leftarrowFlutter Speed:',... num2str(roundn(vflutter4,-2)),'m/s'],'fontsize',12,'fontweight','bold')

fprintf('The flutter speed is %5.2f m/s.',vflutter4 ) fprintf('\nReduced frequency, k= %5.4f',red_freq1) fprintf('\nFlutter frequency, f= %4.2fHz',flutter_freq)

188434519 Aeroelastic Flutter

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