Analysis of the vibratory excitation of gear systems: Basic theory

Formulation of the equations of motion of a generic gear system in the frequency domain is shown to require the Fourier-series coefficients of the components of vibration excitation; these components are the static transmission errors of the individual pairs of meshing gears in the system. A general expression for the static transmission error is derived and decomposed into components attributable to elastic tooth deformations and to deviations of tooth faces from perfect involute surfaces with uniform lead and spacing. The component due to tooth-face deviations is further decomposed into appropriately defined mean and random components. The harmonic components of the static transmission error that occur at integral multiples of the tooth-meshing frequency are shown to be caused by tooth deformations and mean deviations of the tooth faces from perfect involute surfaces. Harmonic components that occur at the remaining multiples of gear-rotation frequencies are shown to be caused by the random components of the tooth-face deviations. Expressions for the Fourier-series coefficients of all components of the static transmission error are derived in terms of two-dimensional Fourier transforms of local tooth-pair stiffnesses and stiffness-weighted weighted deviations of tooth faces from perfect involute surfaces. Results are valid for arbitrary, specified tooth-face contact regions and include spur gears as the special case of helical gears with zero helix angle.