Rotor Dynamic Response of a High-Speed Machine Tool Spindle

Transcription

1 Rotor Dynamic Response of a High-Speed Machine Tool Spindle Nagaraj Arakere, Assoc. Prof., Tony L. Schmitz, Asst. Prof., Chi-Hung Cheng, Grad. Student University of Florida, Department of Mechanical and Aerospace Engineering 237 MAE-B, Gainesville, FL ABSTRACT 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 Chip width Stable zone Spindle Speed Unstable zone (Chatter) Figure 1: Example stability lobe diagram. 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]. 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.

2 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,

3 the spindle used in this study was a 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) A B C D E Because ball bearing stiffness is a nonlinear function of many, difficult to measure factors (such as radial and thrust loads, contact angle, ball diameter, number of balls, operating internal clearance, and spindle speed), it is difficult to predict reliably. Additionally, in our case, the tool holder assembly interface (CAT-40) stiffness must be considered. We have therefore chosen to estimate the bearing stiffness and spindle-holder interface stiffness (translational and rotational) by matching the first four (two rigid body and two bending modes) modal frequencies predicted by the FE rotordynamics model to the experimental natural frequencies of the non-rotating spindle. Figure 3 shows a schematic of the FE rotordynamics model. The four bearings are represented by translational springs. The spindle-holder is modeled as two components with a total of 17 elements, 18 nodes, and 72 degrees of freedom. The main shaft (outer portion) is supported by four ball bearings. The tool holder (inner portion at left of figure) is attached to the main shaft at node 15 with a Figure 3: Finite element rotordynamics model schematic. translational/rotational stiffness connection. Holder Hammer impact Spindle axis Spindle Capacitance probe Figure 4: Photograph of experimental setup for FRF measurements. Figure 5: Time-domain force and displacement voltages prior to filtering (runout removal).

4 EXPERIMENTAL SETUP The rotating tool FRF measurement setup is shown in Fig. 4. The CAT-40 shrink fit tool holder (no tool inserted) was mounted in the direct drive spindle described in the previous section. An instrumented hammer was used to excite the holder at its free end and the response was recorded using a capacitance probe (25 µm/v sensitivity). The force and vibration signals were amplified and then digitally recorded at a sampling frequency of 83 khz. The force excitation bandwidth for all cases was greater than 5 khz. Because runout was present in the holder displacement, the rotating measurement data was filtered in the timedomain. The average of ten periods of runout prior to the hammer impact (pre-trigger sampling was applied) was subtracted from the response measured after the impact. As shown in Fig. 5, the ten runout periods were selected by identifying the last data point before the hammer impulse and then storing the appropriate time interval based on the nominal spindle speed. After removing runout, the FRF was determined by taking the ratio of the frequency-domain complex displacement to the force input. Multiple tests were completed and averaged to obtain the final result. Measurements were performed at 1,000 rpm increments from 0 rpm (non-rotating) to 25,000 rpm. To minimize potential variations in assembly dynamics, the tool and holder were not removed from the spindle during testing. RESULTS AND DISCUSSION Figure 6 shows the experimental zero speed frequency response function of the spindle. The first rigid body mode (cylindrical) is seen at 69 Hz; the second rigid body mode (conical) is found at 177 Hz; the first bending mode is located at 764 Hz; and the second bending mode is seen at 1277 Hz. We tailored the stiffness of the four ball bearings (that support the spindle shaft) and the translational and rotational stiffness of the spindle-holder interface such that the nonrotating FE rotordynamics model predicts the same rigid body and bending natural frequencies. Typically, we would expect that the two rigid body modes would be controlled by the bearing stiffness, while the bending modes would be a function of the flexural rigidity of the shaft. We did observe that the two rigid body modes were exclusively controlled by the bearing stiffness, as expected. However, we found that the spindle-holder interface translational stiffness controlled the first bending frequency, while the rotational stiffness largely controlled the second bending frequency. Figure 6: Measured FRFs for 0 rpm (dotted) and 25,000 rpm (solid). Rigid mode 2 at 180 Hz Shaft Potential Energy = 2.8 % Bearing Potential Energy = 97.2 % Rigid mode 1 at 69 Hz Shaft Potential Energy = 0.2% Bearing Potential Energy = 99.8 % Bending mode 1 at 790 Hz Shaft Potential Energy = 72 % Bearing Potential Energy = 28 % Figure 7: Distribution of potential energy for the rigid body modes and first bending mode.

