COLLEGEOF ENGINEERING MECHANICAL ENGINEERING Ph.D. Preliminary Qualifying Examination Signature Page Vibration Examination January 26, 2009 (Monday) 9:00 am – 12:00 noon Room 2145 Engineering Building For identification purposes, please fill out the following information in ink. Be sure to print and sign your na me. This cover pag e is for attendance p urposes only , and will be separated from the rest of the exam before the exam is graded. Write your student number on all exam pages. Do NOT write your name on any of the other exam pages besides the cover page. Name (print in INK) Signature (in INK) Student Number (in INK) COLLEGEOF ENGINEERING
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For identification purposes, please fill out the following information in ink. Be sureto print and sign your name. This cover page is for attendance purposes only, andwill be separated from the rest of the exam before the exam is graded. Write yourstudent number on all exam pages. Do NOT write your name on any of the otherexam pages besides the cover page.
9:00 am – 12:00 noonRoom 2145 Engineering Building
GENERAL INSTRUCTIONS:
This examination contains five problems. You are required to select and solve fourof the five problems. Clearly indicate the problems you wish to be graded. If youattempt solving all of them without indicating which four of your choice, the four
problems with the worst grades will be considered. Note that Problem number 5 is mandatory.
Do all your work on the provided sheets of paper. If you need extra sheets, pleaserequest them from the proctor. When you are finished with the test, return theexam plus any additional sheets to the proctor.
Mechanical Engineering Ph.D. Preliminary Qualifying ExaminationVibration – January 26, 2009
You are required to work four of the five problems, one of which is Problem No. 5. Clearlyindicate which problems you are choosing. Show all work on the exam sheets provided and write
your student personal identification (PID) number on each sheet. Do not write your name on anysheet.
Your PID number:____________________________
Question #1
A uniform bar of length L and weight W is suspended symmetrically by twounstrechable strings as shown in the figure. If the bar is given small initial rotation aboutthe vertical axis,
a. Draw the free body diagram of the bar during its free oscillation. b. Write down the equation of motion for small angular oscillation about axis O-O.c. Determine the period of the free oscillation of the bar.
You are required to work four of the five problems. Clearly indicate which problems you arechoosing. Show all work on the exam sheets provided and write your student personal
identification (PID) number on each sheet. Do not write your name on any sheet.
Your PID number:____________________________
Question #2The system shown in the figure is in its static equilibrium position (SEP). It consists of auniform rod of mass m and length L and is supported by spring of stiffness k anddashpot of coefficient c .
a. Draw the free body diagram of each system as it oscillates about the SEP. b. Derive the equation of motion of each system using Newton’s second law.
c. Determine the undamped natural frequency.d. Determine the damping ratio, the critical damping coefficient, and the dampednatural frequency.
You are required to work four of the five problems. Clearly indicate which problems you arechoosing. Show all work on the exam sheets provided and write your student personalidentification (PID) number on each sheet. Do not write your name on any sheet.
Your PID number:____________________________
Question #3The system shown consists of a cylinder of mass m with a piston, which impartsresistance proportional to the velocity of a linear viscous damping c , the cylinder isrestrained by a spring of stiffness k
(a) draw the free body diagram of the cylinder,
(b) write down the equation of motion using Newton’s second law, and
(c) determine the response amplitude and phase angle using Complex Algebra.
( ) sin y t Y t → = Ω( ) sin y t Y t → = Ω( ) sin y t Y t = Ω
( ) sin y t Y t → = Ω( ) sin y t Y t → = Ω( ) sin y t Y t = Ω
You are required to work four of the five problems. Clearly indicate which problems you arechoosing. Show all work on the exam sheets provided and write your student personalidentification (PID) number on each sheet. Do not write your name on any sheet.
Your PID number:____________________________
Question #4
The system shown below consists of two rotors coupled by a discontinuous shaft of
modulus of rigidity is 6 211.5 10 / /G lb in rad = × :
• Draw the free-body diagram of each rotor,• Derive the equations of motion,• Determine the natural frequencies of free torsional oscillations and provide the
physical meaning of each value,• Draw the normal mode shape and evaluate the value of the twist at the junction of
MANDATORY PROBLEM (EVERYONE IS REQUIRED TO SOLVE THIS PROBLEM)Your PID number:____________________________
Question #5
Consider a rigid body of mass m and mass moment of inertia J c with respect to center of gravityC g . Suppose that the body is supported by two springs of stiffness k that are attached at distances2l and l with respect to the center of gravity C g as shown in Figure 5a. Let m = 10 kg, J c = 5 kgm2
k = 100 N/m, and l = 1 m.
