Further Maths and Simulation (SC2153)
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
Assignment submission date: 28th February 2018
Assignment feedback: 21st March 2018
You must attempt all questions and show all working.
Please state if you are rounding numbers, and include units where applicable.
1: The admittance, 𝑌, in a circuit is given by
𝑌 =
1
𝑍
where 𝑍 is the impedance of the circuit. If Z = 150 − j22, find 𝑌.
5 marks
2: By applying Kirchoff’s laws on a circuit, the following mesh equations were obtained:
18𝐼1 + 𝑗12𝐼1 − 𝑗25𝐼2 = 200
20𝐼2 + 𝑗40𝐼2 + 15𝐼2 − 𝑗25𝐼1 = 0
Determine, in polar form, the primary and secondary currents, 𝐼1and 𝐼2 respectively,
through the circuit.
12 marks
3: If 6𝑥
2 + 3𝑦
2 + 𝑧 = 1, find the rate at which 𝑧 is changing with respect to 𝑦 at the
point (3, 5, 2)
4 marks
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
4: The stream function, 𝜓 (𝑥, 𝑦),has a circular
function shape as shown, and is related to the
velocity components 𝑢 and 𝑣 of the fluid flow by
𝑢 =
𝜕𝜓
𝜕𝑥 and 𝑣 =
𝜕𝜓
𝜕𝑦
a) If 𝜓 = 𝑙𝑛√𝑥
2 + 𝑦
2, find 𝑢 and 𝑣.
b) The flow is irrotational if 𝜓 satisfies Laplace’s equation,
𝛿
2 𝜓
𝛿𝑥2 +
𝛿
2𝜓
𝛿𝑦2 = 0
Determine if the flow is irrotational.
15 marks
5: Determine the force, F, which has a magnitude of 64kN in the direction of the vector
𝐴𝐵 where A = (2, 4, 6) and B = (6, 3, 4).
8 marks
6: An object is dropped and its position vector, r, is given by
𝑟 = (𝑡
5 − 2𝑡
2
)𝑖 + (7𝑡
3 − 6𝑡
2
)𝑗
a) Find the velocity, 𝑣, and the acceleration, 𝑎.
b) What is the angle between 𝑣 and 𝑎 when 𝑡 = 2?
12 marks
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
7: If 𝛷 = 𝑥𝑧
2 + 3𝑥𝑦
2 + 𝑦𝑧
2
a) Determine grad 𝛷 at the point (3, 5, 1).
b) Find the direction from the point (1, 1, 0) which gives the greatest rate of
increase of the function of 𝜙.
9 marks
8: Consider a twostorey building subject to earthquake oscillations as shown:
Oscillations
The period, T, of natural vibrations is given by 𝑇 =
2𝜋
√(−𝜆)
where 𝜆 is the eigenvalue of the matrix A.
Find the period(s) if A = (
−30 10
10 −20)
10 marks
The equations of motion are
expressed as
𝑥̈= 𝐴𝑥
𝑤ℎ𝑒𝑟𝑒 𝑥 = (
𝑥1
𝑥2
)
𝑥1
𝑥2
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
9: Below is a screenshot of a mathematical function written in MATLAB, along with its
corresponding vector diagram.
What does this code do? Explain with respect to each line in the code and the figure
shown.
10 marks
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
10:
a) Model Q6 using MATLAB. Include a screenshot of the code and explain what
each line of code is doing.
b) Give two advantages of MATLAB over manual calculations and support your
answers with examples
15 marks
Total available marks – 100
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EG260 Continuous Assessment 1
Warning: Failure to follow these instructions will result in zero mark for the entire assessment. These instructions are purposefully very detailed and highlight common mistakes that have been seen in the past. My goal is to make sure that you communicate your answers to this assessment in the correct manner so that I can assign you the correct marks. It is therefore crucial that you read, understand and follow these instructions.
Instructions:
 This assessment must be solved and submitted individually. The submission deadline is:
23.59 on Thursday 9 March 2017
 Use numerical values for the parameters corresponding to your student number from xls file available on blackboard.
 Solve all of the questions in this assessment using the parameters that are assigned to your student number. Remember: Each answer that you will obtain WILL be a numerical value. When solving the questions, maintain highest possible decimal points. Your final answer should be rounded to 4 decimal places (done automatically in the xls file).
