Up boundaries were colored blue in Figure 19.56. Unstable zones lie above the up line and stable zones beneath. An example shows a stable zone marked with a “+” sign and an unstable zone
marked with a “-” sign. Up boundaries were determined by calculating the angle x for each point on the boundary from Equation 19.83 and checking that the point satisfied Equation 19.82.
Up boundaries lie mainly in the top positive region of Figure 19.56 but are occasionally found in the low negative region. Negative boundaries seem irrelevant at first since grinding stiffness and static machine stiffness are always positive. However, the small areas where up boundaries extend into the lower negative region must be avoided. These are up boundaries where if the corresponding speed conditions are applied the process is absolutely unstable. Avoiding the unwanted speed condition is straightforward and is discussed further in Section 19.13.5