Finite-element dynamic analysis of a rotating shaft with or without nonlinear boundary conditions subject to a moving load

A C-0 continuity isoparametric finite-element formulation is presented for the dynamic analysis of a rotating or nonrotating beam with or without nonlinear boundary conditions subject to a moving load. The nonlinear end conditions arise from nonlinear rolling bearings (both the nonlinear stiffness and clearance(s) are accounted for) supporting a rotating shaft. The shaft finite-element model includes shear deformation, rotary inertia, elastic bending, and gyroscopic effect. Lagrange's equations are employed to derive system equations of motion which, in turn, are decoupled using modal analysis expressed in the normal coordinate representation. The analyses are implemented in the finite-element program 'DAMRO 1'. Dynamic deflections under the moving load of rotating and nonrotating simply supported shafts are compared with those obtained using exact solutions and other published methods and a typical coincidence is obtained. Samples of the results, in both the time and frequency domains, of a rotating shaft incorporating ball bearings are presented for different values of the bearing clearance. And the results show that systems incorporating ball bearings with tight (zero) clearance have the smallest amplitude-smoothest profile dynamic deflections. Moreover, for a system with bearing clearance, the vibration spectra of the shaft response under a moving load show modulation of the system natural frequencies by a combination of shaft rotational and bearing cage frequencies. However, for a simply supported rotating shaft, the first natural frequency in bending dominates the response spectrum. The paper presents the first finite-element formulation for the dynamic analysis of a rotating shaft with or without nonlinear boundary conditions under the action of a moving load.