DEPARTMENT OF MECHANICAL AND AEROSPACE ENGINEERING
Pro-forma to accompany assignment / coursework 2023/2024
This pro-forma should be the first page to any set assignment / coursework. A full assignment brief
should accompany this pro-forma.
Module
Leader:
Module Title: Dynamics and Modal Analysis
Module Code: ME5557
Dr Cristinel
Mares
Assessor: Dr Cristinel
Mares
Assessment Title: Assignment
Weighting: 25%
Main objectives of the assessment: To ensure students have understood the methods of defining
analytical models for vibrating systems, deriving the ordinary differential equations of motion, and
solving them analytically for the dynamic response Multi-body rigid body dynamics and interactive
rigid body systems are considered where equations of motion must be derived using the methods
of Lagrangian mechanics.
Brief Description of the assessment: Modal analysis and dynamic response of MDoF systems are
considered and rigid bodies in relative motion are studied.
The assessment helps students acquire a practical knowledge of the subjects as covered in the
lectures.
Learning outcomes for the assessment:
Assessment and marking criteria
Students will demonstrate knowledge and skills Students are required to:
required to reduce an actual physical system to
Submit a single individual report that follows
an analytical model, use Newton’s laws,
Lagrange’s equations to derive the equations of the objectives and meets the format described
motion and to solve them analytically when the in this assignment brief by the specified
deadline.
system is linearized.
Each problem has a unique solution which if
obtained correctly and using a valid method
the student will deserve the full mark. In cases
where the student has made mistakes in
derivation and/or solution process(es) some
marks will be deducted.
Assessment method by which a student can demonstrate learning outcomes:
The derivation and solution procedures help the student acquire an understanding of the process.
This will be further assessed during the exam.
Format for the assessment/coursework (Guidelines on the expected format and length of
submission):
The assignment represents 25% of the final module mark.
The report format and marking scheme are described in the proforma.
The report will be uploaded on WiseFlow at submission of the assignment.
It is recommended that you use Arial font 11 for the “body text” and 1.5 line spacing.
You must fill in and sign the provided electronic cover (e-cover) sheet and affix this as the front
page of your submission on Wiseflow.
Academic Misconduct and Plagiarism
This report is an individual assignment and will be assessed as such. Plagiarism–making use of and
portraying someone else’s work, inventions, writings, thoughts or ideas as one’s own–is NOT
accepted and will result in a zero mark. Where this is detected, further investigation will follow,
and consequences might ensue. You must familiarise yourself with the university’s policy on
plagiarism here.
Engaging with Artificial Intelligence (AI)
Generative AI tools can be helpful for extracting information from the internet, improving
grammar, restructuring, reviewing and critically analysing written materials, and reading and
debugging codes.
In using AI tools, you must be mindful that AI and human intelligence are different; they do not
understand anything they produce neither do they understand what the text they produce means
in a wider engineering or societal context.
Presenting AI-generated text or images as your own work constitutes a form of plagiarism. You,
therefore, must acknowledge and describe how you have used AI in your work. This should be in
an included “Acknowledgement Section”. You must also familiarise yourself with the university’s
policy on Using AI in your studies here.
The submitted assignment will be assessed by an AI detection software package. Any
unacknowledged use of AI identified will be further investigated, and consequences might ensue.
Distribution date to students: TBC
Indicative Reading List:
Submission Deadline: TBC
Mechanical Vibrations – Singiresu Rao
Further information:
All the required taught material is uploaded in Brightspace module ME5557. The assignment is
uploaded in the directory “Assignment”.
Marking scheme
Problem 1.
a) EOM:
50%
b) Natural Frequencies: 20%
c) Normal Modes:
20%
d) Orthogonality:
10%
Problem 2.
a) Sinusoidal approximation: 30%
b) Polynomial approximation: 30%
c) Critical velocity:
40%
1. Antisymmetric Vibration Modes of an Airplane
A simplified model of an aircraft wing in free-free conditions
is presented in the Figure 1.
