why study maths?
TRANSCRIPT
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> amc.edu.au/whystudymathsStudy maths and shape your future
Design ships that travel across oceans supporting global
trade, or work on renewable solutions of the future.
Why Study Maths?
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The Australian Maritime College (AMC), a specialist institute of the University of Tasmania, has developed “Why Study Maths?” as an interactive, practical program that demonstrates how maths can be applied to the field of maritime engineering.
There are thousands of jobs around the globe which require knowledge in mathematics. Through mathematics, we learn how to problem solve and this is of great importance in many industries, including engineering. Engineers are curious about the way things work, they like to analyse and solve problems, they are goal oriented and are good at mathematics and science.
The “Why Study Maths?” program has been developed to show you how mathematics can be applied to maritime engineering and give you a snapshot of what your career pathway might look like.
>In this booklet, you will learn:
• How to apply the mathematics you are currently studying to real-life engineering problems.
• About the types of ships, yachts, oil rigs and renewable energy systems you could be designing and building.
AMC is Australia’s national centre for maritime training, education and research. Our degrees are developed in collaboration with industry and government bodies to ensure our course offerings remain relevant to global demands.
Why Study Maths?
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Why Study Maths?
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Survey of Similar High-Speed Patrol Boats
An application of curve fitting
Similar Vessel Study
Beam WL (m)
Leng
th W
L (m
)
50
45
40
35
30
25
204 5 6 7 8 9 10
Draw line of best fit.
a) Is the association positive or negative?
a) Is the association strong, weak, or somewhere in between?
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Length 112m
Depth18m
beam 30m
Engineers often use scale models of things to help them understand how a finished design will work. A model ship, for instance, might cost $3,000 (AUD) to build and $10,000 (AUD) to test - which is cheap when the full size ship might be millions of dollars.But how can we scale a vessel down?
We use scaling laws:
a) If the model shown to the right is 2.5m long, what is the scale ratio, R?
What would be the model’s beam and depth?
If the Wetted Surface Area (WSA) of the full size ship is 2000m2, what will it be for the model?
An application of algebra
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Scaling Laws
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An application of superposition
0 10 20 30 40 50 60 70 80 90 100
-2.5
2.5
0 10 20 30 40 50 60 70 80 90 100
-2.5
2.5
k = wave numberλ = wavelength
a) How can we describe the surface of the ocean?
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Ocean Waves
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b) Why is your answer to the above question only an estimate?
An engineer is sometimes required to graphically determine approximate solutions to problems by simply using a ruler and protractor. You will now attempt to do the same.
a)
An application of graphical analysis
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Rescue at Sea
The Spirit of Tasmania leaves the Mersey River in Devonport and travels N 10o W (10o from the North axis) for 18km, at which point the ship receives a distress call from a sinking yacht. The captain responds to the call and heads 9km N 60o W. The Spirit of Tasmania rescues the yachtsman and travels west for 2km. The captain of the ship is told to bring the yachtsman back to the Mersey River for medical assessment. In order to save time, the captain takes an immediate turn and heads back. By using a ruler, protractor, and setting a scale (i.e. 10mm of drawing equals 2km at sea), determine the total extra distance the Spirit of Tasmania needs to travel (from the moment it responds to the distress call) in order to get back to the Mersey River.
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10
300 2 4 6 8 10 12 14 16 18 20 22 24 26 28
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
wind speed (metres per second)
Prob
abili
ty
Wind speed (m/s)
Pow
er (K
W)
Power curve V90-3.0 MW
In this example, a company is planning to install a wind farm, or collection of wind turbines, 23 nautical miles (42km) off the east coast of Tasmania. The benefit of a wind farm is the power it produces which can be sold.
a) What would some of the costs of the wind farm be?
Power curve: To the right is the power curve for the wind turbine being investigated - the Vestas V90 3MW turbine, provided by Roaring 40’s. Dotted lines coincide with the mean of each wind range.
Wind speeds: This plot shows the average wind speeds over a year at this location.
Feasibility Studies look at the costs and benefits of new projects. Is the project worthwhile? What profit might it make?
