Using Matlab to analyze shuttle launch into orbitA shuttle launches and I must create a function fitting the given table of values and write a summary of the code. I need help with the code. I am in a time crunch and anything you can do to help would be greatly appreciated. Would pay well for your best attempt at the code, it does not necessarily need to work. I would still be appreciative of a well built attempt. Thank you!
Linear Algebra and Numerical MethodsME 364Final ProjectDue December 9, 2020BackgroundNASA uses numerical methods extensively during the design, launch, orbit and operation ofspacecraft. While many of the calculations that are performed are too complex to be consideredin an introductory course on numerical methods, there are several applications that we canexamine to see how the topics we have introduced in this course are applied in the real world.This project consists of two problems, one required and one optional, each having several parts,which will highlight a numerical method we have discussed in this class. You will need to completeeach section and compile your results into a brief report. Report requirements are given at theend of this document.Section 1 (Required): Modeling the Ascent of the Space ShuttleBackground InformationThe ascent is a very dangerous andimportant phase of any space shuttlelaunch. This phase begins at liftoff andends as soon as the spacecraft reachesorbit. The main propulsion system,which consists of the external tank (ET)and the orbiter engines, together withthe solid rocket boosters (SRBs) supplythe force needed to accelerate to thevelocity of approximately 7.85 km/sthat is required to obtain orbit. Thissystem is so powerful that within 1minute the space shuttle breaks thesound barrier. As the velocity continuesto increase, the stress on the spaceshuttle and the crew increases. Becauseof the immense power of the engines,the acceleration of the space shuttle must be kept below 3 g, or 29.43 m/s2, to ensure the safetyof the vehicle and the crew.The space shuttle continues on its trajectory until Main Engine Cutoff (MECO), soon after whichit reaches orbit. It takes approximately 8 minutes and 30 seconds for the space shuttle to reachMECO, which occurs around 104 km above the surface of the earth. The ET is dropped away tobreak up and land in the ocean, and the orbiter performs a final burn approximately 30 minuteslater to reach orbital altitude of around 320 km.The space shuttle experiences changes in altitude, velocity, and acceleration during the ascentinto space. These changes can be seen by taking a closer look at the entire ascent phase. Theascent process begins with the liftoff from the launchpad. As the SRBs burn their propellant andthe Main Engines burn fuel from the ET the space shuttle accelerates very rapidly. This high-rateof acceleration causes a rapid increase in dynamic pressure, known as Q. As the space shuttlebreaks the sound barrier, its structure can only withstand a certain level of dynamic pressurebefore suffering damage. Before this critical level is reached, the engines of the space shuttle arethrottled down to about 67% of full power to avoid damage. About 50 seconds after liftoff, thedynamic pressure reaches its maximum aerodynamic load (Max Q). The air density then dropsrapidly due to the thinning of the Earth’s atmosphere, and then the space shuttle can be throttleto full power without fear of damage since atmospheric drag is essentially zero at this point.As the space shuttle climbs, the velocity is increasing and the density of the air is decreasing.About 2 minutes after liftoff the atmosphere is so thin that the dynamic pressure drops to zero.The SRBs, having used all their propellant, are commanded by the space shuttle’s onboardcomputer to separate from the ET. The spent SRBs fall into the ocean and are recovered,refurbished, reloaded with propellant, and reused for several missions. The jettisoning of thesebooster rockets marks the end of the first ascent stage and the beginning of the second, whichwe will focus on for this project. The second phase of ascent lasts about six and a half minutes,at the end of which, the space shuttle maneuvers into orbit. Just prior to orbit, ET separationoccurs, and the ET re-enters Earth’s atmosphere, breaking up before falling into the ocean. Theentire ascent phase is summarized in the velocity and acceleration histories below. For thisproject we will focus on the portion of the ascent following SRB separation until the maximumacceleration is reached and levels off (120-460 seconds).Problem Statement:The velocity of the space shuttle from 120 to 460 seconds after liftoff is shown in the tablebelow at 20 second increments.
