Design Parameter Sensitivity Analysis of High-Speed Motorized Spindle Systems Considering High-Speed Effects

Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation August 5-8, 2007, Harin, China Design Parameter Sensitivity Analysis of High-Speed Motorized Spindle Systems Considering High-Speed Effects Chi-Wei Lin Department of Industrial Engineering and Management Minghsin University of Science and Technology Hsin-Chu, Taiwan chiwei@must.edu.tw Astract With increasing demands for higher productivity and lower production costs, high-speed machine tools have een widely utilized in the modern production facilities. Meanwhile, to meet the requirements of higher spindle speeds and more versatile machining capailities, it is ecoming necessary for engineers to thoroughly realize how the spindle system design would influence the system dynamics with considering highspeed effects. With this understanding, machine tool designers can enhance the performances of automatic high-speed machine tools such as spindle speeds, metal removal rates, and machining quality. In this paper, I first develop a design flow chart to represent the overall spindle design prolems. Based on this flow chart, eight design parameters are identified. A design sensitivity analysis of these eight design parameters is then conducted ased on an integrated finite element method model to investigate their influence on the of the spindle system. The results show that the most important design parameters to the system dynamics include spacing etween the front and rear earing sets, spacing etween middle line of the two earing set and the free end of cutter, and length of the spindle shaft. Index Terms high-speed machine tools, motorized spindle system design, system dynamics, FEM model, sensitivity analysis. I. INTRODUCTION With increasing demands for higher productivity and lower production costs, high-speed machine tools have een widely utilized in the modern production facilities. Meanwhile, to meet the requirements of higher spindle speeds and more versatile machining capailities, it is ecoming necessary for engineers to thoroughly realize how the spindle system design would influence the system dynamics with considering high-speed effects. With this understanding, machine tool designers can enhance the performances of automatic highspeed machine tools such as spindle speeds, metal removal rates, and machining quality. Although theories of high-speed metal cutting were reported in the 1930s [1], machine tools capale of achieving these cutting speeds did not exist commercially until the 1980s. This late acceptance was mostly due to the performance and reliaility prolems caused y complicated highspeed effects, such as chatter and premature spindle earing failures [2]. Therefore, to realize the advantages of highspeed machining, it is of advantage to consider these high speed effects in the design stage. The main design requirement of machine tools is concerned with achieving desired surface finish and machining accuracy of the part without sacrificing machine s reliaility and integrity. Important design factors include weight, cutting forces, forced virations, self-excited virations, and thermal expansions [3]. The prolems caused y forced and selfexcited virations during operating are more difficult to predict, particularly at high speeds. The forced viration is primarily due to the unalanced mass of the rotating spindle, while the self-excited viration is induced y the cutting process [4, 5]. These two processes are all highly related to the dynamic characteristics of machine tool spindle systems. Regarding the spindle system design, Al-Shareef and Brandon [6] constructed a simplified multi-stepped spindleearing system model to investigate the effects of design parameters to the static stiffness in the cutting zone for short overhang spindles. Their results showed that the most effective parameter was the earing spacing. In another paper, they used influence coefficient method to investigate the effects of design parameters on the dynamic performance of spindleearing systems [7] and the results showed that the front earing stiffness only slightly influenced the lower modes. Kang et al. [8] analyzed the effects of design parameters on static and dynamic performance of spindle-earing systems y using Finite Element Method (FEM). The parameters considered in their case studies included journal diameter and span ratio, and earing stiffness for static performance, ut for dynamic performance, only the ering stiffness was considered. The only high speed effect included in their FEM was gyroscopic moments. Li and Shin [9] pulished a paper to investigate the effects of earing configuration on the dynamics of high speed spindles. With the constraints of satisfying chatter viration free, Maeda et al. [10] proposed a earing spacing optimization strategy for the spindles configured with an expert system. The dynamics mechanism of high-speed motorized spindles is highly complicated. An integrated FEM model has een developed y Lin et al. [11] to comine the changes 1-4244-0828-8/07/$20.00 2007 IEEE. 2087

of the earing stiffness and shaft rigidity to determine the overall spindle-earing system dynamics. The thermal model adapted in the integrated dynamic model was developed y Bossmanns and Tu [12, 13]. Based on this temperature field analysis, an extended thermal preload model was developed also y Lin et al. [11] to predict thermally induced preload and its influence on earing stiffness. For design purposes, Fig. 1 shows the essential physical models required to descrie the dynamic properties of spindle systems, as well as major independent parameters, which are further divided into design parameters (DPs) and operation parameters (OPs). Fig. 2. The high-speed spindle earing system Fig. 1. The mechanism of integrated dynamic FEM model. In this paper, the integrated FEM model developed y Lin et al. [11] is applied to study the sensitivities of design parameters of the motorized spindle system as shown in Fig. 2 with consideration of high speed effects, where the front earing pairs are rigidly preloaded with spacers of specific sizes. In the following sections, the FEM model of the motorized spindle system is introduced first. Then, the effects of design parameters, including earing preload, earing spacing, distance etween earing sets, distance of the middle line of earing sets to the end of the cutting tool, material of spindle, and dimensions of spindle, on the dynamics of the motorized spindle system are investigated considering high-speed effects. Finally, the relative importance analysis of these eight design parameters is investigated with radar charts and followed with a conclusion. II. THE FEM MODEL FOR SENSITIVITY ANALYSIS The models in Fig. 1 include a spindle integrated dynamic FEM model, a spindle shaft FEM model, a earing stiffness model, a thermal preload model, a thermal expanding model, and a heat transfer model. The spindle integrated dynamic FEM model for the motorized spindle in Fig. 2 can e expressed as ([M T ]+[M R ]){ q} Ω[G]{ q}+ ([K] Ω 2 ([M T ] [M R ])){q} = Q(t)+Ω 2 {B} (1) and [K] =[K S ]+[K B ] (2) where {q} is the gloal node displacement of the spindleearing system in rotational frame coordinates; Ω is the spindle speed; [M T ] is the transitional mass matrix and [M R ] is the rotational mass matrix; [K] is the stiffness matrix of the spindle earing system; [G] is the gyroscopic matrix of the spindle shaft; {Q(t)} is the external load vector and {B} is the unalancing force vector; [K S ] and [K B ] are the spindle shaft and radial earing stiffness matrices. The matrices [M T ], [M R ],and[g] are all determined ased on the size and material of the spindle shaft. No structural damping and axial forces are considered in this model. The stiffness matrix [K] in (1) is formed y the stiffness terms of spindle shaft [K S ] and earings [K B ], as shown in (2). The spindle shaft stiffness matrix, which is derived from ending and shear energy forms, is a function of the size and material elasticity of the spindle shaft. For the earing stiffness matrix, if a earing is located at node n of the finite element system, then the earing stiffness matrix can e expressed as [K B ]=[k ij ] and { kn if i = j =4n k ij = 3 or 4n 2 (3) 0 otherwise. where k n is the radial stiffness of the earing locating on node n. From here we can find that the location of earings affect the dynamics of the spindle system. On the other hand, the earing stiffness k n is a function of earing preload and earing configuration, which is discussed as follows. The stiffness ehavior of angular contact earings is complex. It primarily depends on the applied loads and the earing layout. The stiffness equations represent the static axial and radial stiffness (k as and k rs ) as a function of axial preload (P a ), all diameter (D ), numer of alls (N )and contact angle (θ) of static angular-contact all earings [14]: k as = c a Pa 1/3 N 2/3 sin 5/3 θd 1/3 (4) k rs =0.64c a Pa 1/3 N 2/3 sin 2/3 θ cos θd 1/3 (5) P a = P a,i + P a,t (6) 2088

