A Screw Approach to Vibration Analysis of Beam Structures With Nonstandard Conditions

Abstract A major advantage of the screw theory is that translations and rotations are treated simultaneously, which can provide greater insight into the vibration phenomena, such as vibration centers and axes. The present study describes how these concepts are extended into beam theory. The stiffness matrix of a beam was derived by incorporating different types of vibrations, such as extension, compression, torsion, and bending, at the same time, which was then used to obtain the equations of motion, including nonstandard forms of boundary conditions. We then presented an analytical method to solve these equations by focusing on two distinct examples, namely the cantilever and robot link. In the first numerical example, the mode shapes of the beam could be regarded as rotations about the vibration centers or axes of the rigid bodies in a discrete system. In the second example, the analytical solutions of mode shapes and natural frequencies of a robot link, for which the revolute joints at both sides are not parallel, were presented to demonstrate the utility of the screw theory. We demonstrated that the screw approach could accurately describe the vibrations of both discrete and continuous systems and that the geometric meaning of the vibration modes of discrete systems can be extended into continuous systems.