Time(s)
VerticalVelocity(m/s)
LongitudinalVelocity(m/s)
VelocityMagnitude(m/s)
120
647
1099
1275
140
590
1250
1382
160
503
1405
1492
180
445
1577
1639
200
366
1755
1793
220
311
1968
1992
240
245
2180
2194
260
186
2403
2410
280
126
2660
2663
300
80
2930
2931
320
33
3178
3178
340
0
3533
3533
360
-35
3840
3840
380
-50
4150
4150
400
-64
4645
4645
420
-70
5209
5209
440
-59
5599
5599
460
-41
6188
6188
Using MATLAB create a program or set of programs that performs the following tasks. You shouldcombine everything into a single script file that calls all the function files you have developed,divided into appropriate sections:(1) (20 points) Develop a function that will perform a 4th order polynomial regression todevelop fits to the data for vertical velocity, horizontal velocity, and total velocity.Determine the standard error of the estimate, sy/x, the coefficient of determination, r2,and the correlation coefficient, r for the fit. You can either develop one function that willread in all three sets of velocity data simultaneously or read the velocity data one at atime to develop the fits. Your input for the function should be vectors of time and velocity(vertical, horizontal, and total). Your output should be the coefficients for your 4th orderpolynomial regression of the form:𝑣𝑣(𝑡𝑡) = 𝑎𝑎0 + 𝑎𝑎1𝑡𝑡 + 𝑎𝑎2𝑡𝑡2 + 𝑎𝑎3𝑡𝑡3 + 𝑎𝑎4𝑡𝑡4Other output should include the standard error of the estimate, sy/x, the coefficient ofdetermination, r2, and the correlation coefficient, r, for the fit. Also, generate a plot of theoriginal data (denoted as open circles) along with the regression line for each velocitydata set. (Note: you may receive warning messages about your matrix being singular orbadly scaled, however you can safely ignore these warnings.)(2) (20 points) Using the 4th order fits obtained in part (1) calculate the ‘analytical’ equationsfor acceleration and distance traveled for both the vertical and horizontal components aswell as the magnitudes of each. Recall the acceleration is given by:𝑎𝑎(𝑡𝑡) =𝑑𝑑𝑣𝑣
𝑑𝑑𝑡𝑡𝑥𝑥(𝑡𝑡) = � 𝑣𝑣(𝑡𝑡)𝑑𝑑𝑡𝑡
and the distance is given by:
where the vertical, horizontal and total distance at 120 sec are 44,726m, 125,047m, and132,805m, respectively.(3) (20 points) Using the velocity equations obtained from part (1) and (2) as input, write aMATLAB program to determine the vertical, horizontal, and total acceleration using thefinite difference equations. For the first point you should use the forward differenceequations, for the last point use the backward difference equation, and for the remainingpoints use the centered difference equation. Use a step size of 20 seconds. Calculate the‘true’ relative error between values calculated using the difference equations and thevalues obtained from the analytical acceleration equations obtained in part (2). Createplots comparing the results of the difference calculations to the analytical results for thevertical, horizontal, and total accelerations. Determine the maximum acceleration overthe given time span (120-460 seconds). How many gs did the astronauts experienceduring this phase of the ascent based on your finite difference analysis? (Gravity = 9.81m/s2)(4) (25 points) Using the velocity equations obtained from parts (1) and (2) as input, write aMATLAB program to determine the vertical, horizontal, and total distance traveled usingEuler’s method. The vertical, horizontal, and total distance at t0 =120 sec are 44,726m,125,047m, and 132,805m, respectively. Once again use a step size of 20 seconds.Calculate the ‘true’ relative error between the values calculated with Euler’s method andthose obtained using the analytical distance equations from part (2). Create plotscomparing the results of the difference calculations to the analytical results for thevertical, horizontal, and total distances traveled.