Application of the equations and rules in Section 19.13.1 allows dynamic stability charts to be plotted. The following case study illustrates the application and interpretation of dynamic stability charts. A dynamic stability chart gives the ratio of grinding stiffness to static machine stiffness at the limit of stability. In grinding, it is usually found that increasing this ratio through higher grinding force stiffness or lower machine stiffness makes the process unstable. Conversely, reducing the ratio stabilizes the process. In centerless grinding, it is not possible to make this general conclusion. The following example includes a dynamic stability chart for centerless grinding and explains how to interpret the chart.
Figure 19.56 is a dynamic stability chart for в = 6.5°. Dynamic stability charts are calculated using Equations 19.78, 19.80, and 19.81. The frequency ratio is calculated from Equation 19.78 or 19.81 and the boundary value of stiffness ratio from Equation 19.80. The frequency ratio is
o)/m0 = n ■ Q lrn0 (19.85)
The frequency ratio is the angular frequency of vibration divided by natural frequency. Since the vibration frequency equals number of waves times angular workspeed in radians per second, frequency ratio can be expressed in terms of number of waves for particular workspeed and natural frequency.
Although the charts are presented in terms of number of waves, it is necessary to remember that the stability limit at a particular point requires the particular workspeed to fulfill the relationship with natural frequency expressed by Equation 19.85.
The dynamic stability chart is presented in terms of number of waves n. This is necessary since each value of waviness has slightly differing behavior from the neighboring pairs of waviness values. The stability for 16 to 17 waves is different from 18 to 19 waves. The boundaries tend to repeat but are shifted vertically and horizontally.
In practice, dynamic instability is experienced where the product n Q is close to the natural frequency. This means that dynamic stability problems will not be experienced if the angular workspeed is selected to avoid potentially unstable waviness conditions.
Two sets of dynamic stability boundaries are shown. On the original figure conventional “up” boundaries were shown in blue and unconventional “down” boundaries were shown in red.