Coursework 1 - ENGF0003
Transcript of Coursework 1 - ENGF0003
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Mathematical Modelling and Analysis I
Coursework 1
Release Date: 08 October 2020
Submission Deadline: 11 November 2020
Estimated Coursework Return: 20 working days
Topics Covered: Topics 1 β 5
Expected Time on Task: 12 hours
This coursework counts towards 20% of your final ENGF0003 grades and
is made up of three questions, totalling 100 marks.
LONDONβS GLOBAL UNIVERSITY
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Guidance for Submissions
I. Submit a single Word document with questions in ascending order.
II. Insert any relevant graphs or figures and provide a description to any figures or tables in your document.
III. Explain in detail your reasoning for every mathematical step taken.
IV. Do not write down your name, student number, or any information that might identify you in any part of the coursework. Your coursework will be marked anonymously.
V. Do not copy and paste the coursework questions into your solution. Simply re-write information when necessary.
VI. If you use any MATLAB code to solve questions, annex those in an Appendix at the end of your document. Whenever showing results from your codes, refer to the page in the Appendix where that code is located.
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Model 1: Elasticity [30 Marks]
The vibration of the atoms or molecules of a solid can be modelled by a
lattice containing atoms or molecules that interact with each other through
spring-like forces. Theoretically, a spring is a type of mechanical link which
is assumed to have negligible mass and damping. In practice, any
elastic/deformable body or member, such as a cable, bar, beam, shaft, or
plate, can be treated as a spring in an engineering model.
Figure 1. Schematic of a two-particle spring system.
The modelling of elasticity in multi-component engineering systems relies
on a basic understanding of the individual deformations on each member,
the interactions of the contributions of each component, and how these
combine to form an overall effect on the system. In Figure 1, two masses
are connected by springs of elasticity π!, π" and π#. These springs have
natural lengths π!, π" and π# [m] and are fixed to the walls separated by a
distance πΏ [m].
Hookeβs law is the simplest and most common model of elasticity in
mechanics. It states that, provided the proportionality limit is not exceeded,
!! !" !#!!!" !! !#
"""!
#
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there is a linear relationship between the force πΉ [N] necessary to cause a
displacement Ξπ₯ [m] in a spring of stiffness π:
πΉ = πΞπ₯, (E. 1)
where Ξπ₯ = π₯ β π is the deformation on the spring, π is the natural length of
the spring and π₯ the position of a certain reference point. The tension [N]
on springs 1, 2 and 3 in Figure 1 can be evaluated following Hookeβs law
as:
π! = π!(π₯! β π!)
π" = π"(π₯" β π₯! β π")
π# = π#(πΏ β π₯" β π#)
(E. 2)
a) [5%] Use dimensional analysis to derive the dimensions of π in E.1.
b) [35%] Assuming that the particles are at rest, π! = π" and π" = π#.
Find the matrix equation that relates the particle displacements π₯!
and π₯" to the spring elasticities π$ and the spring natural lengths π$
for π β {1, 2, 3}. Find π₯! and π₯" for the simplest situation, which is π! =
π" = π# and π! = π" = π#.
c) [30%] Develop the model in question b so that it can represent the
equivalent matrix model for three particles connected by four
springs. Assuming that the particles are at rest, we can say that π! =
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π", π" = π#, and π# = π%. Solve this system for the simplest case,
where π! = π" = π# = π% and π! = π" = π# = π%.
d) [30%] A system of five particles and six springs is such that the spring
constants are π! = 10, π" = 15, π# = 9, π% = 6, π& = 12 and π' = 19
N.m-1, πΏ = 1.5 m and π! = π" = β― = π& = π' = 0.13 m. Assuming that
this system is at rest, you are required to calculate the particle
positions π₯!, π₯", π₯#, π₯% and π₯&.
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Model 2: Stability [40 Marks]
The 1940 Tacoma Narrows Bridge [Video Resource] was a suspension
bridge measuring 1,810.2 m in length built in the U.S. state of Washington.
From the moment the deck was built, the bridge would oscillate vertically
even in mild wind conditions, as shown in Figure 3. On the morning of 7
November 1940, 4 months after its opening to traffic, the deck of the bridge
started to oscillate in a twisting motion that increased gradually until the
deck collapsed.
Figure 2. Left: Torsional motion on the bridge deck. Right: longitudinal motion on the bridge deck.
This was an example of a dynamically unstable system undergoing self-
excited structural motion. A system is said to be dynamically stable if the
amplitude of its natural oscillations decreases or remains steady with time.
If the amplitude of natural oscillatory displacement increases continuously
with time, the system is said to be dynamically unstable. The
aerodynamically induced alternating motion in the Tacoma Bridge induced
further oscillation in the deck: a snowball process which caused oscillations
to grow uncontrollably. In other words, the bridge behaved as if it was a
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shock-absorber system in a car that would amplify shocks resulting from
irregularities on the road rather than absorbing them.
