Dynamics of low speed geared shaft systems mounted on rigid bearings

Analytical model for the linear vibration analysis of a pair of coupled spur geared shaft system is developed. The model is a hybrid discrete-continuous one in which the gears are modeled as rigid disks mounted on the elastic shafts having transverse as well as torsional flexibilities and supported by rigid bearings. The non-dimensional governing equations along with the natural boundary conditions are developed using the Hamilton's principle. An extended operator formulation is employed to simplify the system to a compact analytical form and demonstrate that the system is self-adjoint. The natural frequencies and vibration modes are determined by discretizing the system using the assumed modes method, where the critical feature is the use of Lagrange multipliers to enforce the matching boundary conditions at the disk-shaft interfaces. The sensitivities of the natural frequencies to various system parameters are examined and correlated with the modal energy distribution. Response due to the loaded static transmission error at the gear mesh is presented.