A mathematical model of the torsional-axial chatter vibrations in drilling is presented. The model considers the exact kinematics of both rigid body, and coupled torsional and axial vibrations of the drill. The drill is modeled as a pretwisted beam that exhibits axial deflections due to torque and thrust loading. A mechanistic cutting force model is used to model the cutting torque and thrust as a function of feedrate, speed, radial depth of cut, and drill geometry. The drill rotates and feeds axially into the workpiece while the structural vibrations are excited by the cutting torque and thrust force. The exact location of the drill edge is predicted using the model, and the generated surface is digitized at discrete time intervals. The distribution of chip thickness, which is affected by both rigid body motion and structural vibrations, is evaluated by subtracting the presently generated surface from the previous one. The model considers nonlinearities in cutting coefficients, tool jumping out of cut and overlapping of multiple regeneration waves. The dynamic chip thickness obtained from the true kinematics model allows simultaneous prediction of force, torque, power and dimensional form errors left on the surface. The time domain simulation model allows prediction of stability lobes. The paper provides details of the mathematical model, supported by experimental results of both stable and unstable cuts.

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