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.

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

Dynamics I 86PK

Mathematical Modelling 86PN

Mathematical Methods

90% Examination, 10% Coursework.

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.

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

1. Formulate and solve problems concerned with the motion of point particles in central force fields, including the equation of the path, the existence of circular orbits and their stability and the two-body problem.

2. Explain when conservation of energy can be used, derive potentials for conservative force fields and deduce aspects of the point of point particles by applying conservation of energy.

3. Formulate and solve the equations of motion for point particles in rotating frames of reference. Derive formulae for velocity and acceleration in accelerating frames of reference.

4. Derive equations of motion for systems of point particles.

5. Define the inertia tensor for a rigid body. Calculate the centre of mass and the inertia tensor for a variety of mass distributions, including bodies with spherical symmetry or axisymmetry and planar and rod-like bodies. Prove results concerning the inertia tensor such as the parallel and perpendicular axis theorems.

6. Formulate and solve for the motion of a rigid body under given forces and torques, including rolling of a cylinder and the torque-free precession of an axisymmetric top.

7. Construct the Lagrangian for the constrained motion of systems of points particles and rigid bodies and use it to derive the Lagrange equations.

8. Derive the equations expressing conservation of mass, linear momentum and energy for a deformable body undergoing motion described by a single spatial variable. Solve one-dimensional problems involving the behaviour of elastic rods, beams and fluids.