Homework 1 AERE355 Fall 2017 Due 9/1(F) NOTE: If your solution does not adhere to the format described in the syllabus, it will be grade as zero.

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1 1 Hmerk 1 AERE355 Fall 217 Due 9/1(F) Name NOE: If yur slutin des nt adhere t the frmat described in the syllabus, it ill be grade as zer. Prblem 1(25pts) In the altitude regin h 1km, e have the flling atmspheric equatins: h (1.9 * ) ; P P g R (1.55) ; g (1 ) R r P R (1.57). (a)(1pts) Write a Matlab cde that ill generate a table ith clumns h P fr the array f altitude values h :1 :1, give the table, and cmment n h it cmpares t able A.1 n p.395 in APPENDIX 1. Slutin: [See 1(a).] Cmment: (b)(1pts) Run yur cde fr K [i.e. the temperature at h kmfr yur table in (a)], and fr 316 K. hen btain a plt f the percent change in air density vs. altitude assciated ith the increased air surface temperature. Slutin: [See 1(b).] able 1(a) Standard Atmsphere able Figure 1(b) Percent change in Air density. (c)(5pts) Based n yur plt in (b) and the definitin f the lift cefficient, explain, quantitatively, explain h the increased temperature influence the lift f a plane flying at an altitude f 6km. Include yur reference in relatin t the definitin f the lift cefficient. Slutin:

2 2 PROBEM 2(25pts) [Related t bk Prblem 1.2] An airplane s altimeter reading is 5,m. (a)(1pts) If the utside ambient temperature is -2 C and the plane s true airspeed is 3m/s, find the plane s indicated airspeed. Slutin: (b)(5pts) Suppse that the altimeter reading fluctuates beteen rughly 4,99m and 5,1m, and that the mean reading is. Assume h has a nrmal distributin ith h 5, m that the fluctuatin. Use the Matlab cmmand nrmpdf t arrive at a plt f the prbability density functin (pdf) fr the altimeter reading, h. Slutin: [See 2(b).] 3m Figure 2(b) Altimeter reading pdf. g (c)(5pts) On p.17 e have equatin (1.54): 1 ( h h1 ) ln( P / P1 ) ln R. Beginning ith this equatin, carry ut 1 g / R h detailed steps t sh that P P. Slutin: (d)(5pts) arrive at the pdf f P : (i) use the Matlab cmmand nrmrnd t generate N=1, simulatins f h. (ii) Use these in the expressin in (c) t btain simulated measurements f P.(iii) Use the cmmand histgram t arrive at a plt f an estimate f the pdf f P. (iv) Use the cmmands mean and std t estimate the mean and standard deviatin f P. NOE: Recall that 2 2 R 287m / K s and.65 K / m. Slutin: [See 2(d).] Figure 2(d) Simulatin-based pdf fr P. 2 2 P 54,12 N / m and 21.6 N / m. P

3 3 Prblem 3(25pts) he plt bel shs bth lift and mment cefficients as functins f the angle f attack fr the NACA 2312 airfil. Als included are straight-line apprximatins f these relatins. Yu are t use these apprximatins in this prblem. Frm: X&ei=WK87UIyzB8OZ2QW4pYCD&ved=CDIQsAQ&bi=128&bih=637 Figure 1. his plt als includes the C ( Z ) line (in black).

4 4 (a)(6pts) Estimate the cefficients f the lift mdel: C ( ) C C using the RED line. hen plt yur mdel fr Slutin: 2 16 t verify its crrectness. Figure 1.1(a) Plt f lift mdel. (b)(6pts) Estimate the cefficients f the mment mdel: C ( ) C C using the BUE line. hen plt m m yur mdel fr Slutin: m 2 2 t verify its crrectness. (c)(3pts) Cmpute the numerical value f Slutin: Figure 1.1(a) Plt f mment mdel. trim fr yur airfil mment mdel in (b). (d)(6pts) Use yur equatins frm parts (a) & (b) t estimate the parameters C m C C h. m Slutin: C and h h hn f the mment mdel: m (e)(4pts) Fr yur mdel in (d) determine hether the ing has negative pitch stiffness) r psitive pitch stiffness). Explanatin:

5 5 PROBEM 4(25pts) [Related t bk prblem 2.1] A given ing mment equatin is: C.8. 15C m cg (a)(5pts) Recall frm the discussin bel Figure 2.5 n p.43 that the trim cnditin is defined as that cnditin such that. (i.e. the trim mment cefficient equals zer). Obtain the crrespnding trim lift cefficient. C mcg Slutin: (b)(5pts) he (scaled) ing center f gravity f the ing is xcg hcg. 3. Use (2.6) n p.45 t find the ing neutral c pint h N Slutin: x N c. Assume that the neutral pint equals the aerdynamic center pint. (c)(1pts) In (2.7) e see that C C C. Suppse that the ing lift equals zer hen i 2 ( /18 ). 349rad, and it is at trim hen 5 ( /18 ).873rad. Find C as a functin f Slutin: 1. ver the range trim C and C. hen plt Figure 2.1 Plt f C versus α. (d)(5pts) Fr a ing-alne cnfiguratin as shn in Figure 2.7 (n p.45), explain hy it is nt statically stable. Explanatin:

6 6 Appendix Matlab Cde %PROGRAM NAME: h1.m %PROBEM 1 %PAR (a): hg = :1:1; hg = hg'; % Altitude array (km) R = 6371e3; %Earth mean radius (m) g = 9.87; %Gravity (m/s2) at sea level R = 287; %Gas cnstant ((m2/k-s2) = ; %emp (K) at sea level P = e5; %Pressure (N/m2) at sea level r = 1.225; %Density (kg/m3) at sea level = -.65; %emp gradient (K/km) INSER YOUR CODE HERE %PAR (b): INSER YOUR CODE HERE %============================================================== %PROBEM 2 %PAR (a): hvals=499:.1:51; % Altitude Range f Values muh = 5; % Altitude Mean stdh = 3; % Altitude Standard Deviatin INSER YOUR CODE HERE % PAR (d): R=287; =-.65; =288.15; P=11325; g=9.87; N = 1^5; INSER YOUR CODE HERE