Dynamic modeling and analysis of an axially moving and spinning Rayleigh beam based on a time-varying element

Abstract

An axially moving and spinning beam, i.e., a beam having simultaneous axial movement and spin rotation, has been widely used in some engineering fields. In general, the force status of an axially moving and spinning beam is quite complex, with the beam simultaneously producing deformations of tension and compression and bending and torsion. Presently, the main method employed in studies concerning the dynamics of axially moving and spinning beams is structural dynamics, which focuses on the vibration and stability of the beam. Nevertheless, to describe the global motion and deformation motion of an axially moving and spinning beam more efficiently and accurately, scholars have proposed an Euler element model that is suitable for a slender beam. In the Euler element, because the moment of inertia of the cross section is ignored, large errors may occur when the element is used in the dynamic modeling and analysis of an axially moving and spinning beam with a small slenderness ratio. To solve this problem, a new element model based on the Rayleigh beam theory, in which the moments of inertia of the cross section around the beam's neutral as well as center axis are naturally considered by integrating the virtual work of the inertial force at each point in the element [herein referred to as the time-varying Rayleigh beam-shaft element (TVRBSE)], is presented in this paper. Dynamic analysis was performed for the axially moving and spinning cantilever Rayleigh beam, and the kinematics results were compared with those of the structural dynamics method reported in the literature. Both results were consistent with each other, which verifies the correctness of TVRBSE. The dynamic analysis based on TVRBSE can also simultaneously obtain the internal forces (axial force, bending moment, and torque) at each section of the beam and the stresses they cause. Moreover, the solution process for the proposed TVRBSE was easier to program and demonstrated good adaptability to changes in the loading and constraint conditions of the beam, a very advantageous feature. Essentially, the results of this study show that TVRBSE can accurately and effectively solve the dynamic problem of the axially moving and spinning beam.