ANALYSIS OF PERIODICALLY VARYING GEAR MESH SYSTEMS WITH COULOMB FRICTION USING FLOQUET THEORY

This article presents a new analytical model of a gear pair with time varying mesh stiffness, viscous damping and sliding friction parameters. Unlike previous models, the excitation consists of three separate terms, namely the unloaded transmission error, time-invariant external torque and the periodically varying sliding friction force. A Coulomb friction model is considered using first a quasi-static mean transmitted load that is represented by the Meissner equation. Then, a truly dynamic force between gear teeth is described that leads to a triangular function, and after appropriate substitutions, this assumes the form of the Bessel equation of the one-third order. For the damped Meissner equation, the forced vibration response is found with the application of Floquet theory. Exact integrals are calculated for the state transition matrix in a piecewise manner, instead of using the Fourier series expansion, thus eliminating the mode truncation errors. From the state transition matrix, unstable zones are identified and the actual forced response of the system is found in terms of dynamic transmission error for these zones. With the aid of an example, the significance of sliding friction on system response and stability is examined. Finally, key advantages and the need for analytical methods are demonstrated for such systems.