Spring 2012 Sample Exam - Mechanical, Materials and Aerospace ...

ENGINEERING ANALYSIS I
Problem 1
Recall that Fermat’s principle of optics asserts that the path of light u(x) is
such that it minimizes the travel time given by
T [u(x)] = 1 c
( ) 2
√ ∂u
n(x, u) 1 + dx,
x 0 ∂x
∫ x1
where c is the constant speed of light in a vacuum, and n(x, u) is the variable
index of refraction. The light travels between the two specified points (x 0 , u 0 )
and (x 1 , u 1 ). Consider the case where the index of refraction only depends
upon the vertical coordinate, i.e. n = n(u).
a) Determine the first-order differential equation for the path u(x).
b) Solve the differential equation obtained in part (a) for the case when the
index of refraction is a constant, i.e. n(u) = n 0 = constant.
c) Solve the differential equation obtained in part (a) for the case when the
index of refraction is given by n(u) = √ u.

ENGINEERING ANALYSIS I
Problem 2
Obtain the solution to the following system of differential equations using
matrix operations
dx 1
dt = 2x 1 + 2x 2 ,
dx 2
dt = 2x 1 + 2x 2 ,
subject to the initial conditions
x 1 (0) = 1, x 2 (0) = 2.

ENGINEERING ANALYSIS II
Solve the integral equation
where a and b are constants.
∫ t
u(t) − a e −aτ u(t − τ)dτ = e −bt ,
0

DYNAMICS I
C
p
B
d
^
b y
è
^
b x
^n y
A
R
O
^n x
The unit basis ˆn x,y,z is fixed in a Newtonian reference frame. A circular disk of radius R and
mass M is able to rotate about the ˆn z direction without friction. A bead of mass m can move along
a massless track at a distance d from O, the center of the disk, without friction. The massless track
is oriented in the ˆb x direction, which is at an angle θ from the ˆn x direction. The bead is connected
to point A on the disk via a massless spring with spring constant k. The unstretched length of the
spring is AC, where C is the midpoint of AB. The distance from C to the bead is given as x. Use
x and θ as generalized coordinates to position the disk and bead.
1. Find the equation(s) of motion of the system.
2. Locate the equilibrium position(s).
3. Assume the disk is forced to rotate at a fixed angular velocity, ω, via an external torque about
the disk’s axis. Derive an expression for the external torque necessary to keep ˙θ = ω. Hint:
determine the external forces acting on the disk.
4. Assume the disk is forced to rotate at a fixed angular velocity, ω, via an external torque about
the disk’s axis. Find under what conditions the equilibrium point(s) of the system are stable.
1

DYNAMICS II
A thin disk B of radius r and mass m rotates with a constant angular rate ω relative to a
hollow, massless arm A of length L. The arm A is pinned at point P to shaft S, which
rotates with respect to an inertial frame about the vertical axis with constant rate Ω.
(a) Find the equation of motion for θ.
(b)
It is given that given that L r = 3 2 , g
! 2 r =1, and ! " = !1 .
Find the non-zero equilibrium value of θ, and determine whether the system is
stable at this equilibrium state.
S
Ω
P
g
θ
L
A
ω
r
B

Fluids Qualifying Problems Spring 2012
Problem 1
A Couette pump consists of a rotating inner cylinder and a baffled entrance and exit,
as shown. Assuming zero circumferential pressure gradient and small gap over
radius ratio, derive formulas for the volume flow and pumping power per unit
depth. Calculate these values for 300 cSt (0.0003 m 2 /s) oil in the pump with a = 10
cm and b = 9 cm when the pump is operated at 600 rpm.

Fluids Qualifying Problems Spring 2012
Problem 2
A viscous, incompressible liquid flowing down a vertical surface is shown in the
figure. A boundary layer develops next to the vertical surface under the effects of
gravity.
a) Find an expression for the free-stream velocity U(x), assuming that U(x=0)=0.
b) Write down the boundary-layer equations for this flow.
c) From these equations obtain the corresponding momentum integral equations,
and utilize an appropriate polynomial approximation to develop an expression
for the boundary layer thickness (x).