 Download the empty answer file (xls) to your computer from the Blackboard page. This is the file where you will enter your answers to the questions. The only change that you should do to this file is to enter your answers and to save it! Do NOT rename this file. Do NOT change the file format, for example to .xlsx. Do NOT change the internal formatting of the file. Do NOT change or add new sheets to the file. Your task is to simply enter ONLY numerical values for your answers to this file and save it! Do not put units as they are already in the questions. Do NOT enter ANY nonnumerical characters (such as “2*10^2”, “10e4”) or incomplete calculations such as “2×2” or “2*100” or “50/3 – 100”.
 Submit the Excel file (xls) in Blackboard.
 IN ADDITION to the xls file, you MUST submit a SINGLE file containing the supporting work. This file should show how you have solved the problems. This is the evidence that you obtained the numerical results yourself. You can type your solution in WORD or SCAN your handwritten work. Either way, submission should be a PDF file. Remember, this FILE NAME must be MySolution.pdf. Do NOT submit separate files for different parts of your solution. Avoid JPG, TIF or other image files if possible.
 Numerical answers in both files must agree with each other. In case of any discrepancies, the answers in xls will be used for marking.
 Unlike the final exam, no method marks is available for this assessment. You have to get correct numerical values and enter it correctly as described above. This is because, unlike the final exam, you have one full week to solve the two problems.
 Please submit the two files ONLY once.
 Question 1: An inverted pendulum oscillator of length L [m] and mass m [kg] is attached by springs. Two springs of stiffness values k_{1} and k_{2 }[N/m] are arranged in parallel and series respectively as shown below:
Important: The values of L, m, k_{1} and k_{2} in SI units are given for your student number in the excel file CA1_Parameters.xls. Use an equivalent spring in deriving the equation of motion and consider the weight of the mass. Take gravitational acceleration constant as 9.8100 [m/s^{2}]. All answers must be in numerical format and in SI units.
Case 1: springs in parallel Case 2: springs in series
 Calculate the equivalent spring stiffness for case 1 and enter the numerical value to the designated cell in the Excel file. (5 Marks)
 Calculate the equivalent spring stiffness for case 2 and enter the numerical value to the designated cell in the Excel file. (5 Marks)
 Assuming the rotation is small, obtain the equation of motion. From this, calculate the natural frequency in rad/sec for case 1 and enter the numerical value to the designated cell in the Excel file. (15 Marks)
 From the equation of motion, calculate the natural frequency in rad/sec for case 2 and enter the numerical value to the designated cell in the Excel file. (15 Marks)
 Assuming k_{2 }= 2k_{1}, obtain the value of k_{1} (in N/m) for the system to be stable for case 1 and enter the numerical value to the designated cell in the Excel file. (5 Marks)
 Assuming k_{2 }= 2k_{1}, obtain the value of k_{1} (in N/m) for the system to be stable for case 2 and enter the numerical value to the designated cell in the Excel file. (5 Marks)
Question 2: A vibrating system consisting of a weight of W [N] and a spring stiffness of k [N/m] is viscously damped such that the ratio of any two consecutive amplitudes is 10 to y. Determine:
 Log decrement () and enter the numerical value to the designated cell in the Excel file. (10 Marks)
 Damping factor () and enter the numerical value to the designated cell in the Excel file. (10 Marks)
 Damped natural frequency () in (rad/sec) and enter the numerical value to the designated cell in the Excel file. (15 Marks)
 Damping constant (c) and enter the numerical value to the designated cell in the Excel file. (15 Marks)
Hint: The values of W, k, and y in SI units are given for your student number in the Excel file CA1_Parameters.xls. Take gravitational acceleration constant as 9.8100 [m/s^{2}]. All answers must be in numerical format and in SI units.
Reminder: Failure to follow the instructions will result in zero marks even if you obtained correct answers! For the sake of fairness, no exceptions will be allowed. Unless you are ABSOLUTELY sure that your submission is according to the instructions, please do not upload it in the blackboard.
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Need helpWind Turbine Investigation
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The following experiment enables you to:
 Measure the energy in the wind.
 Assess a commercially available wind turbine in an environmental wind tunnel.
 Determine the power curve of a wind turbine and obtain cutin speeds
 Calculate the coefficient of performance of a turbine
 Calculate the Solidity and Tipspeed ratio.
 See how the energy is converted stored and utilised.
 Examine the Beaufort wind scale.
Introduction:
The power available to a wind turbine is the kinetic energy passing per unit time in a column of air with the same cross sectional area A as the wind turbine rotor, travelling with a wind speed U_{0}. Thus the available power is proportional to the cube of the wind speed. See the figure below.
