MEEN363_HW03_Problems_Spring2023_Final (2)
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Date
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MEEN 363 Dynamics and Vibrations (Spring 2023)
Dr. Yong Joe Kim and Ms. Wanyu Xu
Assigned on 02/09/2023
Due:
11:59 pm on 02/16/2023 (Thursday)
Homework Submission Instructions:
An electric
PDF
copy of your homework needs to be submitted
through Canvas by 11:59 PM on each due date.
A scanned PDF copy of your handwritten homework is
acceptable.
Use the following file naming convention:
HWXX_MEEN363_ Sp2023_LastName_FirstName.pdf
where XX represents the homework number (e.g., 01 for homework #1).
If you are requested to submit your Matlab or Python code along with the PDF copy
, your code should
be named as
HWXX_MEEN363_Sp2023_LastName_FirstName_PYY.m
where YY represents the problem (or task) number (e.g., 02 for problem #2).
Then, all the files need to be compressed in a single ZIP file with the following filename. You do not need
to compress your file if you are submitting only one file.
HWXX_MEEN363_Sp2023_LastName_FirstName.zip
Do not use nicknames, as they can make it difficult to discern whom the grade is assigned to.
No late
submission will be accepted even in case of wrong file submissions
.
Homework 3: Particle Kinetics (3 Problems)
1.
Figure 1
shows a block of mass,
࠵?
that is attached to a wall through two ideal springs and a viscous
damper.
(a)
Derive the equation of motion (EOM) for the block, assuming that
࠵?
is measured from its
unstretched spring position (USP).
(b)
Find the symbolic expressions of the undamped natural angular frequency, critical damping
coefficient, and damping ratio.
(c)
Given
࠵? = 0.2࠵?࠵?
,
࠵?
!
= 15࠵?/࠵?
,
࠵?
"
= 35࠵?/࠵?
,
࠵?̇(࠵? = 0) = 4 ࠵?/࠵?
, and
࠵?(࠵? = 0) = 0
, derive the
time solutions to the EOM in the underdamped, critically damped, and overdamped cases.
In
each case, clearly show the expressions of the unknown coefficients in the homogeneous time
solution.
(d)
Draw the vibration responses during the three undamped cycles (i.e.,
3࠵? = 3 ×
"#
$
!
)
in a single
plot
using the damping ratios of
࠵? =
0.25, 1, and 2.5. Discuss the vibration characteristics in terms
of the damping ratios, temporal decaying rates, and oscillatory behaviors.
Figure 1
: 1-DOF mass-damper-spring system.
2.
A rotary air compressor with a mass of
࠵? = 100 ࠵?࠵?
is rigidly mounted on a cart with a mass of
࠵? = 600 ࠵?࠵?
as shown in
Figure 2
.
The undamped, natural angular frequency is measured as
࠵?
%
= 314 ࠵?࠵?࠵?/࠵?
.
A free vibration response in the horizontal direction (i.e.,
࠵?
-direction) is
measured and shown in
Figure 3
.
Find the approximate viscous damping coefficient of the
system in
࠵? ∙ ࠵?/࠵?
using the Logarithmic Decrement.
Figure 2
: Rotary air compressor on a cart.
Figure 3
: Free vibration response.
0
0.05
0.1
0.15
0.2
0.25
0.3
Time [s]
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Displacement [m]
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