Part I:
(a) Derive the equations of motion for this body using coordinates x and θ .
(b) Determine the natural frequencies of the system.
(c) Draw the natural mode shapes of the system.
(5a) (5b)
Next, consider the same rigid body as shown in Figure 5b.
Part II:
(d) Derive the equations of motion for this body using coordinates x1 and x2.
(e) Determine the natural frequencies of the system.
(f) Draw the natural mode shapes of the system.
Part III:(g) State if there are differences in the equations of motion, natural frequencies and mode
shapes obtained in each case and explain why.
(h) What are the couplings in equations of motion, respectively, in these two cases?
A uniform bar of length L and weight W is suspendedsymmetrically by two strings as shown in the figure. If the bar is given small initial rotation about the vertical axis,d. Draw the FBD of the bar during its free oscillation.e. Write down the equation of motion for small angular oscillation about axis O-O.f. Determine the period of the free oscillation of the bar.
Figure 5.Solution:
FBD
From the static equilibrium position we write
2T mg=
(1)Under free vibration of the bar and in an arbitrary position θ , the bar will be raised up
slightly, and will be displaced by a distance ( / 2)a θ from the its suspended string. The
string also be tilted by an angle ϕ from the vertical such that, ( / 2) ha θ = ϕ . This
geometric relation gives ( / 2h)aϕ = θ .
Now writing the equation of motion by taking moments about axis OO, gives
0I 2(Tsin )( / 2) T T2h
aa a a
θ = − ϕ ≈ − ϕ = − θ
&&
(2)Using equation (1), equation (2) takes the form
(b) From the free-body diagram, we write the equation of motion from the staticequilibrium position using Newton’s second law of moments about the hinge axis O
( ) ( )2
2 2
0 03
mL I cL L ka a cL kaθ θ θ θ θ θ = − − → + + =&& & && & (1)
(c) The undamped natural frequency is obtained by dividing both sides of the equation of
motion (1) by the coefficient of θ &, i.e.,
2 2
2 23 3 0 3 3n
c ka ka a k
m mL mL L m
θ θ θ ω + + = → = =&& & (2)
(d) The damping ratio is obtained by writing the equation of motion in the form22 0n nθ ζω θ ω θ + + =&& & (3)
Thus we can write km
3 3 32 3
2 2 32 3
n
n
c c c cL
m m a k kmam
L m
ζω ζ ω
= → = = =(4)
The critical damping coefficient is obtained from (4) as
3 2 3
32 3cr
cr
c cL kmac
c Lkmaζ = = → =
The damped natural frequency, nd ω , is written in terms of the undamped natural
The response must oscillate at the same frequency of the excitation in the steady state atamplitude and phase angle to be determined, thus one can write the response in the form
( ) wherei t i i t i x t X Ime X Ime X X Imeϕ ϕ Ω − Ω −= = = (5)
We need the first and second time derivatives of ( ) x t , i.e.
2( ) , ( )i t i t x t i X Ime x t X ImeΩ Ω= Ω = −Ω& && (6)
Substituting expressions (5) and (6) into equation (4), gives
2 22 2i t
i t i t i t n n n X Ime i X Ime X Ime Y Ime
π
ζω ω ζω
Ω + Ω Ω Ω + Ω + = Ω (7)
Canceling out i t Ime Ω from both sides of equation (7) and rearranging, gives
2 2 2 22 2 2 , wherei i
n n n n X i X X Y Ime Y Y Y Ime
π π
ζω ω ζω ζω −Ω + Ω + = Ω = Ω = (8)
Rearranging
( )2 2
2
2
n
n n
X
Y i
ζω
ω ζω
Ω=
− Ω + Ω (9)
Multiplying and dividing by the conjugate of the denominator, gives
Part III: Natural frequencies remain the same, but equations of motion and mode shapesare different. The first case is static coupling and the second is both dynamic and staticcoupling.