Office Use Only
Semester One 2016
Examination Period
Faculty of Engineering
EXAM CODES:
MEC3453
TITLE OF PAPER:
DYNAMICS II- PAPER 1
EXAM DURATION:
3 hours writing time
READING TIME:
10 minutes
THIS PAPER IS FOR STUDENTS STUDYING AT: (Tick Where Applicable)
 Berwick
Clayton
Malaysia
 Off Campus Learning
 Caulfield
 Gippsland
 Peninsula
 Monash Extension
 Parkville
 Other (specify)
 Open Learning
 Sth Africa
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OPEN BOOK
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permitted)
SPECIFICALLY PERMITTED ITEMS
if yes, items permitted are:
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2-sided A4 sheet of paper with written and/or typed notes
Candidates must complete this section if required to write answers within this paper
STUDENT ID:
__ __ __ __ __ __ __ __
DESK NUMBER:
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Page 1 of 6
1. The pendulum shown in Figure Q1 consists of 2 rods. AB is pin supported at A
and swings only in the Y-Z plane, whereas a bearing at B allows the attached
rod BD to spin about rod AB. At a given instant, the rods have the angular
motions shown.
Collar C, located 0.2 m from B, has a velocity of 3 m/s and an acceleration of 2
m/s2 along the rod.
Consider the observer to be located at B and the coordinate (x,y,z) is allowed
to rotate with the system such that Ω = ω1 (note that they are vectors).
We want to find the velocity and acceleration of the collar C.
a. Using the rotational reference as above, write down the relative velocity
equation to find the absolute velocity of collar C.
(2 marks)
b. Using this equation find the absolute velocity of collar C
(8 marks)
c. Using the rotational reference as above, write down the relative
acceleration equation to find the absolute acceleration of collar C.
(2 marks)
d. Using this equation find the absolute acceleration of collar C (8 marks)
Figure Q1
Page 2 of 6
2. (a)
A uniform slender rod of mass m and length l is hinged at point A and is
attached to 4 linear springs and one torsional spring (see figure Q2a). We wish
to determine the vibration response of this system.
i.
Identify on a clearly labelled diagram, what generalised coordinate
system you will use to derive the equation of motion.
(2 marks)
ii.
Derive the equation of motion and the expression for the natural
frequency for this system.
(8 marks)
1
(Note: Moment of inertia about centre of mass of a slender bar = 𝑚𝑚𝑙𝑙 2 ,
12
where m = mass of the bar, 𝑙𝑙 =length of the bar)
Figure Q2a.
(b)
A spring-mass system with a mass of 2 kg and stiffness 3,200 N/m has
an initial displacement x(0) = 0.
i.
ii.
Calculate the undamped natural frequency of the system
(2 marks)
If the amplitude of the free vibration response must not exceed 0.1 m,
what is the maximum initial velocity that the system can be subjected
to?
(8 marks)
Page 3 of 6
3. (a) The equation of motion below represents a 2D spring mass system.
i.
ii.
1
�
0
0
1
2 −1
� 𝑥𝑥̈ + �
� 𝑥𝑥 = � � sin(0.618𝑡𝑡)
1
−1 1
0
Calculate the natural frequency of the system
Determine the modal equations for the system
(4 marks)
(6 marks)
(b) Consider the 2-degrees of freedom system whose equation of motion
for free vibration is as below;
i.
ii.
iii.
1
�
0
0
0.3 −0.1
3
� 𝑥𝑥̈ + �
� 𝑥𝑥̇ + �
1
−0.1 0.3
−1
−1
0
� 𝑥𝑥 = � �
3
0
Re-write this equation of motion in its de-coupled form. (5 marks)
Calculate the undamped natural frequency of the system
(2 marks)
Calculate the damped natural frequency of the system (3 marks)
Page 4 of 6
4. A particular machine shown in Figure Q4 is modelled as a mass spring damper
system with mass m, spring constant k and damping constant c. The machine is
placed on a base which is subjected to harmonic oscillation y(t) whose
frequency causes the machine’s oscillation x(t) to have its maximum amplitude
for the experienced damping ratio. It is also known that m= 250 kg, k=1000
N/m, and it is observed that when the base oscillates with an amplitude of 0.5
cm, the machine oscillates with an amplitude of 1 cm. The system is known to
be in steady state.
a.
Find the equation of motion and damping ratio of the described system
(5 marks)
b.
Determine the amplitude of the displacement of the machine relative
to the base
(5 marks)
c.
Find the expression of the force experienced by the base based on the
known variables and calculate its value
(5 marks)
d.
If the system is not yet in steady state, write down the expression of
the oscillation of the machine relative to the base and show which
parameters whose value cannot be found based on the provided
information
(5 marks)
Page 5 of 6
5. A simply supported (pin-pin) beam with length L, cross sectional area A, cross
sectional moment of inertia I and mass density ρ is shown in Figure Q5.
a.
Based on the known parameters, derive the general solution for both
the continuous-time and spatial equation of motion for the transverse
vibration y(x,t) of the beam
(8 marks)
b.
Determine the boundary conditions at x=0 and x=L for the transverse
vibration y(x,t) of the beam
(8 marks)
c.
By using the results found in (i) and (ii), derive the beam’s natural
frequency and normal mode expression
(4 marks)
Figure Q5
END OF EXAM
Page 6 of 6