Flight and Orbital Mechanics. Exams

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Candace Stevens
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1 1 Flight ad Orbital Mechaics Exas

2 Exa AE : Flight ad Orbital Mechaics (2 Noveber 2012, ) Please put your ae, studet uber ad ALL YOUR INITIALS o your work. Aswer all questios ad put your ae o each page of your exa. This exa cosists of questios: 1a-b, 2a-g, 3a-e, 4a-e, 5a-h Derive the expressios for each required calculatio (uless etioed i the file equatios by heart, for the orbital echaics part). The way the aswer is obtaied should be clearly idicated by visibly substitutig the ubers i the forulas. Oly etioig the fial aswer will NOT result i ay credits. Use of pecils to write the exa is NOT peritted. Scrap paper ay ot be added to your exa work (please take the scrap paper with you after the exa). It is ot peritted to have ay pre-prograed iforatio o your calculator. The eory of your calculator should be erased prior to the start of the exa. Failure to do so will be see as fraud. I total 100 poits ca be eared. (50 poits for flight echaics ad 50 poits for orbital echaics). At least 55 poits are required to pass the exa. Good luck! Questio 1 (Flight Mechaics - Cruise) [18 poits] A large trasport aircraft with four egies is perforig a cruise flight at 11k altitude. At the start of the cruise flight, the followig data are available for this aircraft: Aircraft Weight (W start ): 3500 [kn] Fuel weight (W f ): 1340 [kn] Wig surface area (S): 520 [ 2 ] True airspeed (V start ) : 936 [k/hr] Air desity at 11k altitude ( ) : [kg/ 3 ] Teperature at 11k altitude (T ): [K] Thrust specific fuel cosuptio (c T ): 0.65 [N/N hr] (costat ad idepedet of airspeed) Gas costat of air (R): [ 2 / s 2 K] The aircraft is perforig the cruise flight at costat altitude ad costat agle of attack. a. (12 poits) Calculate the iiu aspect ratio (A) that will allow this aircraft to fly a distace (rage) of 10000k i the cruise phase with the aout of fuel give above. The lift drag polar of the aircraft is the followig: C C D D0 2 CL CD 0 Ae e 0.85 A ukow

3 b. (6 poits) Next, calculate the required spa of this aircraft ad the Mach uber at the start ad ed of the cruise flight. Questio 2 (Flight Mechaics - Take-off, Equatios of otio) [32 poits] Aircraft carriers ake use of catapult systes to lauch aircraft fro the liited distace available o the deck. Durig the lauch, axiu thrust is also applied by the aircraft. I geeral, the ship will have a forward speed ito the directio of the wid (as idicated i the picture), to iprove the take-off perforace. The followig data are available: Wid speed (V wid ) 10 [kots] (= 5.1 /s) Ship speed (V ship ) 35 [kots] (= 18 /s) Distace of ruway (s deck ) 100 [] Catapult force (T catapult ): 100 [kn] (costat force) Aircraft weight (W): 200 [kn] Maxiu aircraft thrust (T ax ): 120 [kn ] (idepedet of airspeed) Wig surface area (S): 46.5 [ 2 ] Lift coefficiet i groud ru attitude: 0.1 Zero lift drag coefficiet (C D0 ) Lift drag polar: C C 2 kc D D0 L Factor k i lift drag polar: Coefficiet of ladig gear frictio (): 0.04 Air desity at sea level (): [kg/ 3 ] Gravitatioal acceleratio (g) [/s 2 ] V wid V ship V ed S deck a. (a + b total 9 poits) Draw a clear free body diagra (FBD) ad kietic diagra (KD) i which all the relevat forces, acceleratios, agles ad velocities are idicated b. Derive the equatios of otio for the aircraft durig the acceleratio over the ship deck You ca assue for this catapult assisted take-off that the groud ru is a uiforly accelerated otio. The ea acceleratio ust be calculated at the characteristic airspeed V ed,air /2 c. (3 poits) Derive a expressio for the groud ru distace s deck i ters of a ea acceleratio a ad the speed at the oet the aircraft leaves the deck (V ed ). Clearly idicate if the velocity i the equatio is expressed relative to the air or relative to the ship. d. (8 poits) Derive a equatio for the average acceleratio as a fuctio of the speed at the oet the aircraft leaves the deck (V ed ). Clearly idicate if the velocity i the equatio is expressed relative to the air or relative to the ship.