A 2-D trough of semi-circular cross-section (i.e., a 2-D half-cylinder) has a radius R and
is irradiated on its convex side by a constant radiant heat flux q” convex . The trough’s
concave side faces a matching 2-D lid with a grooved surface. The lid is positioned a
distance h above the trough. The grooves in the lid run parallel to the axis of the trough
and have width w, depth d, and separation pitch p. The surfaces of the lid not facing the
trough are insulated. The ambient environment surrounding the lid and the trough is at
temperature T ∞ .
a) If trough has unspecified radiant properties and the grooves in the lid have a
reflectivity ρ = 1, develop expressions for the temperature of the concave surface
of the trough and its net radiant flux to the ambient environment using whatever
additional assumptions you deem appropriate. Discuss how these quantities are
influenced by the radiant properties of the trough’s concave surface and the
geometry of the lid (w, d, p, h).
b) Now assume that the grooves in the lid have emissivity ε ( > 0). Obtain
expressions for the trough temperature and net radiant flux to the ambient
environment using whatever additional assumptions you deem appropriate.
Discuss how these values differ from those in a).
c) If the number of grooves could vary as 2n, where n is a positive integer, discuss
how the results of a) and b) would or would not be affected.
d
w
Lid
p
h
R
Trough
q” convex

Heat Transfer Qualifying Problems Spring 2012
Problem 2
An air jet, exiting from a round pipe, is impinging perpendicularly on a heated large but thin horizontal
plate. The plate is heated with a heater (the same size as the plate surface) that is installed underneath
the plate and it generates heat uniformly. The heater and the plate are perfectly insulated from below
and the sides. The air exiting the pipe is at ambient temperature. Assume that the distance between
exit of the pipe and the plate is H and the pipe diameter is D.
a- Develop the governing equations and the boundary conditions for this problem that if solved,
will provide the temperature distribution on the upper surface (facing the jet) of the plate.
Clearly state all your assumptions. Do not solve the governing equations.
b- Discuss how to determine the local convective heat transfer coefficient on the upper surface of
the plate.
Assume that the jet flow is laminar and steady. Clearly state all your assumptions.

PhD Qualifying Exam Questions
Solid Mechanics – Spring 2012
Problem 1. Given the homogeneous displacement field in a body
0.06 0.05 0.01
0.01 0.03
0.02 0.01
mm
mm
mm
(a) Calculate the strain field,
(b) If there is a line segment 10 mmm long parallel to the axis in the undeformed geometry,
what will be the new length of this line segment?
(c) If there is a line segment 10 mmm long oriented at angles 40, 75 and 54 with the
, and axes respectively in the undeformed geometry, calculate the new length of this
line segment and also the rotation it undergoes after deformation,
(d) Calculate the percent volume change of the body,

Problem 2. Stalactites hang from the ceilings in caves.
They form as the dissolved salts get deposited over long
periods of time. While the cross sections are usually
arbitrary, we shall assume that they are circular, with a
radius r varying with position x measured from the ceiling
as illustrated in the figure. The objective is to determine an
expression for the variation of the radius r(x). An
intuitively appealing assumption on the survival of a
stalactite is that everywhere in the stalactite the stress is
uniform and equal to some value close to the breaking
stress of the stone, o . Stalactites where the deposition is
such that this is violated will fail.
Take the mass density ρ of the stalactite as a constant
everywhere.
r(x)
x
Hint: Ignore higher order terms in the derivation of
equilibrium equation (but do it carefully). In case you
need, the volume of a truncated cone is given by
3

MMAE569 PhD Qualifier Spring 2012
Answer all parts of the question
1. a) Define homogeneous and heterogeneous nucleation.
b) Taking 108 J/m 3 as a typical value of the chemical free energy of a second phase
particle, 1 J/m2 as the surface energy per unit area and neglecting any strain energy,
calculate the radius of a spherical particle for which the surface energy is 1% of the
chemical energy.
c) Determine the degree of undercooling that would result in a chemical free energy of
108 J/m3 assuming the Turnbull approximation:
is valid and given that:
latent heat of melting is 15000 J/mole
Melting point is 727°C
molar volume is 6.4x10 -6 m 3 /mole
d) Using the triangular coordinate paper plot the following three phase field:
composition 25%B, 10%C
composition 55%B, 15%C
L composition 25%B, 35%C
e) Plot alloy X which has a composition of 35%B and 20%C. Determine the relative
amounts of the , and L phases in alloy X.
f) From the attached Ni-Cr-Ta isothermal section determine the following:
i) the equilibrium phases in the alloy compositions 1 and 3.
ii) The phase fractions in alloy 3
f = 1- exp( -kt n
)
g) The JMA equation is:
How would you plot a series of data of volume fraction transformed, f, versus time, t, to
obtain the rate constant k and the time exponent n?

