A Screw Theory of Timoshenko Beams

Abstract

In this work, the classic theory of Timoshenko beams is revisited using screw theory. The theory of screws is familiar from robotics and the theory of mechanisms. A key feature of the screw theory is that translations and rotations are treated on an equal footing and here this means that bending, torsion, and extensions can all be considered together in a particularly simple manner. By combining forces and torques into a six-dimensional vector called a wrench, Hooke’s law for the Timoshenko beam can be written in a very simple form. From here simple expressions can be found for the kinetic and potential energy densities of the beam. Hence equations of motion for small vibrations of the beam can be easily derived. The screw theory also leads to a new understanding of the boundary conditions for beams. It is demonstrated that simple boundary conditions are closely related to mechanical joints. In order to set up the boundary conditions for a beam attached to a joint, a system of wrenches dual to the screws representing the freedoms of the joint must be found. Finally, a screw version of the Rayleigh–Ritz numerical method is introduced. An example is investigated in which the boundary conditions on the beam lead to vibrational modes of the beam involving bending, torsion, and extension at the same time.