During vertical spindle surface grinding, adjustments are

Chapter 6, Problem 59

(choose chapter or problem)

During vertical spindle surface grinding, adjustments are made on a multi-axis computer numerical control (CNC) machine by measuring the applied force with a dynamometer and applying appropriate corrections. This feedback force control results in higher homogeneity and better tolerances in the resulting finished product. In a specific experiment with an extremely high feed rate, the transfer function from the desired depth of cut (DOC) to applied force was

\(\frac{F(s)}{D O C(s)}=\frac{K_C}{1+\frac{K_C}{m s^2+b s+k}-\frac{K_C}{K_f}\left(\frac{1}{T s+1}\right)}\)

where \(k=2.1 \times 10^4 \mathrm{~N} / \mathrm{m}, b=0.78 \mathrm{Ns} / \mathrm{m}, m=1.2 \times 10^{-4} \mathrm{~kg}, K_C=1.5 \times 10^4 \mathrm{~N} / \mathrm{mm}\), and \(T=0.004 \mathrm{~s}\). The parameter \(K_f\) is varied to adjust the system. Find the range of \(K_f\) under which the system is stable (Hekman, 1999).