Posted: March 16th, 2022

Mathematical Modelling

Mathematical Modelling
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 1
Faculty of Engineering, Environment and Computing
7015MAA
Mathematical Modelling in Aerospace Engineering
Assignment Brief

Module Title
Mathematical Modelling in Aerospace Engineering Individual Cohort
SEP-JAN Module Code
7015 MAA
Coursework Title
CW1 Resit / Deferral: Data-Driven Modelling Hand out date
(dd/mm/yyyy)
Aula: 28/01/2022
Lecturer
Mauro Sebastián INNOCENTE Due date and time
Aula: 04/04/2022
18:00 hrs.
Estimated Time
15 hrs.
Page Limit: 9 (excluding cover page, table of contents
and references). Coursework type
Written Credit value assessed
5
(33.33% of the module)
File types: 1 pdf file (report)
1 zip file (all codes)
Mark and Feedback date (dd/mm/yyyy): 18/04/2022
Mark and Feedback method: Written via Aula
AssignmentTutorOnline

Module Learning Outcomes Assessed
1. Understand how the scientific method works and the process of constructing a mathematical
model.
2. Be aware of different modelling techniques, including their strengths and weaknesses and when
to use them.
3. Develop suitable models to extract the underlying pattern of behaviour of a system, sub-system
or component from physical measurements, observed data, or generated data.
4. Differentiate between mechanistic modelling and data-driven modelling, including grey-box and
black-box models.
6. Critically evaluate the validity of the developed models in terms of their ability to represent the
phenomena under study as well as to clearly articulate their inherent simplifying assumptions
and limitations.
7. Evaluate, communicate and interpret the performance of a given system, sub-system or
component based on the insight gained through modelling.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 2

Task and Mark distribution
1. Report presentation 10%
2. Question 01: Constructing a Mathematical Model. 10%
3. Question 02: Taylor Polynomial for Data-Driven Modelling. 10%
4. Problem 01: Heat Transfer Data-Driven Modelling. 35%
5. Problem 02: Spring Stiffness Identification (Load-Controlled). 35%
General Notes
1) I need help writing my essay – research paper notify your registry course support team and module leader for disability support.
2) Any student requiring an extension or deferral should follow the university process as outlined
here.
3) The University cannot take responsibility for any coursework lost or corrupted on disks, laptops
or personal computer. Students should therefore regularly back-up any work and are advised to
save it on the University system.
4) If there are technical or performance issues that prevent students submitting coursework
through the online coursework submission system on the day of a coursework deadline, an
appropriate extension to the coursework submission deadline will be agreed. This extension will
normally be 24 hours or the next working day if the deadline falls on a Friday or over the
weekend period. This will be communicated via your Module Leader.
5) You are encouraged to check the originality of your work by using the draft Turnitin links in Aula.
6) Collusion between students (where sections of your work are similar to the work submitted by
other students in this or previous module cohorts) is taken extremely seriously and will be reported
to the academic conduct panel. This applies to both courseworks and exam answers.
7) A marked difference between your writing style, knowledge and skill level demonstrated in class
discussion, any test conditions, and that demonstrated in a coursework assignment may result in
you having to undertake a Viva Voce in order to prove the coursework assignment is entirely your
own work.
8) If you make use of the services of a proof-reader in your work, you must keep your original version
and make it available as a demonstration of your written efforts.
9) You must not submit work for assessment that you have already submitted (partially or in full),
either for your current course or for another qualification of this university, unless this is
specifically provided for in your assignment brief or specific course or module information. Where
earlier work by you is citable, i.e. it has already been published/submitted, you must reference it
clearly. Identical pieces of work submitted concurrently will also be considered to be self
plagiarism.
Submission Notes
1) Written reports are to be submitted in pdf format via Aula.
2) Programming codes and/or supporting files to be submitted via Aula as one zip-file.
3) Paper size: A4. Maximum size of report is 9 pages, excluding cover page, table of contents, and
reference list.
4) Font type & size: Times, 12pt.
5) Single line spacing and body text justified.
6) Paragraph spacing: 6pt (before and after paragraph).
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 3

