MAST20029

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The theme for MAST20029 Engineering Mathematics is SUSTAINABILITY, with a specific focus on Sustainable Development (SD) – efforts and strategies to achieve a sustainable world. Front and centre of SD is the global race to Net Zero.

According to the International Energy Agency, achieving Net Zero by 2050 would require a massive leap in clean energy innovation and technologies worldwide, dominated by

clean energy exceed triple the current value – to $4 trillion by 2030 – if the world is to have a fighting chance to limit the global temperature

rise to 1.5 °C and avert the worst of climate change disasters.

solar and

wind renewables in place of fossil fuels. The bar has been set (or at least estimated). Estimates

are that the necessary rise in

investment must

This Assignment explores concepts and techniques in Topic 2 – Systems of ODEs and Topic 3 Laplace Transform, as they relate to two types of SD technologies to decarbonize anthropogenic activities: (a) energy harvesting and (b) electric cars.

Energy harvesting (EH) is the method of powering devices by harvesting energy from environmental sources such as solar, wind, thermal, vibrations and electomagnetic waves. These technologies have emerged to resolve the issues with battery powered systems (e.g., limited lifespan, high maintenance, not to mention batteries are toxic, flammable and heavy polluters if not properly disposed). EH find broad applications in autonomous and self- powered devices including: medical implants (e.g., pacemakers), wearable electronics (e.g., wristwatches), automotive components, and environmental monitoring especially in remote to inaccessible sites.

So what is the basic idea behind EH and how does this relate to Eng Maths? In Calculus 2, you learned that many natural phenomena are oscillatory (eg your heart beating, your walk etc.) and can be modelled using a mass-spring-damper system.

Now one can harvest the mechanical (kinetic) energy accumulated from this oscillatory motion and turn it into useful electrical energy. For example, Zurbuchen et al (2013) show how this is done using a sheep’s beating heart: see Figure 1 but maybe don’t look if you are squeamish at the sight of blood. Instead google their paper and see the maths there which should all be familiar from Calculus 2!

Figure 1: From Zurbuchen et al (2013) “Energy harvesting from the beating heart by a mass imbalance

oscillation generator.” Ann Biomed Eng. (left) The evolution of the mechanical energy E for a damped mass-

spring oscillator: https://commons.wikimedia.org/wiki/File:Energy_of_a_Mass_Spring_Damper_System (right)

[NOTE: IN WHAT FOLLOWS, ALL PARTS HIGHLIGHTED LIKE THIS IN BLUE TEXT REFER TO TASKS THAT ARE NOT MARKED BUT ARE ESSENTIAL PRACTICE FOR MASTERY OF THE

TOPIC & EXAM PREPARATION]

Question (1)

From first year Calculus, you learned that the oscillations of a mass-spring-damper system is governed by a single 2nd order ODE with constant coefficients for ��(��):

��̈ + ��̇ + �� = 0, (1)

[Part (a), (b) i-ii are revision from first year Calculus and thus have no allocated marks. You are expected to do these to gain the marks from the rest of the question, since these prerequisite skills underpin Nonlinear Systems in Topic 2. Check your own work on these parts later when

the solutions and feedback are made available on LMS.]

where

(a) Express the system of coupled first order ODEs

�� = �� + ��, �� = �� + �� (2) ��

in the form of equation (1) for ��(��). Confirm that you get the same ODE for ��(��).

(b) Consider the system in Exercise 8 pg 2.25 of Notes.

i. Find the corresponding second order ODE in the form (1) and solve it using the method you learned in Calculus 2.

ii. Classify the oscillation into one of the following types: underdamped, critically damped, overdamped.

iii. Use MATLAB to plot the oscillatory displacement ��(��) versus �� from part i.

B. critically damped

C. overdamped

D. simple harmonic motion

Identify the stability and type of critical point (0,0) that represents each type of oscillation and match it the relevant dynamical phase portrait(s) below.

��(��) = ��̇. ��

Question (2)

We consider a simplified version of an oscillating system, namely, the pendulum in Graves et al (2022) and Graves and Zhu (2022). In particular, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length L is

�� = ��, �� = −��

where �� = �� is the angular position, �� = !" is the angular velocity, and �� is the acceleration

due to gravity (see Figure 2 (left)).

(a) Determine all of the critical points for the system.