MMAE 468 Introduction to Ceramics
Answer one question only
QUESTION 1
1. The ternary phase diagram for calcium aluminosilicate (CAS), calcium
magnesium silicate (CMS), and manganese silicate(MS) is shown below.
Answer the following questions based the ternary phase diagram
How many binary eutectic points are there? Mark them BEx.
What is the lowest binary eutectic temperature in the phase diagram?
Mark the ternary eutectic point with a “TE”.
What is the melting temperature of the ternary eutectic point?
What is the bulk composition of the ternary eutectic?
Mark the composition (40% CMS, 40% MS, 20% CAS) with an
“X” on the phase diagram.
What is the liquidus temperature for that composition?
Mark the composition (20% CMS, 20% MS, 60% CAS) with a
“Y” on the phase diagram.
What is the liquidus temperature for that composition?
QUESTION 2
A high strength silicon nitride ceramic was produced with very fine (2 micron) starting
powders and carefully sintered to produce a very well controlled grain size and flaw
distribution. Tensile test specimens were prepared and then surface finish ground with a
200 grit diamond wheel, perpendicular to the long axis of the test specimen.
The mean room temperature tensile strength of a 200 grit ground silicon nitride specimen
was measured as 280 MPa.
SEM measurement gave a critical flaw size of 110 microns for this tensile specimen.
With that 280 MPa tensile strength value and the 110 micron measured flaw size,
what is the fracture toughness value ( K IC ) for this silicon nitride ?
(Show your calculations and your units –

MMAE 554 Electrical Magnetic and Optical Properties of Materials
Open book
Answer one question only
1.. You are asked to produce a voltage of at least 15,000 volts using a PZT ceramic
for which g33 = 0.025
V.m/N. The PZT is available in the form of a bar with square cross-section, 1.0 mm on
a side. The z axis is parallel to the long axis of the bar. A compressive force of 400N
maximum along the z axis can be applied without crushing the PZT. What is the
minimum length (in mm) for the PZT element?
2. In the semiconductor indium phosphide (InP) Eg=1.30eV, the electron effective mass
= 0.067 mo and the hole effective mass = 2.0 mo. How far above the top of the valence
band is the Fermi energy at 300K in intrinsic InP?
(mo = actual rest mass of electron).

PhD Exam February 24, 2012
Metallurgical and Materials Engineering Program S. Mostovoy
Department of Mechanical, Materials and Aerospace Engineering
1. Lightweight automobiles are being designed using high strength aluminum alloys. As an alternative to
forming prior to heat treating, one of the techniques being considered is one where the alloy is shipped to the
manufacturer in the fully heat-treated condition but then locally temporarily softened using high rate induction
heating and water quenching (retrogression). It is found that a slower rate heating/cooling cycle does not give
satisfactory softening/rehardening properties found using the induction treatment. In the flash heated and
quenched condition (optimum retrogression treatment) the alloy can be easily formed. After forming, the
alloy regains it initial strength in less than an hour. If not worked, initial strength returns after about a week at
room temperature (or several hours at 350°F). Detail the likely micromechanisms responsible for mechanical
properties in each stage in the heat treatment and or working schedule. Use a phase diagram for a typical agehardenable
alloy (e.g., 6061-T6) and a time-temperature-transformation sketch to illustrate these effects.
Suggest a set of experiments that would provide optimum heat-treating, mechanical working and aging cycles
for manufacture.
2. Fracture Toughness
The compliance versus crack length data for a 1 inch thick steel
structure (σ o = 70 ksi; E = 30,000 ksi) is as shown in the table. The
load applied to the structure in service is 10,000 lbs and the room
temperature value for K IC is 40 ksi-√in. However service stresses
range from -50°F to 400°F and the minimum flaw size detectable is
0.4 inches.
Crack
Length a,
inches
Compliance, C, -
10 -9 in/lb
0.1 5.0
0.2 10.0
0.3 26.7
A. Calculate the room temperature factor of safety and comment on
0.4 35.6
this value in light of the structure dimensions and the temperature
range to which the structure is exposed.
0.5 44.8
0.6 53.3
B. This structure may be exposed to high rate loading and it is
proposed that a dynamic toughness criterion be used. Assuming
0.7 80.0
the curve for temperature versus K IC is available for this material and that the value of K IC at -50°F
is known to be 20 ksi-√in. and the value of K IC is 100 ksi-√in. at 100°F; describe a procedure for
estimating the critical value of toughness for high rate loading. Again, comment on the likely
safety factor for this structure.

Design/Manufacturing (Spring 2012)
This is a closed-book, closed-notes exam. However, students are allowed to bring in one-sheet
of equations.
1 Bezier surface
Derive the conditions for C 0 and C 1 continuity of a cubic Bezier composite surface of two patches.
2 Bezier form of a curve on a given Bezier surface
Consider a curve on a bi-quadratic Bezier surface S(u, v) =
2∑
i=0 j=0
2∑
P ij B i,2 (u)B j,2 (v).
In the
parametric (u, v) domain of the surface, this curve corresponds to a straight line between (0.25, 0.25)
and (.5, .5). Find the Bezier form of this curve on the surface.
1