Equipment
The equipment is provided by Marlec and the following information is from their web page but has been modified slightly for this labsheet.
The Rutland 913 is designed for marine use on board coastal and ocean going yachts usually over 10m in length. This unit will generate enough power to serve both domestic and engine batteries on board.
The Rutland 913 is a popular sight in marinas, thousands are in use worldwide, boat owners like it’s clean, aerodynamic lines and its quiet and continuous operation. Without doubt this latest marine model accumulates more energy than any other comparable windcharger available, you’ll always see a Rutland spinning in the lightest of breezes!
 Low wind speed start up of less than 3m/s
 Generates 90w @ 37m/s, 24w @ 20 m/s
 Delivers up to 250w
 Modern, durable materials for reliability on the high seas
 SR200 Regulator – Shunt type voltage regulator prevents battery overcharge
Theory:
During this experiment you will make use of the following equations to calculate key parameters
Key formulae
Energy in the wind E = (watts)
Swept area of rotor A=πR^{2}
Electrical power output P=VxI (watts)
Coefficient of performance
Tip speed ratio
Solidity = blade area/swept area
R is the rotor radius (m)
ρ is air density say 1.23 kg/m^{3}
U_{o} is the wind speed (m/s)
V is voltage (volts)
I is current (amps)
ω (rads/sec) is the angular velocity of the rotor found from
where N is the rotor speed in revs/min
Procedure:
Step 1 Ensure that everything is setup for you and switch on the tunnel.
Step 2 Adjust the wind speed and let it stabilize
Step 3 Measure the wind speed, voltage and current
Step 4 If available measure the rotor speed with the stroboscope.
Repeat steps 2 – 4 for other wind speeds up to a maximum of 10m/s if achievable.
Gather your data by completing tables 1 and 2
Wind speed
U_{o} (m/s) 
Beaufort
number 
Effect on land  Output voltage
V (volts) 
Output current
I (amps) 
Rotor speed
N (revs/min) 
Table 1 measured data
Calculate the following
Rotor radius use a ruler to measure from center to tip of turbine  R = 
Swept area A=πR^{2}

A = 
Blade area = blade area + hub area
do your best! 
= 
Solidity = blade area / swept area.  = 
Table 2 measured data
Now analyse your data by completing table 3.
Energy in the wind  Electrical power  Coefficient of performance  Tip speed ratio 
E = (watts)  P = V x I
(watts) 
P/E
(or column 2 /column 1) 

Table 3 Analyse your data
Present your data:
Now present your results in graphical format to give you a better understanding of the data you have gathered and analysed.
Use excel and the xy scatter chart for this.
Graph 1
Plot the values U_{o} (xaxis) against P (y1axis) and E (y2axis).
Graph 2
Plot the values of U_{o} (xaxis) against C_{p} (yaxis).
What conclusions do you draw?
How efficiently are you converting the kinetic energy in the wind into electrical energy that is stored chemically in the batteries?
Write up the laboratory formally and submit to turnitin. Please ensure presentation is clear and quote fully any references.
The Beaufort Wind Speed Scale 

Beaufort Number 
Wind Speed at 10m height  Description  Wind Turbine effects 
Effect on land 
Effect at Sea 

m/s  
0  0.0 0.4  Calm  None  Smoke rises vertically  Mirror smooth  
1  0.4 1.8  Light  None  Smoke drifts; vanes unaffected  small ripples  
2  1.8 3.6  Light  None  Leaves move slightly  Definite waves  
3  3.6 5.8  Light  Small turbines start – e.g. for pumping  Leaves in motion; Flags extend  Occasional breaking crest  
4  5.8 8.5  Moderate  Start up for electrical generation  Small branches move  Larger waves; White crests common  
5  8.5 11.0  Fresh  Useful power Generation at 1/3 capacity  Small trees sway  Extensive white crests  
6  11.0 14.0  Strong  Rated power range  Large branches move  Larger waves; foaming crests  
7  14.0 17.0  Strong  Full capacity  Trees in motion  Foam breaks from crests  
8  17.0 21.0  Gale  Shut down initiated  Walking difficult  Blown foam  
9  21.0 25.0  Gale  All machines shut down  Slight structural damage  Extensive blown foam  
10  25.0 29.0  Strong gale  Design criteria against damage  Trees uprooted; much structural damage  Large waves with long breaking crests  
11  29.0 34.0  Strong gale  Widespread damage  
12  >34.0  Hurricane  Serious damage  Disaster conditions  Ships hidden in wave troughs 
Supplementary Theory
The power available to a wind turbine is the kinetic energy passing per unit time in a column of air with the same cross sectional area A as the wind turbine rotor, travelling with a wind speed u_{0}. Thus the available power is proportional to the cube of the wind speed.