TABLE I THE DPS CONSIDERED IN THIS RESEARCH. Fig. 3. Elements of the spindle system where c a is an empirical data decided y the experimental results; P a,i and P a,t are the initial and thermally induced preload in the axial direction, respectively. The total deformation of the earing in the direction of line of contact Δ 0 can e expressed as the following equation Δ 0 =Δ 1 +Δ 2 cos θ Δ 3 sin θ (7) where Δ 1 is the axial thermal expansion difference, which is mostly caused y the shaft and the spacer; Δ 2 is the thermal expansion difference of the inner ring and outer ring in the radial direction; and Δ 3 is the expansion of the rolling elements. Then, the thermally induced preload can e otained as P a,t = k t Δ 1.5 0 (8) where k t is an equivalent spring constant, which is otained empirically. Eqs. (7) and (8) are then used to determine P a,t and susequently the earing stiffness suject to thermal preload which can e determined. As shown in Fig. 3, the spindle shaft is represented y a 23-element finite element model in this paper. A program is written using MATLAB to convert the aove system data into a set of equations of motion. Detailed derivation and validation of these equations can e found in [11]. III. THE DESIGN SENSITIVITIES OF DPS ONTHE NATURAL FREQUENCIES The influences of DPs on the system, mainly the first mode (NF1) and second mode (NF2) frequencies, are explored in this section. The DPs, introduced in Fig. 1, are further depicted in Fig. 3 and their definitions can e found in Tale I. These DPs can e separated into three categories, for which DP1 elongs to the initial preload category, DP2, DP3, DP4, and DP5 elong to the earing location category, while DP6, DP7, and DP8 elong to the spindle specification category. The following results are derived y changing all the design parameters one at a time while the other parameters remain unchanged. The focused DP is varied from 90 percents to 110 percents of its initial values and the system are otained y using the integrated FEM model for rotating speeds from 0 rpm to, with a 5,00 increment. To facilitate demonstration and analysis, only the results of forward modes are provided. DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8 1230 1210 1190 Bearing initial preload Spacing etween the earings of front earing set Spacing etween the earings of rear earing set Spacing etween the front and rear earing sets Spacing etween the middle line of front and rear earing sets and the free end of cutter Material of the spindle shaft Diameters of the spindle shaft Length of the spindle shaft Ratio to the Initial Value of DP1 (340 N) 5,00 10,00 15,00 20,00 5,00 10,00 15,00 20,00 Ratio to the Initial Value of DP1 (340 N) Fig. 4. The effects of the earing initial preload (DP1) on spindle system A. Initial Preload Design Parameter For the spindle system of Fig. 2, the front earing set of the target spindle system is preloaded with flexile devices, while the rear earing set is preloaded rigidly. Therefore, only the rear earing preload is treated as a design parameter and is denoted as DP1. The front earing preload ecomes a operating parameter ecause it can e adjusted during operation. Fig. 4 shows the effects of DP1 on the system. As expected, y increasing the rear ering preload, oth NF1 and NF2 increase at all speeds. Note that the stiffness of the spindle system softens at higher speeds as discussed in [11]. Fig. 4 indicates that y increasing the rear earing preload, the NF1 and NF2 can e recovered to some extent ut are still lower than the values at 0 rpm. Note that rear earing preload can not e increased indefinitely ecause it can cause thermal instaility [15, 16]. B. Bearing Location Design Parameters It is assumed that the center lines of the front earing set, the rear earing set, and the front and rear earing sets remain unchanged during variation processes of DP2, DP3, and DP4 respectively; therefore, the right-hand side and left-hand side 2089

