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AE202 Problem Set 5 Solutions

# AE202 Problem Set 5 Solutions - Problem Set 5 Spring 2011...

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Unformatted text preview: Problem Set 5: Spring 2011 DUE: 20 April 2011 1. Three astronauts are about to leave Mars, departing in their Martian lander vehicle. a) They need to leave the surface (v=0) for low Martian orbit, i.e. orbit at a constant altitude of 200km. What velocity do they need to achieve to enter this orbit? Mars mass = 0.1074 Earth mass, Martian equatorial radius = 3393km. Their spacecraft has an empty (unfueled) mass of 4000 kg. It contains 6500 kg of fuel. Each astronaut has a mass of 80kg. The specific impulse of the rocket motor is 408 s. Ignoring gravity loss and atmospheric drag, can their spacecraft reach orbit? You should have found that the rocket’s performance is more than sufficient to reach orbit. So the astronauts could bring some rock samples along as payload and still get to orbit. What is the maximum mass (in kg) of samples they can bring and still reach this 200 km altitude orbit? 5 q W‘W‘v‘ ””5 1 IOSM ‘Ma 7, wag: .\Oq’L‘JJL = 9138-10 )3: Sad 4k) 33 = 63;; (26, 2 334,3 14m ¢\‘\' = 90C) Mvvx \[ = .loi‘r4 J'L W1-“ = 3,451 mm (m 3513 _ b> iwfla v'haAS = 4606 My C: 4 “WIS ¥uc\ : tQI$OO [4% gas-Laws '1 80 Kb e-LJA 3.45‘ = 4 MM ﬁn ‘11:). 5‘7 M9 .-: as; 0 — C) AU = C, 9m #ifiwwmwvw 1: 3'44; (no rod/.4.) a) 4240 m afﬁx“ +qu 1% Ms‘ «w. I”, We .. 3aL\‘Sl : 4k“ ‘014‘49; MR .... - L‘aLlO + VWR {DAN/4 [0,440 +Mg = = 9.346;; 8 auto +VV‘g “9'4”" + ““2 -5‘- (All—‘04 mg) magi—7. [01140 - Lllll6(g,3mq) .2: Me :' 504; K}, / L3444W‘a => 2. We solved an example problem in class for the performance of a one-stage rocket having a total mass (incl. payload) of 25,000 kg. Then we solved a similar problem but dividing the rocket mass into 3 stages (all with the same structural coefficient and exhaust velocity) and saw an improvement in performance — but with a (arbitrary) selection of mass ratios: m2_m3_1 m1 m2 5 a) Write a Matlab function to solve for final rocket velocity for any given ratios m2/m1 and m3/m2. Exhaust velocity c = 3.3 km.s‘1 Total mass = 25000 kg Payload mass = 500 kg Structural coefficient s=0.1 function dv = threeStageRocket(massRatio,payloadMass,totalMass,structCoeff,exhaustVel) computes the delta v produced by a three state rocket with mass ratios m2/ml = massRatio(l), m3/m2 = massRatio(2) totalMass — total mass of the rocket = ml+m2+m3 structCoeff — vector of structural coefficients for each stage exhaustVel — vector of exhaust velocity for each stage % calculate the masses of each stage ml:(totalMass—payloadMass)/(l+massRatio(l)+massRatio(2)*massRatio(l)); m2=massRatio(l)*ml; m3=massRatio(2)*m2; mp=payloadMass; % delta v of stage one dvl=exhaustVel(l)*log(totalMass/(structCoeff(1)*ml+m2+m3+mp)); % delta v of stage two dv2=exhaustVel(2)*log((m2+m3+mp)/(structCoeff(2)*m2+m3+mp)); % delta v of stage three dv3=exhaustVel(3)*log((m3+mp)/(structCoeff(3)*m3+mp)); % total dv dv=dv1+dv2+dv3; In this function the values for payload mass, total mass, structural coefficient and exhaust velocity are passed in, they could also be hardcoded. b) Produce a figure of final velocity versus the two mass ratios. It can be a 3D graph (surf , mesh, etc), a contour plot (contour, contour3 , etc), or any other graph that you think best represents the data. Explain your choice. % plot final velocity vs mass ratios for a 3 stage rocket payloadMass=500; % kg totalMass=25000; kg structCoeff=0.l; exhaustVel=3.3; % km/s [X,Y]=meshgrid(x,y); Z=zeros(size(x)); % get dv for each combination of m2/ml and m3/m2 for i=lzlength(x); for