The waves generated on the workpiece as well as on the grinding wheel surfaces are the envelope of the relative vibration between them. To start with, the waves generated on the workpiece surface will be considered. In this case, the waves are the envelope of the periphery of the grinding wheel. As far as the following conditions are satisfied, the amplitudes of the relative vibration and the waves generated on the workpiece surface are identical: low vibration frequency, small relative amplitude, and low workpiece velocity. However, once the critical limit, determined with the above-mentioned parameters, is exceeded, the amplitude of waves generated on the workpiece surface becomes smaller than that of the relative vibration. In other words, the envelope curve is attenuated.
Assuming that the amplitude of the relative vibration and the waves are y and aw respectively, the following relationship can be derived:
where
is a critical amplitude, vw is the workpiece speed, is the angular chatter frequency, dw is the workpiece diameter, and ds is the grinding wheel diameter. The plus sign in Equation 8.2 is for cylindrical external grinding, the minus sign for internal grinding, and dw = ^ for surface grinding. When ycr < y the amplitude of waves becomes smaller than that of the relative vibration. Otherwise, both amplitudes are identical. As for the waves generated on the grinding wheel, the critical amplitude can be obtained by replacing the workpiece speed vw with the grinding wheel speed vs. Therefore, the critical amplitude is much larger for the waves generated on the grinding wheel because the wheel speed is much higher than the workpiece speed.
The calculated results from Equation 8.1 are given in Figure 8.4 [Inasaki 1975]. The geometrical interference is a strong nonlinear term in the grinding dynamics [Inasaki et al. 1974]. It is clear from Equations 8.1 and 8.2 that waves with large amplitude and higher frequency cannot be generated on the workpiece surface because the critical amplitude becomes small.