The nonlinear equation for the rotational vibration of a pair of spur gears has a restriction that the analytical solution of the equation cannot be obtained. In this paper, the linear equation of vibration is derived theoretically and its physical model, i.e., the linear model of vibration is presented. The analytical solution of the linear equation, which is derived by analytical method, agrees well with the numerically calculated result by the nonlinear equation. By analyzing the analytical solution of the linear equation in detail, we clarified the relation between the waveforms of the vibration and the profile error of gear pairs, and also found that the effect of the contact ratio to the vibration is large and complex. The equivalent error, accounting for effects of the static load, the time-varying stiffness, and the profile error of gear pairs, is proposed in this paper. It can be considered as promising for evaluating the profile error, because the vibration of gear pairs is excited mainly by the equivalent error. Finally, for confirming the above results, the vibration of two tested gear pairs has been measured by an experimental set-up for this purpose.

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