## 1 Introduction
### Ball End Milling  
In the past, the dynamic time responses were determined by considering the local cutting force coefficients, measured transfer function, and instantaneous regenerative chip load, using a lumped model of the tool/spindle with a one-degree-of-freedom or two-degree-of-freedom linear system, but the dynamic behavior of a flexible rotor-bearing system for ball end milling is seldom discussed. [PERSON] and [PERSON] [14] modeled the cutting coefficients for ball end milling based on the shear component of the orthogonal cutting separating flanks. [PERSON] and [PERSON] [15] further modified the model to include the static and dynamic processes of ball end milling. The variation of shear coefficient with chip thickness and cutting point position can be obtained from orthogonal cutting experiments with tilted transformation. In high-speed milling, a classical regenerative vibration model is proposed by [PERSON] et al. [16] to generate a simplified discrete model of the nonlinear vibration. [PERSON] et al. [17] presented the identification of system parameters including cutting force coefficients and equivalent parameters. The cutting forces predicted using the recursive least squares method were almost in agreement with the experimental results. By using the Fourier transform filter and the extended Kalman filter, the identified parameters, such as equivalent mass, damping, and stiffness, allow the prediction of the displacements of tool-tip similar to experimental measurements. In [PERSON] et al. [18], the critical speed analysis of the rotor-bearing system for ball end milling and nonlinear effects were also investigated. The solved critical speeds of the four proposed cases were verified with two sets of experimental data. In addition, the relationship between the natural frequencies and the critical speeds for various feeds per tooth, with linear and nonlinear cutting forces in linear and nonlinear rotor systems, was discussed. However, the rotor was assumed to be rigid so that the gyroscopic effects of the system cannot be included. [PERSON] et al. [19] proposed a method for predicting chatter stability lobes in high-speed milling considering the dynamics of variable-speed spindles. Using the dynamic finite element model of the high-speed spindle system, the influence of speed on the spindle and bearings was systematically investigated through simulation and experiment. The whirl map has not been established. [PERSON] and [PERSON] [20] presented an optimal spindle speed determination method for reducing vibration during ball end milling. They suggested an original method of spindle speed optimization based on the Liao-Young criterion to reduce the vibration situation. The gyroscopic matrix was not found in their study. An online identification method, that considers the stiffness characteristics of the tool holder-spindle system, was proposed by [PERSON] et al. [21]. The results showed that machine tool stiffness effects can be identified and removed in real time using the new method. Only a two-degree-of-freedom vibration model of cutter was established. [PERSON] et al. [22] first established a model characterized by multiple delays for a five-axis system of ball end milling. They analyzed the tool orientation effects on the prediction of the stability for ball end milling. The correctness of the proposed model was verified with the relevant experiment. Though the milling system was modeled as a mass-spring-damper system with multiple degrees of freedom, the gyroscopic effect of the system has not been considered. A method for identifying the milling force coefficients while taking vibration into account, in order to accurately predict milling forces for slender end mills, was presented by [PERSON] et al. [23]. Combined with the frequency response function measured at the end of the cutter handle, and the Timoshenko beam model of the cutter using a finite element, they identified the interface dynamic parameters between the cutter and the cutter handle.