MAST20029工程数学计算

MAST2029工程数学的主题是可持续性，特别关注可持续发展（SD）努力和实现可持续世界的战略。可持续发展的前沿和中心是全球净零排放竞赛。根据国际能源署的数据，如果世界要有机会将全球气温上升限制在1.5摄氏度并避免最严重的气候变化灾难，到2050年实现净零排放将需要全球清洁能源创新和技术的大幅飞跃，到2030年，以清洁能源为主的全球清洁能源的创新和技术将超过目前的三倍，达到4万亿美元。太阳能和风能替代化石燃料。标准已经设定（或者至少估计）。据估计，必须增加必要的投资。本任务探讨了主题2 ODEs系统和主题3拉普拉斯变换中的概念和技术，因为它们与两种类型的人为活动脱碳SD技术有关：（a）能源收集和（b）电动汽车。能量收集（EH）是一种通过从太阳能、风能、热能、振动和电磁波等环境源收集能量来为设备供电的方法。这些技术的出现是为了解决电池供电系统的问题（例如，寿命有限、维护成本高，
The theme for MAST20029 Engineering Mathematics is SUSTAINABILITY, with a specific focus on Sustainable Development (SD) efforts and strategies to achieve a sustainable world.

1. Front and centre

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 sheeps beating heart: see Figure 1 but maybe dont 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_Spri... mper_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]

2. Question

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. (c) Consider each of the following four types of oscillations: A. underdamped 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. (c) Consider the critical points at the origin and (, 0): 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 dont 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 transports 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). (c) Suppose = '( , = 2 and the force () represents an impulse force. Write down the ) 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
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