Higher National Certificate/Diploma in Construction and the Built Environment Assignment Brief
Unit Number and Title UNIT: 28 Further Mathematics for Construction Academic Year 2021/22 Unit Tutor
Assignment Title Issue Date Submission Date IV Name & Date
Submission Format This assignment is to be a time-controlled assessment. You are to complete the following tasks, submitting your working calculations (including any diagrams or sketches) with your completed solution clearly indicated. All work is to be submitted on A4 sheets.
Scenario: You are working as an Engineering Research Technician in a research and development laboratory where good analytical skills and understanding of mathematics is essential for everyday work. Your line manager has informed you that you need to improve and update your present Mathematical skills to work effectively & contribute to the research team. The essential skills include Number theory, Matrices, graphical and numerical methods, Models of engineering systems using ordinary differential equations . To test your suitability for this position in research, the following tasks have been compiled for you to complete. Unit Learning Outcomes LO1
Apply instances of number theory in practical construction situations.
Assignment Brief and Guidance Task 1 a. Convert each number into denary, • 11001.01 • 4D b) calculate the following in both binary and denary • 1101+1001
Task 2 Apply de’Moivre’s theorem or otherwise to solve for Zo and C from these expressions given below : Z0=Z/Y and C=Z*Y Where: • Z is a complex number. • Y is also a complex number. • Re (Z0) >0 and Re (C) >0 Find Z0 and C when: Z = 1 + 5 j,Y = 1 − 3 j
Task 3 a. Simplify the following equation:
G = 1× e j2π × 2 × e j0.5 × 0.5 × e j0.75
b. Express the following expression in complex exponential form: v=20sin (1000t-30°)
Task 4 Create an identity and find the formula for cos (3θ) in terms of cos (θ) and sin (θ) using de Moivre’s Theorem and test the identity with real values to confirm its validity.
L02 Solve systems of linear equations relevant to construction applications using matrix methods Assignment Brief and Guidance
Task 1 cos(θ ) sin (θ ) 1 0 R θ = −sin(θ ) cos(θ ), X = 0,Y = 1 a) Determine the vector Z when θ = π , Z = R ×( X − Y ) 2 θ b) Determine the determinant of the matrix R when θ = π θ 4 c) Determine the inverse of Rθ When θ =π 4 d) Solve the following equation for 1 = R × X π 0 4 Task 2 You have been asked by the structural engineering department to find the determinant and inverse of the following matrix 3 0 2 2 0 −2 0 1 1
Task 4 You have been asked to the following set of equations that have been obtained from the structural engineering Department and verify your calculations using computer methods 2×2 + x3 = −8 x1 − 2×2 − x3 = 0 − x1 + x2 + 2×3 = 3
*Access HN Global for additional resources support and reading for this unit. For further guidance and support on report writing please refer to the Study Skills Unit on HN Global. Link to www.highernationals.com
LO3 Approximate solutions of contextualised examples with graphical and numerical methods
Assignment Brief and Guidance
Task 1 The engineering department has developed the following equation for the bending moment of a beam and you have asked to investigate its behaviour M ( x) = x3 − 3×2 − 4 The Beam is 4m long and the design team suspect the is problem if the bending moment is zero in the range between 3-4m and you have been asked to a) Plot the bending moment at 0.5m interval for the range 0 ≤ x ≤ 4m 0 ≤ x ≤ 4 and determine if the bending moment is zero in range 3m ≤ x ≤ 4m b)
Use the graph to estimate where the bending moment is zero c) Use the bisection method to numerically estimate the exact location where the bending moment is zero d) Newton-Raphson method to obtain the required location e) Compare the results of the above method to determine which gives a best solution
Task 2 The following offsets are taken from a chain line to an irregular boundary towards right side of the chain line.
chainage 0 25 50 75 100 125 150 Offset ‘m’ 3.6 5.0 6.5 5.5 7.3 6.0 4.0 Common distance d =25m You have been asked to estimate the area using the following methods and compare and comment on their difference and accuracy. a) Trapezium Rule b) Simpson’s Rule
Also, critique the numerical estimation methods including Newton-Raphson, bisection method, Trapezium and Simpson’s Rule with the same application and comments on the accuracy of them, i.e., which of these methods is more accurate in your perspective providing exhaustive rationale. Task 3 The equation governing a body travelling in a water channel is given by the following equation dv = 1 − v 2 dt Plot the velocity time graph for the object and determine the final velocity and the time taken to reach this velocity
LO4 Review models of construction systems using ordinary differential equations
Assignment Brief and Guidance
TASK 1 The equation of catenary is given by the following second order differential equation y ” = 5 x = 0, y = 100 x = 100, y = 100
Solve the above differential equation and plot the curve at 10m intervals. Task 2 The differential equation governing the motion of a particle is given by the following differential equation
y ”+ 5 y = 0 t = 0, y = 20 Solve the above and plot the results and determine the amplitude and frequency of the oscillations Task 3
A new series of tests is carried out and the equation modified to y ”+ 2 y + 5 = 0 y = 20, t = 0
Use Laplace transforms or any other method to solve the new equations and plot the function and comment on the results Assignment Brief and Guidance Solve the above differential equation and determine the frequency and amplitude of the vibrations
HNC/HND Construction a
Note that Example Assessment Briefs are for guidance and support only. They can be customised and amended according to localised needs and requirements. All assignments must still be moderated as per the internal verification process. Learning Outcomes and Assessment Criteria Pass Merit Distinction LO1 Apply instances of number theory in practical construction situations.
P1 Apply addition and multiplication methods to numbers that are expressed in different base systems.
P2 Solve engineering problems using complex number theory.
P3 Perform arithmetic operations using the polar and exponential form of complex numbers.
M1 Deduce solutions of problems using de Moivre’s Theorem.
D1 Test the correctness of a trigonometric identity using de Moivre’s Theorem.
LO2 Solve systems of linear equations relevant to construction applications using matrix methods P4 Ascertain the determinant of a 3×3 matrix.
P5 Solve a system of three linear equations using Gaussian elimination.
M2 Determine solutions to a set of linear equations using the inverse matrix method.
D2 Validate all analytical matrix solutions using appropriate computer software.
LO3 Approximate solutions of contextualised examples with graphical and numerical methods P6 Estimate solutions of sketched functions using a graphical estimation method.
P7 Identify the roots of an equation using two different iterative techniques.
P8 Determine the numerical integral of construction functions using two different methods.
M3 Solve construction problems and formulate mathematical models using first-order differential equations.
D3 Critique the use of numerical estimation methods, commenting on their applicability and the accuracy of the methods.
LO4 Review models of construction systems using ordinary differential equations
P9 Determine first-order differential equations using analytical methods.
P10 Determine second- order homogeneous and non-homogenous differential equations using analytical methods.
P11 Calculate solutions to linear ordinary differential equations using Laplace transforms.
M4 Evaluate how different models of construction systems use first-order differential equations to solve structural or environmental problems.
D4 Evaluate first- and second-order differential equations when generating the solutions to construction situations.