# Mathematics BSc/MA/MSci/MA(SocSci)

## 3H: Mechanics of Rigid and Deformable Bodies MATHS4078

• Academic Session: 2022-23
• School: School of Mathematics and Statistics
• Credits: 20
• Level: Level 4 (SCQF level 10)
• Typically Offered: Semester 2
• Available to Visiting Students: Yes
• Available to Erasmus Students: Yes

### Short Description

This course considers the two-dimensional motion of interacting particles, the one-dimensional motion of deformable bodies and the three-dimensional motion of deformable bodies using Cartesian tensors.

### Timetable

34 x 1hr lectures and 12 x 1hr tutorials in a semester.

### Excluded Courses

Dynamics I 86PK

Mathematical Modelling 86PN

### Co-requisites

Mathematical Methods

### Assessment

90% Examination, 10% Coursework.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? Not applicable

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

### Course Aims

The aim of the course Modelling Rigid and Deformable Bodies is to extend the work on the one dimensional motion of particles covered in 2C, first, to the planar motion of particles acted on by a central force, second, to the motion of rigid body using the Lagrangian methodology, third, to the one-dimensional motion of deformable bodies, and fourth, to the three-dimensional motion of deformable bodies using Cartesian tensors. Conservation laws of linear momentum, angular momentum and energy will be considered for rigid and deformable bodies. The concept of entropy will be introduced and used to develop constitutive relations for fluids and solids leading to the solution of simple one-dimensional problems for these classes of materials.

### Intended Learning Outcomes of Course

By the end of this course students will be able to:

1. derive conservation laws of linear momentum, angular momentum and energy for systems of particles interacting along their line of centres;

2. solve two-dimensional motions involving central forces and identify what quantities are conserved in the motions;

3. derive the vectorial relationship between linear and angular velocity for the motion of the particles in a rigid body;

4. calculate inertia tensors for various simple rigid bodies;

5. prove and apply the parallel-axis and perpendicular-axis theorems;

6. construct the Lagrangian function for simple motions of particles and rigid bodies;

7. formulate and solve the Lagrangian equations for simple motions of particles and rigid bodies;

8. derive the equations expressing conservation of mass, linear momentum and energy for a deformable body undergoing motion described by a single spatial variable;

9. derive the entropy inequality and use it to particularise constitutive equations;

10. define Helmholtz free energy, internal energy and specific entropy and establish expressions for these functions given an equation of state for a perfect fluid;

11. derive the Rankine-Hugoniot conditions and apply them to the propagation of shocks in a motion described by a single spatial variable;

12. solve simple one-dimensional problems involving the behaviour of fluids and beams;

manipulate Cartesian tensors and use them to derive the conservation equations for a three-dimensional deformable body.

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