1 Exercise 3: Lift and Airfoils The first part of this week's assignment is to choose and research a reciprocating engine powered (i.e. propeller type) aircraft. You will further use your selected aircraft in subsequent assignments, so be specific and make sure to stay relatively conventional with your choice in order to prevent having trouble finding the required data during your later research. Also, if you find multiple numbers (e.g. for different aircraft series, different configurations, and/or different operating conditions), please pick only one for your further work, but make sure to detail your choice in your answer (i.e. comment on the condition) and stay consistent with that choice throughout subsequent work. In contrast to formal research for other work in your academic program at ERAU, Wikipedia may be used as a starting point for this assignment. However, DO NOT USE PROPRIETARY OR CLASSIFIED INFORMATION even if you happen to have access in your line of work. 1. Selected Aircraft: Pilatus PC-12 For the following part of your research, you can utilize David Lednicer's (2010) Incomplete Guide to Airfoil Usage at http://m-selig.ae.illinois.edu/ads/aircraft.html or any other reliable source for research on your aircraft. 2. Main Wing Airfoil (if more than one airfoil is used in the wing design, e.g. different between root and tip, pick the predominant profile and, as always, stay consistent): Please note also the database designator in the following on-line tool (see picture below): Find the appropriate lift curve for your Airfoil from 4. You can utilize any officially published airfoil diagram for your selected airfoil or use the Airfoil Tool at http://airfoiltools.com/search and text search for NACA or other designations, search your aircraft, or use the library links to the left of the screen. Once the proper airfoil is displayed and identified, select the \"Airfoil details\" link to the right, which will bring up detailed plots for your airfoil similar to the ones in your textbook. Text search input Library links Search result display Airfoil details tab This document was developed for online learning in ASCI 309. File name: Ex_3_Lift&Airfoils Updated: 06/23/2015 Please note the airfoil database designator (in parenthesis) in your answer to 2 above. 2 Concentrate for this exercise on the Cl/alpha (coefficient of lift vs angle of attack) plot. Start by de-cluttering the plot and leaving only the curve for the highest Reynolds-number (R e) selected (i.e. remove all checkmarks, except the second to last, and press the \"Update plots\" tab). Details Link \"Update plots\" tab 3. From the plot, find the CLmax for your airfoil (Tip: for a numerical breakdown of the plotted curve, you can select the \"Details\" link and directly read the highest C L value, i.e. the highest number within the second column, and associated AOA in the table, i.e. the associated number in the first column): 1.9024 CLmax 4. Find the Stall AOA of your airfoil (i.e. the AOA associated with C Lmax in 3.): 5. Find the CL value for an AOA of 5 for your selected airfoil: 16.750 1.0384 6. Find the Zero-Lift AOA for your airfoil (again, the numerical table values can be used to more precisely interpolate Zero-Lift AOA, i.e. the AOA value for which C L in the second column becomes exactly 0): 0.4536 This document was developed for online learning in ASCI 309. File name: Ex_3_Lift&Airfoils Updated: 06/23/2015 3 7. Compare your researched airfoil plot to the given plot of NACA 4412 (http://airfoiltools.com/airfoil/details?airfoil=naca4412-il). a) How do the two CLmax compare to each other? Describe the differences in airfoil characteristics (i.e. camber & thickness) between your airfoil and the given NACA 4412, and how those differences affect CLmax. (Use your knowledge about airfoil designation together with the airfoil drawings and details in the on-line tool to make conclusions about characteristics.) The overall difference is minute comparing the two for CLmax with the 4412 CLmax being 1.6706 at 16.250 with only a .5 difference at the angle of attack. For the PC-12 max thickness of 17% is at 30.3% chord where the 4412 is only 12% at 30% chord. When an airfoils camber increases the CLmax decreases. b) How do the two Stall AOA compare to each other? Explain how the differences in airfoil characteristics (i.e. camber & thickness) between your airfoil and the given NACA 4412 affect Stall AOA. There is only a slight difference in the AOA for the stall of each aircraft with the PC-12 being 0.500 degrees higher than the 4412. With an increase in camber the AOA decreases in correlation. c) How do the two Zero-Lift AOA compare to each other? Evaluate how the differences in airfoil characteristics between your airfoil and the given NACA 4412 affect Zero-Lift AOA. The PC-12 has a 0 angle CLmax of .4536 compared to the 4412 at .4833, with a higher camber the zero lift point AOA will decrease as well. 8. Compare your researched airfoil plot to the NACA 0012 plot. a) How do the two Zero-Lift AOA compare to each other? Evaluate how the differences in airfoil characteristics between your airfoil and the given NACA 0012 affect Zero-Lift AOA. The Zero-lift AOA for 0012 is at 0.000 which suggests that this airfoil is symmetrical is design. There is a significant amount more of camber in the PC-12 airfoil compared to the 0012 design. A 17% max thickness shows that there will be more lift with the PC-12 compared This document was developed for online learning in ASCI 309. File name: Ex_3_Lift&Airfoils Updated: 06/23/2015 4 to the 12% at 30% chord for the 0012. The max camber for the 0012 is 0% which tells us that the airfoil is symmetrical in design. b) What is special about the design characteristics of NACA 0012? How and where could this airfoil design type be utilized on your selected aircraft? Describe possible additional uses of such airfoil in aviation. For the second part of this assignment use your knowledge of the atmosphere and the Density Ratio, (sigma), together with Table 2.1 and the Lift Equation, Equation 4.1, in your textbook (remember that the presented equation already contains a conversion factor, the 295, and speeds should be directly entered in knots; results for lift will be in lbs): L = CL * * S * V2 / 295 Additionally, for your selected aircraft use the following data when applying Equation 4.1: 9. Research the Wing Span [ft]: 53ft 3 in 10. Find the Average Chord Length [ft]: Note: Average Chord = (Root Chord + Tip Chord) / 2 found 5.23 Ft (if no Average Chord is directly in your research) 11. Find the Maximum Gross Weight [lbs] for your selected aircraft: 10,450 lbs A. Calculate the Wing Area 'S' [ft2] based on your aircraft's Wing Span (from 9.) and Average Chord Length (from 10.): 277.8 ft ^2 12. Use the CL value for an AOA of 5 for your airfoil found in 5. above to simulate cruise conditions in the following exercise B. (Note it here for easier reference): 1.0384 B. Prepare and complete a table of Lift vs. Airspeed at different Pressure Altitudes utilizing the given Lift Equation and your previous data. (For the calculation of Density Ratio ' ' you can assume standard temperatures and neglect humidity.) You can utilize MS?? Excel (ideal for repetitive application of the same formula) to populate table fields and examine additional speeds and altitudes, but as a minimum, include six speeds (0, 40, 80, 120, 160, & 200 KTAS) at three different altitudes (Sea Level, 10000, 40000 ft), as shown below: Calculate LIFT (lb) Airspeed: 0 KTAS 40 KTAS 80 KTAS 120 KTAS 160 KTAS 200 KTAS 0 0 1,565 6,258 14,081 25,033 39,114 Pressure Altitude (PA) ft 10,000 0 1,155 4,622 10,399 18,487 28,886 This document was developed for online learning in ASCI 309. File name: Ex_3_Lift&Airfoils Updated: 06/23/2015 40,000 0 385 1,541 3,467 6,163 9,630 5 I) What is the relationship between Airspeed and Lift at a constant Pressure Altitude? Evaluate each Altitude column of your table individually and describe how changes in Airspeed affect the resulting Lift. Be specific and mathematically precise, and support your answer with the relationships expressed in the Lift Equation. As the airspeed increase, the lift increase by a square. Each are directly proportional to each other. II) What is the relationship between Altitude and Lift at a constant Airspeed? Evaluate each Airspeed row of your table individually and describe how changes in Altitude affect the resulting Lift. Be specific and mathematically precise, and support your answer with the relationships expressed in the Lift Equation. Altitude and lift are inversely proportional. Lift and density are directly proportional. However, as altitude increases, the density decreases to lift III) Estimate the Airspeed required to support the Maximum Gross Weight of your selected airplane (from 11. above) at an Altitude of 10000 ft and flying at the given AOA of 5. (As initially indicated, a more detailed table/Excel worksheet is beneficial precision for this task. To support the Weight of any aircraft in level flight, an equal amount of Lift has to be generated - therefore, you can also algebraically develop the equation to yield a precise Airspeed result, i.e. substituting L=W and solving for V in the lift equation. Remember that conditions in this question are not at sea level.) 88.87 kts C. In B.III) above, we noted that lift has to equal weight in order to sustain level flight. Using the same Maximum Gross Weight (from 11.), and the same Wing Area (from A.), calculate required AOA for level flight at the different airspeeds in your table under standard, sea level conditions (i.e. =1). You can start a new table or expand your existing one. (See also step by step instructions below the table.): Airspeed (KTAS) 0 40 80 120 160 200 Required Lift = Weight Required CL Corresponding AOA for your airfoil 0 10,450 10,450 10,450 10,450 10,450 1.37 1.73 0.77 0.43 0.28 8.000 12.250 2.750 -0.250 -1.500 THIS PLOT IS AN EXAMPLE First and similar note in B.III) above, develop the lift equation algebraically toONLY yield C L AND NOT results based oninAirspeed inputs (i.e. substitute Lift with the aircraft Weight and solve the Lift the APPLICABLE Equation for the vertical, Coefficient CL; then insert the different Airspeeds into V, calculate theFOR corresponding Cleft L values, and note them in your table). YOUR AIRFOIL - scale PLEASE USE YOUR Finally, use your researched airfoil Cl/alpha plot (from 3. through 8.) to find RESEARCHED corresponding LIFT AOA to your calculated CL values (enter the plot in the left scale with each calculated CL value, trace CURVE FROM 3. horizontally to intercept the graph for that CL value, then move down vertically to find the THROUGH 8. This document was developed for online learning in ASCI 309.ABOVE. Enter to withthe CL File name: Ex_3_Lift&Airfoils Updated: 06/23/2015 Read corresponding AOA on the bottom scale 6 corresponding AOA and note it in your table (alternatively, you can also look up values in the detailed table): I) Comment on your results. Are there airspeeds for which you could not find useful results? Describe where in the step by step process you've got stuck and why. Explain what it aerodynamically means for your airfoil if a required C L value is greater than the CLmax that you found in 3. For my results there were no CL values that were higher than my CL Max, also a 0 kts there is no useful CL as the math will always result is equating to 0. II) What is the standard sea level Stall Speed for your selected aircraft at its Maximum Gross Weight? (Utilize above data and the Stall Speed Equation on page 44 of \"Flight Theory and Aerodynamics\"). Stall speed at Maximum Gross Weight is 76.38 kts This document was developed for online learning in ASCI 309. File name: Ex_3_Lift&Airfoils Updated: 06/23/2015 1 Exercise 4: Drag and Applications The first part of this week's assignment is to revisit our reciprocating engine powered (i.e. propeller type) aircraft from last week. 1. Selected Aircraft (from last week's module): Pilatus PC-12 Make sure to review your data and results from last week and any feedback that you may have received on your work, in order to prevent continuing with faulty data. 2. Main Wing Airfoil type & on-line database designator (from last week's module): NASA LS(1)-0417MOD 3. Aircraft Maximum Gross Weight [lbs] (from last week's module): 10,450 Pounds 4. Wing Span [ft] (from last week's module): 53 feet 5. Average Chord Length [ft] (from last week's module): 5.24 feet 6. Wing Area 'S' [ft2] (from last week's module): 277.8 Sq ft 7. Find the Aspect Ratio 'AR' for your selected aircraft wing. (Use the wing span and average chord length from last week's module/from above. See also page 63 in your textbook.): AR- 53/5.24= 10.11 8. CLmax for your airfoil (from last week's module): 1.9024 9. Standard sea level Stall Speed 'Vs' for your aircraft [kts] (from last week's calculation): 76.38 Find the appropriate drag polar curve for your airfoil selection (2. above; from last week's module). You can utilize any officially published airfoil diagram for your selected airfoil or use again the Airfoil Tool at http://airfoiltools.com/search . This document was developed for online learning in ASCI 309. File name: Ex_4_Drag&Applications Updated: 07/11/2015 2 Concentrate for this exercise on the Cl/Cd (coefficient of lift vs coefficient of drag) plot, i.e. the so called drag polar. Use again only the curve for the highest Reynolds-number (Re) selected (i.e. remove all checkmarks, except the second to last, and press the \"Update plots\" tab). How to find the minimum Cd 10. From the polar plot, find the CDmin value for your airfoil, i.e. the lowest value that the coefficient of drag 'Cd' (bottom scale in the online tool depiction) reaches. (Tip: for a numerical breakdown of the plotted curve, you can again select the \"Details\" link and directly read the lowest CD value in the table - third column, labeled \"CD\"): 0.00661 What we've just found (...with some degree of simplification...) is the parasite drag coefficient for our airfoil, i.e. the drag that exists due to skin friction and the shape of our airfoil, even when little or no lift is produced. However, this value will only represent the airfoil, i.e. main wing portion of our aircraft; therefore, let us for the remainder of our calculations assume that our aircraft is a Flying Wing type design and the total CDP for the aircraft is the same as the CDmin that we've just found. Let us also assume that we are at standard sea level atmospheric conditions and that our wing has an efficiency factor of e = 0.82. This document was developed for online learning in ASCI 309. File name: Ex_4_Drag&Applications Updated: 07/11/2015 3 A. Prepare and complete the following table for your aircraft (with the data from 1. through 8. above). Start your first row with the Stall Speed 'V s' (from 7. above) and start the second row from the top with the next higher full twenty knots above that stall speed. Then increase speed with every subsequent row by another 20 knots until reaching 300 kts. You are again encouraged to utilize MS Excel as shown in the tutorial video and can also increase your table detail. However, the below depicted, and above described, interval is the minimum required for this assignment. V (KTAS) q (psf) 76.38 21.7 100 33.9 120 48.81 140 66.44 160 86.78 109.8 3 135.5 9 164.0 7 195.2 5 229.1 5 265.7 6 305.0 9 180 200 220 240 260 280 300 CL 1.90 2 1.10 9 0.77 1 0.56 6 0.43 4 0.34 3 0.27 7 0.22 9 0.19 3 0.16 4 0.14 2 0.12 3 CDP 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 0.0066 1 CDI 0.139 0.047 0.023 0.012 0.007 0.005 0.003 0.002 0.001 4 0.001 0 0.000 7 0.000 6 CD 0.14 6 0.05 4 0.03 0 0.01 9 0.01 4 0.01 2 0.01 0 0.00 9 0.00 8 0.00 8 0.00 7 0.00 7 CL/CD DP (lb) 13.027 36.77 20.537 62.25 25.700 89.63 122.0 0 159.3 5 201.6 8 248.9 8 301.2 7 358.5 3 420.7 8 488.0 0 560.2 2 29.789 31.000 28.583 27.700 25.444 24.125 20.500 20.286 17.571 DT (lb) DI (lb) 837.92 8 442.61 9 874.70 504.87 311.867 221.48 4 168.75 2 152.55 4 401.50 343.48 328.10 354.23 113.001 361.98 91.157 392.43 75.937 434.47 63.658 484.44 51.680 539.68 50.852 611.07 Equations for Table: W CDi =[1/ (eAR)] CL 2 q= CL = CD = CDP + CDi CD = CDP + [1/ ( e AR)] CL 2 qS Di = CDi q S = [1/ ( e AR)] CL2 q S Dp = CDp q S Dt = Di + Dp = CD q S Answer the following questions from your table. This document was developed for online learning in ASCI 309. File name: Ex_4_Drag&Applications Updated: 07/11/2015 4 I) Determine the minimum total drag 'Dmin' [lbs] (i.e. the minimum value in the total drag 'DT' column): 328.10 II) Determine the airspeed at which this minimum drag occurs 'V Dmin' [kts] (i.e. the speed associated with the row in which 'D min' was found): 160 kts III) Compare parasitic 'DP' and induced 'DI' drag at VDmin. What is special about this point in your table? Dp is greater than the Di. Additionally this is also the max glide range. IV) Determine the maximum CL/CD value in your table (i.e. the maximum value in the CL/CD column) and the speed at which it occurs. 31.00/ 160 knots V) Compare your results in IV) with II) and comment on your findings. CL/CD can be found at the same velocity. VI) Explain which values in your table will directly allow glide performance prediction and how (Tip: Reference again the textbook discussion pp. 61-63). Cl/Dl Max= 31.00 which takes place at the efficient AOA. At this juncture there is minimum drag, and thus is the best engine out glide ratio. B. If the gross weight of your aircraft is decreased by 10% (e.g. due to fuel burn), how would the stall speed change? Support you answer with calculation as well as written assessment. (Remember, stall speed references and discussions can be found pp. 4345 in your textbook.) Vs= The square root of (295)(9405)/(1.9024)(1)(277.8)= 72.256 knots For the second part of this assignment use the given figure below (Figure 1.13 from Aerodynamics for Naval Aviators [1965]) to answer the following questions. (This assignment is designed to review some of the diagram reading skills required for your midterm exam; therefore, please make sure to fully understand all the diagram information and review book, lecture, and/or tutorials if necessary.): This document was developed for online learning in ASCI 309. File name: Ex_4_Drag&Applications Updated: 07/11/2015 5 Figure 1.13 from Aerodynamics for Naval Aviators (1965). C. What is the Angle of Attack at Stall for the aircraft in Figure 1.13? AOA stall is at CLmax= 20 degrees D. What Angle of Attack is associated with Best L/D? L/D max= AOA 6 degrees E. What would be the best Glide Ratio for this aircraft? 12.5, the intersection of CL, L/D, and Cd F. What is the maximum coefficient of lift (C Lmax) value? CLmax=1.5 This document was developed for online learning in ASCI 309. File name: Ex_4_Drag&Applications Updated: 07/11/2015 1 Exercise 5: Aircraft Performance For this week's assignment you will revisit your data from previous exercises, therefore please make sure to review your results from the last modules and any feedback that you may have received on your work, in order to prevent continuing with faulty data. 1. Selected Aircraft (from module 3 & 4): 2. Aircraft Maximum Gross Weight [lbs] (from module 3 & 4): Jet Performance In this first part we will utilize the drag table that you prepared in module 4. Notice that the total drag column, if plotted against the associated speeds, will give you a drag curve in quite similar way to the example curves (e.g. Fig 5.15) in the textbook. (Please go ahead and draw/sketch your curve in a coordinate system or use the Excel diagram functions to depict your curve, if so desired for your own visualization and/or understanding of your further work.) Notice also that this total drag curve directly depicts the thrust required when it comes to performance considerations; i.e. as discussed on pp. 81 through 83, in equilibrium flight, thrust has to equal drag, and therefore, the thrust required at any given speed is equal to the total drag of the airplane at that speed. Last but not least, notice also that, so far, in our analysis and derivation of the drag table in module 4, we haven't at all considered what type of powerplant will be driving our aircraft. For all practical purposes, we could use any propulsion system we wanted and still would come up with the same fundamental drag curve, because it is only based on the design and shape of the aircraft wings. Therefore, let's assume that we were to power our previously modeled aircraft with a jet engine. A. What thrust [lbs] would this engine have to develop in order to reach 260kts in level flight at sea level standard conditions? Notice again that in equilibrium flight (i.e. straight and level, unaccelerated) thrust has to be equal to total drag, so look for the total drag at 260kts in your module 4 table. (In essence, this example is a reverse of the maximum speed question - expressing it graphically within the diagram: We know the speed on the X-axis and have the thrust required curve; that gives us the intercept point on the curve through which the horizontal/constant thrust available line must go.) B. Given the available engine thrust from A. above, what is the Climb Angle [deg] at 200kts and Maximum Gross Weight? (Notice that climb angle directly depends on the available excess thrust, i.e. the difference between the available thrust in A. above and the required thrust from your drag curve/table at 200kts. Then, use textbook Eq. 6.5b relationships to calculate climb angle). C. What is the Max Endurance Airspeed [kts] for your aircraft? Explain how you derived at your answer. This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 2 Prop Performance In this second part we will utilize the same aircraft frame (i,e, the same drag table/graph), but this time we will fit it (more appropriately and closer to its real world origins) with a reciprocating engine and propeller. D. To your existing drag table, add an additional column (Note: only the speed column, the total drag column and this third new column will be required - see below). To calculate the Power Required in the new column, use textbook p. 115 equation and the V and D values that you already have: Pr = D*Vk / 325 V (KTAS) VS 80 100 120 140 160 180 190 200 220 240 260 DT = Tr (lb) Pr (HP) Pr maxE maxR V E. Draw/sketch (or plot in an Excel diagram) your Power Required curve against the speed scale from the table data in A. above. (Note: This step is again solely for your visualization and to give you the chance to graphically solve the next questions in analogy to the textbook and examples. See sketch above.) F. Find the Max Range Airspeed [kts] for your aircraft. Remember from the textbook discussion pp. 125 through 127 that Maximum Range Airspeed for a reciprocating/propeller driven aircraft occurs where a line through the origin is tangent to the power required curve (see textbook Fig. 8.9 and sketch above). However, as per the textbook discussion, it is also the (L/D) max point, which we know from our previous