An application of probability
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Wind Farm Feasibility Study
3,5003,2503,0002,7502,5002,2502,0001,7501,5001,2501,00750500250
00 5 10 15 20 25
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b) What is the average continuous power produced?
However, power is bought and sold in units of watt-hours (Wh); one Wh is a continuous power of 1 watt for 1 hour.
c) How much power (in Wh) will the turbine produce over one year?
d) If the maintenance boat can only go out if the wind is less than 17.5 knots, what is the probability that it can service the wind turbines on any given day? HINT: 1 knot = 0.51444 m/s.
Tera: 1012
Giga: 109
Mega: 106
Kilo: 103
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0
1
2
3
4
5
6
7
8
9
10
11
0.8
2.0
3.6
5.6
7.8
10.2
12.6
15.1
17.8
20.8
24.2
28.0
0.0
1.4
2.8
4.6
6.7
9.0
11.4
13.8
16.5
19.3
22.5
26.1
-
-
-
-
-
-
-
-
-
-
-
-
1.4
2.8
4.6
6.7
9.0
11.4
13.8
16.5
19.3
22.5
26.1
30.0
0.019
0.048
0.105
0.181
0.212
0.185
0.120
0.076
0.033
0.014
0.006
0.001
Beaufort Number
Wind Speed (m/s)
Probability
RANGEMEAN
Power (kW)
Prob. x Power (kW)
Total: 1.000
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Extra tricky question
375
3000
67.875
18
12
a) What happens close to shore and what does this do to the wave?
Wavelength λ is given by the equation:
where g is acceleration due to gravity, 9.81m/s2, T is the period of the wave (the time it takes for the wave to travel one wavelength), and d is the depth of water.
b) What makes this a difficult equation to solve for λ? How could we get around this?
Wav
elen
gth,
λ, in
met
res
Depth, d, in metres
0
5
10
15
20
25
0 2 4 6 8 10 12 14 16 18 20
Wavelen
gth,
λ, in m
etres
Depth, d, in metres
Often when we see waves close to shore they are bent or curved. Why?
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An application of trigonometry
Wave Refraction
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c) If λ1 is 25m, and α1 is 20° and the depth changes from 20m to 0.5m, what are the values of λ2 and α2?
a) The ocean floor does not usually have steps in it; it gradually slopes. What would that do to the wave?
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How does the speed (celerity) change?
Deep - 20m
Shallow - 0.5m
h
α= Angle of wave, C = Celerity (speed)
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e) This time, there are three different depths: d1 = 20m, d2 = 2m, d3 = 0.5m Determine the subsequent angles and wavelengths.
Extra tricky question
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In maritime engineering it is important to understand how long a ship will be at sea for.
a) Why is this important for engineers to know?
b) A vessel is travelling in a river, sailing in a direction directly opposing the tide. The tide is travelling at a velocity of 1.1 knots. If the vessel’s speedometer records it as having a speed of 30km per hour, disregarding tidal velocity, how fast would the vessel theoretically travel in the tide? (Give your answer in km per hour)
NOTE: 1knot = 0.51444 metres per second 1km per hour = metres per second
c) If the vessel has to travel 200m in order to leave the river, how long will it take against the tide?
d) Once the vessel reaches the ocean it is exposed to an ocean current acting at 30 degrees to the heading of the boat, at a velocity of 0.8 m/s. At what velocity is the vessel being pushed off course, i.e. sway velocity?
HINT: This is the current velocity perpendicular to the vessel heading
An application of physical science
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Tidal Velocities
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It is important for Naval Architects to calculate the resistance of ships and floating structures.
a) Why is this important for Naval Architects to know? In order to calculate the resistance, the Naval Architect must first calculate the volumetric displacement (s) measured in m3. An important formula Naval Architects use to accomplish this is r=sρ, where r is the mass displacement in kg and ρ is the density in kg/m3.
b) The mass displacement of a motorised floating barge is 2000 tonnes, and the dimensions of the underwater portion of this barge are:
• Length (L) = 50m
• Breadth (B) = 25m
• Draft (d) = 2m
Calculate the volumetric displacement of the barge. Show that the units are in m3. HINT: Recall that the density of sea water is 1025kg per cubic metre.
c) After attaining the volumetric displacement the Naval Architect needs to calculate the block coefficient (CB), given by For a given displacement and installed power, the lower the block coefficient, the higher the speed of the vessel.