(b) Determine the linearised system for each critical point in part (a), and find the corresponding eigenvalues of the general solution.

i. State the expected type and stability of each critical point, given the

eigenvalues of the linearized system

ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system

(d) Set $ = 1 and use MATLAB or any other plotter to sketch the global phase portrait %

of the non-linear system in a region that includes the critical points at the origin and (��, 0). Given initial values (��&, ��&) = @��(0), ��(0)A, provide a physical interpretation of the orbits by describing the corresponding oscillations for the case:

i. where ��& and ��& are both small;

ii. where ��& and ��& are both just large enough to almost reach the top;

iii. where ��& and ��& are large enough to swing over the top.

Clearly relate your interpretation to the orbits in the phase portrait.

Figure 2: Schematic of simplified pendulum (left). Images from Graves et al (2022) “Pendulum energy harvester with torsion spring mechanical energy storage regulator” Sensors and Actuators A: Physical and Graves & Zhu (2022) “Design and experimental validation of a pendulum energy harvester with string-driven single clutch mechanical motion rectifier” Sensors and Actuators A: Physical (middle; right).

Question (3)

If you don’t like blood covered living organs, then this second SD technology may be more palatable you. It is about electric cars. There are many reasons why we should care about this. First the transportation sector is responsible for at least 20% of global greenhouse gas (GHG) emissions. One way to reduce transport’s CO2 emissions is by switching to electric cars. Not only are they better for the environment, they are also better for our health. Studies have shown that in 2015 alone, vehicle exhaust emissions were responsible for ~385,000 premature deaths worldwide costing to the tune of US$1 trillion.

Figure 3: Australia's first 100% electric small SUV https://www.whichelectriccar.com.au/hyundai/the-new- hyundai-kona-electric-long-range-zero-emissions-suv-has-arrived-in-australia/(left); check out the phase portrait art on this car https://www.drive.com.au/news/2022-volkswagen-id-5-gtx-teased/ (right)

The following dynamical systems model was recently proposed by Garcia and Redondo (2022) “Dynamical systems approach in automobiles technological transition from environmental drivers” to represent the technological transition of automobiles from conventional (��) to electric (��):

�� = �� − ��, �� = −�� + ��

The change over time of the number of automobiles of a given type is modelled as the difference between the automobiles of that type entering the market and coming out of the market per year. Phase portraits involving a saddle-node at ( ��& , ��& ), ��& > 0, ��& > 0 were analysed to better understand scenarios where both technologies grow initially but with one type growing much faster than the other (e.g., electric cars dominating over conventional cars) with time.

(a) Solve the linear system below using Laplace Transforms

�� = −�� + ��, �� = �� − ��

for some real constant ��, and ��(0) = 0, ��(0) = 1.

(b) Find the range of values of �� where (��, ��)=(0,0) is an unstable saddle.

(c) If �� = 0.1, what initial conditions ��(0) > ��&, ��(0) > ��& near (��&, ��&) will guarantee a faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the phase portrait!]

Question (4)

Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement of a mass-spring oscillator resting on a frictionless surface,

governed by the ODE

��̈ + �� = ��(��) .... (1)

(a) Suppose �� = �� = 1, and the force ��(��) represents a push to the left of the mass over a time period from �� = 2�� to �� = 2�� + �� for some �� > 2��. Let ��(��) be

��(��)= −[��(��−2��)−��@��−(2��+��)A]

such that a small �� corresponds to a push of short duration, while a large �� is a push of

long duration. The function ��(��) is the unit step function. Use Laplace Transform to solve the initial value problem

��(0) = 1, ��̇(0) = 0.

(b) Use MATLAB to plot the solution in part (a).

governing equations for the displacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and then struck by 4 force units at �� = 2�� time units later. [Solve the governing equation for practice!]

Consider now the more realistic scenario of a frictional surface, so that the displacement of the mass-spring oscillator is damped.

(d) Find an expression for the underdamped displacement ��(��) in terms of a convolution integral for some real constants ��& and ��' given

��̈+2��̇+2��=��(��), ��(0)=��&, ��̇(0)=��'.

Other oscillating systems (e.g., circulating-fuel reactors) are governed by integro-differential

equations like that below, with ��(��) being the unit step function �� #

��+��+��O[��(��)−��(��−��)]�� =��(��−��). &

(e) Find��(��), given��(0)=0, ��=4, ��=��and��=1. END OF ASSIGNMENT 2