An analytical expression for end milling forces and tool deflection using Fourier series

In this research, a novel and generalized analytical expression of cutting force and tool deflection in end milling is presented as a function of tool rotational angle and other cutting parameters. The discontinuous cutting force function caused by periodic tool entry and exit is changed to an integrable continuous function using Fourier series expansion. Tool deflection is also formulated explicitly by the direct integration of the distributed loads along the helical cutting edges. Cutting conditions, tool geometry, runout components, and the stiffness of tool clamping part are considered in estimating the cutting force and tool deflection. Cumbersome computational procedures needed to check whether segmented cutting edges are engaged in cutting or not are eliminated by this proposed method. The presented analytical approach has advantages in flexibility, prediction time, and accuracy as compared with other numerical techniques. In addition, the effects of cutting conditions and run-outs, such as eccentricity and tilting on the cutting force and tool deflection, can be analyzed quantitatively in the time domain or frequency domain. The validity and effectiveness of the suggested method are verified through a series of cutting tests. The model presented in this research can be used in real-time machining error estimation and cutting condition selection for error minimization since the form accuracy is easily estimated from the acquired tool deflection curve.