Would you expect this barge to be able to move fast for a given power? NOTE: Values of CB vary from about 0.35 for a high-speed ferry to about 0.85 for a low-speed cargo ship.
d) How could the barge be made faster?
An application of resistance
CB = sL x B x d
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Block Coefficient
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Students studying their second year of a Bachelor of Engineering degree at the Australian Maritime College at the University of Tasmania, are required to design a bridge made from commercially-bought pasta, and modify it to withstand a large external force.
A common pasta bridge design resembles an arch.
a) Considering the arch to be a parabola, develop a mathematical equation which represents the arch if the bridge legs are 1m apart and the bridge has a maximum height of 400mm (such as the pasta bridge image below).
b) Now, consider the arch to be a half-circle with a maximum height of 500mm. What length of cannelloni pasta will be required to form the arch?
An application of parabola
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Pasta Bridge Design
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Certain large cargo tankers have a number of ballast tanks which are used to take in or let out water, depending on the vessel’s hydrostatic requirements. The aft peak ballast tank of a double hull TI-class supertanker has a length of 70m, width of 8m, and height of 3m.
a) When full, how many litres of water will fit inside the aft peak ballast tank?
NOTE: One cubic metre of volume can hold approximately 1000 litres of water, i.e. 1m3 = 1000L.
b) The density of the water is 1025 kilograms per cubic metre. Prove that the mass of water in the ballast tank is 1,722,000kg.
HINT: , where ρ is the density in kg/m3, m is the mass in kg, and V is the volume in m3.
c) Excluding the mass of ballast, the TI-class supertanker has a mass of 509,000 tonnes. What is the mass of the supertanker after it takes on ballast water? NOTE: 1 tonne = 1000kg.
d) Given that the supertanker propellers provide a thrust force of 100,000 Newtons, calculate the acceleration of the vessel after it has taken on ballast.
HINT: Recall Newton’s laws of Motion.
e) The ballast pumps of the TI-class supertanker are capable of pumping ballast water back into the ocean at a flow rate of 3000 cubic metres per hour. How long will it take to empty the ballast tank at this rate? (Give your answer in minutes and seconds)
An application of volumes
ρ = mV
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Ballast Tanks
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An application of differentiation
In a following sea the waves are travelling in the same direction as the vessel. How would a following sea change the vessel’s velocity?
In this example, the Bluefin is travelling at a mean speed of 6m/s in a following sea.
AMCAMCAMC
AMC
AMC AMC
AMC
t = 0s
gg
400
350
300
250
200
150
100
50
0
Dis
pla
cem
ent
(m)
Time (s)
The Bluefin’s position
Using motion capture equipment, the position of the Bluefin model (pictured right) is recorded as it travels through following seas in AMC’s Model Test Basin.
Horizontal displacement
The graph of the horizontal displacement (the distance travelled) in full scale is shown to the right:
Vessel Speed in Waves
HINT:
AMCAMCAMC
AMC
AMC AMC
AMC
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AMCAMCAMC
AMC
AMC AMC
AMC
a) This data matches the curve: where y is the distance travelled, and t is time in seconds. We can differentiate the displacement which gives us the change in distance over time, :
What is this called?
a) What is the second derivative?
What is this called?
If we arrange all of this data in a graph, we can see how it corresponds to the movement of the vessel:
AMCAMCAMC
AMC
AMC AMC
AMC
displacement
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400
350
300
250
200
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50
0
Dis
pla
cem
ent
(m)
0 2 4 6 8 10 12 14 16 18 20
Time (s)
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An application of integration
Many hydrostatics calculations involve finding the area of the waterplane - the shape of the hull at the waterline.
Shown below is the lines plan for a Wigley hull:
This hull is a theoretical hull used for research. The hull shape is defined by an equation at any point on the hull, , given by:
where:
and B, L and D are the beam, length and draft of the hull. Substituting gives:
a) How can we describe the waterline?
