Proportional gain Kp usual culprit on CNC machining centers for your problem.
You might also may want too look into adjusting the Integral Gain also. I have also seen spindles do similar things if the orientation ring has been missaligned or has bad teeth, damaged sensor, bad orientation card on spindle drive and even once I had bad transistor modules that caused extreme oscillation. Remember the usual gap on the orientation sensor should be .005"
Chris