# MECHANICAL ENGINEERING DEPARTMENT

```MECHANICAL ENGINEERING DEPARTMENT
Ph.D. Qualifying Exam
Part I
Mathematics and Fundamentals
Closed Book/Notes
September 9, 2010
Thursday
1:00 pm - 5:00 pm
EMS E250
Enter problem number(s)
that you selected
The problems are:
Ordinary Differential Equations
2 problems
(
)
(
)
Partial Differential Equations
2 problems
(
)
(
)
Linear Algebra
2 problems
(
)
(
)
Calculus
2 problems
(
)
(
)
Numerical Methods
2 problems
(
)
(
)
The student should select 8 problems out of 10 problems given here. Write the
problem numbers that you selected in ( ) above. Also circle the problems that you
choose on each problem sheet. Choose at least one problem from each category
listed above. Use one exam book (blue book) for each problem. Include your
assigned number and NOT your name on each book. Submit both exam books and this
problem sheet when you leave.
QualExam-F10/Fall 2010
Ordinary Differential Equation #1
Solve the differential equation below for y (x)
y&quot;6 y '9 y 
e 3 x
x5
Ordinary Differential Equation #2
The position from equilibrium, y, at a time, t, in a damped spring-mass system is
described by the equation
d2y
dt
2
5
dy
y0
dt
The momentum for this system is given by p = dy/dt
For the initial conditions of y(0) = 2, p(0) = -2, find
(a) the equation describing the position, y, as a function of time,
(b) the equation describing the momentum, p, as a function of time, and
(c) the total energy of the system at t = 1.5.
Partial Differential Equation #1
Solve
2Z
x 2

2Z
y 2
0
where 0 &lt; x &lt; and
0&lt;y&lt;
with the boundary conditions:






Z (x,0) = 4 sin2 x , 0 &lt; x &lt; 
Z (x, = 0, 0 &lt; x &lt; 
Z (0,y) = cos2y, 0 &lt; y &lt; 
Z (,y) = 0, 0 &lt; y &lt; 
Partial Differential Equation #2
The differential equation to describe the transient heat conduction is given as:
0 x  1
u
 2u
 2
t
x
0  x 1
with the initial conditions of u linearly varying from 0 to 1 from x=0 to x=1. Solve the
above equation for the boundary condition:
B.C.:
u0
and
u
0
x
at x  0
at x  1
Data:
Thermal diffusivity:
=constant=1
Linear Algebra #1
Find the eigenvalues and eigenvectors of the following matrix:
1 2 3
A  2 1 3


 3 3 6
Linear Algebra #2
1  2 2 
Given the matrix A  2  3 6, find the inverse A1.
1 1 7
Calculus #1
Evaluate the volume, V generated when the area bounded by the curve y  ax 2 , the
x  axis and the line y  2b is revolved about the y  axis.
Calculus #2
1 
 x
lim



x

1
ln
x


Numerical Methods #1
Using the second-order terms of the Taylor series expansion and the forward-time and
centered-space differencing, show that the advection equation
f
2 f
 2
t
x
can be formulated as:
fi n 1  fi n 1
f n  2 fi n  fi n1
  i 1
2t
x 2
where α=constant
What is the order of the accuracy?
Using von Neumann stability analysis, determine the stability condition.
Numerical Methods #2
The following is a set of two simultaneous nonlinear equations with two unknowns:
x1  x1 x 2  10
2
x 2  3x1 x 2  57
2
Use the multiple-equation Newton-Paphson method to determine the roots of the above equations.
Initiate the computation with guesses of x1 = 1.5 and x2 = 3.5.
MECHANICAL ENGINEERING DEPARTMENT
Ph.D. Qualifying Exam
Part II
Area of Concentration
Thermal Science Stem
Open Book/Notes
September 10, 2010
Friday
1:00 pm – 5:00 pm
EMS E250
Enter problem number(s)
that you selected
The problems are:
Fluid Mechanics
3 problems
(
) (
) (
)
Thermodynamics
3 problems
(
) (
) (
)
Heat Transfer
3 problems
(
) (
) (
)
The student should select 6 problems out of 9 problems given here. Write the problem
numbers that you selected in ( ) above. Also circle the problems that you choose on
each problem sheet. Choose at least one problem from each category listed above.
Use one exam book (blue book) for each problem. Include your assigned number and
NOT your name on each book. Submit both exam books and this problem sheet when
you leave.
QualExam-F10/Fall 2010
Fluid Mechanics #1
Consider long surface waves that propagate in a channel with a constant crosssection. The frequency is high enough, so that (v/)1/2 is small comparing with the
width of the channel and its depth. Find the damping coefficient  for he waves.
Given: the size of the channel (a and h) and v/.
Hint: Use the expression for average energy dissipation E  
the energy dependence on time E=E0exp(-2t)
surface
h
liquid
a
1
2

2
 V  df and
2
0
Fluid Mechanics #2
A horizontal flow with an inlet diameter of 1m is divided and has equal flow rates
from each outlet. The outlets have diameter of 0.5 m. The inlet pressure is 200 kPa
and the inlet flow rate is 5 m3/s.
(a) Calculate the centerline pressure at the outlets assuming no friction losses.
(b) Calculate the reaction forces on the divided flow that must be absorbed by a
support system
(c) Calculate the reaction forces on the divided flow if the friction loss between the
inlet and either outlet is give by h= 1.6V2/2g, where V is the inlet velocity.
