EAS 3406AEROELASTICBYDR. MOHAMMAD YAZDI HARMIN
ASSIGNMENT 2:FLUTTER
NAME: ONG THIAM CHUNNO. MATRIC: 158347
DISCUSSIONa) 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 also neglected. 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.
b)Figure 2: Effect of changes in flexural axis on flutter
speedThe 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.
c)
Figure 3: Effect of flap stiffness on flutter speed
Figure 4: Effect of pitch stiffness on flutter speed
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 plotFigure 6: Vg plot
Figure 7: Output of the calculationIn 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 aircraft may 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. REFERENCEBabister, 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.
APPENDIXAeroelastic Equation
where=
Frequency Matching k method
MATLAB codingclear all; ; clc%Parametersm = 100; % unit mass /
area of wingc = 1.7; % chord in ms = 7; % semi span in mxcm =
0.5*c; % position of centre of mass from nosexf = 0.5*c; % position
of flexural axis from nosee = xf/c-0.25;% eccentricity between
flexural axis and aero centre (1/4chord)K_k = 1.2277e7; % Flap
Stiffness in Nm/radK_t = 7.2758e5; % Pitch Stiffness in
Nm/radMthetadot = -1.2; % unsteady aero damping termroll = 1.225; %
air density in kg/m^3aw = 2*pi; % 2D lift curve slopeM =
(m*c^2/(2*xcm))-m*c; % leading edge mass termb=c/2; %Inertial,
Damping and Stiffness Matrixa11=(m*s^3*c)/3 + M*s^3/3; % I kappa
based on pg 179a12 = ((m*s^2)/2)*(c^2/2 - c*xf) - M*s^2*xf/2; %I
kappa thetaa21 = a12;a22= m*s*(c^3/3 - c^2*xf + xf^2*c) +
M*(s*xf^2); % I thetaA=[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
Ratioii=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 = 1:4im(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);freq(j,ii)
= freq(j,ii)/(2*pi); endVel(ii) = V;ii=ii+1;end %Find Flutter
speedfor dd=1:1:4for 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 endendvflutter=min(vflut); %Display
resultfigure(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
ontitle('Vg plot','fontsize',15);xlabel ('Air Speed (m/s) ');
ylabel ('Damping Ratio (%)'); grid on; hold
onplot(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*ce=xf/c-0.25; %Inertial, Damping and
Stiffness Matrixa111=(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];
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 lambdakk=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 ratiofor
jj = 1:4im1(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 Speedkk=kk+1;for ddd=1:1:4for
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
endendendxff(iii)=xf/c;vflutter1(iii)=min(vflut1);iii=iii+1;end
%Display resultfigure(2)plot(xff,vflutter1)title(['Flutter speed,
V_f_l_u_t_t_e_r (m/s) vs Flexural axis,'... 'x_f at x_c_m =
0.5c'])grid onxlabel('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 valuexf=0.5*c;e = xf/c-0.25;%Find Eigenvalue of Lambda and
Flutter Speedii=1;for K_k=0.5e7:0.1e7:5e7%Structural
Stiffnesse211=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 = 1:4im2(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 hertzend k=k+1; for dd=1:1:4for
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
endendendK_kappa(ii)=K_k;vflutter2(ii)=min(vflut2);ii=ii+1;end
%Display Resultfigure(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 onxlabel('Flap Stiffness, K_\kappa
(Nm/rad)')ylabel('Flutter Speed, V_f_l_u_t_t_e_r (m/s)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Reset
ValueK_k=1.2277e7; %Find Eigenvalue of Lambda and Flutter
Speedii=1;for
K_t=1e5:0.1e5:10e5e311=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 = 1:4im3(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 hertzendk=k+1;for dd=1:1:4for 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
endendendK_theta(ii)=K_t;vflutter3(ii)=min(vflut3);ii=ii+1;end
%Display Resultfigure(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 onxlabel('Pitch Stiffness, K_\theta
(Nm/rad)')ylabel('Flutter Speed, V_f_l_u_t_t_e_r
(m/s)')%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%d)
K-method%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Reset
ValueK_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=1:1:2for 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; endendend
vflutter4=min(vflut4)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 Resultfigure (5)title('Vf plot','fontsize',15)xlabel
('Air Speed (m/s) '); ylabel ('Frequency (Hz)'); grid on; hold
onplot(V4(1,:),freq4(1,:),'b','linewidth',1.5); hold
onplot(V4(2,:),freq4(2,:),'g','linewidth',1.5); hold
onplot(Vel4,Freq,'--');hold onlegend('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 endend 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
onplot(V4(2,:),g(2,:),'g','linewidth',1.5); hold
onplot(vflutter4,0,'or','markerfacecolor','r');hold onlegend('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)
