Unit 35: Further Analytical Methods for Engineers


Unit 35:

Further Analytical Methods for

 

 

Engineers


Unit code:
J/601/1465

QCF level:
5


Credit value:
15






Aim

This unit aims to further develop the analytical knowledge and techniques necessary to analyse and solve a variety of engineering situations and problems.

Unit abstract

This unit has been designed to enable learners to use number systems, graphical and numerical methods, vectors, matrices and ordinary differential equations to analyse, model and solve realistic engineering problems.

Learners will use estimation techniques and error arithmetic to establish realistic results from experiments and general laboratory work. They will then consider the conversion of number systems from one base to another and the application of the binary number system to logic circuits. Complex numbers and their application to the solution of engineering problems are also studied.

Learners will look at the use of graphical techniques together with various methods of numerical integration (for example Simpson’s rules) and estimation (for example Newton-Raphson). They will then go on to analyse and model engineering situations using vector geometry and matrix methods.

Finally, learners will study both first and second order differential equations and their application to a variety of engineering situations dependant upon the learner’s chosen discipline.

Learning outcomes

On successful completion of this unit a learner will:

1       Be able to analyse and model engineering situations and solve problems using number systems

2       Be able to analyse and model engineering situations and solve problems using graphical and numerical methods

3       Be able to analyse and model engineering situations and solve problems using vector geometry and matrix methods

4       Be able to analyse and model engineering situations and solve problems using ordinary differential equations.

Unit content



1      Be able to analyse and model engineering situations and solve problems using number systems

Error arithmetic: significant figures and estimation techniques; error arithmetic operations; systematic and random errors; application to experimentation and general laboratory work

Number systems: natural, integer, rational, reals, dinary, binary, octal and hexadecimal number systems; conversion from dinary to numbers of other bases and vice versa; two-state logic systems, binary numbers and logic gates, logic gate tables, application to logic circuits

Complex numbers: real and imaginary parts of complex numbers, complex number notation; Cartesian and polar forms; modulus, argument and complex conjugate; addition, subtraction, multiplication and division of Cartesian and polar forms; use of Argand diagrams; powers and roots and the use of de Moivre’s theorem

Engineering applications: applications eg electric circuit analysis, phasors, transmission lines, information and energy control systems

2      Be able to analyse and model engineering situations and solve problems using graphical and numerical methods

Graphical techniques: Cartesian and polar co-ordinate systems and representation of complex number operations; vector representation; standard curves; asymptotes; systematic curve sketching; curve fitting; irregular areas and mean values of wave forms; use of phasor and Argand diagrams; application to engineering situations

Numerical integral: determine the integral of functions using mid-ordinate; trapezoidal and Simpson’s rules

Numerical estimation methods: method of bisection; Newton-Raphson iteration method; estimates of scientific functions

3      Be able to analyse and model engineering situations and solve problems using vector geometry and matrix methods

Vector notation and operations: Cartesian co-ordinates and unit vectors; types of vector and vector representation; addition and subtraction; multiplication by a scalar; graphical methods

Matrix operations and vectors: carry out a range of matrix operations eg vectors in matrix form, square and rectangular matrices, row and column vectors, significance of the determinant, determinant for 2x2 matrix, the inverse of a 2x2 matrix; use Gaussian elimination to solve systems of linear equations (up to 3x3)

Vector geometry: determine scalar product, vector product, angle between two vectors, equation of a line, norm of a vector, dot and cross products; apply vector geometry to the solution of engineering problems eg velocity vector and mechanisms, acceleration vector and mechanisms, forces in static frameworks and structures, evaluation of static joint structures using dot product, phasors

4      Be able to analyse and model engineering situations and solve problems using ordinary differential equations

First order differential equations: engineering use; separation of variables; integrating factor method, complementary function and particular integral

Numerical methods for first order differential equations: need for numerical solution; Euler’s method; improved Euler method; Taylor series method

Application of first order differential equations: applications eg RC and RL electric circuits, time constants, motion with constant and variable acceleration, Fourier equation for heat transfer, Newton’s laws of cooling, charge and discharge of electrical capacitors, complex stress and strain, metrology problems

Second order differential equations: engineering use; arbitrary constants; homogeneous and non-homogeneous linear second order equations

Application of second order differential equations: applications eg RLC series and parallel circuits, undamped and damped mechanical oscillations, fluid systems, flight control laws, mass-spring-damper systems, translational and rotational motion systems, thermodynamic systems, information and energy control systems, heat transfer, automatic control systems, stress and strain, torsion, shells, beam theory

Engineering situations: applications eg heat transfer, Newton’s laws, growth and decay, mechanical systems, electrical systems, electronics, design, fluid systems, thermodynamics, control, statics, dynamics, energy systems, aerodynamics, vehicle systems, transmission and communication systems

Learning outcomes and assessment criteria




Learning outcomes
Assessment criteria for pass


On successful completion of
The learner can:


this unit a learner will:










LO1 Be able to analyse and model

1.1
use estimation techniques and error arithmetic to


engineering situations and


establish realistic results from experiment


solve problems using number

1.2
convert number systems from one base to another,


systems





and apply the binary number system to logic circuits









1.3
perform arithmetic operations using complex





numbers in Cartesian and polar form




1.4
determine the powers and roots of complex





numbers using de Moivre’s theorem




1.5
apply complex number theory to the solution of





engineering problems when appropriate








LO2 Be able to analyse and model

2.1
draw graphs involving algebraic, trigonometric and


engineering situations and


logarithmic data from a variety of scientific and


solve problems using


engineering sources, and determine realistic


graphical and numerical


estimates for variables using graphical estimation


methods


techniques




2.2
make estimates and determine engineering





parameters from graphs, diagrams, charts and data





tables




2.3
determine the numerical integral of scientific and





engineering functions




2.4
estimate values for scientific and engineering





functions using iterative techniques







LO3 Be able to analyse and model

3.1
represent force systems, motion parameters and


engineering situations and


waveforms as vectors and determine required


solve problems using vector


engineering parameters using analytical and


geometry and matrix methods


graphical methods




3.2
represent linear vector equations in matrix form and





solve the system of linear equations using Gaussian





elimination




3.3
use vector geometry to model and solve appropriate





engineering problems








Learning outcomes
Assessment criteria for pass

On successful completion of
The learner can:

this unit a learner will:







LO4 Be able to analyse and model
4.1
analyse engineering problems and formulate

engineering situations and

mathematical models using first order differential

solve problems using ordinary

equations

differential equations
4.2
solve first order differential equations using





analytical and numerical methods


4.3
analyse engineering problems and formulate



mathematical models using second order differential



equations


4.4
solve second order homogeneous and non-



homogenous differential equations


4.5
apply first and second order differential equations to



the solution of engineering situations.






Guidance

Links

This unit builds on and can be linked to Unit 1: Analytical Methods for Engineers and can provide a foundation for Unit 59: Advanced Mathematics for Engineering.

Essential requirements

There are no essential requirements for this unit.


Employer engagement and vocational contexts

This unit will benefit from centres establishing strong links with employers who can contribute to the delivery of teaching, work-based placements and/or detailed case study materials.




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