弹性机翼飞机的模态分析

url = 'https://www.mathworks.com/supportfiles/ident/flexible-wing-GVT-data/FlexibleWingData.mat';
websave('FlexibleWingData.mat',url);
load FlexibleWingData.mat MeasuredData

fs = 2000;                % data sampling frequency
Ts = 1/fs;                % sample time
y = MeasuredData.Exp1.y;  % output data (2 columns, one for each accelerometer)
u = MeasuredData.Exp1.u;  % input force data
t = (0:length(u)-1)' * Ts;
figure
subplot(211)
plot(t,y)
ylabel('Outputs (m/s^2)')
legend('Leading edge','Trailing edge')
subplot(212)
plot(t,u)
ylabel('Input')
xlabel('Time (seconds)')

ExpNames = fieldnames(MeasuredData);
Data = cell(1, 10);
for k = 1:10
   y =  MeasuredData.(ExpNames{k}).y;
   u =  MeasuredData.(ExpNames{k}).u;
   Data{k} = iddata(y, u, Ts, 'InterSample', 'bl');
end

Data = merge(Data{:})

G = cell(1, 10);
N = 24000;
for k = 1:10
   % Convert time-domain data into a FRF using ETFE command
   Data_k = getexp(Data, k);
   G{k} = etfe(Data_k, [], N); % G{k} is an @idfrd object
end

G = cat(1, G{:});     % concatenate outputs of all estimated FRFs
G.OutputName = 'y';   % name outputs 'y1', 'y2', ..., 'y20'
G.InterSample = 'bl';

opt = bodeoptions;           % plot options
opt.FreqUnits = 'Hz';        % show frequency in Hz
opt.PhaseVisible = 'off';    % do not show phase
OutputNum = [1 15];          % pick outputs 1 and 15 for plotting
clf
bodeplot(G(OutputNum, :), opt)   % plot frequency response
xlim([4 35])
grid on

f = G.Frequency/2/pi;           % extract frequency vector in Hz (G stores frequency in rad/s)
Gs = fselect(G, f>6 & f<=32)    % "fselect" selects the FRF in the requested range (6.5 - 35 Hz)

FRF = squeeze(Gs.ResponseData);
Weighting = mean(1./sqrt(abs(FRF))).';
n4Opt = n4sidOptions('EstimateCovariance',false,...
   'WeightingFilter',Weighting,...
   'OutputWeight',eye(20));
sys1 = n4sid(Gs,24,'Ts',0,'Feedthrough',true,n4Opt);
Fit = sys1.Report.Fit.FitPercent'

[~,Imin] = min(Fit);
[~,Imax] = max(Fit);
clf
bodeplot(Gs([Imin, Imax],:), sys1([Imin, Imax],:), opt);
xlim([6 32])
title('Worst (top) and best (bottom) fits between data and model')
grid on
legend('Data', 'Model')

ssOpt = ssestOptions('EstimateCovariance',true,...
   'WeightingFilter',n4Opt.WeightingFilter,...
   'SearchMethod','lm',...     % use Levenberg-Marquardt search method
   'Display','on',...
   'OutputWeight',eye(20));
sys2 = ssest(Gs, sys1, ssOpt);  % estimate state-space model (this takes several minutes)
Fit = sys2.Report.Fit.FitPercent'

[~, Imin] = min(Fit);
[~, Imax] = max(Fit);
clf
h = bodeplot(Gs([Imin, Imax],:), sys2([Imin, Imax],:), opt);
showConfidence(h, 1)
xlim([6.5 35])
title('Worst (top) and best (bottom) fits between data and refined model')
grid on
legend('Data', 'Model (sys2)')

f = Gs.Frequency/2/pi;     % data frequencies (Hz)
fn = modalfit(sys2, f, 12); % natural frequencies (Hz)
disp(fn)

FRF = permute(Gs.ResponseData,[3 1 2]);
f = Gs.Frequency/2/pi;
%{
 figure
 pf = modalsd(FRF,f,fs,'MaxModes',20,'FitMethod','lsrf');
%}

