
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
NewtonRaphson). 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; twostate 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
coordinate 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 midordinate; trapezoidal
and Simpson’s rules
Numerical estimation methods: method of bisection; NewtonRaphson
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 coordinates 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 nonhomogeneous
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,
massspringdamper 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, workbased placements and/or detailed case study materials.
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