## MECH2610 Engineering Mechanics

### 20 creditsClass Size: 300

Module manager: Dr Qingen Meng
Email: Q.Meng@leeds.ac.uk

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2022/23

### Pre-requisites

 MECH1230 Solid Mechanics MECH1520 Engineering Mathematics

Module replaces

MECH2110MECH2540

This module is not approved as a discovery module

### Objectives

On completion of this module students should be in a position to model mechanical, structural and fluid systems in terms of one, or more, ordinary or partial differential equations and solve these equations using appropriate analytical or numerical techniques. The mathematical techniques will be demonstrated and applied to particular problems of estimating stresses and strains in mainly elastically loaded engineering solid structures.

By the end of the course students should be able to derive force equilibrium equations, strain-displacement equations and elastic stress-strain equations and be able to combine them:
(i) to address how the bending of a beam is influenced by the beam cross-section, its loading and its modulus of elasticity and yield stress; and
(ii) to use the mathematical theory of elasticity to solve problems of stressing cylinders and discs.
(iii) use statistical theory to test hypotheses about properties of materials.

Learning outcomes
On successfully completing this module, students will have learned how to:
1. Model the behaviour of second order mechanical and electrical systems, and solve the associated ordinary differential equations analytically
2. Use finite differences to formulate numerical solutions of ordinary differential equations representing engineering systems
3. Apply the concepts and notation of vector calculus to analyse relevant scalar and vector fields and represent partial differential equations efficiently
4. Use Fourier Series to capture the behaviour of periodic systems
5. Predict how bending and stability of a beam is affected by its loading, geometry and physical properties
6. Predict stresses in cylinders and discs and understand how properties of materials affect failure conditions

Upon successful completion of this module the following UK-SPEC learning outcome descriptors are satisfied:

A comprehensive knowledge and understanding of the scientific principles and methodology necessary to underpin their education in their engineering discipline, and an understanding and know-how of the scientific principles of related disciplines, to enable appreciation of the scientific and engineering context, and to support their understanding of relevant historical, current and future developments and technologies (SM1m)
Knowledge and understanding of mathematical and statistical methods necessary to underpin their education in their engineering discipline and to enable them to apply a range of mathematical and statistical methods, tools and notations proficiently and critically in the analysis and solution of engineering problems (SM2m)
Ability to identify, classify and describe the performance of systems and components through the use of analytical methods and modelling techniques (EA2)
Ability to apply relevant practical and laboratory skills (P3)
Apply their skills in problem solving, communication, information retrieval, working with others and the effective use of general IT facilities (G1)

Skills outcomes
Mathematical modelling, structural mechanics, numerical solution of Ordinary Differential Equations.

### Syllabus

- Classification of Ordinary Differential Equations (ODEs) and use of non-dimensionalisation to highlight dominant physical processes.
- Solution of 1st order ODEs using direct integration, separation of variables, homogeneous functions and integrating factors.
- Introduction to Laplace Transforms and their use in solution of initial value ODEs.
- Double integration and its applications to calculate moments of area
- For bending beam stresses, relationship between stresses, bending moments and cross-section size and shape. Bending moment diagrams and calculating stresses in such beams.
- Calculation of (i) additional stresses in asymmetric beam sections and warping that occurs in such beams, (ii) shear stress distributions in symmetric beams bent with a bending moment varying along its length.
- Derivation of relationships between bending moment and curvature of symmetric beams. Mathematical techniques for the solution of these and higher order ODEs via the methods of homogeneous equations, and the determination of Particular Integrals for inhomogeneous equations via the methods of undetermined coefficients and variation of parameters.
- Derivation and solution of equations for limit loads on structures and the instability and collapse of axially loaded beams.
- Partial differential equations in the context of the mathematical theory of elasticity: derivation of radial and hoop stresses in an asymmetric stress field and relations between radial and hoop strains and displacements. Mathematical solution of these equations with application to problems of pressurizing a thick walled cylinder, shrink fits and stresses in rotating discs and cylinders.
- Taylor series and Finite Difference methods for the numerical solution of differential equations arising in beam bending problems.
- Introduction to Vector Calculus.
- Fourier Series.
- Hypothesis testing for material properties.

### Teaching methods

 Delivery type Number Length hours Student hours Example Class 10 1.00 10.00 Class tests, exams and assessment 1 1.00 1.00 Class tests, exams and assessment 1 2.00 2.00 Lecture 40 1.00 40.00 Practical 2 1.00 2.00 Tutorial 4 1.00 4.00 Private study hours 141.00 Total Contact hours 59.00 Total hours (100hr per 10 credits) 200.00

### Private study

- Reviewing lecture notes, solving example sheets, preparing for tutorials, class test and assignment.
- Revising for final exam.
- Students to spend 1 hour preparing for each lecture; 2 hours preparing for each tutorial; 2 hours preparing for each practical; 15 hours preparing for class test; 74 hours exam preparation.

### Opportunities for Formative Feedback

An online discussion board will be monitored during specified times each week.
Minerva/TopHat quiz after each of the 7 units.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Practical Assignment based on practical session 20.00 In-course Assessment Class test 20.00 Total percentage (Assessment Coursework) 40.00

1) Coursework marks carried forward and 60% resit exam OR 2) 100% exam

Exams
 Exam type Exam duration % of formal assessment Unseen exam 2 hr 60.00 Total percentage (Assessment Exams) 60.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated