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EM/APR 2007/KJM457/421
UNIVERSITI TEKNOLO GI M ARA
FINAL EXAMINATION
COURSE
COURSE CODE
EXAMINATION
TIME
: VECTOR DYNAMICS AND VIBRATION
: KJM457/421
: APRIL 2007
: 3 HOURS
INSTRUCTIONS TO CANDIDATES
1 . This question paper consists of two (2) parts: PAR T A (3 Questions)
PART B (3 Questions)
2.
Answer ALL questions in PART A and any two (2) question in PART B.
3. Answer ALL questions in the Answer Booklet. Start each answer on a new page.
4 . Do not bring any material into the examination room unless permission is given by the
invigilator.
5. Please check to make sure that this examination pack consists of:
i) the Question Paper
ii) an Answ er Booklet - provided by the Faculty
iii) a one - page Appe ndix
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
This examination paper consists of 8 printed pages
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PART A 40 Marks)
Q U E S T I O N 1
The assembly which consists of a sphere of mass 0.5 kg, attached to a hydraulic arm A Bof
negligible mass spins about its axis at a constant rate of C 2= 4 rad/s. At the same instant,
the hydraulic arm AB moving outward at a constant rate of b = 2 m/s and rotates at a
constant rate of , = 2 rad/s about its V-axis shaft. For the instant represented in Figure
Q1 , determine
a) the angular momentum of the assembly with respect to point A , and
b) the percentage of angular kinetic energy of the assembly.
(10 m arks)
(5 marks)
Given:
r = 30m m
/ = 120 mm
Figure Q1
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Q U E S T I O N 2
The plate which has a moving belt is rigidly fitted to the rotating arm ABC. At the instant
shown, the arm rotates at the constant ratea = 5 rad/s while the belt moves at the constant
rate/? = 120 m m/s relative to the plate in the direction as shown in Figure Q2.
a) Determ ine, at this instant, the absolute velocity and acceleration of point E on the link of
the belt.
(10 marks)
b) Does the a cceleration of point on the link of the belt have the Coriolis s terms?
Discuss or explain about your answer.
(2 marks)
c) If we were to determine the acceleration of point H on the link of the belt, will you have
the Coriolis s terms in this acce leration? Discuss or exp lain about your answer. If your
answer is yes, also compu te its value.
(3 marks)
a
= 240 mm
b=
100 mm
c=80 mm
r
= 60 mm
Figure Q2
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Q U E S T I O N 3
A machine element having a mass of 600 kg is supported by two springs and one damper
as shown. Each spring has a constant of 50 kN/m and the coefficient of damping is 1500
N-s/m.
a) If a periodic force of maximum magnitude eq ual to 200 N is applied to the element at a
frequency of 3.2 Hz, determine the am plitude of vibration at steady state.
(5 marks)
b) If the damping element is now being rem oved, determine the critical mass of this
machine element for which the amplitude ratio is most excessive at this same operating
frequency of 3.2 Hz.
(5 marks)
Figure Q3
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PART B 60 Marks)
QUESTION 1
At the instant shown in Figure Q 1 , collar P is sliding outward at the rate of 1.5 m/s and is
decreasing at 20 m/s
2
relative to arm BC which is rotating at the constant rate 6 = 1 rad/s
relative to arm ABwhich itself is rotating at the constant rate of 4.5 rad/s relative to arm OA.
Knowing that at the same instant, arm OA is rotating at 3 rad/s and is increasing at 5 rad/s
2
,
determine at this instant,
a) the absolute angular velocity and acceleration of arm OA,arm A B,and arm BC,and
(10 marks)
b) the velocity and acceleration of collarP.
(15 marks)
c) the kinetic energy of collar P if the mass of the collar is 0.45 kg. Does this kinetic energy
change with time? Explain.
(5 marks)
Given:
BP= 100 cm
9=30
Figure Q1
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QUESTION 2
The 2-kg plate which has a moving belt is rigidly fitted to the rotating arm ABC of negligible
mass. At the instant shown, the arm rotates at the constant rateco = 5 rad/s while the belt
moves at the constant ratep = 120 mm/s relative to the plate in the direction as shown in
Figure Q2.
a) Determ ine, at this instant, the absolute velocity and acceleration of point
D
on the link of
the belt.
(8 marks)
b) Do pointsDand Hon the link of the belt have the same velocity? Explain.
(2 marks)
c) Determ ine the m ass mom ent of inertia of the plate with respect to the centroidalx axis
i.e. through the centroid C) parallel to the Xaxis and also all products of inertia with
respect to the centroidal planes i.e.planes through the centroid C).
(8 marks)
d) Determine the dynamic reaction at the support at A. Neglect the mass of the moving
belt.
(10 marks)
e) Does the dynamic reaction at the support atA depend on the speed of rotation of arm
ABC and/or the mass of the plate? Explain.
(2 marks)
a
a=240 mm
b=1 0 0 mm
c=80 mm
r=60mm
Figure Q2
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QUESTION 3
a) The square plate ABCD is confined within the slots at A and 6 as shown in
Figure Q3(a). At the instant whenG= 30, slot A has a velocity of 8 m/s to the right.
Determine at this instant the angular velocity of the plate ABCD and the velocity of
slot B. Discuss your selection of the location of the origin O of the global coordinate
system
OXYZ.
(10 marks)
iii) Do you think that we can use the sam e angular velocity found in part (i) to calculate
the velocity of point C? Discuss your answer. If so, then calculate the velocity of
point C with respect to pointA .
(5 marks)
Y
0.3 m
Qm/s
Figure Q3 a)
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b) The 2.1-kg slender rodACE is supported at A as shown in Figure Q3(b). A spring and a
dashpot are attached to the rod at point C and its free end Eand together they hold the
assembly at rest in the position shown. Use / = 600 mm, k = 0.25 kN/m, and
b = 1 0 N s /m .
i) Determine the period of vibration of the assembly when end E of the rod is given a
small displacement and released.
(10 marks)
ii) Classify the type of vibration induced i.e. free or forced, dam ped or unda mped , etc.
(1 m ark)
iii) If a constant torque of magn itude 10 N-m is applied to the rod, determ ine the system
response and calculate the percent maximum overshoot.
(4 marks)
D
Figure Q3 b)
END OF QUESTION PAPER
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APPENDIX
EM/APR 2007/KJM457/421
Kinematics
MainFormulasin ynamics
R
= Ru
R=Ru+QAR
R=Ru
+
2QA(Ru)
+
aAR
+
QA(QAR)
Moment
of
Inertia
Iv
=
I
x
u\ +I
r
uj +I
z
uj -
Angular Momentum
where
H
o
= R
G
A mv
G
+ H
KG =
H
x
i
H
y
j
+
H
z
k
+Izxu
z
u
x
)
H
H
H
X
y
z
Ix
- I yx I y
- I zx 1zy
-Ixv -I
y
i xy
lyz
Iz
Moments
where
M_
G
=
M
x
I +
My
j
M
z
k
M
x
= / ;
My j
M
z
= U
Kinetic Energy
x
\Iy
^ - { 1 2 -
:(X
J
-(Ix-
Iz
ix
Oy O
z
)a>
z
G)
x
o
x
o
y
IzCO
2
z
) - IxyO)
x
O)y+ Iy~,0)
y
C0
2
+ I zxG
Z
G
X
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