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11 SEA SPECTRUM AND MARINE VEHICLE PITCH RESPONSE 2111 Sea Spectrum and Marine Vehicle Pitch Response1. Make a plot of the spectrum for about one hundred frequencies from zero to 4 rad/s,with modal frequency ωm= 1 rad/s, and significant wave height H1/3= 0.90m.See the top graph in Figure 2 for the wave spectrum.2. Confirm that the area under the spectrum is equal to ES, by making a numericalintegration. You can then take this ESto double-check H1/3.The area under the curve E is 0.0502; it was supposed to be 0.0506. Note in theMATLAB code that I randomize the actual frequencies used; this is to avoid relatedfrequencies (as seen in the first problem above). The corresponding significant heightis 0.896m; pretty close to the desired value of 0.90m.3. Make ten minutes worth of this wave-like data, using a sampling period of 0.1 seconds,and show a plot, with the original H1/3maximum and minimum levels indicated.See the middle graph in Figure 2 for a time-domain realization of this spectrum.4. Compute the parameter EYfrom the area under Y (ω), and so estimate the ”significantheight” of the pitch motion.The significant height of the pitch motion is 0.26 radians, or about fifteen degrees. Thelower graph in the figure shows the multiplication in frequency space.%-------------------------------------------------------------------------% Bretschneider Sea Spectra and Vehicle Pitch Responseclear all;dfreq = .04; % frequency resolution for creating waves, rad/smaxfreq = 3.99 ; % highest wave frequency made, rad/sdt = .1 ; % sampling period, stff = 600 ; % final time, swmodal = 1 ; % modal frequency (used for Bretschnieder construction)E = .05 ; % spectra parameter for Bretschnieder:% E = sig_height^2 / 16%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% no user parameters below this point%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%time = 0:dt:tff; % time vectorn = length(time) ; % number of samplesfreq = dfreq:dfreq:maxfreq; % imposed frequencies% (near zero to maxfreq)11 SEA SPECTRUM AND MARINE VEHICLE PITCH RESPONSE 22 freq = freq + (.5-rand(1,length(freq)))*dfreq ; % add random components in freq vector disp(sprintf(’Imposed dfreq/available resolution: %g (best if below one)’,... dfreq/(2*pi/dt/length(time)))) ; if (pi/dt < max(freq)), disp(’Must have higher sampling rate to avoid aliasing! -- ABORT.’); break ; end; % make up a Bretschneider spectrum and then get out the amplitudes % for the example case B = wmodal^4*1.25 ; H = 4*sqrt(E) ; A = 4*B*E ; sBret = A./freq.^5.*exp(-B./freq.^4); figure(1);clf;hold off; subplot(’Position’,[.2 .2 .5 .2]); plot(freq,sBret,’LineWidth’,2); xlabel(’rad/s’); title(’Bretschneider Spectrum, H_{1/3} = 0.90m’); print -deps bret_veh1.eps disp(sprintf(’E: %g vs. %g.’, E, sum(sBret)*dfreq)); amp = sqrt(2*sBret*dfreq) ; % Note that sBret is invariant with dfreq, but amp definitely changes % with dfreq, according to PNA def. of spectrum given in the problem. % make up the time series, with random phase phase = 2*pi*randn(length(freq),1); x = zeros(size(time)); for i = 1:length(freq) x = x+amp(i)*cos(freq(i)*time+phase(i)); end figure(2);clf;hold off; subplot(211); plot(time,x); xlabel(’Time, seconds’) hold on; plot([0 tff],[1 1]*H/2,’r--’,[0 tff],-[1 1]*H/2,’r--’); legend(’Simulation’,’+/- H_{1/3}/2’);11 SEA SPECTRUM AND MARINE VEHICLE PITCH RESPONSE 23 print -deps bret_veh2.eps % Here is the transfer function, and the pointwise frequency multiply of % F*F’ with S i = sqrt(-1); F = (.4*i*freq + .3) ./ (-freq.^2 + i*freq*1 + 3) ; FF = F.*conj(F) ; figure(3);clf;hold off; subplot(’Position’,[.2 .2 .5 .5]); plot(freq,FF,freq,sBret,’--’,freq,FF.*sBret*20,’:’,’LineWidth’,2) ; legend(’FF*’,’S’,’20 x S x FF*’,1); xlabel(’rad/s’); print -deps bret_veh3.eps % Compute the area under the curve and from it get the significant % response height FFS = FF.*sBret ; EFFS = sum(FFS)*dfreq ; FFSsig = 4*sqrt(EFFS) ; disp(sprintf(’Significant response height: %g.’, FFSsig)); %-------------------------------------------------------------------------MIT OpenCourseWarehttp://ocw.mit.edu 2.017J Design of Electromechanical Robotic Systems Fall 2009 For information about citing these materials or our Terms of Use, visit:

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