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Unformatted text preview: Massachusetts Institute of Technology Department of Aeronautics and Astronautics Cambridge, MA 02139 16.03/16.04 Unified Engineering III, IV Spring 2004 Problem Set 3 Time Spent (min) F7 Name: F8 F9/F10 Due Date: 2/24/04 M7/M8 M9 Study Time Announcements: Spring 2004 Unified Engineering Fluids Problems F7–F10 F7. The profile drag of a particular wing is assumed to be some given constant over the expected range of operating CL ’s. cd � constant For an elliptically-loaded wing of some aspect ratio AR . . . a) Determine the operating CL at which the lift/drag ratio CL /CD is maximized. This is the desirable operating p oint for maximum range. Determine how the CD at this operating p oint compares to cd . 3/2 b) Determine the operating CL at which the “power coefficient” CL /CD is maximized. This is the desirable operating p oint for maximum endurance. Determine how the CD at this operating p oint compares to cd . F8. A wing is to have an elliptic circulation distribution. �(y ) = �0 � 1− � 2y b �2 The planform is to b e a straight taper, with root and tip chords defined in terms of the average chord cavg and the taper ratio r = ct /cr . cr = cavg 2 1+r ct = cavg 2r 1+r a) Define the chord distribution c(y ) in terms of cavg and r . Assuming cavg /b = 0.125, draw the planforms for r = 0.75, 0.5, 0.25. b) Determine the spanwise c� (y ) distribution, and plot for r = 0.75, 0.5, 0.25. Note: Only the shape of the c� (y ) curve is of interest. All scaling constants like �0 , cavg , etc. can b e set to unity for plotting purposes. c) Local stall is obviously undesirable. If the airfoil is the same across the span, which taper ratio appears to b e most attractive for the purpose of giving the largest stall margin everywhere on the wing? F9+F10. The circulation distribution on a wing is �(� ) = 2bV� (A1 sin � + A2 sin 2� ) where A1 = 0.05, and A2 = 0.01. a) Determine and plot �i (y ). b) Determine the rolling moment on the entire wing. Mroll = b/2 −b/2 � V� � y dy Unified Engineering Spring 2004 Problem M7 and M8 (this is a two hour question) A simply supported aluminum alloy beam is 3 m long and has a cross-section which is an “I” cross-section 200 mm high and 100 mm wide. A uniform distributed load of 6 kN/m acts on the left hand two thirds of the beam. The Young’s modulus of the aluminum alloy is 70 GPa. The yield stress is 300 MPa. 6 kN/m 1m 1m 1m a) Determine the loading, shear force and bending moment as functions of the distance x measured from the left end of the beam. Draw the appropriate diagrams. b) Determine the maximum deflection(s) of the beam and its (their) location(s). c) Determine the magnitudes and locations of the maximum axial stress, sxx and the maximum shear stress, sxz. Will the aluminum alloy yield? Problem M9 A beam of length L and flexural rigidity EI is clamped at each end. The beam has a continuous load of magnitude qo applied along the beam. Using the “standard solutions” below, or by other means, solve for the reactions at A and C. q0 z C A L Standard solutions for deflections of beams under commonly encountered loading Configuration End slope End deflection, Central deflection, dw/dx (x=L) w(L) w(L/2) ML EI ML2 2 EI PL2 2EI PL3 3 EI q0 L3 6 EI q0 L4 8EI PL2 16 EI q0 L3 24 EI PL3 48 EI q0 L4 384 EI 2 ...
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