If the beam B = 0.304m and the length L = 3.048m, and letting ƶ = 0 gives
The graph of y is shown below:
Hydrostatics is the study of the behaviour of a vessel: will it float or capsize?
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-1
-0.5
0.5
1
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Ship Hydrostatics
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b) We can integrate this function to find the purple area in the graph below:
c) Is your answer correct? How can you check it?
d) How else could we find the area of a waterplane?
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-1
-0.5
0.5
1
0.152
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Bring mathematics to life in our world-class facilities.
Our students enjoy access to the southern hemisphere’s most advanced collection of maritime facilities including: ship simulators, a fleet of training vessels, towing tank, model test basin and unique cavitation laboratory.
>Towing TankAs the largest and only commercially operating facility of its type within Australasia, the Towing Tank was commissioned in the mid 1980’s and has since undertaken tests on over 500 models of ships and other ocean structures.
>Model Test BasinThe Model Test Basin is a state-of-the-art facility used by Engineering students and national research organisations. The Model Test Basin is used to conduct hydrodynamic experiments to simulate maritime operations within shallow water environments.
>BluefinAMC’s 35m flagship training vessel, Bluefin, cruises Australian waters with up to 25 students and staff on board for training voyages. Studies include habitat monitoring, fish sampling, fishing technology, machinery operation and maintenance, environmental assessment, oceanographic instrument mooring, and ship design and function.
AMC's Facilities
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AMC's Facilities
>Cavitation Research LaboratoryThis research laboratory is unique in Australia and one of the few experimental laboratories in the world. The laboratory is used to test hydrodynamic behaviour of submerged structures such as submarines and ship hulls.
>Survival CentreCombining a heated pool and mock ship’s superstructure, this facility is blacked out for simulated night exercises, and can also create water turbulence, rain, wind noise and simulated storm effects.
>Centre for Maritime SimulationsThis state-of-the art facility offers real-time maritime simulation technology that includes a full-mission ship’s bridge, a tug simulator and six ship operations bridges. It is used for research and investigation into port development, ship manoeuvring, and improving ship and port safety and efficiency.
>AUVAMC’s AUV is an underwater vehicle that is programmed to survey the ocean’s depths, and collects necessary data on research missions.
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Each of our Bachelor of Engineering specialisations are four year degrees, including honours. Our three engineering specialisations include:
• Naval Architecture
• Ocean Engineering
• Marine and Offshore Engineering
>Naval ArchitectureCan you picture yourself designing luxury yachts and big cruiseliners, or working on the design and construction of submarines?
There are many design and construction fields for a naval architect to work within including: high-speed craft, leisure craft, sailing and power craft, super yachts, destroyers and patrol boats for the defence industry, underwater vehicles and submarines.
>Ocean EngineeringYou would be designing and managing the installation of offshore, subsea and coastal structures for the oil and gas industry, renewable energy industry and also consultancy firms specialising in coastal engineering, underwater vehicles, and port and harbour design.
>Marine & Offshore EngineeringMarine systems focuses on the selection, deployment and commissioning of machinery, mechanical and electrical systems and operational systems designed to support the ship and underwater vehicle industry. As with marine systems, offshore systems support the offshore oil and gas industry.
You could be working within the defence industry, oil and gas industry or the alternative energy and the power generation sectors, both in Australia and internationally.
Study Maritime EngineeringBachleor of Engineering (Specialisation) with Honours
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This program allows you to combine your studies with practical experience in your chosen specialisation.
Working under the supervision of professional engineers, the program allows you to evaluate your career choice and gain industry experience. You will alternate periods of full-time study with periods of full-time paid work experience but you must maintain a credit average throughout your degree to remain in the program.
Further information
For more information, please visit: amc.edu.au/whystudymaths amc.edu.au/study
Or email: [email protected]
For more information on courses at the University of Tasmania, please visit: utas.edu.au/courses
To find out if you are eligible for a scholarship, please visit: utas.edu.au/scholarships
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