work on drag happens where total drag is at a minimum (therefore, you can also reference the total drag column in your table and find the airspeed associated with the minimum total drag value). G. Find the Max Endurance Airspeed [kts] in a similar fashion. (Tip: The minimum point in the curve will also be visible as minimum value in the P r column of your table.) H. Let's assume that the aircraft weight is reduced by 10% due to fuel burn (i.e. similar to the gross weight reduction in Exercise 4, problem B). This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 3 I) Aircraft Weight [lbs] for 90% of Maximum Gross Weight (i.e. the 10% reduced weight from above). Simply apply the factor 0.9 to your aircraft Maximum Gross Weight from number 2. above: your II) Find the new Max Range Airspeed [kts] for the reduced weight. Remember (from textbook reading and Exercise 4, B.) that the weight change influence on speed was expressed by Eq. 4.2 in the textbook. Landing Performance For this last part of this week's assignment you will continue with your reciprocating engine (i.e. prop) powered aircraft and its reduced weight. Let's first collect some of the data that we already know: 3. Stall Speed for 90% of Maximum Gross Weight (i.e. the stall speed for 10% decreased weight from above, which we already calculated in Exercise 4, problem B.): I. Find the Approach Speed [kts] for your 90% max gross weight aircraft trying to land at a standard sea level airport. Approach speed is usually some safety margin above stall speed -.let's assume for our case a factor of 1.2, i.e. multiply your stall speed from number 3. with a factor of 1.2 to find the approach speed: J. Determine the drag [lbs] on the aircraft during landing roll. I) For simplification, start by using the total drag value [lbs] for stall speed (for the full weight aircraft) from your module 4 table: the II) Adjust the total drag (from I) above) for the new weight (from H. I) above) by using textbook Equation 7.1 relationship: D2/D1 = W2/W1 the III) Find the average drag [lbs] on the aircraft during landing roll. A commonly used simplification for the dynamics at play is to use 70% of the total drag at touchdown as average value. Therefore, find 70% of your II) result above. K. Find the frictional forces during landing roll. The Total Friction is comprised of Braking Friction at the main wheels and Rolling Friction at the nose/tail wheel. For this example, let's assume that, in average, there is 75% of aircraft weight on the main wheels and 25% on the nose/tail wheel over the course of the landing roll. The Average Friction Force is then the product of respective friction coefficient and effective weight at the wheel/wheels (see p. 209 textbook): F = *N This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 4 I) If the rolling friction coefficient is 0.02, what is the Rolling Friction [lbs] on the nose/tail wheel? (Remember that only 25% of total weight are on that wheel and that the weight was reduced by 10% from maximum gross weight - see H I)): II) If the main wheel brakes are applied for an optimum 10% wheel slippage (as discussed on textbook pp. 209/210), what is the Braking Friction [lbs] on the main wheels during landing roll on a dry concrete runway? Use textbook figure 13.9 to determine the friction coefficient. (Remember that the weight on the main wheels is only 75% of total aircraft weight). III) Find the total Average Friction [lbs] during landing by building the sum of I) and II): L. Find the Average Deceleration [ft/s2] during landing roll. Use the same rectilinear relationships as in module 1, applying the decelerating forces of friction and drag from J. III) & K. III) above. Assume that residual thrust is zero. (Keep again in mind that for application of Newton's second law, mass is not the same as weight. Your result should be a negative acceleration value since the aircraft decelerates in this case.): M. Find the Landing Distance [ft] (Remember that we start from a V0 at approach speed and want to slow the aircraft to a complete stop, applying the negative acceleration that we found in L. Also, remember to convert approach speed from I. above into a consistent unit of ft/s.): N. If your aircraft was to land at a higher than sea level airport (e.g. at Aspen, Co) what factors would change and how would it affect your previous calculations, especially your landing distance. Explain principles and relationships at work and support your answer with applicable formula/equations from the textbook. You can include example calculations to support your answer: This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 126 PROPELLER AIRCRAFT BASIC PERFORMANCE - o 2@ .u4{m e 5 o ! = B o J PROP AIRPLANE'OATA wElGHt 20,(M *- lbs , wlNG AREA (Sl 5oo ftSEA LEVEL DETISITY d = I ui 1500 Hsm c. ul = 2@ rom .g lt tt l r E 500 alH l o lrJ g o ld , MAX ENOURANCE FF mln az HPP min -v? ) mox , Dv mln ' .{!t 'vD d, c, !J 1000 -o tr u G, o a lnwer-producing occur at (I MAX ENDURANCE -Tz tlt .LD t ,rs tE tf;t 3 a, ,,.rn ru 1H$R1 min pof,1 a Maximum engine' L Maximum range .in. i tf,t ,',o, pilot should reali I o = J E J UI f vELoclrY - {(NOT5) Efiect on Specifi" ts flying into a should be headwind. Conv and range' Fig. 8.9. Finding the maximum endurance t miles flown t* : Nautical To*a, orf"tGta is way and shou nt. The amc be defined by the following relationship: L10. Airspeed (knots) Fua ftow (tbAr) For maximum sPecific range, or SR-"* occur'"t [T].,, ,*-".: [*-l Lt F l-u* a straight line drawn from th The point on the P, curve in Fig' 8'9 where velocit-v'the maximum specific-range origin is tangent to ttre curve will i-ndicate Io obtain the valu ind, by plotting a new tangent l ind is done simila tangent line is d R.ange total range for 1 ine aircraft (set 6 Jet Aircraft Basic Performance The performance capabilities of an aircraft depend on the relationship of the forces acting on it. The principal forces are lift, weight, thrust, and drag. If these forces are in equilibrium, as shown in Fig.6.1, the aircraft will maintain steady-velocity, constant-altitude flight. If any of the forces acting on the aircraft change, the performance of the aircraft will also change. To better understand the relationship between these forces and performance, we will study aircraft performance curves in this chapter. Some of the items of performance that can be obtained from these curves are 1. Maximum level flight velocity 2. Maximum climb angte 3. Velocity for maximum climb angle 4. Maximum rate of climb 5. Velocity for maximum rate of climb 6. Velocity for maximum endurance 7. Velocity for maximum range This chapter and the next one show the construction and use of the performance curves for thrust-producing aircraft. Power-producing aircraft (propellerdriven aircraft) are discussed in Chapters 8 and 9. THRUST-PRODUCING AIRCRAFT Some aircraft produce thrust directly from the engines. Turbojet, ramjet, and rocket-driven aircraft are examples of thrust producers. Fuel consumption is roughly proportional to the thrust output of these aircraft and, because range and endurance performance are functions of fuel consumption, the thrust required is of prime interest. Thrust-Required Curve Each pound of drag requires a pound of thrust to offset it. In Chapter 5 we discussed how the drag curve for an aircraft is drawn. We can now call this same curve a thrust-required curue. The curves used to illustrate the drag or thrust-required (4) curves in this chapter are for the T-38 supersonic turbojet trainer. Because this aircraft encounters a large amount of compressibility drag 8l JET AIRCRAFT BASIC PERFORMANCE oRAG 4000lb THRUST 4(NO Ib E x wEIGHT 10,000 lb - c = a ! Fig' 6'1' n'it"'uft in equilibrium flight' a I at high speeds, a sharp increase in in the thrust required will be noted a this t r il region' r^- the +L^ 'r ?Q The drag dras curve curve shows the - for T-38' Figure 6'2 shows trre drag'curve that the ui uutiou' airspeeds' Figu're 9:' :hoYt amount of drag or tn* Ili'u?t 485 knos ihe buffet timli(stall speed). At about aircraft has a drag "f |- 2fi;il;t fu h atso thic r & thrus 6Jbid &ag o; .