Down boundaries were colored red in Figure 19.56. Unstable zones for down boundaries lie beneath the line and stable zones above the line. Down boundaries indicate unstable zones for increasingly negative grinding forces and stable zones for more positive forces. In a case where a down boundary lies in the negative region of the chart, the system cannot go unstable at the corresponding waviness condition. This is reassuring, since there are large areas of red on the chart. Most of these areas represent areas of absolute stability.
However, there are small areas where down boundaries indicate unstable regions for low positive grinding forces and stable regions for high positive forces. This needs further examination.
An example of a down boundary for low positive grinding forces occurs close to n = 26 suggesting a region affected by poor geometric stability.
The geometric setup в = 6.5° is generally considered to be favorable as this is one of the most stable set-up geometries and is close to в = 6.4°, the setup recommended as optimum by Hashimoto and Lahoti [2004]. Geometric stability for a tangent angle of 6.5° is shown in Figure 19.57 for integer waves. This may be compared with Figure 19.50 for в = 7.5°. In both cases, the waviness n = 26 is only just stable.
Unfortunately, Figure 19.57 does not clarify why the red boundary goes into the positive force domain because the waviness n = 26 is shown as stable. Figure 19.58 is more helpful in this regard, showing geometric stability for noninteger wave numbers.
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Figure 19.58 reveals marginal geometric instability at several waviness values. These values correspond precisely with the conditions where the dynamic stability charts show red boundaries in the positive force region. The conclusion is clear. At these values of waviness, it is possible to experience dynamic instability. A further conclusion is that if instability is experienced at one of these waviness values, the grinding force would need to be increased to stabilize the system. This is the opposite of the normal situation in machining where the system is stabilized by reducing the force.
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