Fluid Mechanics #3
A long cylinder in an infinite fluid with a constant density, , and constant viscosity, .
Let the radius of the cylinder be R and the angular velocity of the cylinder be . Find:
(a)
(b)
The velocity distribution of the infinite fluid
The torque per unit length of the cylinder.
Thermodynamics #1
Liquid water fills a tank with a volume of 0.002 m3. The water is initially at 40&deg;C, with
a pressure of 100 kPa. A pressure-relief valve is fitted onto the tank, such that
saturated water vapor can escape from the tank to maintain a pressure of 100 kPa inside
the tank. 1250 kJ of heat is added to the water. Determine how much water remains in
the tank after the heating process is complete.
Thermodynamics #2
Air enters the compressor of an ideal air-standard Brayton cycle at 1.01bar, 310K. The
compressor pressure ratio is 12 and the temperature at the turbine exit is 770K.
The velocity at the inlet of the compressor is 10m/s and the cross sectional area
of compressor inlet is 0.5m2. Determine:
(a) The mass flow rate in kg/s.
(b) The net power developed in kW.
(c) The thermal efficiency
(d) The back work ratio (compressor work/turbine work).
Use “Air-standard Analysis”, which means the specific heats are treated as a variable.
[Hint] The cycle is a steady state with negligible kinetic and potential energy changes
across each component.
Thermodynamics #3
Superheated steam at 100 bar and 520 ◦C leaves
the steam generator of a vapor power plant and
enters a turbine with an isentropic efficiency of
90%. The pressure of the turbine at the exit is
0.08 bar. Saturated liquid leaves the condenser at
0.08 bar. The isentropic efficiency of the pump is
88%. Compressed liquid leaves the pump at 100
bar. The mass flow rate is 106 kg/hr.
a) Draw a T-S diagram. Be sure to label clearly
states 1, 2, 2s, 3, 4, and 4s.
b) Using the isentropic efficiency of the turbine,
find the specific enthalpy at state 2.
c) Using the isentropic efficiency of the pump,
find the specific enthalpy at state 4.
d) The net power output, in MW.
Heat Transfer #1
In a double pipe heat exchanger, oil (Cp=1.9 kJ/kg&middot;K) at a flow rate of 5 kg/min is
cooled from 120 &deg;C to 80 &deg;C by water which enters at 20 &deg;C and exits at 50 &deg;C. If
the overall heat transfer coefficient is 160 W/m2&middot;&deg;C, calculate the required heat
exchanger area for
a). Parallel flow
b). Counterflow
Heat Transfer #3
An electronic device, for example a power transistor mounted on a finned heat sink can
be modeled as a spatially isothermal object with internal heat generation and an
external convection resistance.
(a) Consider such a system mass M , specific heat capacity c, and surface area As , which
is initially in equilibrium with the environment at T . The electronic device is suddenly

energized so that a constant heat generation E g W  occurs. Show that the temperature
response of the device is:

t
 exp(
)
i
RC
Where   T  T () and T () is the steady state temperature;  i  Ti  T (); R is thermal
resistance and C is thermal capacitance.
(b) An electronic device which generates 60W of heat, is mounted on an aluminum sink
weighing 0.25 kg and attains a temperature of 100 C in ambient air at 20 C under steady
state conditions. If the device is initially at ambient temperature what will be its
temperature 10 minutes after power is switched on?
MECHANICAL ENGINEERING DEPARTMENT
Ph.D. Qualifying Exam
Part II
Area of Concentration
Machine Design Stem
Open Book/Notes
September 10, 2010
Friday
1:00 pm - 5:00 pm
EMS E250
Enter problem number(s)
that you selected
The problems are:
Machine Design
3 problems
(
) (
) (
)
Kinematics &amp; Dynamics
3 problems
(
) (
) (
)
Controls &amp; Vibration
3 problems
(
) (
) (
)
The student should select 6 problems out of 9 problems given here. Write the problem
numbers that you selected in ( ) above. Also circle the problems that you choose on
each problem sheet. Choose at least one problem from each category listed above.
Use one exam book (blue book) for each problem. Include your student ID number and
NOT your name on each book. Submit both exam books and this problem sheet when
you leave.
QualExam-F10/Fall 2010
Machine Design #2
An Acme-threaded power screw with a crest diameter of 1.125 in. and single thread
is used to raise a load of 25,000 lb. The screw has 5 turns per inch. The collar mean
diameter is 1.5 in. The coefficient of friction is 0.12 for both the thread and the
collar. Determine the following:
(a) Pitch diameter of the screw.
(b) Screw torque required to raise the load.
Controls &amp; Vibration #2
A unity feedback system has the forward loop being KG( s) 
K 10( s  1)
( s  2)( s  4)
1) Please sketch the Nyquist plot for this system with K = 1 assess the stability
of the closed loop system for K = 1
2) Determine the range of K for securing the closed-loop system stability using
the Nyquist stability criterion.
Controls &amp; Vibration #3
The block diagram of the simplified control of the Yaw of the Lockheed Y-22 experimental
fighter jet is shown below:
Where  (t ) is the desired yaw angle and  (t ) is the actual angle. The feedback transfer function is
that of a low pass filter that eliminates any high frequency noise.
a) Sketch the Root Locus as the gain K is varied from 0 to  (Apply
all 9 rules)
b) Find the gain K where the system is critically damped.
c) Find the gain K where the system is marginally stable. What is the frequency of
oscillation?
d) Find the gain where the system is underdamped.
e) Find the gain where the system is overdamped.