- The t mnfu o o o I (, d o fitrt pox md thn Simpl ln o the of tI 9*F ts ,!9usEglgErc_ 3OO VELOCITY 4O0 _KNOTS TA5 Fie.6.2. T-38 drag curve' r trh rcactr( (tl-'l)O:l:'4 (r'q) NOISTOdoud.{o sf,ldlJ ueelt fep prepuels uo Islol eos 1e uotlernEguoc eqJ salrnc ut pae Z'g's8tg u^\\Jp eft "'peJln ur eueldrte qf000'0l 3 roJ E'g t":':;tT lsnrql ,, sB pelaqsler ,, ,1n" Ieque^ "{t il'^-?:"i'r1"^:':"::,1i;;, sr 'z's:att'ul u^\\ot{s euo esr 01 l?3l}uapl s'e'E1g [Ji-i;;il,I'lo'l'fi; ?n'p u1(oqs a^rnc otIJ p""o. rLr 1n "q1 :' ]:-:b,:.:i:::i31i:':Xl.'rol,''i: sql roJ sr pus 'lietcrtu lqErein il"ol;.I.i,I#to ,iot, t'3o 'n1nn 1'nq 'ql000z oslu st Eerp urnutullN Eerp otz fi sI gotq^\\ "'-@iii '''n'"o 'perrnber lsnrqr 8t-I ''9'a!.{ (svr sroN)l) All30llA rYrc (rrl OOt SlOu)I oqt -t I, C o -{ :o m o g ! m o u x t 8 NOIS'INdOUd JO SA"IdIJNIUd 1 eq1 sq1 ul SPECIFIC FUEL CONSUMPTION 85 t.o .9 .8 .? .5 qo.3!L 'Vz*Vr ,3 'le .4 .3 12 .t o .3 .4 .5 .t VrA/z .7 .8 .9 .6 ratio r.O . Fig. 6.5. Propulsion efficiency. type by isht area second haust The thrust available is nearly a constant with airspeed and is considered to be a constant in this discussion. Thrust varies, however, if the rpm of the engine fo changed from the 1007o value. The reduction of thrust is not linear with the roduction of rpm (Fig.6.6). Thrust is also reduced with an increase in altitude. Equation 6.1 indicates that the mass flow decreases as the density of the air decreases. Temperature is also involved. Lower temperatures at altitude improve efficiency, so the loss in thrust is not as great as the decrease in density (Fie.6.7). SPECIFIC FUEL CONSUMPTION (6.2) Specific fuel consumption c, is the fuel flow per pound of thrust developed by 6.5. of the This tles it MILITARY 1(x) F al, I SEA LEVEL :) S80 F *i M= G It o-. 0.1- M=0.5- F I = = tvt f40 = 0.P-* F 2 820 c, U G nearly OL 30 UI ry 'r/ T / o 6.1, 'mass llt 40 50 60 70. 80 PERCENT RPM Fig. 6.6. T-38 installed thrust. 90 MIL|TARY a ofjet 8 at Propeller Aircraft Perforrnance Basic - 12,000 weights .{ll aircraft in flight must produce thrust to overcome the drag of the aircraft. rn turbojets and other thrust-producing aircraft thrust is produced directly throttle is is allowed from the engine. In aircraft that have propellers (or rotors), the engine does not produce thrust directly. These aircraft are called power producers because the engine produces power, which turns the propeller. The propeller then develops an aerodynamic force as it turns through the air. This force is thrust. Fuel consumption, of power-producing aircraft is roughly proportional to the power produced, instead of the thrust produced. Range and endurance performance are functions of fuel consumption, and so the power required to fly the aircraft is of prime importance. The relationship between thrust and power was discussed in Chapter 1: Compare fu flying at Thrust: Force: A push or pull Work Power - : work Time 0b) Force x Distance (ft-lb) - Force x Distance Time Distance ___=:_: I velocity rme Power: Force x Velocity - pVrr po*aa' Horsepower: If I/ (ft-lb/sec) 550 : TVro" (ftJb/sec) IY{* s50 is in knots, rtP: TVN 32s PIOWER-REQUIRED CURVES Equation 1.13 allows us to convert the drag (or 4) curve into the horsepowerrequired, P., curve. Equation 1.13 can be rewritten as D- "- DY, 325 1I5 ITEMS OF AIRCRAFT PERFORMANCE 125 AIRPLANE DATA wETGHT 20.000 lb wlNG AREA (sl 5OO ftz ffi SEA LEVEL DENSITY o= rlo 1P,* 6 U E 5 o ut 1 GC fr zmo o = o ul v, E o I r00 200 VELOCITY (KNOTSI " Fig. 8.8. Finding the maximum rate of climb' Endurance To obtain maximum endurance, minimum fuel flow is required. Time in flight, not distance covered, is the objective: Specific endurance : Fuel available (lb) Fuel flow (lb/hr) (hours) Figure 8.9 is used to show the velocity to fly for both maximum endurance and maximum specific range for a propeller aircraft. Because fuel flow is roughly proportional to the power required, minimum P, will produce minimum fuel ilow. As mentioned earlier, this is not (L/D)-., for a propeller aircraft. It is merely called minimum power required. Mathematically it is the point where c?t'lc, is maximum. Specific Range To get maximum range, the aircraft must fly the maximum distance with the fuel available. The specific range must be a maximum. Specific range, SR, can 126 PROPELLER AIRCRAFT- BASIC PERFORMANCE PBOP AIRPLANE DATA lb3 , 5oo rtsEA LEVEL OEi{SITY t wElGHt 2o,ooo wlNG AREA (s| =I alH l o MAX t! d ) o ENDURANC' FF min az HPtr min az DV mrn ' ; ^t/, {!f, ) nror ld e E lr, o = c U G, o For a po, MAX 'ENDURANCE I -3/z {lfi t performa '* 1. Ma min 2. Ma az 1-olt min. ; {f,t ,no' Every pil tf;t E g ,,.rn avlSt I airspeed. o = ) ul ) L J VELOCITY _ {KNOTS} Wind Ef L and range' Fig. E.9. Finding the maximum endurance An aircri the airsp to the ht on its be defined bY the following relationshiP: SR: Nautical miles flown _ Airspeed (knots) - Fuel flow (lb/hr) Pounds of fuel used For maximum sPecific range, SR*u*: [#]-,. or SR-u* occur"t [T]-," straight line drawn from the The point on the P, curve in Fig' 8'9 where a will iirdicate the maximum specific-range velocity' origin is tangent to the curve t environr Fig.8.10 Too headwin Draw a tailwind a new tz Total Rr The tot produci -tsnrql'go le{} ot 'QZ'g'Etg, aes) gercrre Eurcnpord rBllrurs sr U?rcrr? surcnpord-re,uod ro3 "r:iHl# '[1rco1ea eEue er{1 ruo{ u^\\I elJnc aql o1 u1("Jp sr auq luoEuul lrreu t pue urErro eql Jo Ual oql ol Jo prel sr purarlll eql 'flrepuls euop sr purapul u ro3 Eu4coJJoC 'elJno 'd eql lurod sql tuo{ out1 luoEuq /hou B ,/r\\B{I "o1 'urEuo aql Jo lqElr aql ol epcs [1rco1err egt uo pum\\ eql Eurllold ,{g 'puranpeoq e ro3 poedsrre paloeJroc eql pug 'peadsrre lsoq Jo senle^ oql urulqo oI .OI'8'EIJ ur u^roqs sr paEuuqc aq plnoqs peeds;re eql lerll lunotue oql 'luouruoJrArn elq"Jo^BJ slr{l Jo oEuluenpe e{Bl ol u^rop plnoqs puu {en slr uo ^rols 'pul&peag eq} ol podleq Eureq sr pur^rlrel r{lr^\\ Surfg l;ercrre ue'flasreluo3 pesodxa sr U?JcJr? eql l"ql arurl orll acnpoJ ol peseaJ3ur aq ppoqs peedsrrz ag1 'erogereql'lueuruoJr^ue elquJo^eJun uB ur sr pur/(peoq e olur Eur,(g UEJcJre uV e8uug rglreds uo tragg pu!& 'poodsrr: slr ^\\ou{ ppogs pue'"-(Ol'I) oqt 1o ocuelrodurr eql az.leil p1nor{s lopd frairg e8uer tunurrxehi 'Z (Eerp urnururru) oqer opgE 1no-eurEuo ununxe141 '1 :lJ?JoJrB Eurcnpord-remod rog tJsJcJr? Jo suelr oirrl '1ulod '"'(g/?) lB Jnoco acuerurograd eql sl srql 'gercrre Surcnpord-reinod e rog 'a8uer uo pur^\\ Jo m l''-u l' l-- pegg '61'g 'E;g oNli 008 "-(ell) q ov3H I ONli (sroN)l Arrcorrl <-f-..-t (xiz mz 09r mt 09 0 I 'llvl t-- ffi(x)9 T cm m0! ra I ro d)Sl = = r : J 0002 LZI IIJNVI^IUOCUAd JJIVUJUIV dO SWAII V 126 PROPELLER AIRCRAFT BASIC PERFORMANCE - o 2@ .u4{m e 5 o ! = B o J PROP AIRPLANE'OATA wElGHt 20,(M *- lbs , wlNG AREA (Sl 5oo ftSEA LEVEL DETISITY d = I ui 1500 Hsm c. ul = 2@ rom .g lt tt l r E 500 alH l o lrJ g o ld , MAX ENOURANCE FF mln az HPP min -v? ) mox , Dv mln ' .