# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 1 Introduction
[15] used a genetic algorithm that makes use of an artificial neural network model of the cutting force to optimize the feed rate and spindle speed in sculptured surface machining. Their results are very accurate and effective. However, their methods can hardly be separated from the corresponding milling condition and need much experimental data. Recently, [PERSON] et al. [16] even proposed a robust optimization in which the machining parameters are chosen based on a combination of methodologies instead of the mechanical mechanism. The influence of noise on the surface roughness can be reduced effectively, but a mass of experiments are needed to be conducted. Various feed rate optimization strategies have different feed rate rescheduling control parameters such as reference cutting force [15; 17], MRR, chip thickness, surface roughness [18; 19], tool deflection [20], etc. Conventionally, the planning of the feed rate variation is to consider only one constraint at all machining segments, such as keeping constant MRR, keeping constant tool deflection value [21] or keeping constant resultant cutting force. [PERSON] et al. [22] compared the influence of different control parameters on the machining time and feed rate changes and proposed strategies combining various feed rate control parameters such as min (MRR, chip, force), max (MRR, chip, force), etc. Furthermore, [PERSON] et al. [23] proposed a proper selection of tilt and feed sense angles to avoid tool deflection due to the cutting forces. Different feeds and cutting speeds were also determined based on different hardness levels of the whole zones of advanced high-strength steels workpiece. [PERSON] et al. [24] and [PERSON] et al. [25] proposed a feed rate optimization by considering the constraint from the feed drive system. In order to preserve drive component life, avoid actuator saturation, and limit the dynamic positioning errors and structural vibrations. The velocity, acceleration,and jerk magnitudes commanded to each actuator were considered, and these methods have been experimentally validated in rough machining of a sculptured surface. It can be observed that most of the research about feed rate optimization is based on the cutting force and MRR. This can be appropriate for a rigid workpiece, and it will be different when the workpiece is flexible. Both static and dynamic analyses are needed to be considered because the flexibility of the workpiece would induce vibration or even instability. When the amplitude of vibration is large enough, the plastic deformation will be created instead of elastic deformation, which will aggravate the machining accuracy. [PERSON] et al. [26] reviewed that the part deformation may be caused by the machining load, residual stress, temperature, cutting sequence, cutting path, fixture conditions or vibration variation, which will result in the dimensional error and affect the surface quality. The machining deformation errors caused by both flexible cutter [27; 28] and flexible workpiece [29] have been studied in much research. With the use of the finite element method (FEM), [PERSON] et al. [29] proposed a machined surface error estimation method by the static analysis and explicit dynamic analysis, the static deformation due to clamping force and the dynamic deformation due to milling forces are considered and calculated. Based on the virtual machining technology withanalytical models, this paper studied and simulated the cutting force and vibration of the whole end milling process and proposed a feed rate variation strategy. A general flow diagram of the proposed feed rate variation strategy is shown in Figure 1. In the vibration analysis, the stability of the cutting process is estimated with the use of the stability lobe diagram (SLD) and the forced vibration response is obtained combining the cutting force simulation from [PERSON]'s method [30] and dynamic performance simulation from the spectro-geometric-Ritz method (SGM) proposed by [PERSON] et al. [31].