5 To determine the frequency response from the FE model, a step function input with an amplitude of 1250 N and duration of sec (to approximate an experimental impact measurement) was applied and the transient vibrations computed. The Newmark method was used to solve the time-dependent equations of motion (1x10-5 s time step). For the model the estimated bearing stiffnesses were: front bearings 7.5x10 5 N/m and rear bearings 2.88x10 6 N/m (due to the close proximity of the front and rear bearing pairs, we assigned the same stiffness value to each bearing in these pairs). The spindle-holder interface translational stiffness was 5.35x10 7 N/m and the rotational stiffness was 1.35x10 6 N-m/rad [28]. Figure 7 shows that the first two modes are indeed rigid body modes and the third mode is a bending mode based on a comparison between the strain energy stored in the bearings and the strain energy stored in the shaft. Table 2 provides a comparison of the measured (Test) and modeled (FE) natural frequencies. Measured values are shown in 1,000 rpm intervals, up to 25,000 rpm. Modeled results are shown at 5,000 rpm increments. The rigid body modes show good agreement between test and FE results and exhibit very little change with speed, indicating that the bearing stiffness variation with speed is minimal. Due to gyroscopic coupling of two planes of motion, the FE model predicts two closely-spaced peaks at each resonance. Typically, only one of these is observed in practice, depending on the damping available for each mode. The measured bending modes show a decreasing trend in natural frequency with increasing speed. In our opinion, this is not a consequence of bearing stiffness variation, since it would also have resulted in a change in rigid body modes. Therefore, we conclude that the variation in bending modes is primarily due to changes in stiffness at the spindle-holder interface. The shaft whirling due to unbalance and the resulting micro slip at the interface may be responsible for the observed variation. These effects will be examined in greater detail in future work. Table 2: Comparison of measured (Test) and modeled (FE) natural frequencies. Spindle speed (rpm) Mode 1 Mode 2 Mode 3 Mode 4 Test FE Test FE Test FE Test FE , , , , , , , , , , , , , , , , , , , , ,

6 Although it was not possible to measure the mode shapes for the spindle shaft-holder by impact testing, the mode shapes were available from the FE simulation and are therefore presented here (see Fig. 8). It is seen that the spindle shaft and holder move together for the two rigid body modes (found at 68.9 Hz and 177 Hz for the non-rotating First rigid body mode at 69.2 Hz spindle). However, there is relative motion for the first bending mode (762 Hz) due to the interface stiffness between the holder and spindle shaft. CONCLUSIONS This paper provides initial results for the development of a comprehensive rotordynamics model used to describe highspeed milling spindle vibrations. A comparison of model predictions with impact tests suggests that the spindle-holder interface is a critical issue in the overall dynamic performance of the assembly. The ability to diagnose this sort of issues makes the rotordynamics model a valuable tool for high-speed spindle design. Second rigid body mode at 180 Hz ACKNOWLEDGEMENTS This material is partially supported by the National Science Foundation (DMI ) and Office of Naval Research (Young Investigator Program). The authors also acknowledge helpful discussions with Dr. J. Ziegert, University of Florida, during the completion of this work. Holder Spindle shaft REFERENCES First bending mode at 790 Hz 1. Arnold, R. N., The Mechanism of Tool Vibration in the Cutting of Steel, Proceedings of the Institution of Figure 8: First three mode shapes from Mechanical Engineers, Vol , pp , finite element model (non-rotating spindle). 2. Tobias, S. A., and Fishwick, W., Theory of Regenerative Machine Tool Chatter, The Engineer, Vol. 205, Tobias, S. A., Machine-Tool Vibration, Blackie and Sons Ltd., Glasgow, Scotland, Merrit, H., Theory of Self-Excited Machine Tool Chatter, Journal of Engineering for Industry, Vol. 87-4, pp , Koenisberger, F., and Tlusty, J., Machine Tool Structures-Vol. I: Stability Against Chatter, Pergamon Press, Halley, J., Helvey, A., Smith, S., and Winfough, W., The Impact of High-Speed Machining on the Design and Fabrication of Aircraft Components, Proceedings of the 17th Biennial Conference on Mechanical Vibration and Noise, 1999 ASME Design and Technical Conferences, Las Vegas, Nevada, September 12-16, Tlusty, J., Smith, S., and Winfough, W., Techniques for the Use of Long Slender End Mills in High-Speed Machining, Annals of the CIRP, Vol. 45/1, pp , Smith, S., Winfough, W., and Halley, J., The Effect of Tool Length on Stable Metal Removal Rate in High- Speed Milling, Annals of the CIRP, Vol. 47/1, pp , Davies, M., Dutterer, B., Pratt, J., and Schaut, A., On the Dynamics of High-Speed Milling with Long, Slender Endmills, Annals of the CIRP, Vol. 47/1, pp , Schmitz, T., and Donaldson, R., Predicting High-Speed Machining Dynamics by Substructure Analysis, Annals of the CIRP, Vol. 49/1, pp , Schmitz, T., Davies, M., and Kennedy, M., Tool Point Frequency Response Prediction for High-Speed Machining by RCSA, Journal of Manufacturing Science and Engineering, Vol. 123, pp , Agapiou, J., Rivin, E., and Xie, C., Toolholder/Spindle Interfaces for CNC Machine Tools, Annals of the CIRP, Vol. 44/1, pp , Weck, M., and Schubert, I., New Interface Machine/Tool: Hollow Shank, Annals of the CIRP, Vol. 43/1, pp , 1994.