It is required to carry out the analysis of the anti-symmetrical
vibration modes for this wing discretized in three zones with
associated masses:
- fuselage, inner wing and nacelle of mass 𝑚2 and located at
𝑙1 = 2𝑚 from the centre plane.
- outer wing of mass 𝑚 and located at 𝑙 = 4.5𝑚 from the centre
plane.
The wing is considered elastic with stiffnesses 3𝑘 and 𝑘, as indicated in the figure.
The degrees of freedom considered for the vibration analysis
1
2
are:
-the angle of roll 𝜑 of the entire wing moving in the vertical
plane, with the roll moment of inertia of the entire wing
being 𝐼.
-the elastic deformations 𝑤 and 𝑤 at 𝑚 and 𝑚 respectively.
Because the anti-symmetric vibration behavior is considered
there is no vertical deformation at the central fuselage.
1
2
1
2
a) Obtain the equations of motion of the antisymmetric vibrations using the method of force equilibrium and the
equations of Lagrange.
b) Calculate the three natural frequencies.
c) Calculate the associated vibration modes shapes and
sketch the vibration modes along the wingspan.
Determine the normalized modes using the displacements at mass 𝑚 .
d) Verify the orthogonality of the vibration modes using
the modal matrix.
1
Numerical data:
𝑚1 = 200𝑘𝑔, 𝑚2 = 500𝑘𝑔, 𝐼 = 18,000𝑘𝑔 ∙ 𝑚2 ,
𝑘 = 105 𝑁⁄𝑚.
Figure 1.
2a. Using the Rayleigh method to determine the natural
frequencies of a beam.
For the beam given in the Figure 2a, having constant
bending stiffness 𝐸𝐼 and the mass per unit length 𝑚.
a) Determine the first natural frequency by approximating the bending vibration deflection curve with:
𝑦 = 𝑠𝑖𝑛 (𝜋 );
𝑥
3𝑙
does this equation satisfy the boundary condition at
𝑥 = 2𝑙 ? what is the value of the moment at 𝑥 = 3𝑙 ?
b) Determine the first natural frequency by approximating the bending vibration deflection curve with:
𝑦 = 𝑥 + 𝑎𝑥 2 + 𝑏𝑥 3 + 𝑐𝑥 4
determine the coefficients so that this equation satisfies the boundary conditions at 𝑥 = 0; 𝑥 = 2𝑙 and that the
moment values are zero at 𝑥 = 0, 𝑥 = 3𝑙;
Figure 2a.
2b. Antenna vibration created by air flow around a
travelling vehicle.
It can often be observed that an antenna mounted on a
passenger car starts to vibrate transversely to the direction of travel at a certain driving speed, the vibration
mode being a fundamental bending mode.
An explanation could be that at that driving speed, vortices are created on the sides of the flow, these vortices
being alternatively released to the left and to the right
(Figure 2b) and the antenna placed in an alternating air
flow direction, starts vibrating. The relation between the
frequency 𝑓 at which the vortices detach, and the driving
speed 𝑉, is given by the dimensionless Strouhal number:
𝑆=
𝑓∙𝑏
𝑉
where 𝑏 is a specific dimension of the frontal area, in
this case, the car width.
Determine the driving speed at which the antenna is
brought into resonance, for the case of an antenna having a beam shape of circular cross section. The natural
frequency of the antenna can be determined using Rayleigh's method by considering a displacement function
approximating the fundamental bending shape of a
clamped uniform beam:
𝑦=
1
𝑥 2
𝑥 3
[3 ( ) − ( ) ]
2
𝐿
𝐿
Numerical data:
-Vehicle: - width 𝑏 = 1.4𝑚;
- the specific Strouhal number 𝑆 = 0.3
-Antenna: - length 𝐿 = 0.9𝑚
- diameter 𝑑 = 2.5𝑚𝑚
- circular cross-section 𝐼 =
𝜋∙𝑑 4
64
- density 𝜌 = 7.8 × 10 𝑘𝑔 ∙ 𝑚
3
−3
- Young’s modulus 𝐸 = 2 × 10 𝑁 ∙ 𝑚𝑚
6
Figure 2b.
−2