Unformatted text preview: Massachusetts Institute of Technology
Department of Aeronautics and
Cambridge, MA 02139 16.03/16.04 Unified Engineering III, IV
Spring 2004 Problem Set 3 Time Spent
F7 Name: F8
Due Date: 2/24/04 M7/M8
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
Standard solutions for deflections of beams under commonly encountered loading
Configuration End slope End deflection, Central deflection, dw/dx (x=L)
2 EI PL2
3 EI q0 L3
6 EI q0 L4
16 EI q0 L3
24 EI PL3
48 EI q0 L4
384 EI 2 ...
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