Theoretical analysis of the natural frequency of a geared system under the influence of a variable mesh stiffness

Abstract

Owing to the change in the number of contact tooth pairs, the mesh stiffness is periodically time varying, which leads to additional time-variant coefficients in the corresponding governing equations of a geared dynamic system. The natural frequency of such a system differs distinctly from that of a time-invariant system. However, it has not gained sufficient attention in geared dynamic analysis, and the natural frequency of a geared system is usually assumed to be the average frequency obtained using the time-average mesh stiffness. Whether it is reasonable to obtain the natural frequency of a geared system in this way should be analysed seriously. Thus, the natural frequency of parametric vibration for a spur-gear-pair system will be researched using the Floquet theory in this paper. The influences of the periodically time-varying mesh stiffness parameters (including the mesh frequency and the contact ratio) on the natural frequency will be discussed in detail. Contrasts between the natural frequency and the average frequency for the system in the stable and unstable regions will be processed and the scope of application for taking the average frequency as a substitution for the natural frequency of a geared system will then be presented.