{!t 'vD d, c, !J 1000 -o tr u G, o a lnwer-producing occur at (I MAX ENDURANCE -Tz tlt .LD t ,rs tE tf;t 3 a, ,,.rn ru 1H$R1 min pof,1 a Maximum engine' L Maximum range .in. i tf,t ,',o, pilot should reali I o = J E J UI f vELoclrY - {(NOT5) Efiect on Specifi" ts flying into a should be headwind. Conv and range' Fig. 8.9. Finding the maximum endurance t miles flown t* : Nautical To*a, orf"tGta is way and shou nt. The amc be defined by the following relationship: L10. Airspeed (knots) Fua ftow (tbAr) For maximum sPecific range, or SR-"* occur'"t [T].,, ,*-".: [*-l Lt F l-u* a straight line drawn from th The point on the P, curve in Fig' 8'9 where velocit-v'the maximum specific-range origin is tangent to ttre curve will i-ndicate Io obtain the valu ind, by plotting a new tangent l ind is done simila tangent line is d R.ange total range for 1 ine aircraft (set a:'v xA '#[ ,n :,n A4l 962= SzAo') e', z'? : I t't 'IroJUtV SNOIIVNOX lcas/rlJ),(ltsocst,r crleuaur;tr (nu) (79) earu Eu116 lnc slols s tu (ssaluorsueurrp) requrnu sp1ou,(ag (sseluorsueurp) rolce; peol Jo anl?A tuntulxery ') Eutsn ,(q D (xeu)1, ,) luolculeo] (sseluorsusrurp) Ee:p Jo lualcgeo3 UII Jo o) Jo [Ereue eJotu eqr STOSI^IAS aqI.I oql Jo 'uor1u;udes s,(e1op pue .re,(e1 ,(repunoq eq1 saztS:aua-eJ JIB Suvrrolq . aq; 'sdeg e8pe-Eurperl oql Jo luo{ ut po}Bool slo1s 3urcu3-pJ,!\\JeoJ olur pedld pue eut3ue ourqJnl eql Jo JosseJdluoJ eql uror; parlddns sI JIU ernssard-q8t11 '(St't'El.{) aor^ep..rre 8ut,tto1q,, otll sI f,Tg Jo ad,{} uouuroc aJou sql 'uorle;edas s,(u1ep srql'sre.{u1 raqEtq oql wo{ le Eur,rour relszg ,(q paceldar oJB puu pelotueJ are ra(u1 (repunoq oql uI JIe go s.ra,(e1 Je^\\ol aql 'perlddu sI uollcns uaq16 'Surrvr eql Jo acBJJns reddn eq1 u1 'a,rrnc r-a3 eq1 uo deg eEpe-EurIerl E re^o J'I8 8uu'rolq Jo ]ceJA olztou3Nl-3u L9 sNotrvnbs '8I', 'El.{ EJ EJ s1r\\ou JI eql 'poqletu -nols 'Eurarr elnoJ ol 104 JET AIRCRAFT APPLIED PERFORMANCE o 1500 e (, tr o o ul o f o z 1000 500 g o 1p0 200 300 3 400 I I AIRSPEED, knots TAS amA G, 5 Fig. 7.1. Effect of weight change on induced drag. o trl t F f c. I F changing the weight changes the drag proportionately and L, :L, D2 Dr E Under lG flight L: W g so I D, :W, Dr Wr (7 l) o J lL J lrl , L Figure 7.2 was drawn by altering Fig.6.2 for different weights. The airspeeds were changed using Eq. 4.2 and the corresponding drags, f, were changed Using Eq. 7.1. Later in this chapter we discuss how the pilot must use this curve to obtain maximum performance from the aircralt as the weight changes. Configuration Changes When a pilot lowers the landing gear (and/or flaps), the equivalent parasite area is greatly increased. This increases the parasite drag of the aircraft, as shown in Fig. 7.3, but it has little elfect on the induced drag. The effect on the thrust-required curve is to move it upward and to the left. The amount that the curve moves is not easily calculated. The equivalent parasite area must be known for the aircraft in the dirty configuration, and the new parasite drag values must be calculated. The { curve can then be lound by adding the new curve values to the old D, curve values and plotting. The thrust-required curves for several configurations of the T-38 are shown in Fig. 7.4. In discussing the effect of weight change we saw that the values of LID remained constant at the same AOA. 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Selected Aircraft (from module 3 & 4): 2. Aircraft Maximum Gross Weight [lbs] (from module 3 & 4): Jet Performance In this first part we will utilize the drag table that you prepared in module 4. Notice that the total drag column, if plotted against the associated speeds, will give you a drag curve in quite similar way to the example curves (e.g. Fig 5.15) in the textbook. (Please go ahead and draw/sketch your curve in a coordinate system or use the Excel diagram functions to depict your curve, if so desired for your own visualization and/or understanding of your further work.) Notice also that this total drag curve directly depicts the thrust required when it comes to performance considerations; i.e. as discussed on pp. 81 through 83, in equilibrium flight, thrust has to equal drag, and therefore, the thrust required at any given speed is equal to the total drag of the airplane at that speed. Last but not least, notice also that, so far, in our analysis and derivation of the drag table in module 4, we haven't at all considered what type of powerplant will be driving our aircraft. For all practical purposes, we could use any propulsion system we wanted and still would come up with the same fundamental drag curve, because it is only based on the design and shape of the aircraft wings. Therefore, let's assume that we were to power our previously modeled aircraft with a jet engine. A. What thrust [lbs] would this engine have to develop in order to reach 260kts in level flight at sea level standard conditions? Notice again that in equilibrium flight (i.e. straight and level, unaccelerated) thrust has to be equal to total drag, so look for the total drag at 260kts in your module 4 table. (In essence, this example is a reverse of the maximum speed question - expressing it graphically within the diagram: We know the speed on the X-axis and have the thrust required curve; that gives us the intercept point on the curve through which the horizontal/constant thrust available line must go.) 150 kts B. Given the available engine thrust from A. above, what is the Climb Angle [deg] at 200kts and Maximum Gross Weight? (Notice that climb angle directly depends on the available excess thrust, i.e. the difference between the available thrust in A. above and the required thrust from your drag curve/table at 200kts. Then, use textbook Eq. 6.5b relationships to calculate climb angle). T = Ta= 375 lbs, Tr = D = 257 lbs, This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 2 W = 6750 lbs, Sin y =(T - D)/W Sin y = ( 375 lbs -257 lbs ) / 6750 lbs = 0.01748Y = 1 deg C. What is the Max Endurance Airspeed [kts] for your aircraft? Explain how you derived at your answer. 150 kts. The minimum required thrust occurs at 140 kts, at this point, L/D reaches the maximum. This point is the max endurance airspeed point. Prop Performance In this second part we will utilize the same aircraft frame (i,e, the same drag table/graph), but this time we will fit it (more appropriately and closer to its real world origins) with a reciprocating engine and propeller. D. To your existing drag table, add an additional column (Note: only the speed column, the total drag column and this third new column will be required - see below). To calculate the Power Required in the new column, use textbook p. 115 equation and the V and D values that you already have: Pr = D*Vk / 325 V (KTAS) VS 80 100 120 140 160 180 190 200 220 240 260 DT = Tr (lb) Pr (HP) 455 372 268 224 210 215 231 100 92 82 83 91 106 128 257 290 330 375 158 197 244 300 Pr maxE maxR E. Draw/sketch (or plot in an Excel diagram) your Power Required curve against the speed scale from the table data in A. above. (Note: This step is again solely for your visualization and to give you the chance to graphically solve the next questions in analogy to the textbook and examples. See sketch above.) This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 V 3 F. Find the Max Range Airspeed [kts] for your aircraft. Remember from the textbook discussion pp. 125 through 127 that Maximum Range Airspeed for a reciprocating/propeller driven aircraft occurs where a line through the origin is tangent to the power required curve (see textbook Fig. 8.9 and sketch above). However, as per the textbook discussion, it is also the (L/D)max point, which we know from our previous work on drag happens where total drag is at a minimum (therefore, you can also reference the total drag column in your table and find the airspeed associated with the minimum total drag value). 108.6 kts G. Find the Max Endurance Airspeed [kts] in a similar fashion. (Tip: The minimum point in the curve will also be visible as minimum value in the Pr column of your table.) 