Screw theory of Timoshenko Beams

The proposed research will produce a relatively simply model of elastic beams and rods. At the physical level the theory is standard and well understood, the theorey of Timoshenko beams. However, the contribution we seek to make is to put this theory on a sound geometrical footing. In this way it will be possible to consider complicated loadings of the beam and constraints on the ends of the beam which are not straightforward. This will by done by using the Lie algebra of the group of rigid body motions, sometimes called Screw theory. In this theory small motions are represented by six dimensional vectors which record the linear and angular displacements along the beam. Forces and torques are also combined into six-vectors called wrenches. In this formalism the six possible type of deflection of the beam, bending and shearing in two possible directions, torsion and extension/compression, are treated in the same fashion and hence combinations of these six types of deflection can be treated together. Using this formalism equations will be developed for small vibrations of the beams. In the case of pure bending these can be compared with the equations from the standard theory.To compliment this geometrical theory we seek to develop numerical methods to approximate the solutions of the equations comprising the theory. The idea here will be to represent the solution as a static shape for the beam modulated by a harmonic function of time. A gain this is a standard technique, the novelty of our approach will be that the static shape of the beam will be represented by a spline curve in the Lie algebra. This is particularly well suited to the problem since the first and last screws defining the curve will correspond with the screws defining the boundary conditions for the beam. In this way the boundary conditions of the problem will be satisfied automatically.