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MATLAB PROGRAMMING COURSEWORK

OBJECTIVES:

Learn to solve engineering problems using MATLAB

Write Euler and Runge-Kutta initial-value ODE solvers

Write a Shooting Method boundary-value ODE solver

Investigate the properties of the solvers

Summarise your work in a short report

2. Assignment Part 2 – Writing a shooting method BVP ODE solver

A test of the Beagle 3 Mars rover is being conducted (on Earth). The rover is

designed to enter the Mars atmosphere, deploy two parachutes, inflate

airbags and bounce on impact before coming to rest. For this test, the rover

will be dropped from beneath an aircraft and its descent will be filmed, so the

release and landing points must be precise. The rover will be released 10km

above the surface, and the target forward distance from the release point is

1km. Your job is to calculate the required velocity of the aircraft at the point of

release, and you will achieve this using the Shooting Method. Further

information is provided below:

Rover

Mass [kg] 50

Frontal area [m2] 0.5

Drag coefficient 0.7

Pilot parachute

Release time 60

Area [m2] 1.0

Drag coefficient 1.0

Main parachute

Release time 120

Area [m2] 5

Drag coefficient 1.0

Airbags

Inflation height [m] 200

Frontal area [m2] 3.5

Drag coefficient 0.8

Stiffness [N/m] 10000

Damping [Ns/m] 300

Other Air density at sea level [kg/m3] 1.207

Suggested plan of attack:

1. Work out the ODE for this system and convert it to state-space form.

2. Modify your Dz.m function from part 1 to calculate the state derivatives.

3. Modify your odeSolver code to compute and plot the trajectory of the

rover, given initial values for the states. You should mark the points of

parachute release. Run this to make sure it is working before moving

on to the shooting method.

4. Create a new function to use the shooting method to calculate the required

aircraft velocity to hit the target distance. This function should repeatedly

call all of the modified functions you have just created. Check the notes

to see how the shooting method works. Think carefully about how

your new function will work before starting – sketch it out.

Just like any real Engineering problem, you will need to make (and justify)

simplifying assumptions to solve the problem.

Once you have solved the problem – well done! However, this is the

minimum requirement to pass the coursework. Successful completion of

this work, combined with neat code and a good report could get you into 2:1

territory. Access to higher marks is gained by taking the problem further. This

exercise is open-ended – what you choose to explore is entirely up to you.

Here are some suggestions to get you started:

Think about the simplifying assumptions you have made in arriving at

your ODE. How can you make the solution more realistic? For

example, the air density changes with height above the ground. How

does this affect the problem?

What about making the target distance a user input?

You may have discovered that it is easy to make your code fail to reach

a solution. Investigate what causes it to fail and find ways to make it

more robust.

How could you make your code more accurate or efficient?

Write a report of maximum length 4 pages containing:

The derivation of your ODE

A graph of the trajectory, showing the point of parachute deployment

The impact speed

Discussion of simplifying assumptions

Discussion of the further work you have completed

Your commented code included in an appendix (not included in the 4 pages limit)

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# MATLAB Matlab programming using shooting method, Euler and Runge Ketta methods

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