Mech2700 assignment 1 2018 PDF

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The University of Queensland School of Mechanical & Mining Engineering

MECH2700 Engineering Analysis I (2018) Assignment 1: Computational Mechanics

1

Introduction

On March 24, 2018 “Mad” Mike Hughes launched himself in the steam powered rocket shown in Figure 1 to a reported height of 572m and maximum speed of 563km/hr, landing 460m away from the launch point.

Figure 1: “Mad” Mike Hughes in his steam powered rocket. See https://www.gizmodo.com.au/2018/03/at-long-last-flat-earth-rocketeer-finally-manages-to-blast-himselfinto-sky-at-god-knows-what-speed/ This assignment requires you to simulate the trajectory of an idealised steam powered rocket. Given the underlying physical conditions, you will write a Python program that describes the state of the rocket throughout its flight.

2 2.1

Theory Rocket Propulsion

The principle of operation of a simple steam rocket is that water is heated within a pressure vessel until its saturated vapour pressure is significantly higher than ambient pressure. The hot water is thenpushed out of the pressure vessel by the high pressure steam and through an expanding nozzle. As the pressure acting on the water drops below the vapour pressure, it flashes to steam significantly increasing its volume and necessitating a corresponding increase in speed. This change in momentum provides the thrust generated by the rocket. A schematic of an idealised simple steam powered rocket is shown in Figure 2 where state S0 is steam, S1 is hot water, S2 is the throat of the nozzle where the water flashes to steam, and S3 is the nozzle exit 1

where pressure is assumed equal to atmospheric.

S0

S1

S2

S3 Figure 2: Steam rocket schematic Assuming incompressible flow for the hot water the mass flow rate through the nozzle may be calculated using Bernoulli’s equation. Problem 1. Show that the mass flow rate of water through the nozzle is given by v u 2ρ (p − p ) 1 3 u q = A2t  2 , A2 1 − A1 where q is the mass flow rate, ρ is the density, p is the pressure, and A is the cross sectional area. Assume that p2 = p3 . To satisfy conservation of mass, the steam exiting the nozzle must have the same mass flow rate as that calculated for the hot water above. The velocity of the steam exiting the rocket nozzle, relative to the rocket, may then be calculated according to v3 = q/ρ3 A3 , where the density of the steam, ρ3 , may be obtained from Table 1. The force acting on the rocket due to the steam rocket engine is then Fthrust = qv3 . This thrust force acts through the centre of mass for the rocket and we will assume that it aligns with the direction of travel. We will assume that the pressure within the rocket remains constant until the propellant is expended. However, atmospheric pressure varies with altitude and is modeled according to the standard atmosphere model as shown in Table 2. This will also influence the drag force experienced by the rocket according to 1 Fdrag = ρair v 2 Ar CD , 2 2

Table 1: Saturation pressure and specific volume of water/steam at various temperatures Temperature ◦ C

Saturation pressure kPa

50 100 150 200 250 300 350

12.35 101.3 475.8 1554 3973 8581 16513

Specific volume, m3 /kg liquid vapour 0.001012 0.001044 0.001091 0.001157 0.001251 0.001404 0.001740

12.03 1.6729 0.3928 0.12736 0.05013 0.02167 0.008813

where ρ is the density of air, v is the speed of the rocket, Ar is the projected area of the rocket in the direction of travel, and CD is the dimensionless drag coefficient for the rocket. The drag force acts in the opposite direction to the direction of travel. In your Python program, you will need to make a decision about how to compute atmospheric pressure at altitudes not given explicitly in Table 2. Make suitable assumptions when the altitude is outside the range of Table 2. Table 2: Standard atmosphere model Altitude, m Pressure, kPa Density, kg/m3 0 100 200 300 400 500 600 700 800 900 1000

2.2

101.3 100.1 98.9 97.7 96.6 95.4 94.3 93.2 92.1 91.0 89.9

1.225 1.213 1.202 1.190 1.179 1.167 1.156 1.145 1.134 1.123 1.112

Motion

The continuous motion of an object in the x,y-plane can be described by specifying the object’s position (x, y) and velocity (vx , vy ) as functions of time, t. If we know the forces (Fx , Fy ) acting on the object, then we can compute the state of motion for all times by integrating the equations of motion, shown here in Cartesian coordinates, max =

X

Fx ,

may =

3

X

Fy .

The object is assumed to have a mass, m, concentrated at its centroid and the components of acceleration are defined as

ax =

d2 x , dt2

ay =

d2 y . dt2

Note that in the case of a rocket, where propellant is expended throughout its flight, the mass will be a function of time and thus the acceleration will be also. In your Applied Mechanics course in first year, you usually had a simple functional form for the force components and integrated the equations of motion analytically to obtain a closed- form solution. For most situations that you encounter in your engineering practice this analytic approach will not be feasible and you will have to resort to some form of numerical scheme to integrate the equations of motion approximately. This is the basis of nearly all simulation packages. In this assignment, you will develop a Python program to compute a discrete approx- imation to the trajectory of a steam powered rocket. Exercise. Read Section 14.6 Numerical Solutions of an old version of Bedford & Fowler’s Engineering Mechanics text. You may have this book from your Applied Mechanics course or find it in the library. In case the numbering is different in your copy of the text, Chapter 14 is labelled “Force, Mass and Acceleration”. Problem 2. Show that the state of motion of the rocket at t = t0 + ∆t (i.e. a short time after some arbitrary time t0 ) can be approximated by 1 ax0 ∆t2 , 2 1 y (t0 + ∆t) = y0 + vy0 ∆t + ay0 ∆t2 , 2 vx (t0 + ∆t) = vx0 + ax0 ∆t, x (t0 + ∆t) = x0 + vx0 ∆t +

vy (t0 + ∆t) = vy0 + ay0 ∆t, where the subscript 0 indicates a value at t = t0 .

