fall_2003_pset15

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Unformatted text preview: Massachusetts Institute of Technology Department of Aeronautics and Astronautics Cambridge, MA 02139 16.01/16.02 Unified Engineering I, II Fall 2003 Problem Set 15 Time Spent (min) F21 Name: F22 M23 Due Date: Not Due M24 M25 Study Time Announcements: Good luck on your finals. Reminder: The unified final is on Monday, 12/15 at 9am Fall 2003 Uniﬁed Engineering Fluids Problems F21–F22 F21. As shown in class, the nonlifting irrotational ﬂow past a circular cylinder can be represented by superimposing the uniform freestream ﬂow and a doublet. An alternative representation is proposed using a source sheet placed on the cylinder surface as shown. The proposed sheet strength distribution about the cylinder is �(� ) = −2V� cos � . There is no vortex sheet, so on the surface, � = 0. �(�) freestream+doublet model (known) actual � freestream + source sheet model (proposed) You are to determine whether the new model is correct. a) Determine the velocity at p oint A from the known exterior surface velocities for the cylinder. V� (� ) = −2V� sin � , Vr = 0 Using the sheet jump relations, �Vn = � , �Vs = � determine the interior velocity at p oint B. b) Again using the known exterior V� (� ), Vr result at point C, use the sheet jump condition to determine the velocity at p oint D. c) Compare velocities at B and D. What appears to be the ﬁctitious velocity inside the cylinder? d) Is the source sheet jump �Vn = � consistent with the exterior and interior normal ﬂows everywhere on the cylinder surface? Is the proposed source-sheet model correct? C D A B ? F22. A long rectangular wing has span b and chord c, and hence the wing area is S = bc. a) The wing airfoil has certain lift and drag coeﬃcients c� and cd which are constant across the span. Determine how these relate to the wing’s overall CL and CD . (Hint: Determine L and D  , then get L and D , then from these determine CL and CD ). The wing airfoil has a drag p olar which can be approximated by cd � 0.01 + 0.015 c3 � in the range c� = 0.1 . . . 1.2. The propulsive power P needed to overcome drag D at ﬂight speed V� is given by P = D V� P cl 1.0 ? 0.5 0.0 0.01 0.02 cd V b) Determine the form of the P (V� ) relation in level ﬂight, and plot it for the range c� = 0.1 . . . 1.2. Any constant multiplicative factors on the P and V� axes are not important – only the shape of the curve is of interest. Hint: Simplest approach is to plot P (c� ) versus V (c� ) with c� as a dummy parameter. (Note: Using only the airfoil’s cd ignores other contributions such as induced drag, which b ecome especially signiﬁcant at low ﬂight speeds!) Unified Engineering Fall 2003 Problem M23 (Materials and Structures) The potential energy, U of a pair of atoms in a solid can be written as: -A B U= m + n r r where r is the separation of the atoms and A, B, m and n are positive constants. Indicate the physical significance of the two terms in this equation. A material has a simple cubic unit cell with atoms placed at the corners of the cubes. Show that, when the material is stretched in a direction parallel to one of the cube edges, Young's modulus E is given by: E= mnkTM W Where W is the mean atomic volume, k is Boltzmann's constant and TM is the absolute melting temperature of the solid. You may assume that U0 ( r0 ) = - kTM ,where r0 is the equilibrium separation of a pair of atoms. Problem M24 Two metals of current and historical interest for aerospace applications, nickel and magnesium, have face centered cubic and close packed hexagonal structures respectively. a) Assuming that the atoms can be represented as hard spheres, calculate the percentage of the volume occupied by atoms in each material. b) Calculate, from first principles, the dimensions of the unit cell in nickel and in magnesium. (The densities of nickel and magnesium are 8.90 Mgm-3 and 1.74 Mgm-3 respectively, the atomic weight of Nickel is 58.69, Magnesium is 24.31, Avogadro’s number is 6.023x1023). Problem M25 In addition to chapters 4-7 of Ashby and Jones Engineering Materials, you may also find the chapters on polymers in Ashby and Jones, Engineering Materials 2, helpful (this is a green covered book, available in the Aero-Astro library). a) Define the term polymer and list three engineering polymers. b) Define a thermoplastic and a thermoset. c) Distinguish between a cross-linked and a non-cross-linked polymer. d) What is the glass transition temperature? e) Explain the change in moduli of polymers at the glass transition temperature. f) What is the range of temperature in which TG lies for most engineering polymers? g) How would you increase the modulus of a polymer? ...
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