7) Margins: 2.5 cm on the top and 2 cm elsewhere.
8) Equations MUST be sequentially numbered between parentheses.
9) Tables and figures MUST be sequentially numbered, placing the labels below them followed by a
period and explanatory text (caption).
10) Font size in labels and captions: 10pt.
11) References and citations as well as numbering and cross-references of equations, tables and
figures MUST be automatic (as opposed to manual).
12) Page numbers MUST be included in the report.
13) If citations and references are provided, use any standard reference style (e.g. IEEE 2006 – Write a paper; Professional research paper writing service – Best essay writers or
Ace my homework – Write my essay – Harvard) generated by any reference manager of your choice (e.g. Word built-in reference
manager, EndNote, JabRef, Zotero).
Important definitions
• Problem Formulation
It consists of the translation of the problem statement from plain words into mathematical
language. The problem formulation defines the mathematical model, stemming from an
interpretation of the real-world problem and from the setting of a number of assumptions.
Typically, a formulation is given in terms of algebraic equations; of algebraic functions with
unknown coefficients (data-driven modelling); or in terms of difference or differential equations
and initial and/or boundary conditions.
• Explanations and Solutions
Solutions refer to the problem formulated. Thus, a correct solution may be found for either a
correctly or an incorrectly formulated problem. Typically, solutions consist of coefficients’ values;
closed-forms that satisfy difference or differential equations; or values of discretised fields for
numerical problems (a model could be fit afterwards). Solutions may also entail quantities derived
from them (i.e. velocity at a given location derived from obtained position field; or heat flux at a
given location once temperature field has been obtained). Detailed explanations and justifications
of what has been done and why throughout the problem-solving procedure –between formulation
and results’ presentation– will also be assessed.
Note: If the problem is solved using a numerical method (e.g. Runge-Kutta, Finite Differences,
etc.), then the submitted code will also be assessed here. Make sure it is neat and commented.
• Results Presentation
This does not refer to the professional look of the report, which is assessed as Report Presentation
(see above). Instead, results’ presentation here refers to the relevance, clarity and completeness
of the results presented (e.g. equations, data, tables, figures, etc.) to describe and help interpret
the solution(s) obtained for the problem at hand. They must be easily accessible and
understandable for their analysis and interpretation.
• Results’ Interpretation (Ace my homework – Write my paper – Online assignment help tutors – Discussion and Conclusions):
This refers to the student exhibiting clear understanding of the meaning of the results obtained
and what they entail, instead of merely enumerating them. This may also include using the model
developed to study the phenomena in question and draw some conclusions. While doing this, it is
quite common to find that results do not make sense and therefore mistakes be identified. The
importance of this component is evidenced by the high weight assigned to it in the marking rubric
below. Different grades will be awarded according to the depth of this analysis. Do not confuse
depth with length. Be clear and concise.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 4

MARKING RUBRIC FOR REPORT PRESENTATION
First
70-100 – Guidelines for the report presentation were followed.
– Outstanding, well-structured and professionally looking report.
Upper Second
60-69 – Guidelines for the report presentation were followed for the most part.
– Well-structured and professionally looking report.
Lower Second
50-59 – Guidelines for the presentation of the report were partially followed.
– Acceptably well-structured report lacking some presentation quality.
Third
40-49 – Guidelines for the presentation of the report were largely ignored.
– Rather poorly structured and/or unprofessionally looking report.
– Nonetheless, results obtained can still be extracted with minimal effort.
Fail
0-39 – Guidelines for the presentation of the report were largely ignored.
– Low quality report, which may be of low standards, poorly structured, poorly presented, and/or
incomplete.
– Extracting and understanding of results obtained are not straightforward.