7 14. Smith, S., Jacobs, P., and Halley, J., The Effect of Drawbar Force on Metal Removal Rate in Milling, Annals of the CIRP, Vol. 48/1, pp , Jones, A., A General Theory for Elastically Constrained Ball and Radial Roller Bearings under Arbitrary Load and Speed Conditions, ASME J. of Basic Engineering, Vol. 82, pp , Harris, T., Rolling Bearing Analysis, 3 rd Ed., Wiley Interscience, Shin, Y., Bearing Nonlinearity and Stability Analysis in High-Speed Machining, ASME J. of Engineering for Industry, Vol. 114, pp , Chen, C., Wang, K., and Shin, Y., An Integrated Approach Toward the Dynamic Analysis of High-Speed Spindles, Part 1: System Model, ASME J. of Vibrations and Acoustics, Vol. 116, pp , Chen, C.H. and Wang, K., An Integrated Approach Toward the Dynamic Analysis of High-Speed Spindles, Part 2: Dynamics Under Moving End Load, ASME J. of Vibrations and Acoustics, Vol. 116, pp , Jorgenson, B. and Shin, Y., Dynamics of Spindle-Bearing Systems at High Speeds Including Cutting Load Effects, ASME J. of Manufacturing Sciences and Engineering, Vol. 120, pp , Schmitz, T., Ziegert, J., and Stanislaus, C., A Method for Predicting Chatter Stability for Systems with Speed- Dependent Spindle Dynamics, SME Technical Paper TP04PUB182, Transactions of NAMRI/SME, Vol. 32, pp , Nelson, H. D., and Chen, W. J., Undamped Critical Speeds of Rotor Systems Using Assumed Modes, ASME Journal of Vibrations and Acoustics, Vol. 115, Chen, W. J., Concise Equations for Rotor Dynamics Analysis, ASME DE-84-2, Vol. 3, Part B, pp , Chen, W. J, and Nelson, H. D., Parameter Sensitivity in the Dynamics of Rotor-Bearing Systems, ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol. 108, pp , Chen, W. J., A Note of Computational Rotor Dynamics, ASME Journal of Vibration and Acoustics, Vol. 120, pp , Chen, W. J., Energy Analysis to the Design of Rotor-Bearing Systems, ASME Journal of Engineering for Gas Turbines and Power, Vol. 119, pp , Nataraj, C., Nelson, H. D., and Arakere, N. K., An Analytical Study of a Rigid Rotor System with a Coulomb Spline, Instability in Rotating Machinery, NASA 2409, pp , Agapiou, J., A Methodology to Measure Joint Stiffness Parameters for Toolholder/Spindle Interfaces, Transactions of NAMRI/SME, Vol. 32, pp , 2004.