Rotor Dynamic Response of a High-Speed Machine Tool Spindle

In high-speed machining, the maximum stable depth of cut at any spindle rotating frequency depends on the spindle-holder-tool dynamic stiffness as reflected at the tool’s free end. Because this dynamic stiffness can vary with rotating frequency, we have modeled the spindle dynamic response using a finite element-based rotordynamics approach. The spindle shaft is modeled using beam finite elements with four degrees-of-freedom at each node. The finite element model incorporates rotatory inertia, shear deformation, gyroscopic effects, transient disturbances, and steady-state synchronous unbalance. The spindle-holder interface is modeled using an effective translational and rotational stiffness. Because ball bearing stiffness is a nonlinear function of radial and thrust loads, contact angle, ball diameter, number of balls, operating internal clearance, and spindle speed, it is difficult to predict reliably. We have therefore estimated the bearing stiffness and spindle-holder interface stiffness by matching the first four natural frequencies predicted by the model with experiment (non-rotating spindle). We conclude that the ball bearing stiffness controls the rigid body modes while the spindle-holder interface stiffness controls the two bending modes. The experimental spindle-holder assembly frequency responses are presented as a function of spindle speed and compared to the rotordynamics model results. INTRODUCTION Early research in the area of milling stability [1-5] led to mathematical process models and the development of stability lobe diagrams, i.e., graphical charts that compactly represent stability information as a function of the control parameters: chip width and spindle speed. See Fig. 1. These studies led to a fundamental understanding of regeneration of waviness, or the overcutting of a machined surface by a vibrating cutter, as a primary feedback mechanism for the growth of self-excited vibrations (chatter) due to the modulation of the instantaneous chip thickness, cutting force variation, and subsequent tool vibration. Many subsequent research efforts have built on this work and the use of stability lobe diagrams, coupled with highspeed/power machining centers and improved cutting tool materials, has been shown to dramatically increase material removal rates (MRR). For example, high-speed machining has been applied in the aerospace industry, where the dramatic increases in MRR have allowed designers to replace assembly-intensive sheet metal buildups with monolithic aluminum components resulting in substantial cost savings [6]. Spindle Speed Stable zone Unstable zone (Chatter) Chip width Figure 1: Example stability lobe diagram. In general, stability lobe diagrams are developed by selecting the cutting parameters, which include the processdependent cutting force coefficients, radial immersion, and system dynamics (as reflected at the tool point of the machine-spindle-holder-tool assembly), and then carrying out the selected simulation algorithm. In general, the tool point frequency response function (FRF) is measured using impact testing, where an instrumented hammer is used to excite the tool point and the response is measured using an appropriate transducer (often a low-mass accelerometer), while the spindle is stationary, or non-rotating. The underlying assumption here is that the spindle dynamics do not change as the spindle speed is increased. However, any variation in the tool point FRF with changes in spindle speed will directly translate into errors in the stability limit predicted by the selected analysis technique. Potential sources for these variations are described in the following section. SPEED DEPENDENT SYSTEM DYNAMICS New spindle designs have significantly contributed to the productivity gains afforded by high-speed milling. These spindles are capable of accommodating tools up to 25 mm in diameter, for example, and rotate at speeds of 40,000 rev/min (rpm) and higher with powers of 40 kW and above. Typically, these spindles are directly driven by brushless motors and the spindle shaft is supported by hybrid angular contact bearings with silicon nitride balls. Bearing axial preload is kept essentially constant during thermally-driven changes to the spindle dimensions by spring or hydraulic arrangements. The tool point FRF for these spindles depends on a large number of factors, including tool length [7-11], holder characteristics [12-13], drawbar force [14], spindle shaft geometry, and the stiffness and damping provided by the spindle shaft bearings. While most of these factors are independent of the rotational speed of the spindle, it is known that the radial and axial stiffnesses of angular contact bearings vary with changes in load and speed. This behavior is generally attributed to changes in the contact angles for the inner and outer races due to applied axial/radial loads, centrifugal and gyroscopic forces on the rotating balls, and variations in the operating internal clearance due to thermal and centrifugal effects. Additionally, we propose that the contact conditions between the holder and spindle taper may change with spindle speed. Therefore, non-rotating measurements of tool point dynamics may not adequately describe the rotating system. Furthermore, stability predictions based on these non-rotating measurements may be in error. Numerous authors have investigated the dynamic behavior of angular contact bearings both analytically and experimentally. Jones [15] developed a general theoretical model for bearing dynamic loads. Harris [16] showed that the radial stiffness of angular contact bearings decreases with increasing radial load and speed. Shin [17] and Chen et al. [18] showed that the dynamic characteristics of the spindle system vary with speed-dependent changes in the bearing stiffness, affecting chatter stability. Chen and Wang [19] demonstrated the effect of changing end loads on the system dynamics and describe a method for including the analytically predicted, speed-dependent dynamics in the computation of stability lobes. Jorgenson and Shin [20] compare analytical predictions of spindle dynamics at speed with experimental measurements. Schmitz et al. [21] presented an experimental method for the prediction of stable cutting regions which incorporates variation in the spindle-holdertool dynamics under rotation. In this work, we build on these previous efforts by developing a finite element (FE)-based rotordynamics model of a milling spindle. The spindle shaft is modeled using beam elements with four degrees-of-freedom at each node. The finite element model incorporates rotatory inertia, shear deformation, gyroscopic effects, and forced vibration due to unbalance. Bearing stiffness may be considered a function of speed and used for rotor critical speed computations. Also, the tool holder is coupled to the spindle internal taper through springs and dampers. Simulation results are compared to both rotating and nonrotating impact test data. The spindleholder model and experimental test setup are described in the following sections. Figure 2: Spindle-holder model. SPINDLE-HOLDER FINITE ELEMENT MODEL The rotordynamics of high-speed flexible shafts is influenced by the complex interaction between unbalance forces, bearing stiffness and damping, inertial properties of the rotor, gyroscopic stiffening effects, aerodynamic coupling, and speed-dependent system critical speeds. For stable high-speed operation bearings must be designed with the appropriate stiffness and damping properties, selected on the basis of a detailed rotordynamic analysis of the rotor system [22-27]. To investigate the influence of these effects on high-speed milling dynamics, the spindle used in this study was a 36000 rpm/36 kW direct drive, rolling element bearing spindle located in the University of Florida Machine Tool Research Center. The spindle shaft is supported by two pairs of hybrid angular contact bearing (silicon nitride balls with steel races); a floating mount carried the rear bearings, with the axial preload provided by a stack of Bellville washers. The spindle shaft-holder geometry and relevant dimensions are provided in Fig. 2 and Table 1. Table 1: Dimensions for spindle finite element model. Cross-section Inner diameter (mm) Outer diameter (mm)