MARKING RUBRIC FOR QUESTIONS
First
70-100 – Question answered clearly, concisely, and correctly.
– Answer demonstrates a high level of understanding of the topic and of the question.
– Evidence of critical thinking and deep knowledge.
Upper Second
60-69 – Answers are mostly correct and understanding of fundamentals is evident.
– Level of accuracy and/or details could be improved.
– Good applied knowledge.
Lower Second
50-59 – Answer is partly correct.
– Answer is mostly descriptive but includes some level of discussion.
– Fundamentals are understood, despite some lack of accuracy and detail.
Third
40-49 – Question answered but without detail, precision or discussion. Alternatively, question answered
extensively but without coherence (as if combining extracts from different sources).
– Some errors and/or inconsistencies may be present.
– Descriptive answers showing some level of understanding of the topic and of the question.
Fail
0-39 – Question answered incorrectly, superficially, and/or largely incompletely.
– Descriptive answers only. A lack of understanding of the topic and of the question is evident.
– If question is unanswered, grade will be set to zero.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 5

MARKING RUBRIC FOR PROBLEMS
WEIGHTS
GRADES PROBLEM
FORMULATION
30% EXPLANATIONS
AND SOLUTION
25% RESULTS
PRESENTATION
15% RESULTS
INTERPRETATION
(DISCUSSION)
30%
First
70-100 – Mathematical
formulation clearly
presented in the report.
– The student shows clear
evidence of understanding
the problem at hand.
– Problem correctly
translated from plain
words to mathematical
formulation (main features
captured).
– Clear explanations of
how the formulation has
been derived.
– Main simplifying
assumptions stated. – Clear, detailed, well
structured, consistent,
concise and yet in
depth step-by-step
explanation of what
has been done, how,
and why.
– If numerical method
used to solve the
problem, the submitted
code runs when
executed, and it is
written neatly and well
commented.
– Entirely appropriate
methods used to solve
the formulated
problem.
– Correct solutions
obtained for the
formulated problem. – Results obtained are
the ones requested and
can be easily found,
read and understood.
– High-quality plots
presented in a
professional manner.
– Entirely appropriate
plots to convey results
obtained. – Interpretation of
results is sound, mostly
correct and consistent.
– Elaborate and
articulate interpretation
of results which goes
beyond the obvious.
– Model developed used
to study the phenomena
of interest, drawing
some conclusions.
Upper
Second
60-69 – Mathematical
formulation clearly
presented in the report.
– The student shows some
evidence of understanding
the problem at hand.
– The mathematical
formulation of the
problem is mostly correct,
with minor mistakes or
incomplete.
– If mistakes were made,
the formulation still makes
engineering sense.
– Some simplifying
assumptions stated. – Clear explanations of
what has been done,
how, and why, despite
some lack of depth.
– If numerical method
used to solve the
problem, the submitted
code runs when
executed, and it is
written neatly and with
some comments.
– Adequate methods
used to solve the
formulated problem.
– Solutions for the
formulated problem
may or may not be
correct. If incorrect,
this is due to minor
mistakes in an
otherwise sound
problem-solving
procedure. – Results obtained are
the ones requested but
are not highlighted and
not so easy-to-identify
in the report.
– Good quality plots
presented in a
professional manner.
– Adequate plots to
convey results
obtained, though some
convenient one(s) may
be missing. – Interpretation of
results is mostly correct
and consistent.
– Evidence of
understanding the
problem at hand, the
solution obtained, and
the implications of the
latter to some extent.
– Solution may be
incorrect, still making
engineering sense.
– Solution may be
evidently incorrect,
which the student
identifies and discusses
in a fair attempt to
identify the reason why.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 6