88.6 kts H. Let's assume that the aircraft weight is reduced by 10% due to fuel burn (i.e. similar to the gross weight reduction in Exercise 4, problem B). I) Aircraft Weight [lbs] for 90% of Maximum Gross Weight (i.e. the 10% reduced weight from above). Simply apply the factor 0.9 to your aircraft Maximum Gross Weight from number 2. above: 2450-10%= 2205 lbs your II) Find the new Max Range Airspeed [kts] for the reduced weight. Remember (from textbook reading and Exercise 4, B.) that the weight change influence on speed was expressed by Eq. 4.2 in the textbook. V2= V1SQRT/ W1 SQRT(2205/2450)= 0.95 108.6 - 0.95= 107.7 Landing Performance For this last part of this week's assignment you will continue with your reciprocating engine (i.e. prop) powered aircraft and its reduced weight. Let's first collect some of the data that we already know: 3. Stall Speed for 90% of Maximum Gross Weight (i.e. the stall speed for 10% decreased weight from above, which we already calculated in Exercise 4, problem B.): Vs new = Vs old weight * (new weight /old weight) Vs new = 48.6* (2205/2450) Vs new = 46.1 I. Find the Approach Speed [kts] for your 90% max gross weight aircraft trying to land at a standard sea level airport. Approach speed is usually some safety margin above stall speed -.let's assume for our case a factor of 1.2, i.e. multiply your stall speed from number 3. with a factor of 1.2 to find the approach speed: This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 4 46.1 * 1.2 = 55.32 J. Determine the drag [lbs] on the aircraft during landing roll. I) For simplification, start by using the total drag value [lbs] for stall speed (for the full weight aircraft) from your module 4 table: 237.8402 lbs II) Adjust the total drag (from I) above) for the new weight (from H. I) above) by using the textbook Equation 7.1 relationship: D2/D1 = W2/W1 237.8402*0.9= 214.0562 the III) Find the average drag [lbs] on the aircraft during landing roll. A commonly used simplification for the dynamics at play is to use 70% of the total drag at touchdown as average value. Therefore, find 70% of your II) result above. 214.0562*70% = 149.8393 K. Find the frictional forces during landing roll. The Total Friction is comprised of Braking Friction at the main wheels and Rolling Friction at the nose/tail wheel. For this example, let's assume that, in average, there is 75% of aircraft weight on the main wheels and 25% on the nose/tail wheel over the course of the landing roll. The Average Friction Force is then the product of respective friction coefficient and effective weight at the wheel/wheels (see p. 209 textbook): F = (*N I) If the rolling friction coefficient is 0.02, what is the Rolling Friction [lbs] on the nose/tail wheel? (Remember that only 25% of total weight are on that wheel and that the weight was reduced by 10% from maximum gross weight - see H I)): 25% of 58950 = 14737.5 14737.5*0.02= 294.75 lbs II) If the main wheel brakes are applied for an optimum 10% wheel slippage (as discussed on textbook pp. 209/210), what is the Braking Friction [lbs] on the main wheels during landing roll on a dry concrete runway? Use textbook figure 13.9 to determine the friction coefficient. (Remember that the weight on the main wheels is only 75% of total aircraft weight). 75% of 58950= 44212.5 According to fig. 13.9 mu is 0.75 44212.5*0.75 = 33159.38 lbs III) Find the total Average Friction [lbs] during landing by building the sum of I) and II): This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 5 Maximum friction coefficient is mu=0.7 at 15% percent slip Minimum friction coefficinet is mu=0.5 at 100% percent slip average braking friction coefficient mu =0.6 Average braking friction on main tires (0.75 Weight on main tires) F f = mu*0.75*W =0.6*0.75*15000 = 6750lbs L. Find the Average Deceleration [ft/s2] during landing roll. Use the same rectilinear relationships as in module 1, applying the decelerating forces of friction and drag from J. III) & K. III) above. Assume that residual thrust is zero. (Keep again in mind that for application of Newton's second law, mass is not the same as weight. Your result should be a negative acceleration value since the aircraft decelerates in this case.): Total Force = W*a Total force = Residual thrust - Rolling Friction on nose tire- Drag - Braking Friction on main tires =500 - 750 - 500 - 6750 = -7500 lb =-7500*0.4536 = -3402 kg =-3402*9.81 = -33373.62 N W =15000 lb =15000*04536 = 6804 kg a = F/W =-33373.62/6804 = -4.905 m/(s^2) = -4.905*3.28 = -16.09 ft/(S^2) M. Find the Landing Distance [ft] (Remember that we start from a V0 at approach speed and want to slow the aircraft to a complete stop, applying the negative acceleration that we found in L. Also, remember to convert approach speed from I. above into a consistent unit of ft/s.): V= Vapproach = 128 KTAS = 128*1.688=216.69 ft/s a= -16.09 ft/s/s s=V2/2a= 1459 ft N. If your aircraft was to land at a higher than sea level airport (e.g. at Aspen, Co) what factors would change and how would it affect your previous calculations, especially your landing distance. Explain principles and relationships at work and support your answer with applicable formula/equations from the textbook. You can include example calculations to support your answer: Lets assume an example Runway is at 5,000 ft Density Altitude. Assume that residual Average Thrust and Average Drag remains the same M=W/g = 15,000lbs. / (32.2ft/sec^ 2 ) = 466 slugs Residual thrust(500lbs.) - Rolling friction nose(750lbs.) - Average drag(288lbs.) - Braking friction main(6750lbs.) = -7288lbs. / 466slugs = a = F/m = -15.6ft/sec^2 = Vf 2/ 2a = [113 (1.69)]^ 2 / 2(-15.6) s = 1169 ft. This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 1 Exercise 5: Aircraft Performance For this week's assignment you will revisit your data from previous exercises, therefore please make sure to review your results from the last modules and any feedback that you may have received on your work, in order to prevent continuing with faulty data. 1. Selected Aircraft (from module 3 & 4): 2. Aircraft Maximum Gross Weight [lbs] (from module 3 & 4): Jet Performance In this first part we will utilize the drag table that you prepared in module 4. Notice that the total drag column, if plotted against the associated speeds, will give you a drag curve in quite similar way to the example curves (e.g. Fig 5.15) in the textbook. (Please go ahead and draw/sketch your curve in a coordinate system or use the Excel diagram functions to depict your curve, if so desired for your own visualization and/or understanding of your further work.) Notice also that this total drag curve directly depicts the thrust required when it comes to performance considerations; i.e. as discussed on pp. 81 through 83, in equilibrium flight, thrust has to equal drag, and therefore, the thrust required at any given speed is equal to the total drag of the airplane at that speed. Last but not least, notice also that, so far, in our analysis and derivation of the drag table in module 4, we haven't at all considered what type of powerplant will be driving our aircraft. For all practical purposes, we could use any propulsion system we wanted and still would come up with the same fundamental drag curve, because it is only based on the design and shape of the aircraft wings. Therefore, let's assume that we were to power our previously modeled aircraft with a jet engine. A. What thrust [lbs] would this engine have to develop in order to reach 260kts in level flight at sea level standard conditions? Notice again that in equilibrium flight (i.e. straight and level, unaccelerated) thrust has to be equal to total drag, so look for the total drag at 260kts in your module 4 table. (In essence, this example is a reverse of the maximum speed question - expressing it graphically within the diagram: We know the speed on the X-axis and have the thrust required curve; that gives us the intercept point on the curve through which the horizontal/constant thrust available line must go.) 150 kts B. Given the available engine thrust from A. above, what is the Climb Angle [deg] at 200kts and Maximum Gross Weight? (Notice that climb angle directly depends on the available excess thrust, i.e. the difference between the available thrust in A. above and the required thrust from your drag curve/table at 200kts. Then, use textbook Eq. 6.5b relationships to calculate climb angle). T = Ta= 375 lbs, Tr = D = 257 lbs, This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 2 W = 6750 lbs, Sin y =(T - D)/W Sin y = ( 375 lbs -257 lbs ) / 6750 lbs = 0.01748Y = 1 deg C. What is the Max Endurance Airspeed [kts] for your aircraft? Explain how you derived at your answer. 