We can see that the power achieved is highly dependent on the wind speed. Doubling the wind speed increases the power eightfold but doubling the turbine area only doubles the power. Thus optimising the siting of wind turbines in the highest wind speed areas has significant benefit and is critical for the best economic performance. Information on power production independently of the turbine characteristics is normally expressed as a flux, i.e. power per unit area or power density in W/m^{2}. Thus assuming a standard atmosphere with density at 1.225kg/s :
Wind speed m/s Power W/m squared 5.0 76.6 10.0 612.5 15.0 2067.2 20.0 4900.0 25.0 9570.3
The density of the air will also have an effect on the total power available. The air is generally less dense in warmer climates and also decreases with height. The air density can range from around 0.9 kg/m^{3} to 1.4kg/m^{3}. This effect is very small in comparison to the variation of wind speed.
In practice all of the kinetic energy in the wind cannot be converted to shaft power since the air must be able to flow away from the rotor area. The Betz criterion, derived using the principles of conservation of momentum and conservation of energy gives a maximum possible turbine efficiency, or power coefficient, of 59%. In practise power coefficients of 20 – 30 % are common. The section on Aerodynamics discusses these matters in detail.
Most wind turbines are designed to generate maximum power at a fixed wind speed. This is known as Rated Power and the wind speed at which it is achieved the Rated Wind Speed. The rated wind speed chosen to fit the local site wind regime, and is often about 1.5 times the site mean wind speed.
The power produced by the wind turbine increases from zero, below the cut in wind speed, (usually around 5m/s but again varies with site) to the maximum at the rated wind speed. Above the rated wind speed the wind turbine continues to produce the same rated power but at lower efficiency until shut down is initiated if the wind speed becomes dangerously high, i.e. above 25 to 30m/s (gale force). This is the cut out wind speed. The exact specifications for designing the energy capture of a turbine depend on the distribution of wind speed over the year at the site.
Performance calculations
Power coefficient C_{p} is the ratio of the power extracted by the rotor to the power available in the wind.
It can be shown that the maximum possible value of the power coefficient is 0.593 which is referred to as the Betz limit.
where
P_{e} is the extracted power by the rotor
V_{¥} is the free stream wind velocity (m/s)
A is area normal to wind (m^{2})
ρ is density of the air (kg/m^{3})
The tip speed ratio (l) is the ratio of the speed of the blade tip to the free stream wind speed.
where
w is the angular velocity of the rotor (rads/sec), and
R is the tip radius (m)
This relation holds for the horizontal axis machine which is the focus of these notes.
The solidity (g) is the ratio of the blade area to the swept frontal area (face area) of the machine
Blade area = number of blades * mean chord length * radius = N.c.R
Mean chord length is the average width of the blade facing the wind.