Freq = [7.8 9.4 15.3 19.3 23.0 27.3 29.2 31.7];

[fn,dr,ms] = modalfit(sys2,f,length(Freq),'PhysFreq',Freq);

AccelPos = [...   % see figure 2
   16.63 18.48;   % nearest right of center
   16.63 24.48;
   27.90 22.22;
   27.90 28.22;
   37.17 25.97;
   37.17 31.97;
   46.44 29.71;
   46.44 35.71;
   55.71 33.46;
   55.71 39.46;  % farthest right
   -16.63 18.48; % nearest left of center
   -16.63 24.18;
   -27.90 22.22;
   -27.90 28.22;
   -37.17 25.97;
   -37.17 31.97;
   -46.44 29.71;
   -46.44 35.71;
   -55.71 33.46;
   -55.71 39.46]; % farthest left

AnimateOneMode(3, fn, ms, AccelPos); 

function AnimateOneMode(ModeNum, fn, ModeShapes, AccelPos)
% Animate one mode.
% ModeNum: Index of the mode.
    
% Reorder the sensor locations for plotting so that a continuous, 
% non-intersecting curve is traced around the body of the aircraft.
PlotOrder = [19:-2:11, 1:2:9, 10:-2:2, 12:2:20, 19];
Fwd = PlotOrder(1:10);
Aft = PlotOrder(20:-1:11); 
x = AccelPos(PlotOrder,1);
y = AccelPos(PlotOrder,2);
xAft = AccelPos(Aft,1);
yAft = AccelPos(Aft,2);
xFwd = AccelPos(Fwd,1);
yFwd = AccelPos(Fwd,2);

wn = fn(ModeNum)*2*pi;  % Mode frequency in rad/sec
T = 1/fn(ModeNum);      % Period of modal oscillation
Np  = 2.25;             % Number of periods to simulate
tmax = Np*T;            % Simulate Np periods
Nt = 100;               % Number of time steps for animation
t = linspace(0,tmax,Nt);
ThisMode = ModeShapes(:,ModeNum)/max(abs(ModeShapes(:,ModeNum))); % normalized for plotting
z0 = ThisMode(PlotOrder);
zFwd = ThisMode(Fwd);
zAft = ThisMode(Aft);

clf
col1 = [1 1 1]*0.5;
xx = reshape([[xAft, xFwd]'; NaN(2,10)],[2 20]);
yy = reshape([[yAft, yFwd]'; NaN(2,10)],[2 20]);
plot3(x,y,0*z0,'-', x,y,0*z0,'.', xx(:), yy(:), zeros(40,1),'--',...
   'Color',col1,'LineWidth',1,'MarkerSize',10,'MarkerEdgeColor',col1);
xlabel('x')
ylabel('y')
zlabel('Amplitude')
ht = max(abs(z0));
axis([-100  100  10  40 -ht ht])
view(5,55)
title(sprintf('Mode %d. FN = %s Hz', ModeNum, num2str(fn(ModeNum),4)));

% Animate by updating z-coordinates.  
hold on
col2 = [0.87 0.5 0];
h1 = plot3(x,y,0*z0,'-', x,y,0*z0,'.', xx(:), yy(:), zeros(40,1),'--',...
   'Color',col2,'LineWidth',1,'MarkerSize',10,'MarkerEdgeColor',col2);
h2 = fill3(x,y,0*z0,col2,'FaceAlpha',0.6);
hold off

for k = 1:Nt
   Rot1 = cos(wn*t(k));
   Rot2 = sin(wn*t(k));
   z_t = real(z0)*Rot1 - imag(z0)*Rot2;
   zAft_t = real(zAft)*Rot1 - imag(zAft)*Rot2;
   zFwd_t = real(zFwd)*Rot1 - imag(zFwd)*Rot2;
   zz = reshape([[zAft_t, zFwd_t]'; NaN(2,10)],[2 20]);
   set(h1(1),'ZData',z_t)
   set(h1(2),'ZData',z_t)
   set(h1(3),'ZData',zz(:))
   h2.Vertices(:,3) = z_t;
   pause(0.1)
end

end