Applied Biomechanical Mathematics

Learning Outcomes

Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Show logical thinking in problem solving.
• Apply a range of numerical and computational methods.

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Mathematical methods used in technical engineering subjects

Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Critically analyse and solve engineering problems.
• Apply appropriate methods of mathematical analysis.

Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Construct mathematical models that capture the key features of an (engineering) problem.
• Interpret mathematical results and their implications in their wider (engineering) context.
• Represent mathematical concepts in sketches and visual forms.
• Record consistent learning and revision activities.

• Oscillating function; Wave function

Calculus

• Integration: double integrals; polar coordinates; triple integrals.

Further Calculus:

• Chain rule for partial derivatives; higher partial derivatives; total differentials and small errors
• Partial Differential Equations.
• Separation of variables
• Use of Fourier Series
• Two-Dimensional Wave Equations
• Rectangular Membrane. Double Fourier Series

Differential Equations

• linear operators; linear inhomogeneous second order ODEs; free and forced oscillators

Vectors:

• Vectors: triple products; differentiation and integration of vectors; vector equations of lines and planes

Vectors: basic properties; Cartesian components, scalar and vector products. Vectors in 2-Space and 3-Space. Inner Product (Dot Product)

• Vector Product (Cross Product)
• Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
• Curves. Arc Length. Curvature. Torsion
• Gradient of a Scalar Field.
• Divergence of a Vector Field
• Curl of a Vector Field
• Directional derivatives.
• Alternative coordinate systems.

Systems

• Systems of ODEs as Models in Engineering Applications
• Basic Theory of Systems of ODEs.
• Matrix Solutions to coupled second order ODEs

Biomechanical Mathematics:

• Particle Dynamics
• Dynamics of Rigid Bodies
• Directional derivatives.
• Stress and strain, Hookes Law, Statics
• Complete Hookes Laws in 3D, cylindrical coordinates
• Shearing stress and strain.
• Statically determinate systems.
• Stress/Strain transformations. Mohrs stress/strain circle.
• Torsion, parallel and cylindrical shafts.

Modelling

• Analysis of a mass-spring system, Mechanical-electrical analogues.
• Coupled oscillators

Teaching and learning methods

Taught using a blended approach, incorporating “flipped mode” teaching and traditional problem classes.

• Comprehensive explanatory video lectures on each topic
• Case study recordings
• Weekly discussion tutorials and seminars
• Case study based problem sheets
• Weekly intensive problem classes – problem sheets worked on
• Self-study notes for each topic
• Self-testing on each block
• Mini-test at end of each block
• Past examination papers and solutions.
• Computing Labs on data analysis and numerical computation

Assessment strategy

The module is assessed by a combination of formative and summative problem sheets, self-testing and numerical laboratories associated with each block and a written examination paper as the final assessment.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.