Lower
Second
50-59 – The mathematical
formulation is not clearly
presented, but relevant
equations and analysis
indicate the student is on
the right track.
– Alternatively, the
mathematical formulation
appears mostly correct but
it is not explicitly written.
– Alternatively, the
mathematical formulation
is incorrect, nonetheless
still making sense within
an engineering context.
– The student shows some
difficulty in understanding
the problem at hand. – Problem-solving
procedure not clearly
presented.
– The student’s train of
thought and process
followed to tackle the
formulated problem is
hard to follow.
– If numerical method
used to solve the
problem, the submitted
code runs when
executed, though it is
hard to read and lacks
explanatory comments.
– Adequate methods
used to solve the
formulated problem.
– Solutions may be
incorrect due to
mistakes in the use or
implementation of the
chosen problem
solving technique.
– If solutions are
correct, there is no
justification or clear
path as to how they
were obtained. – Results obtained are
hard to identify in the
report. Some of the
requested ones may be
missing.
– Plots of rather low
quality and hard-to
read.
– The choice of plots to
convey the results
obtained may not be
entirely adequate or
insufficient. – Interpretation of
results is mostly correct,
yet not very deep.
– Evidence of
understanding the
problem at hand, the
solution obtained, and
the implications of the
latter to a lesser extent.
– Some relatively
important implications
of the solution obtained
may be missing.
– Solution may be
incorrect, still making
engineering sense.
– Solution may be
evidently incorrect,
which the student
identifies and discusses
in a fair attempt to
identify the reason why.
Third
40-49 – The mathematical
formulation is not clearly
presented or incorrect,
though still making some
sense within an
engineering context.
– The student shows
strong difficulties in
understanding what the
problem consists of.
– Lack of clarity in
important concepts. – Problem-solving
procedure not clearly
presented and
discussed in little to no
detail.
– The problem-solving
procedure followed by
the student is unclear
and/or inconsistent.
– If numerical method
used to solve the
problem, the submitted
code runs when
executed.
– Adequate methods
used to solve the
formulated problem.
– Solutions may be
incorrect due to
mistakes in the use or
implementation of the
chosen problem
solving technique.
– If solutions are
correct, there is no
justification or clear
path as to how they
were obtained. – Some of the
requested results are
very hard to identify in
the report, or simply
missing.
– Low quality plots
which are hard to
interpret.
– Some important plots
to convey the results
obtained are missing,
and/or the choice of
plots is inadequate. – Interpretation of
results is rather poor in
quality and/or quantity,
showing weaknesses. It
may be inaccurate
and/or incomplete.
– Lack of clarity in
understanding the
problem and/or the
implications of the
results obtained.
– Some important
implications of the
solution obtained may
be missing.
– Incorrect solutions
make little sense within
an engineering context
and hence should have
been identified by the
student during this
exercise.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 7

Fail
1-39 – The mathematical
formulation is not clearly
presented or incorrect,
making little to no sense
within an engineering
context.
– The student does not
evidence sufficient
understanding of what the
problem at hand is about. – The problem-solving
procedure is not
discussed in sufficient
details. Major
Inconsistencies can be
identified.
– The methods used to
solve the formulated
problem are not
appropriate or are not
presented in any detail.
– If numerical method
used to solve the
problem, the submitted
code does not run
when executed or no
code was submitted.
– Solutions are
incorrect due to
choosing an
inadequate problem
solving method, or due
to misuse / incorrect
implementation of the
chosen method.
– If solutions are
correct, there is no
justification or clear
path as to how they
were obtained. – Some of the
requested results are
either very hard to find
or plainly missing.
– Low quality plots,
which are hard to
interpret.
– Some important plots
to convey the results
obtained are missing,
and/or the choice of
plots is inadequate. – Interpretation of
results is poor in quality
and/or quantity,
showing strong
weaknesses.
– Interpretation of
results largely
inaccurate and/or
incomplete.
– Lack of clarity in
understanding the
problem and the
implications of the
results obtained.
– Incorrect solutions
make no sense within an
engineering context,
which the student fails
to identify.
– If no results’
interpretation or
discussion is offered,
this section will be
graded as zero.
0 Late submission.
NOTE: By regulation, one minute past the deadline is considered late submission.
Example of Coursework Grade Calculation
Partial markings are manually entered in the white cells.
Problem
Formulation
Explanations
and Solutions
Results’
Presentation
Results’
Interpretation
(Ace my homework – Write my paper – Online assignment help tutors – Discussion &
Conclusions)
0.30 0.25 0.15 0.30
75 0.10 7.50
Q1. Constructing a Mathematical Model 80 0.10 8.00
Q2. Taylor Polynomial for Data-Driven Modelling 65 0.10 6.50
P1. Heat Transfer Data-Driven Modelling 85 72 75 20 61 0.35 21.26
P2. Spring Stiffness Identification (Load-Controlled) 70 70 70 0 49 0.35 17.15
60
Report Presentation
Prob. Quest.
GRADE
ITEMS ASSESSED for CW01 – RESIT
(SEP-JAN 2021/22) Marks Weights Weighted Mark
Weights
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module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
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In order to further clarify the important definitions in page 3, which apply to the marking rubric, a
simple worked example is presented below to show what is expected from the students.
Example of How to Solve the Two Problems in this Coursework
Problem statement
Given the pairs shown in the table below, obtain the polynomial of lowest degree that interpolates
all points exactly.

