During vertical spindle surface grinding, adjustments are made on a multi-axis computer numerical control (CNC) machine by

Question:

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

where k = 2.1 × 10⁴ N/m; b = 0.78 Ns/m; m = 1.2 × 10^-4 kg; K_C = 1.5 × 10⁴N/mm; and T = 0.004s: The parameter K_f is varied to adjust the system. Find the range of K_f under which the system is stable (Hekman, 1999).