A digital simulation model is developed to describe a loaded gear system consisting of a drive unit, a spur-gear pair and a load connected to a flexible shaft. This gear system vibrates in the 3-directions(circumferential, radial and axial direction) due to a relatively unstable excitation sources such as variation of tooth stiffness, tooth profile errors, backlash, eccentricity, etc. Furthermore this spur gear system is affected by a gyroscopic effect due to the axial vibration of the gear.
In general, the circumferential vibration is dominant in the spur gear system, but the radial vibration and the axial one can not be neglected, while vibrations in each direction are closely correlated with each other.
In this study, the equations of motion are set up by considering both the 3-directional vibrations including the gyroscopic effect due to the axial vibration and the variation of the dynamic load associated with the tooth profile errors. The tooth profile error is assumed to be a periodic function of time and tooth mesh frequency in the form of cosine series, and the tooth stiffness is approximated as a periodic function of time and tooth mesh frequency, i.e., $K_s(t)=K_s(1-C_1cosω_ft)$, to account for the single and double tooth contact in the mesh. Therefore, the dynamic load can be described as $K_s(t)(η(t)-g+e(t))$, where g is the amount of backlash.
The equations of motion formulated in a differential form are integrated numerically using Runge-Kutta 4th order method.
The magnitude of the radial vibration is one half of that of the circumferential vibration, and the axial vibration is much smaller than that of the other directions, and then the gyroscopic effect is negligible. Therefore if circumferential vibration is stable, the radial vibration and the axial one are also stable.
Experimentally obtained time signal and frequency spectra of the vibration are compared with the numerical results. It is shown that the theoretical and experimental results agree fairly well. From the numerical calculation it is found that the tooth profile error is the major sources of vibration.