xi yi
0 5
1 3
2 5
3 23
Table 1. Data points
Problem Formulation
The polynomial of lowest degree that interpolates four points exactly is given by a third-degree
polynomial. Therefore, the interpolation function is as follows:
( ) = 0 + 1 ⋅ + 2 ⋅ 2 + 3 ⋅ 3 (1)
where the coefficients in red are the unknowns whose values need to be found so that the function
passes through all the data points.
Explanations and Solution
By evaluating the polynomial in Eq. (1) for each of the four data-points, ( ) = , a system of
four linear equations with four unknowns is obtained:
{
0 + 1 ⋅ 0 + 2 ⋅ 02 + 3 ⋅ 03 = 5
0 + 1 ⋅ 1 + 2 ⋅ 12 + 3 ⋅ 13 = 3
0 + 1 ⋅ 2 + 2 ⋅ 22 + 3 ⋅ 23 = 5
0 + 1 ⋅ 3 + 2 ⋅ 32 + 3 ⋅ 33 = 23
Operating:
{ 0 0 0 0 + + + + 0 1 2 3 ⋅ ⋅ ⋅ ⋅ 1 1 1 1 + + + + 0 1 4 9 ⋅ ⋅ ⋅ ⋅ 2 2 2 2 + + + + 0 1 8 27⋅ ⋅ ⋅ ⋅ 3 3 3 3= = ==5 3 523
In matrix form:
⋅ =
= (1 1 1 1 0 1 2 3 0 1 4 9 27 0 1 8 ) ; = (23 5 3 5 ) ; = ( 10 2 3)
Solving the system (you are allowed to use commercial software to solve systems of equations):
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
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= ( 20 1 3=== =-05 24)
Results Presentation
The polynomial function that interpolates the four data points exactly is given by:
( ) = 2 ⋅ 3 – 4 ⋅ 2 + 5 (2)
The colour-coded data-points are shown in Table 2.

x y
0 5
1 3
2 5
3 23
Table 2. Colour-coded data points.
A plot of the interpolant in Eq. (2) passing through the colour-coded data points is shown in Fig. 1.
Fig. 1. Third degree polynomial interpolant fitting the four data-points provided exactly.
Interpretation of Results
(Example of easy and obvious interpretation)
Clearly, the interpolant passes through the data-points, and therefore it can be inferred that it has
been obtained accurately. As a sanity check, it has been evaluated at the points to confirm this.
Thus, the trend of the data has been captured with a cubic polynomial.
(Example of interpretation beyond the obvious)
In addition to the above, by observing a plot of the data-points alone, it is not evident that the
interpolant should exhibit such steep and positive slopes to the left of the data range. It is possible
that this is not the actual trend of the phenomenon underlying this data but a side effect of choosing
a 3rd degree polynomial (as required for exact interpolation). Therefore, it may be safe to use this
interpolant within the interpolated data (i.e. ∈ [0,3]) but not to extrapolate outside that range.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 10
Assignment Brief Introduction
This is an individual coursework which is aimed at assessing the student’s ability to understand a
given engineering or scientific problem formulated in plain words, choose or develop an appropriate
mathematical formulation to describe it, and then solve such formulation.
The mathematical formulations of the problems in this coursework are given by parametric functions,
whilst their solutions consist of the values of the parameters which maximise the fitting to the
available data.
Problems are purposely kept simple to prevent the student from getting lost in the complexity of a
problem-solving technique and instead focus on the whole process of understanding the problem,
formulating it mathematically, and finally attempting to find a solution and interpreting the results.
The student is expected to identify a model’s strengths and limitations, use it to study the physical
phenomena of interest (being modelled), and prepare a professional report.
I need help writing my essay – research paper, read the submission details, marking rubrics, and important definitions thoroughly. It is
a common recurrence that students forget about interpreting the results, therefore losing 30% of the
marks even if they had solved the problem correctly. Thus, for each of the two problems, it is
recommended that the student use the following sub-headings (or others along these lines):
• Mathematical Formulation
• Solution
• Results Presentation
• Results Interpretation (or Ace my homework – Write my paper – Online assignment help tutors – Discussion and Conclusions)
1. Q1: Constructing a Mathematical Model
In your own words,
a) Ace my homework – Write my paper – Online assignment help tutors – Discuss the process of constructing a Mathematical Model of a real system or phenomenon.
Use a diagram (of your own) to help you describe the process.
b) Clearly differentiate between Mathematical Formulation and Solution (see definitions in page
3 and solved example).
c) Define verification and validation.
2. Q2: Taylor Polynomial for Data-Driven Modelling
a) Mathematically derive the Taylor Polynomial and explain the process.
b) Explain its terms.
c) Explain how it can be used for data-driven modelling.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 11
3. P1: Heat Transfer Data-Driven Modelling
You are required to obtain the temperature profile of a bar experimentally. However, your budget is
limited and can only carry out 3 experiments. Setting up the bar horizontally for the experiment, its
left end is at x = 0 meters whilst its right end is at x = 8 meters. The temperature is thus measured at
three locations, as shown in Table 1.