4 e. (3 poits) Write a expressio that gives the relatio betwee V ed relative to the air (airspeed) ad V ed relative to the ship deck. f. (4 poits) Calculate the both the airspeed ad the speed of the aircraft relative to the ship at the oet the aircraft reaches the ed of the ship deck. g. (5 poits) The iiu airspeed of this fighter aircraft for the give aircraft weight is 55 /s. Is the catapult force still sufficiet to lauch this aircraft i case the ship is ot ovig ad whe there is o wid preset? Give a thorough explaatio with your aswer. Questio 3 (Orbital Mechaics) [18 poits] A essetial aspect of a satellite issio is the occurrece of solar eclipses. a. (5 poits) For what (3) subsystes plays the presece (ad absece) of solar radiatio a role? Discuss each situatio briefly (2-3 lies each). b. (4 poits) For a eclipse to occur, 2 coditios have to be satisfied. What are they? Illustrate i a sketch. c. (4 poits) Cosider a 2-diesioal situatio, where the Su is located i the orbital plae of the satellite. What is the legth of the eclipse period for a satellite i a circular orbit at 800 k altitude (expressed as percetage of the orbital period)? What is it for a geostatioary satellite? d. (2 poits) Give the defiitio of a su-sychroous orbit. e. (3 poits) Is it possible to select a satellite orbit such that the satellite is i full sulight throughout its issio lifetie (e.g., 10 years)? If so, what are the (3) coditios for this? Is this solutio applied? Data: R e = k, μ earth = k 3 /s 2, h GEO = k. Questio 4 (Orbital Mechaics) [17 poits] The gravity field of the Earth is doiat for the otio of spacecraft orbitig Earth. It is give by the followig equatio: Where Re Re,,, U [1 J P (si ) J P (si ) cos( ( )) ] r r r P ( x) (1 x ), 2 /2 1 d ( 2)! dx d P ( x) dx 2 ( ) (1 x ) P x a. (2 poits) What is the geeral defiitio for the East-West acceleratio, based o the potetial forulatio for the gravity field of the Earth? b. (7 poits) Derive the geeral equatio for the East-West acceleratio due to the ter (2,2) for a arbitrary satellite.

c. (2 poits) Copute the orbit radius of a geostatioary satellite. d. (3 poits) What is the (uerical) expressio for the East-West acceleratio due to J 2,2 for a geostatioary satellite? e. (3 poits) The resultig equatio has the followig for: acc EW;2,2 = costat si(2(λ-λ 2,2 )).

5 c. (2 poits) Copute the orbit radius of a geostatioary satellite. d. (3 poits) What is the (uerical) expressio for the East-West acceleratio due to J 2,2 for a geostatioary satellite? e. (3 poits) The resultig equatio has the followig for: acc EW;2,2 = costat si(2(λ-λ 2,2 )). What are the locatios of the equilibriu poits? Data: μ Earth = k 3 /s 2 ; T E = 23 h 56 4 s ; J 2,2 = ; λ 2,2 = ; R e = k. Questio 5 (Orbital Mechaics) [15 poits] Cosider the situatio where the cargo vehicle Drago is to dock with the Iteratioal Space Statio (ISS). Both are orbitig Earth i a circular (coplaar) orbit at 400 k altitude. The ISS is 90 ahead of the Drago vehicle. You are resposible for desigig the trasfer betwee the two, where Drago is the active vehicle ad the ISS the (passive) target. I order to catch up with the ISS, Drago will be take to a elliptical (i.e. o-circular) trasfer orbit with a differet orbital period. a. (1 poit) I order to redezvous with the ISS, would you lower or raise the orbit of the Drago vehicle? Give a arguetatio o your choice. b. (1 poit) What is the orbital period at a altitude of 400 k? c. (2 poits) What is the circular velocity at a altitude of 400 k? d. (2 poits) Cosider the situatio that oe wats to have Drago coplete 15 revolutios i its elliptical trasfer orbit, before it does the redezvous with the ISS. What is the required shift per revolutio of Drago i its trasfer orbit, w.r.t. the ISS? e. (3 poits) What is the orbital period of this trasfer orbit, where the total trasfer is to be copleted after exactly 15 revolutios? If you were uable to aswer questio (b), use a value of 100 iutes for the orbital period i the origial orbit. f. (2 poits) What is the sei-ajor axis of this trasfer orbit? g. (2 poits) What is the velocity i the trasfer orbit at the origial altitude of 400 k? I case you could ot copute a aswer for questio (f), use a value of the sei-ajor axis of k. h. (2 poits) What is the required total velocity chage for this trasfer sceario? Data: μ Earth = k 3 /s 2 ; R e = k.

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