Swept frontal area is pR^{2}
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HelpProject INSE 6110:Foundations of Cryptography
HelpProject INSE 6110:Foundations of Cryptography Due:Last class
1 Project paper Choose one of the preapproved topics below,or you may suggest other topics (either as a survey or a novel contribution) but they must be approved by me.Projects are to be done individually or in groups of 2.You may suggest a group project that involves 3 or more people but it must be approved by me.All group members receive the same mark for the project. For this project,research the topic and write a paper (max 8 pages) explaining the subject,with ref erences to the related literature,using the following template:
http://www.springer.com/computer/lncs?SGWID=016467933410 Your paper should summarize the subject with an introduction, explaining very clearly what the research problem is and how the subject addresses it. You should then explain the solution with technical detail. You should understand and cite at least 3 academic papers that appear at good quality venues. To ﬁnd papers and understand the concepts,try:
http://scholar.google.com http://link.springer.com/referencework/10.1007%2F9781441959065 If the paper does not appear at a conference in the ﬁrst 50 on this list,it is not likely a good quality venue:
http://academic.research.microsoft.com/RankList?entitytype=3&topdomainid=2&subdomainid= 2&last=0&orderby=1 In all cases,you can use your discretion (e.g.,papers at specialized workshops can be high quality, nonacademic resources can be as well) and if you have any questions,ask me during the lecture break or during ofﬁce hours. Be sure to cite all sources you use. You may do citations in a conversational way (e.g., “Boneh et allist the ﬁve essential properties of blah as follows [9].”) Under no circumstance can you use someone else’s text as your own (even if you modify the grammar).Review Concordia’s plagiarism policy and understand it:
http://www.concordia.ca/students/academicintegrity/plagiarism.html https://www.concordia.ca/content/dam/encs/docs/ExpectationsofOriginalityFeb142012. pdf
2 Preapproved Topics Cryptographic Primitives and Protocols • Attributebased Encryption • Blind Signatures • Bilinear Pairings • Bitcoin • Cryptographic Accumulators • Direct Anonymous Attestation (used by TPMs) • Dining Cryptographers • Fair Exchange • Fully Homomorphic Encryption • Garbled Circuits • GCMMode of Operation • Group Signatures • Indistinguishability Obfuscation • Identitybased Cryptography • Mix Networks • Oblivious Transfer • OfftheRecord Messaging • PostQuantum Cryptography • Ring Signatures • TimedRelease Encryption • Universal Composability Cryptanalysis and Attacks • Differential Cryptanalysis • Boomerang Attack • Biclique Cryptanalysis Newsworthy Events • RC4 biases in SSL/TLS • NSAbackdoor in Dual ECDRGB
2
Engineering Problems 14
Problem 1
Use the random function and a rounding function described in this chapter to simulate the rolling of a die ten times. The function should produce a vector of ten values corresponding to ten die rolls. The die should have a range of [1 , 6]. Do the following operations on the vector with the die results. a. Compute the mean. b. Compute the median. c. Compute the standard deviation. d. Sort the vector values in ascending order. Note: The outcome of each operation will vary each time the random function is executed.
Problem 2
In MATLAB, use the built in complex function to generate a vector of complex numbers ranging from 0 + 0𝑖𝑖 to 10 + 10𝑖𝑖 with increments of 1.
Problem 3
Using the following complex number, 𝑍𝑍 = 3 + 5𝑖𝑖, use MATLAB to do the following operations: a. Compute the absolute value. b. Compute the angle. c. Obtain the real part. d. Obtain the imaginary part. e. Obtain the conjugate.
Problem 4
The Euler formula establishes a relationship between complex exponentials and trigonometric functions. It is defined as the following: 𝑒𝑒𝑖𝑖𝑖𝑖 = cos(𝜃𝜃) + 𝑖𝑖sin(𝜃𝜃) Where 𝑖𝑖 represents the imaginary number √−1 𝜃𝜃 represents an angle in radians. A modified version of this equation is used in electrical engineering to aid in the analysis of periodic signals like a cosine wave: 𝑒𝑒𝑗𝑗(2𝜋𝜋𝑓𝑓𝑜𝑜𝑡𝑡+𝜃𝜃) = cos(2𝜋𝜋𝑓𝑓𝑜𝑜𝑡𝑡 + 𝜃𝜃) + 𝑗𝑗sin(2𝜋𝜋𝑓𝑓𝑜𝑜𝑡𝑡 + 𝜃𝜃) Where 𝑗𝑗 is the same as 𝑖𝑖 (electrical engineers use 𝑖𝑖 for current and instead use 𝑗𝑗 to represent complex values) 𝑓𝑓𝑜𝑜 is the frequency of the wave in Hertz. 𝜃𝜃 is called the phase angle in radians (For this problem assume 𝜃𝜃 = 0) 𝑡𝑡 is time in seconds Create a vector 𝒕𝒕⃗ of time values ranging from [0, 2] with an increment of . 01 for the vector. Let the frequency of the wave be 𝑓𝑓𝑜𝑜 = 1. Use the exponential function in MATLAB to obtain a vector of complex values. In MATLAB, you can use 1𝑖𝑖 to create 𝑗𝑗 inside the exponential function. Your function should look like this: 𝒗𝒗�⃗ = 𝑒𝑒1𝑖𝑖2𝜋𝜋𝑓𝑓𝑜𝑜𝑡𝑡 Once you have the vectors 𝒗𝒗��⃗ and 𝒕𝒕⃗. Use the built in ‘real’ function in MATLAB to obtain only the real values of 𝒗𝒗��⃗. Finally, put the following in your code to see the plot of this function: 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(𝑡𝑡⃗, 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝒗𝒗��⃗))