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MECH4305 PROBLEM SET 1
Fourier Analysis and Convolution
Due date
5pm Friday Week 3 (September 27)
Submission instructions
Submit your problem sets on Moodle by the due date using the template provided.
In your submissions you will need to show FULL working (handwritten or typed), state any
assumptions, and include any discussion, figures, and a copy of any MATLAB Code.
Question 1
Sojourner landed on Mars in 1997 as part of the Pathfinder
mission and was the first rover to operate on a planet other than
Earth. Prior to its deployment, vibration testing was performed at
the Jet Propulsion Laboratory (JPL) Environmental Test
Laboratory to determine whether Sojourner could withstand the
expected dynamic forces it would encounter during its mission.
The rover can be modelled as the single degree-of-freedom
system shown in Figure 1, with a mass of 11.5 kg and equivalent
stiffness of 126 N/m. During the first phase of testing, Sojourner was subjected to the repeated
forces (which are periodic for all t) shown in Figure 2, where one period of each force is given.
Figure 1.
For each force (Force A & B, separately), complete the following:
a) Derive the Fourier series representation of the force.
b) Plot the original force (as shown in Figure 2) against your Fourier series representation of
the force from t = -20 to 20 s with n = 100 terms.
c) Starting with the equation of motion for the system shown in Figure 1, derive the steady
state response of the rover.
d) Plot the steady state response of the rover from t = 0 to 20 s with n = 100 terms.
e) If displacement amplitudes of > 20 mm damage onboard equipment, does the applied force
present a problem? Use your response plotted in d) and answer in a sentence or two.
= 0
(t )
(t )
2
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Figure 2. One period of Force A, () and Force B, ().
Question 2
In the second phase of testing, the rover (see Figure 1) was subjected to the non-periodic forces in
Figure 3.
For each force (Force A & B, separately), complete the following:
a) Derive the response of the rover using the convolution integral.
b) Find the response using a numerical integrator such as MATLAB’s ode45 and compare the
integral and numerical solutions on a plot for t = 0 to 10 s.
c) Damping material is now added to the rover to influence its vibrational response. Repeat the
numerical solution in b) (there is no need to calculate another convolution integral) with a
damping constant ζ = 0.25.
Compare the undamped and damped response on a plot for t =
0 to 10 s.
d) In a sentence or two, describe the effect of damping on the response.
() = { , − ≤
≤ 0 2, 0 ≤
≤
where
= 2
() = { 0, − ≤
≤ 0 cos() , 0 ≤
≤
where
=
3
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Figure 3. Non-periodic Force A, () and Force B, ().
Question 3
“We do not learn from experience. We learn from reflecting on experience”
(attributed to John Dewey, Educator and Philosopher)
Reflective writing gives you an opportunity to think about what you've learned so far and the
relationship between this course and the acquisition of knowledge for your future career. I want you
to consider the link between theory (what you study in class) and practice (the application of the
theory in the real world).
Reflecting on your experiences in this course over the first 3 weeks, write a reflection (approx. ½
page in length) that addresses the following questions:
• What have you learned so far and why is it important?
• How much did you know about the subject before we started?
• What do you think is the link between industry practice and this course?
• What have you found to be clear/unclear, easy/difficult, interesting/surprising/unexciting,
and why?
• How will you enhance your learning in this course moving forward?
() = { 0,
≤ 0 −4,
> 0
() = { 0,
≤ 0
+ sin(10),
> 0
4
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Reflection Marking Rubric
Criteria/Mark 1 2 3
Describing
Experience
Unclear and vague. Clear but general. Clear and focused on
the specific aspects that
challenge or interest
the student.
Reflection Minimal reflection - No
personal reflection or
discussion is limited to
description of general
opinions. Connections
are not drawn between
class content and
personal experience
and/or industry
practice.
Some reflection -
Connects class
content/activities with
personal experience or
industry experience but
remains superficial.
Limited/superficial
insight about self or a
particular issue.
Critical reflection –
Demonstrates superior
connections between
class content/activities,
personal experience
and/or industry
practice. Articulates
new
understanding/insights
about self or particular
issue/concept.
Writing Quality Unfocused,
unorganized, vague,
and sloppy. May feature
many grammatical or
spelling errors.
Some focus and
organization. Language
has some precision.
Reflection was
proofread.
Discussion is clear and
specific; descriptions
use concrete and
precise language and
insights are precise and
clear. Organization is
apparent and effective.
Reflection has been
proofread for grammar,
punctuation, and
spelling errors.
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