3

Simulation

The simulation is to calculate the trajectory of the rocket from the initial condition shown in Table 3 until the rocket impacts the ground at an altitude of zero metres. The simulation will calculate the forces of thrust and drag acting on the rocket and, with the inclusion of gravity, update the state of the rocket (position and velocity) in small increments of time until its terminal moment. The thrust calculation will take into account the changing mass of the rocket as propellant is expended along with nozzle exit pressure variation due to increasing altitude, which will also impact on the drag calculation. Problem 3. Draw a free-body diagram showing the forces acting on the rocket during the powered stage of the trajectory. It will be helpful to consider the forces when the rocket is at some flight path angle of γ. The flight path angle, γ, is measured from horizontal. Problem 4. Write expressions for the acceleration components ax and ay . Write expressions for each stage of the flight: 1. During the boost phase when the rocket is providing thrust. 2. During free flight. 4

Variable mr mw Tw α Ar A1 A2 A3 CD

Table 3: Rocket flight initial conditions Value Description 300 kg 700 kg 300 ◦ C 85 deg 1.767 m2 0.5 m2 0.0028 m2 0.0314 m2 0.3 -

Mass of rocket Mass of water Temperature of water Inclination (from horizontal) Projected area of rocket Cross sectional area of pressure vessel Nozzle throat area Nozzle exit area Drag coefficient

Problem 5. Write a Python program that computes the trajectory of the rocket from launch site to ground impact. The program should start with the initial condition provided in Table 3 and integrate the equations of motion with small increments in time while saving the rocket position, velocity, and acceleration at each time increment. Problem 6. Calculate, by hand, the speed of the rocket at the point in time when all of the propellant is expended. For this hand calculation, neglect drag and assume that the thrust at lift-off is constant over the flight (until the water is all used up). Also, assume the inclination angle is 90◦ to simplify the analysis. Using these same assumptions, verify that your program can duplicate this result. Problem 7. Determine a time step that gives a sufficiently accurate estimate of the trajectory for the case of nonzero aerodynamic drag. Use the time of flight, maximum altitude, and distance down range as the test criteria. Explain why you consider this to be good enough. What was the maximum altidue and how long did the flight take? Problem 8. Produce plots of the rocket flight: 1. A plot of the rocket trajectory (x vs y) 2. A plot of velocity vs time with time on the x-axis and velocity on the y-axis. This should have a line for each component of velocity. 3. A plot of acceleration vs time. As above but with acceleration on the y-axis. Problem 9. The rocket range is limited in size and so your rocket must land less than 400m from the launch site. You also have somewhat sensitive instrumentation on the rocket that cannot endure more than 10 g of acceleration. By varying the initial temperature and volume of the water in the rocket, what is the maximum altitude that the rocket can reach whilst satisfying these constraints?

4

Submission Notes and Assessment Criteria

This is an individual assignment, however, discussing your formulation and code development with your peers is a useful activity, especially when working out details of how to formulate code from the mathematical model. Submit a report, detailing your analysis, code and results for Problems 1 to 9. This report should be brief but well structured, so that the tutor can easily interpret your results. Include your Python code and produce a single PDF file for your report. It is fine to scan hand-written elements (such as equations and schematic figures) and include them in your report. The plots in Problem 8, however, 5

should be computer-generated and appropriately labelled. As well as the report PDF, you will need to submit the text .py files containing all of your Python code. For assessment of your work, the tutors will be looking for answers to the following questions. • Have you done what the tasks require? Note that your report should be well structured and tidy so that the tutor can easily find your answers. Your plots having correct titles and axes labels is quite helpful. • Are your answers accurate? • Is your code well-structured and clearly written, with suitable documentation comments? • Does your code correspond to the results shown in your report?

4.1

Submission instructions

Your assignment will be submitted through Blackboard. Submit a single PDF file for your report and the Python files used to produce the output.

4.2

PDF format notes

Assignments will be marked through Blackboard. To make this is as error free as possible you must make sure that your submitted PDF contains all of your figures, all of the pages are correctly oriented, all of the pages are in the correct order and any fonts and line styles used in figures are readable. You may scan diagrams and detailed working, but it is your responsibility to make sure what you submit is readable. If it is not legible, it cannot be marked.

4.3

Software submission

Submit the Python files used to generate all of the results (especially the figures) in your report. Leave all of the files as text files with the .py extension. If you have many files, please zip them into a single file for upload to blackboard. Include documentation in a README file so that the marker knows how to generate your results from the code you have provided.

4.4

Report structure

Your submission is to be a brief technical report. You should structure it in the way you feel is most appropriate (a good starting point is the IMRAD Format, see http://en.wikipedia.org/wiki/IMRAD). While you must address all of the problems required in your submission, structuring your submission as a series of headings Problem 1 through Problem N is not a technical report: you must make the effort to synthesise your work for the reader. Your report should have only enough detail such that another engineer would be able to recreate your results from the report provided — or in other words, another engineer should be able to critically evaluate your work using only the material you provide.

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