On the mathematical modelling of beams rotating about a fixed axis

Abstract

In this paper, the equations of motion of a rotating beam encountered in various mechanical systems are given in the Euler-Newtoian form using four different dynamic modelling approaches. These models differ from each other in that they use different elastic displacements to define the state of deformed beam, i.e. the longitudinal and transversal deflections model, the axial and transversal deflections model, the axial, transversal deflections and slope angle model, and finally the transversal deflection with normal force model, which are abbreviatedly denoted by the UVM, the SVM, the SVφM, and the VMM. Following a brief discussion about geometric relationship among three linear elastic displacements,-that is, the longitudinal u-, the transversal v-, and the axial s-displacements- geometric stiffening effect, and the choice of spatial functions consistent with boundary conditions to discretize equations of motion, simulation results found from these models are presented in graphics, and comparatively evaluated. It is concluded that the SVM and the SVφM is most reliable models provided that the comparison functions for axial displacement s are the eigenmodes of a fixed-free, longitudinally vibrating rod. The VNM appears to be a prudential model, because it always gives results above those obtained by the SVM.