(5) (15 points) Comment on the accuracy of the computational methods developed in thissection. How could your models be improved? Can you use the models you havedeveloped to extrapolate to times less than 120 seconds or greater than 460 seconds?Looking at the graphs of acceleration and velocity in the background information above,would it be possible to accurately develop a single equation to represent the entire ascentprocess from 0 to 550 seconds? Why or why not?Extra Credit (+20 Points – all or nothing): Using Root Finding Methods on the Launch PadDuring liftoff the space shuttle has four forces acting on it, as shown in thefree body diagram. The combined weight of the two solid rocket boostersand the external fuel tank is WB = 1.663 x 106 lb. The weight of the orbiterwith a full payload is WS = 0.23 x 106 lb. The combined thrust of the two solidrocket boosters and the external fuel tank is TB = 5.30 x 106 lb. The combinedthrust of the three liquid fuel orbiter engines is TS = 1.125 x 106 lb.At liftoff the orbiter engine thrust is directed at an angle θ that makes theresultant moment acting on the entire craft assembly (external tank, solidrocket boosters, and orbiter) equal to zero. With the resultant momentequal to zero, the craft will not rotate about its mass center G at liftoff. Withthese forces, the craft will have a resultant force with components in boththe vertical and horizontal directions. The vertical resultant forcecomponent is what allows the craft to lift off from the launch pad and flyvertically. The horizontal resultant force component causes the craft to flyhorizontally. The resultant moment acting on the craft will be zero when θis adjusted to the appropriate value. If this angle is not adjusted properly,and there is some resultant moment acting on the craft, the shuttle willbegin rotating about its mass center and might potentially lead to anincorrect orbit or even an emergency abort of the launch if θ is too far off.FBD of the SpaceShuttle(1) Resolve the orbiter thrust TS into horizontal and vertical component, and then sum themoments about point G, the center of mass of the craft. Set the resulting momentequation equal to zero. This equation can now be solved for the value of θ required forliftoff.(2) Derive an equation for the resultant moment acting on the craft in terms of the angle θ.Plot the resultant moment as a function of the angle θ over a range of -5 radians to +5radians. Use this plot to estimate the location of the root that gives the most logical valuefor θ.(3) Using the MATLAB function files we developed in class solve for the root using thebisection method, false position, fixed point iteration, and the Newton-Raphsonmethod. Continue iterations until the relative approximation error in θ is accurate to sixsignificant figures. Create a table listing the method, the calculated root (if one is found),the initial guess or bounds, the number of iterations, and the final error. Make sure youchoose an appropriate range for the bracketed methods, since the plotted range of -5radians to +5 radians will contain several roots.(4) Create a plot of Percent Relative Error vs Number of Iterations for each of the methodslisted in part 3 (on a single graph). You will need to modify the MATLAB functions to adda vector of length ‘iter’ that stores the calculated error for each iteration and is saved asan output variable. Based on the graph, comment on the efficiency of the selectedmethods. Make sure you label your axes and provide a legend and some distinguishingmarkings to identify each method on the plot.Report RequirementsThe ability to present your work in a clear and concise written report is essential to your successas an engineer. Your report should include the following sections:1) A title page2) A brief background on the problems being solved. This can come from the problemstatement or your own research.3) A brief technical description of the problems being solved and an explanation of yourcode and your approach to solving the required tasks. (1-2 paragraphs for each section)4) A brief description of your troubleshooting process. (Note: if you are unable to get yourcode to work this section should be fairly extensive and outline the steps you took inattempting to resolve your issues)5) Results from each section. Displaying your data graphically can be very powerful andgive the reader better insight into your results than you might be able to with words.Include both tables and plots as you see fit. Make sure at a minimum that you show theplots required in the problem statements.6) Briefly draw some conclusions about the results you saw and the functionality of yourcode.7) Include an appendix containing your code, as well as the generic functions you called.Make sure your code is well commented. Also include the published file from MATLAB.
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