Vibration of high speed helical geared shaft systems mounted on rigid bearings

Analytical model for linear vibration analysis of a pair of meshing helical gears mounted on compliant spinning parallel shafts is developed. The gears are modeled as rigid disks while the shafts are modeled as flexible rotating cantilever beams. The rotational speed is high such that the gyroscopic effect is non-negligible. Hamilton's principle is used to obtain the non-dimensional governing equations. Longitudinal and rotational mesh stiffnesses and corresponding elastic energies are incorporated in the total strain energy formulation. Extended operator formulation is used to express the system equations in a consolidated form and later, discretized using Galerkin method. Gyroscopic system structure is apparent in the system equations. Natural frequency sensitivity to various system parameters such as the rotational speed and mesh stiffnesses is determined. Response to loaded static transmission error at the mesh is obtained by performing modal analysis after reducing the system to a first order form. The study shows that gyroscopic effect is present even for short lengths of the shafts. Natural frequency veering between flexural frequencies is present at low speeds and rare at high speeds. For coupled frequencies, response of the system is found to be approximated by modal superposition of smaller number of modes.