j=l:1ength(y); Z(i,j)=threeStageRocket([X(i,j);Y(i,j)],payloadMass,totalMass,structCoeff,exhaustVel); end end % plot results figure; surf(X,Y,Z); xlabel('$\frac{m_2}{m_l}$','interpreter','latex'); ylabel('$\frac{m_3}{m_2}$','interpreter','latex'); zlabel(‘\Delta v (km.sA{—l})'); figure; [C,h] =contour(X,Y,Z, [9:0.2:lO.4 10.5:0.02:10.75]); clabel(C,h); xlabel('$\frac{m_2}{m_l}$','interpreter','latex'); ylabel('$\frac{m_3}{m_2}\$','interpreter‘,'latex'); c) Use fmincon (or any other optimization routine) to find the optimal mass ratios, that is, the mass ratios that maximize the rockets final velocity. find optimal mass Ratios % Rocket parameters payloadMass=500; totalMass=25000; structCoeff=0.1; exhaustVel=3.3; N=3; % Initial objective function [email protected](x)multiStageRocketObjFun(x,payloadMass,totalMass,structCoeff,exhaustVel); fminconoptions=optimset('Display','none','Algorithm','interior—point‘); set Initial guess and bounds for optimal mass ratios x0=ones(N—l,l)*l/5; lb=0.0l*ones(size(xO)); ub=l*ones(size(x0)); [x,fval,exitflag]=fmincon(objFun,xO,,,,,lb,ub,,fminconoptions); % compute the masses from the mass ratios massRatioMat=tril(ones(N—l,l)*x'); % replace the zeros on the upper triangle with ones massRatioMat(massRatioMat== )=1; % create mass vector mass=zeros(N,1); mass(l)=(totalMass—payloadMass)/(l+sum(prod(massRatioMat,2))); for i=2:N mass(i)=x(i—l)*mass(i—l); end You want to maximize delta-v, fmincon will minimize the objective function. Therefore you need the objective function to output —Av . You can either write an entirely new function function f: multistageRocketObjFun(massRatio,payloadMass,totalMass,structCoeff,exhaustVel) computes the delta v produced by a multi state rocket massRatio — the mass ratio between each adjacent stage m(i+l)/mi totalMass - total mass of the rocket structCoeff — vector of structural coefficients for each stage exhaustVel — vector of exhaust velocity for each stage if structCoeff or exhaustVel are scalars it is assumed that every stage has the same exhaust velocity or structural coefficient. dv = multiStageRocket(massRatio,payloadMass,totalMass,structCoeff,eXhaustVel); f=—dV; function dv = multiStageRocket(massRatio,payloadMass,totalMass,structCoeff,exhaustVel) computes the delta v produced by a multi state rocket massRatio — the mass ratio between each adjacent stage m(i+l)/mi totalMass - total mass of the rocket structCoeff — vector of structural coefficients for each stage exhaustVel - vector of exhaust velocity for each stage if structCoeff or exhaustVel are scalars it is assumed that every stage has the same exhaust velocity or structural coefficient. Number of stages N=length(massRatio)+l; ensure structCoeff and exhaustVel are vectors if length(structCoeff)==l structCoeff=ones(N,l)*structCoeff; end if length(exhaustVel)==l exhaustVel=ones(N,l)*exhaustVel; end % need massRatios as a row vector [m n]=size(massRatio); if n<m massRatio=massRatio'; end % calculate the masses of each stage massRatioMat=tril(ones(N—l,l)*massRatio); % replace the zeros on the upper triangle with ones massRatioMat(massRatioMat== )=l; %keyboard % create mass vector mass=zeros(N,l); mass(l)=(totalMass—payloadMass)/(l+sum(prod(massRatioMat,2))); for i=2:N mass(i)=massRatio(i—l)*mass(i—l); end % initialize vector of deltav's deec=zeros(length(mass),1); % calculate the delta v for each stage for i=l:N deec(i)=exhaustVel(i)*log((sum(mass(N:—l:i))+payloadMass)/(structCoeff(i)*mass(i) +sum(mass(N:—l:(i+1)))+payloadMass)); end % total dv dv=sum(deec); ...
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AE202 Problem Set 5 Solutions - Problem Set 5 Spring 2011...

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