The phenomenon of self-exciting vibration can be modelled with the same
mathematical tools used to analyse the instability of rotating shafts, the
response of resonant electrical circuits, the stability of controllers, the
flutter of turbine blades, the flow-induced vibration of pipes and the
automobile wheel shimmy. These models show how the systemβs
components act towards absorbing energy and limiting its motion. Models
of damping are a natural progression from models of elasticity. While
elastic components conserve the kinetic energy in a system, (viscous)
damping components dissipate mechanical energy mostly by turning it into
heat.
The vibrational displacement in a system at rest is given by:
π₯(π‘) = πΆ! exp *+βπ + /π" β 11ππ‘3 + πΆ" exp *+βπ β /π" β 11ππ‘3, (E. 3)
where π = <π/π is the natural frequency of the system, that depends on
its mass π [kg] and elasticity π[N.m-1]. π = (")*
is the ratio between the
damping constant π and the inertial and elastic constants of the system. π‘
[s] is time, and πΆ! and πΆ" are constants.
a) [5%] Knowing that π is dimensionless, use dimensional analysis to
find the dimensions of the damping constant π.
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b) [10%] Find an expression for the velocity of vibrational displacement
οΏ½ΜοΏ½(π‘) = +,+-
.
c) [10%] Knowing that π₯(π‘ = 0) = π₯. and οΏ½ΜοΏ½(π‘ = 0) = οΏ½ΜοΏ½., find an
expression for the constants πΆ! and πΆ" for the cases where: (i) The
system is overdamped (ΞΆ > 1), and (ii) The system is critically damped
(π = 1).
When π < 1, the system is said to be underdamped. In this case, E.3 takes
the complex form
π₯(π‘) = πΆ! exp *+βπ + π/1 β ΞΆ"1ππ‘3 + πΆ" exp *+βπ β π/1 β ΞΆ"1ππ‘3. (E. 4)
d) [15%] Show that E.4 can be expressed as
π₯(π‘) = exp(βπππ‘) 8πΆ!# cos *+/1 β π"1ππ‘3 + πΆ"# sin *+/1 β π"1ππ‘3>, (E. 5)
where πΆ!/ = πΆ! + πΆ" and πΆ"/ = π(πΆ! β πΆ").
e) [10%] Knowing that π₯(π‘ = 0) = π₯. and οΏ½ΜοΏ½(π‘ = 0) = π₯.Μ, find an
expression for πΆβ²! and πΆβ²" and express E.5 in a fully explicit form.
f) [25%] Assuming π = 4π radian.s-1, π₯. = 1 m and οΏ½ΜοΏ½. = 1 m.s-1, plot the
response π₯(π‘) for the following values of π: 1.5, 1, 0.5, 0 and -0.5.
Based on your results, describe the effect of the constant π on the
stability of the system.
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g) [25%] The conjugate frequencies of damped vibration π ! =
Lβπ + π<1 β ΞΆ"Mπ and π " = Lβπ β π<1 β ΞΆ"Mπ can show the systems
locus of stability on the complex plane when calculated for β1 β€ π β€
1. Plot an Argand diagram showing how π affects the position of π !
and π " in the complex plane. Indicate in the complex plane the
regions of stability and instability of the system. Knowing that the
vibration of the 1940 Tacoma Bridge was self-excited and grew
unbounded, in which region of this diagram do you believe the
response could be placed?
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Model 3: Flow [30 Marks]
Figure 3 displays a conical container filled with water with an outlet at the
bottom. By assuming that water is an incompressible fluid with zero
viscosity, we can use Bernoulliβs equation to obtain an expression for the
height of water in the container as a function of time β(π‘).
Figure 3. A conical water container of height β. and apex angle π β 2π.
Both the height β(π‘) and radius π(π‘) of water in the container are
functions of time.
This function is given as:
β(π‘) = Rβ.&" β
5π0<2π
2 tan" Xπ2 β πYπ‘Z
"&
(E. 6)
where π [m.s-2] is the acceleration due to gravity and π‘ [s] is the time. β. [m]
is the height of water in the container at π‘ = 0 and π [m] is the radius of the
β(#)
%(#)
β&
'
Outlet
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containers outlet. In the derivation of E. 6, we assume that π βͺ π(π‘), and
that π(π‘)/β(π‘) β tan(1"β π) for all possible values of π‘.
a) [10%] If all quantities are expressed in SI units, find the value of π in
E.6.
b) [10%] Create a mathematical model of the time-dependent volume of
water in the containerπ(π‘) in terms of the height of water β(π‘).
c) [10%] Find an expression for the time π that it takes for the container
to empty. Check the validity of your expression by calculating π for
π = 1 cm and β. = 30 cm and plotting β(π‘) from 0 < π‘ < π.
d) [20%] Express β(π‘) in terms of π, then find an expression for βΜ(π‘) =+2+-
.
e) [50%] Find the volumetric flow rate οΏ½ΜοΏ½(π‘) = +3+- [m3.s-1] at which water
leaves the container. Discuss how the design parameters π, β. and
π affect the outflow in this container.