x [m] T [°C]
2.00 182.5
4.70 272.5
6.50 210.0
Table 1. Temperatures measured at three locations of a bar.
You suspect that the temperature profile is quadratic and are requested to obtain:
a) The temperature profile along the bar.
b) The maximum temperature along the bar.
c) Extra budget comes along, and you are able to measure the temperature at the right end of the
bar, which returns 81.5 °C. Assuming this measurement is accurate, what is the error of the
original model at this location?
You MUST implement your own code to solve the regression or interpolation problem. You are
encouraged to use off-the-shelf (commercial) tools to verify your results.
NOTE: Calling an external optimiser or a built-in function to solve a system of equations within your
implementation will still be considered “your own code”.
Bear in mind that you are expected to formulate the problem, solve it, explain the problem-solving
procedure, and finally interpret/discuss the results obtained. The quality of the results’ presentation
and that of the report overall will also be assessed (see marking rubrics).
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 12
4. P2: Spring Stiffness Identification (Load-Controlled)
Springs store energy when stretched or compressed by a force and release stored energy when the
force is removed. Within their elastic limit, massless springs can be modelled as obeying Hooke’s
law. Hence the force required to vary the length of the spring from l0 to lf = l0 + x is given by:

= ⋅ , (1)
where W is the acting force [N]; x is the deflection [m]; and k [N/m] is the spring constant/stiffness.
Let us have a massless spring fixed at one end to the ceiling and holding an object at the other end,
as shown in Fig. 1 (right). The object’s self-weight W [N] stretches the spring thus producing a
deflection x = lf ‒ l0 [m]. The equilibrium length l0 corresponding to W = 0 N is shown on the left.
Fig. 1. Object hanging from massless spring fixed to ceiling.
Suppose that you are provided with a spring whose unknown stiffness (k) you wish to find. You
proceed to carry out some experiments obtaining the following data:

W [N] x [cm]
0 -0.08
10 0.30
20 1.10
30 1.50
40 1.90
50 2.60
60 2.80
70 3.60
80 3.90
90 4.40
100 5.20
110 5.60
120 6.10
Table 2. Experimental data to calculate spring stiffness.
You are asked to:
1- Estimate the spring stiffness k(x).
2- Evaluate the spring stiffness at x = 6 cm.
3- Evaluate the mechanical potential energy (U) of the spring at x = 6 cm.
You MUST implement your own code to solve this data-driven modelling problem. You are
encouraged to use off-the-shelf (commercial) tools to verify your results such as Matlab’s built-in
functions and Toolboxes (‘polyfit’, ‘csape’, ‘spline’, ‘Curve Fitting Toolbox’, etc.).
NOTE: Calling an external optimiser or a built-in function to solve a system of equations or
optimisation problem within your implementation will still be considered ‘your own code’.
Bear in mind that you are expected to formulate the problem, solve it, explain the problem-solving
procedure, and finally interpret/discuss the results obtained. The quality of the results’ presentation
and that of the report overall will also be assessed (see marking rubrics).

l0 k [N/m] lf
k [N/m]

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