MODULE DESCRIPTOR
MECH2007 – Modelling and Analysis II
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Level:
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Taught by:
MECH2007
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Modelling and Analysis II
2
0.5/7.5
September
June
Dr Mehrdad Zangeneh (Module Coordinator)
Dr Eral Bele (Maternity Cover for Dr Rebecca Shipley)
Prerequisites
Students considering registering for this course would normally be expected to have completed an
introductory course in engineering mathematics.
Co-requisite
None
Course Aims and Objectives
Aims
To introduce students to:
 Basic concepts of vector calculus such as gradient, divergence and curl and their practical
 applications to derive governing equations from conservation laws.
 Fourier series and its application to forced damped periodic oscillations.
 Partial differential equations and their solution by using method of separation of variables.
 Basic concepts of matrix representation.
 Solution of systems of linear equations.
 The application of matrices to dynamical systems.
 Solution of linear ODEs with various types of forcing using Laplace theory.
 Solution of a set of linear equations using iterative or direct methods.
Objectives
At the end of the course students should be able to:
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Formulate and solve integral problems involving 2 and 3 space variables.
Formulate the equation of motion of a particle in 3D space in parametric form and compute the
work
done by a force in moving a particle in 3D space.
Apply Divergence Theorem to conservation laws (mass, energy etc) to derive governing
equations
for velocity ( continuity equation) or temperature ( heat conduction equation).
Compute moment of inertia of a lamina or 3D volume with variable density.
Represent any periodic function with arbitrary interval by Fourier series.
Solve the equation of motion representing the forced damped oscillation subject to a periodic
forcing function.
Solve wave equation, unsteady 1D heat conduction equation and steady 2D conduction equation
by
using the method of separation of variables.
Solve systems of linear equations.
Solve multiple degree of freedom dynamic systems using matrices.
Solve linear ODEs with various types of forcing using Laplace theory.
Solve a set of linear equations using iterative or direct methods.
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Method of Instruction
This is a lecture based class supported by tutorial classes.. Lecture materials and tutorial sheets
provided on Moodle in advance.
Assessment
The course has the following assessment components:
 Written examination (3 hours, 75%)
The examination rubric is:
 2 tests (25%); one in each term.
To pass this course, students must:
 Obtain an overall pass mark of 40% for all sections combined
The examination rubric is:
Answer FIVE questions (from eight offered) and answer no more than THREE questions from EITHER
section (A or B). All questions carry equal weight.
Resources:
The timing of the course makes it not suitable for affiliate students who are only at UCL for one
semester.
Content
1. Matrix notation and operations
 Definition
 Notation
 Operations: Addition, Scalar multiplication, Matrix multiplication, Transposition
2. Determinants and their properties
 Definitions
 Cofactors
 Second and third order determinants
3. Inverse matrices
 Inversion method
 Singularity
 Uniqueness
4. Matrix transformations
 Method
 Examples: Axes stretch and rotation
 Double transformation
5. Orthogonality
6. Sparse matrices in physical systems
 Partitioning methods
7. Systems of linear algebraic equations
 Definitions: Homogeneous system, Augmented coefficient matrix, Consistent/Inconsistent
equations
 Row operations
 Rank of a matrix
 Consistency test
 Uniqueness test
 Methods for solving linear equations
8. Eigenvalues and eigenvectors
 Definitions: Eigenvector, Eigenvalue
 Procedure for solving an eigenvalue problem
 Example of an eigenvalue problem: Vibrations of physical systems
9. Numerical methods
 Direct and iterative methods
 Direct method: Gaussian elimination, Matrix factorisation, Performing the LU decomposition,
Simplification using Gaussian elimination
 Iterative solution methods and strategies
10. Laplace Transforms
 Definition
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Linearity Property
Existence and Uniqueness of Laplace transforms
Strategy for applying Laplace transform techniques
Transform of derivatives
Solving ODEs
Special Functions: Heaviside step function, Dirac delta function
Transforms of an integral: 1st shift property, 2nd shift property, Convolution property
11. Multiple Integrals - 4 Lectures
• Geometric interpretation and evaluation of double and triple integrals.
• Velocity distribution and volume flow rate in pipes of different cross section. Mass and moment
of inertia of arbitrary shaped lamina. Volume of three dimensional objects.
• Jacobians and variable transformations. Cylindrical & Spherical polar coordinates.
12. Vector Calculus I -- 5 Lectures
• Revision of vectors. Parametric representation of curves. curves and arc lengths.
• Velocity and acceleration.
• Vector and scalar fields. Directional derivatives and gradient
• Line integral and its evaluation. Conservative vector fields. theorem in a plane.
13. Vector Calculus II - 5 lectures
• Surface Integrals and their evaluation.
• Flux of a vector, Divergence operator, Gauss’s Divergence theorem.
• Applications in fluid mechanics and heat conduction.
• Vorticity and curl operator, rectilinear shear flow, circulation, Stokes theorem, Applications in
Electromagnetism.
14. Fourier Series – 4 lectures
• Forced Damped vibrations with harmonic forcing function.
• Definition of Fourier series, trigonometric preliminaries. Evaluation of
• Fourier coefficients. Odd and even functions. Periodic extension and half range series.
• Fourier series with arbitrary intervals. Differentiation and integration of
• Fourier series.
• Forced damped vibrations with periodic arbitrary forcing function.
15. Partial Differential Equations – 5 lectures
• Method of separation of variables.
• Vibrations of a stationary string - Zero initial velocity.
• Unsteady heat conduction in a bar - constant temperature /insulated ends.
• Steady state temperature distribution in rectangular plates.
General Learning Outcomes
Knowledge and understanding
Matrix methodology - analytical and numerical solutions, eigenvalues and eigenvectors, Laplace and
other transforms, periodic phenomena including damped vibrations, heat flow, plate vibration, fields in
mechanical systems, partial differential equations.
Skills and Attributes:
(I) Intellectual
Application of mathematical analysis to the engineering sciences; translation of real engineering
problems into mathematical representations and solving these using appropriate methods.
(ii) Practical
Manipulation of mathematical expressions and the application of numerical analysis to the solution of
engineering problems.
(iii) Transferable
Mathematical skills applied in both understanding and application to courses involving a quantitative
dimension.
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