Produce a sine wave of the same frequency

Assignment Help Engineering Mathematics

Reference no: EM131544530 , Length: word count:7500

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.

Reference no: EM131544530

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Reviews

len1544530

6/26/2017 5:12:45 AM

Distinction Criteria D1 Use critical reflection to evaluate own work and justify valid conclusions: realistic improvements have been proposed against defined characteristics for success Critical reflection are successfully used to evaluate own work and justify valid conclusions within task 5.1. D2 Take responsibility for managing and organising activities: autonomy and independence has been demonstrated Autonomy and independence are clearly demonstrated within task 4.3. D3 Demonstrate convergent/lateral/creative thinking: self-evaluation has taken place Convergent and lateral thinking are obviously demonstrated: self-evaluation has shown to have been taken placed within task 3.4.

len1544530

6/26/2017 5:12:38 AM

Merit Criteria M1 Identify and apply strategies to find appropriate solutions: an effective approach to study and research has been applied. The strategies are clearly identified and applied to find appropriate solutions: an effective approach is correctly applied to study and research within task 1.3. M2 Select/design and apply appropriate methods/techniques: relevant theories and techniques have been applied. The appropriate methods/techniques are correctly selected/ designed and applied: relevant theories and techniques are clearly shown to have been applied within task 1.2. M3 Present and communicate appropriate findings: appropriate structure and approach has been used. The appropriate findings are clearly and correctly presented and communicated: the appropriate structure and approach are clearly shown to have been used within task 2.2.

len1544530

6/26/2017 5:12:25 AM

P3.3 Determine integrals The integrals are correctly determined P3.4 Calculate areas and volumes using definite integrals The areas and volumes are correctly calculated using definite integrals P4.1 Select appropriate differential equation models The appropriate differential equation models are correctly selected P4.2 Solve problems using initial and boundary value conditions The problems are correctly solved using initial and boundary value conditions P4.3 Solve differential equations numerically using mathematical software The differential equations are correctly solved numerically using mathematical software P5.1 Determine lines of best fit The probability distributions for discrete data are correctly analysed P5.2 Analyse probability distributions for continuous data The probability distributions for continuous data are correctly analysed P5.3 Solve engineering problems using standard statistical distributions The engineering problems are correctly solved using standard statistical distributions

len1544530

6/26/2017 5:12:14 AM

Pass Criteria P1.1 Sketch graphs for trigonometric functions of the form Rsin(wt + a) The graphs of trigonometric functions of the form Rsin(wt + a) are correctly sketched P1.2 Analyse the sum of waves at the same frequency The sum of waves at the same frequency are correctly analysed P1.3 Evaluate problems using trigonometric identities The problems are correctly evaluated using the trigonometric identities P2.1 Construct algebraic equations to represent engineering problems The algebraic equations are correctly constructed to represent the engineering problems P2.2 Solve simultaneous linear equations The simultaneous linear equations are correctly solved P2.3 Solve non-linear equations The non-linear equations are correctly solved P3.1 Determine partial derivatives of functions The partial derivatives of functions are correctly determined P3.2 Classify stationary points of functions with one and two variables The stationary points of functions with one and two variables are correctly classified

len1544530

6/26/2017 5:11:48 AM

1. Do the whole assignment 2. Submission Requirement: 3. Font Times New Roman 4. Font Size 12 5. Spacing Single Line Spacing Margin 2 Centimetres (Left-side of page only) 6. Printing Single Sided with Page No. 7. Number of Words (Approx.) 7,500 words & at least 3 pages of answers To achieve the diffident criteria that the student should fulfil the related requirement as shown as follow: Pass Criteria and Related requirement – A pass grade is achieved by meeting all the requirements in the assessment criteria for pass for each task. Merit Criteria and Related requirement –A merit grade is achieved by meeting all the criteria for M1, M2 and M3 plus all pass criteria. Distinction Criteria and Related requirement–A distinction grade is achieved by meeting all the criteria for D1 , D2 and D3 plus all pass criteria and all merit criteria.

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