150 kts. The minimum required thrust occurs at 140 kts, at this point, L/D reaches the maximum. This point is the max endurance airspeed point. Prop Performance In this second part we will utilize the same aircraft frame (i,e, the same drag table/graph), but this time we will fit it (more appropriately and closer to its real world origins) with a reciprocating engine and propeller. D. To your existing drag table, add an additional column (Note: only the speed column, the total drag column and this third new column will be required - see below). To calculate the Power Required in the new column, use textbook p. 115 equation and the V and D values that you already have: Pr = D*Vk / 325 V (KTAS) VS 80 100 120 140 160 180 190 200 220 240 260 DT = Tr (lb) Pr (HP) 455 372 268 224 210 215 231 100 92 82 83 91 106 128 257 290 330 375 158 197 244 300 Pr maxE maxR E. Draw/sketch (or plot in an Excel diagram) your Power Required curve against the speed scale from the table data in A. above. (Note: This step is again solely for your visualization and to give you the chance to graphically solve the next questions in analogy to the textbook and examples. See sketch above.) This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 V 3 F. Find the Max Range Airspeed [kts] for your aircraft. Remember from the textbook discussion pp. 125 through 127 that Maximum Range Airspeed for a reciprocating/propeller driven aircraft occurs where a line through the origin is tangent to the power required curve (see textbook Fig. 8.9 and sketch above). However, as per the textbook discussion, it is also the (L/D)max point, which we know from our previous work on drag happens where total drag is at a minimum (therefore, you can also reference the total drag column in your table and find the airspeed associated with the minimum total drag value). 108.6 kts G. Find the Max Endurance Airspeed [kts] in a similar fashion. (Tip: The minimum point in the curve will also be visible as minimum value in the Pr column of your table.) 88.6 kts H. Let's assume that the aircraft weight is reduced by 10% due to fuel burn (i.e. similar to the gross weight reduction in Exercise 4, problem B). I) Aircraft Weight [lbs] for 90% of Maximum Gross Weight (i.e. the 10% reduced weight from above). Simply apply the factor 0.9 to your aircraft Maximum Gross Weight from number 2. above: 2450-10%= 2205 lbs your II) Find the new Max Range Airspeed [kts] for the reduced weight. Remember (from textbook reading and Exercise 4, B.) that the weight change influence on speed was expressed by Eq. 4.2 in the textbook. V2= V1SQRT/ W1 SQRT(2205/2450)= 0.95 108.6 - 0.95= 107.7 Landing Performance For this last part of this week's assignment you will continue with your reciprocating engine (i.e. prop) powered aircraft and its reduced weight. Let's first collect some of the data that we already know: 3. Stall Speed for 90% of Maximum Gross Weight (i.e. the stall speed for 10% decreased weight from above, which we already calculated in Exercise 4, problem B.): Vs new = Vs old weight * (new weight /old weight) Vs new = 48.6* (2205/2450) Vs new = 46.1 I. Find the Approach Speed [kts] for your 90% max gross weight aircraft trying to land at a standard sea level airport. Approach speed is usually some safety margin above stall speed -.let's assume for our case a factor of 1.2, i.e. multiply your stall speed from number 3. with a factor of 1.2 to find the approach speed: This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 4 46.1 * 1.2 = 55.32 J. Determine the drag [lbs] on the aircraft during landing roll. I) For simplification, start by using the total drag value [lbs] for stall speed (for the full weight aircraft) from your module 4 table: 237.8402 lbs II) Adjust the total drag (from I) above) for the new weight (from H. I) above) by using the textbook Equation 7.1 relationship: D2/D1 = W2/W1 237.8402*0.9= 214.0562 the III) Find the average drag [lbs] on the aircraft during landing roll. A commonly used simplification for the dynamics at play is to use 70% of the total drag at touchdown as average value. Therefore, find 70% of your II) result above. 214.0562*70% = 149.8393 K. Find the frictional forces during landing roll. The Total Friction is comprised of Braking Friction at the main wheels and Rolling Friction at the nose/tail wheel. For this example, let's assume that, in average, there is 75% of aircraft weight on the main wheels and 25% on the nose/tail wheel over the course of the landing roll. The Average Friction Force is then the product of respective friction coefficient and effective weight at the wheel/wheels (see p. 209 textbook): F = (*N I) If the rolling friction coefficient is 0.02, what is the Rolling Friction [lbs] on the nose/tail wheel? (Remember that only 25% of total weight are on that wheel and that the weight was reduced by 10% from maximum gross weight - see H I)): 25% of 58950 = 14737.5 14737.5*0.02= 294.75 lbs II) If the main wheel brakes are applied for an optimum 10% wheel slippage (as discussed on textbook pp. 209/210), what is the Braking Friction [lbs] on the main wheels during landing roll on a dry concrete runway? Use textbook figure 13.9 to determine the friction coefficient. (Remember that the weight on the main wheels is only 75% of total aircraft weight). 75% of 58950= 44212.5 According to fig. 13.9 mu is 0.75 44212.5*0.75 = 33159.38 lbs III) Find the total Average Friction [lbs] during landing by building the sum of I) and II): This document was developed for online learning in ASCI 309. File name: Ex_5_Aircraft Performance Updated: 07/19/2015 5 Maximum friction coefficient is mu=0.7 at 15% percent slip Minimum friction coefficinet is mu=0.5 at 100% percent slip average braking friction coefficient mu =0.6 Average braking friction on main tires (0.75 Weight on main tires) F f = mu*0.75*W =0.6*0.75*15000 = 6750lbs L. Find the Average Deceleration [ft/s2] during landing roll. Use the same rectilinear relationships as in module 1, applying the decelerating forces of friction and drag from J. III) & K. III) above. Assume that residual thrust is zero. (Keep again in mind that for application of Newton's second law, mass is not the same as weight. Your result should be a negative acceleration value since the aircraft decelerates in this case.): Total Force = W*a Total force = Residual thrust - Rolling Friction on nose tire- Drag - Braking Friction on main tires =500 - 750 - 500 - 6750 = -7500 lb =-7500*0.4536 = -3402 kg =-3402*9.81 = -33373.62 N W =15000 lb =15000*04536 = 6804 kg a = F/W =-33373.62/6804 = -4.905 m/(s^2) = -4.905