Assignment title Applied Mathematics for Complex Engineering Problems

The purpose of this assignment is to:

1. Understand the use of trigonometric functions
2. Be able to solve algebraic equations representing engineering problems
3. Be able to apply calculus to engineering problems
4. Be able to apply differential equations to engineering problems
5. Be able to apply statistical techniques to engineering problems

Task 1

In this task, you will understand the use of trigonometric functions.

Scenario: A complex voltage is usually represented by the waveform expressed in trigonometric function. The waveform of voltage representing the input voltage in the circuit by the expression in sinusoidal waveform as v = 3 sin ωt + 5 cos ωt as an analogue circuit. The voltage has the frequency ω.

Task 1.1

If v = 3 sin ωt + 5 cos ωt in the form R sin (ωt + α ) to perform

i. Sketch graphs of the waveform on the axes

ii. Explain the meaning of R, ω and α.

Task 1.2

a) The value of voltage in an AC circuit at any time t second is given by above waveform R sin (ωt + α), determine:
i) The amplitude, period, frequency and phase angle (in degrees)
ii) The value of the voltage when t=0
iii) The value of the voltage when t=10ms
iv) The time when the voltage first reaches 300V

b) The conversion of a sinωt + b cosωt into R sin(ωt + α ) to prove the sine and cosine function of the same frequency added produce a sine wave of the same frequency.

Task 1.3

Evaluate the following problems using trigonometric identities to prove that:

cos(3Π/2 + ωt)/cos(2Π - ωt) = tan ωt

Verify the above identities using the numerical values.

Task 2
Scenario: In the electrical circuit design and analysis, there is an unbalanced, three-phase, star-connected, electrical network. If we consider three closed loop network with Kirchoff's Voltage Law, we can get the simultaneous equations as follows:

2.4Ι₁ + 3.6I₂ + 4.8I₃= 1.2
-3.9 Ι₁ + 1.3I₂- 6.5I₃ = 2.6
1.7 Ι₁ + 11.9I₂ + 8.5I₃ = 0

Task 2.1

Construct the above algebraic equations to represent the engineering problems in matrix form.

Task 2.2

Solve the above simultaneous linear equations for I₁, I₂ and I₃ by using matrices and determinants.

Task 2.3

Given that the critical speeds of oscillation, , of a loaded beam are given by the non-linear equation

λ³ - 3.250λ² + λ - 0.063 = 0

Solve the non-linear equation by Newton's method, correct to 4 decimal places, to determine the values of λ.

Task 3

In this task, you will be able to apply calculus to engineering problems.

Scenario: In the civil and building engineering problem, calculus is usually used in the analysis. For instance, we can use it to state the location and classification of stationary point, calculation of bounded area and volume.

Task 3.1

Given z = x2sin(x - 2y), determine partial derivatives of functions

∂²Z/∂x∂y

Task 3.2

a) Show that the curve

y = 2/3 (t-1)³ + 2t(t-2)

Classify stationary points of the above function with one variable.

Task 3.3

Determine the following integrals.

∫(5x² - 30x +44)/(x-2)³ dx

You need to clearly make justification to the selection of methods and techniques applied. You also need to show the justification of your results.

Task 3.4

Use definite integral to find out the area of the ellipse.

x²/a² + y²/b² =1.

Task 4

In this task, you will be able to apply differential equations to engineering problems.

Scenario: It is widely use differential equations with boundary condition to model the electrical systems. In this task, analytical and numerical methods to solve the solution of differential equation with initial boundary condition need to be applied.

Task 4.1

In an alternating current circuit containing resistance R and inductance L the current i is given by:

R_i + L.di/dt = E₀sinωt

Given i = 0 when t = 0, select appropriate differential equation model to show that the solution of the equation is given by: (hint: partial differential model to set up second order differential equation or others)

i =(E₀/(R² + ω²L²)) (Rsinωt - ωLcosωt) + (E₀ωL/(R² + ω²L²))e^-Rt/L

Task 4.2

Ld²q/dt2 + R.dq/dt +1/c.q = V^osinωt

represents the variation of capacitor charge in an electric circuit.

Given that R = 40 Ω, L = 0.02H, C = 50 10^-6 F, V_o = 540.8V and ω = 200 rad/s and also given the boundary conditions that when t = 0, q =0 and dq/dt= 4.8.

Solve problem using initial and boundary value conditions to determine an expression for q at t second.

Task 4.3

Obtain a numerical solution of the differential equation dy/dx = 3(1+x)-y

Given the initial conditions that x=1 when y=4 for the range x=2.0 with intervals of 0.2. Solve differential equations numerically using mathematical software.

ii) Find the value of y if x = 1.58, dy/dx = 0.5

Task 5

In this task, you will be able to apply statistical techniques to engineering problems.

Scenario: Statistical techniques are commonly used to study the engineering problems such as material testing, quality control etc. Engineering problems with different discrete and continuous distribution are stated in this task to illustrate the practical use of statistical techniques mentioned in the course.

Task 5.1

a) The points with coordinates (0, 6), (2, 7), (4, 8) and (6, 9) lie on a straight line. Draw the line and determine its equation.

b) The following graph shows a scatter plot and a line of best fit:

Determine the equation of the line of best fit. Use the equation to estimate y when x = 4.

Use the equation to estimate x when y = 18

Task 5.2

Analyse the probability distributions for discrete and continuous data with the following situations.

a) Concrete blocks are tested and it is found that, on average, 7% fail to meet the required specification. For a batch of 9 blocks, determine the probabilities that less than three blocks will fail to meet the specification. This is to determine the distributions for discrete or continuous data, please explain it.

The data sheet is provided on the next page.

Task 5.3

A package contains 50 similar components and inspection shows that 4 have been damaged during transit. If 6 components are drawn at random from the contents of the package determine the probabilities that in this sample a) one and b) less than 2 are damaged.

The data sheet is provided on the next page.