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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 – 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.

LONDON’S 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 𝑘!, 𝑘" and 𝑘#. These springs have

natural lengths 𝑙!, 𝑙" and 𝑙# [m] and are fixed to the walls separated by a

distance 𝐿 [m].

Hooke’s 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 𝐹 [N] necessary to cause a

displacement Δ𝑥 [m] in a spring of stiffness 𝑘:

𝐹 = 𝑘Δ𝑥, (E. 1)

where Δ𝑥 = 𝑥 − 𝑙 is the deformation on the spring, 𝑙 is the natural length of

the spring and 𝑥 the position of a certain reference point. The tension [N]

on springs 1, 2 and 3 in Figure 1 can be evaluated following Hooke’s law

as:

𝑇! = 𝑘!(𝑥! − 𝑙!)

𝑇" = 𝑘"(𝑥" − 𝑥! − 𝑙")

𝑇# = 𝑘#(𝐿 − 𝑥" − 𝑙#)

(E. 2)

a) [5%] Use dimensional analysis to derive the dimensions of 𝑘 in E.1.

b) [35%] Assuming that the particles are at rest, 𝑇! = 𝑇" and 𝑇" = 𝑇#.

Find the matrix equation that relates the particle displacements 𝑥!

and 𝑥" to the spring elasticities 𝑘$ and the spring natural lengths 𝑙$

for 𝑖 ∈ {1, 2, 3}. Find 𝑥! and 𝑥" for the simplest situation, which is 𝑘! =

𝑘" = 𝑘# and 𝑙! = 𝑙" = 𝑙#.

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 𝑇! =

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𝑇", 𝑇" = 𝑇#, and 𝑇# = 𝑇%. Solve this system for the simplest case,

where 𝑘! = 𝑘" = 𝑘# = 𝑘% and 𝑙! = 𝑙" = 𝑙# = 𝑙%.

d) [30%] A system of five particles and six springs is such that the spring

constants are 𝑘! = 10, 𝑘" = 15, 𝑘# = 9, 𝑘% = 6, 𝑘& = 12 and 𝑘' = 19

N.m-1, 𝐿 = 1.5 m and 𝑙! = 𝑙" = ⋯ = 𝑙& = 𝑙' = 0.13 m. Assuming that

this system is at rest, you are required to calculate the particle

positions 𝑥!, 𝑥", 𝑥#, 𝑥% and 𝑥&.

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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 system’s

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:

𝑥(𝑡) = 𝐶! exp *+−𝜁 + /𝜁" − 11𝜔𝑡3 + 𝐶" exp *+−𝜁 − /𝜁" − 11𝜔𝑡3, (E. 3)

where 𝜔 = <𝑘/𝑚 is the natural frequency of the system, that depends on

its mass 𝑚 [kg] and elasticity 𝑘[N.m-1]. 𝜁 = (")*

is the ratio between the

damping constant 𝑐 and the inertial and elastic constants of the system. 𝑡

[s] is time, and 𝐶! and 𝐶" are constants.

a) [5%] Knowing that 𝜁 is dimensionless, use dimensional analysis to

find the dimensions of the damping constant 𝑐.

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b) [10%] Find an expression for the velocity of vibrational displacement

�̇�(𝑡) = +,+-

.

c) [10%] Knowing that 𝑥(𝑡 = 0) = 𝑥. and �̇�(𝑡 = 0) = �̇�., find an

expression for the constants 𝐶! and 𝐶" for the cases where: (i) The

system is overdamped (ζ > 1), and (ii) The system is critically damped

(𝜁 = 1).

When 𝜁 < 1, the system is said to be underdamped. In this case, E.3 takes

the complex form

𝑥(𝑡) = 𝐶! exp *+−𝜁 + 𝑖/1 − ζ"1𝜔𝑡3 + 𝐶" exp *+−𝜁 − 𝑖/1 − ζ"1𝜔𝑡3. (E. 4)

d) [15%] Show that E.4 can be expressed as

𝑥(𝑡) = exp(−𝜁𝜔𝑡) 8𝐶!# cos *+/1 − 𝜁"1𝜔𝑡3 + 𝐶"# sin *+/1 − 𝜁"1𝜔𝑡3>, (E. 5)

where 𝐶!/ = 𝐶! + 𝐶" and 𝐶"/ = 𝑖(𝐶! − 𝐶").

e) [10%] Knowing that 𝑥(𝑡 = 0) = 𝑥. and �̇�(𝑡 = 0) = 𝑥.̇, find an

expression for 𝐶′! and 𝐶′" and express E.5 in a fully explicit form.

f) [25%] Assuming 𝜔 = 4𝜋 radian.s-1, 𝑥. = 1 m and �̇�. = 1 m.s-1, plot the

response 𝑥(𝑡) for the following values of 𝜁: 1.5, 1, 0.5, 0 and -0.5.

Based on your results, describe the effect of the constant 𝜁 on the

stability of the system.

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g) [25%] The conjugate frequencies of damped vibration 𝑠! =

L−𝜁 + 𝑖<1 − ζ"M𝜔 and 𝑠" = L−𝜁 − 𝑖<1 − ζ"M𝜔 can show the systems

locus of stability on the complex plane when calculated for −1 ≤ 𝜁 ≤

1. Plot an Argand diagram showing how 𝜁 affects the position of 𝑠!

and 𝑠" 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 Bernoulli’s equation to obtain an expression for the

height of water in the container as a function of time ℎ(𝑡).

Figure 3. A conical water container of height ℎ. and apex angle 𝜋 − 2𝜃.

Both the height ℎ(𝑡) and radius 𝑟(𝑡) of water in the container are

functions of time.

This function is given as:

ℎ(𝑡) = Rℎ.&" −

5𝑎0<2𝑔

2 tan" X𝜋2 − 𝜃Y𝑡Z

"&

(E. 6)

where 𝑔 [m.s-2] is the acceleration due to gravity and 𝑡 [s] is the time. ℎ. [m]

is the height of water in the container at 𝑡 = 0 and 𝑎 [m] is the radius of the

ℎ(#)

%(#)

ℎ&

'

Outlet

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containers outlet. In the derivation of E. 6, we assume that 𝑎 ≪ 𝑟(𝑡), and

that 𝑟(𝑡)/ℎ(𝑡) ≈ tan(1"− 𝜃) for all possible values of 𝑡.

a) [10%] If all quantities are expressed in SI units, find the value of 𝜙 in

E.6.

b) [10%] Create a mathematical model of the time-dependent volume of

water in the container𝑉(𝑡) in terms of the height of water ℎ(𝑡).

c) [10%] Find an expression for the time 𝜏 that it takes for the container

to empty. Check the validity of your expression by calculating 𝜏 for

𝑎 = 1 cm and ℎ. = 30 cm and plotting ℎ(𝑡) from 0 < 𝑡 < 𝜏.

d) [20%] Express ℎ(𝑡) in terms of 𝜏, then find an expression for ℎ̇(𝑡) =+2+-

.

e) [50%] Find the volumetric flow rate �̇�(𝑡) = +3+- [m3.s-1] at which water

leaves the container. Discuss how the design parameters 𝜃, ℎ. and

𝑎 affect the outflow in this container.