1280 1450 1260 1160 5,00 10,00 15,00 20,00 1400 1350 1300 1250 1150 1100 5,00 10,00 15,00 20,00 1140 Ratio to the Initial Value of DP2 (83.7 mm) 1050 Ratio to the Initial Value of DP4 (371.3 mm) 950 5,00 10,00 15,00 20,00 900 750 700 5,00 10,00 15,00 20,00 Ratio to the Initial Value of DP2 (83.7 mm) 650 Ratio to the Initial Value of DP4 (371.3 mm) Fig. 5. The effects of the spacing etween the earings of front earing set (DP2) on spindle system Fig. 7. The effects of the spacing etween front and rear earing sets (DP4) on spindle system 1230 1210 1190 Ratio to the Initial Value of DP3 (43.3 mm) 870 830 810 790 5,00 10,00 15,00 5,00 10,00 15,00 20,00 20,00 Ratio to the Initial Value of DP3 (43.3 mm) 1450 1400 1350 1300 1250 1150 1100 1050 1000 Ratio to the Initial Value of DP5 (408.9 mm) 900 740 5,00 10,00 15,00 20,00 5,00 10,00 15,00 20,00 720 Ratio to the Initial Value of DP5 (408.9 mm) Fig. 6. The effects of the spacing etween the earings of rear earing set (DP3) on spindle system Fig. 8. The effects of the spacing of the middle line of front and rear earing sets and the free end of the cutter (DP5) on spindle system natural frequencies earings or earing sets move inward or outward with same distances simultaneously. We investigate the effects of earing spacing for the front earing set and rear earing set separately, and the results are shown in Figs. 5 and 6, respectively. It can e found from the figures that of oth modes increase with respect to the increases of DP2 and DP3. As expected, DP2 for the front earing set is a much more effective design parameter to change NF1 and NF2 drastically. DP4 represents the spacing etween the middle lines of front and rear earing sets, as Fig. 3 shown. The effects of DP4 on system are demonstrated in Fig. 7. From these figures we can see that DP4 affects NF1 and NF2 in opposite ways. NF1 decreases with increases of DP4, while NF2 increases y contrast. It can also e found that the relation etween these two and DP4 are highly nonlinear. In Fig. 3, DP5 stands for spacing etween the free end of the cutter and the middle line of front and rear earing sets. From Fig. 8, we can clearly oserve that the natural frequencies of oth modes decrease as DP5 increases. This is consistent with the practical experience in high speed end milling, for which, in general, long tool with large overhang should e avoid to prevent chatter. C. Spindle Specification Design Parameters DP6, DP7, and DP8 are classified as spindle specification design parameters since they express the material and features of the spindle shaft. DP6 denotes the material of the spindle shaft, where only the modulus of elasticity is considered in this research. The effects of DP6 on the system are shown 2090

1280 1450 1260 1160 5,00 10,00 15,00 20,00 1400 1350 1300 1250 1150 1100 1050 5,00 10,00 15,00 20,00 1140 Ratio to the Initial Value of DP6 (2.1x10 11 N/mm 2 ) 1000 Ratio to the Initial Value of DP8 (692.1 mm) 1000 5,00 10,00 15,00 20,00 950 900 750 700 5,00 10,00 15,00 20,00 Ratio to the Initial Value of DP6 (2.1x10 11 N/mm 2 ) 650 Ratio to the Initial Value of DP8 (692.1 mm) Fig. 9. The effects of material of spindle shaft (DP6) on spindle system Fig. 11. The effects of the length of spindle shaft (DP8) on spindle system 1230 1210 1190 5,00 10,00 15,00 20,00 1170 Ratio to the Initial Value of DP7 (Diameters) 870 830 810 790 5,00 10,00 15,00 20,00 Ratio to the Initial Value of DP7 (Diameters) Fig. 10. The effects of diameter of spindle shaft (DP7) on spindle system Fig. 12. The radar charts for the first mode natural frequency frequencies decrease with increases of DP8, which confirms that longer shafts result in lower. Note that the overhang ratio of the spindle system remains constant during the variations of DP8. in Fig. 9. From the figures it can e seen that the natural frequencies increase with DP6. Therefore, high strength materials are preferred. DP7 represents the sizes of spindle shaft in diameters. The effects of DP7 are shown in Fig. 10. From these two figures, we can discover NF2 possesses local maxima around ratio 0.98 of the original value of DP7. This nonlinear phenomenon results mainly from the fact that the system s mass and stiffness matrices contriuted from the spindle shaft, ([M T ]+[M R ]) and [K S ] shown in (1) and (2), increase simultaneously with increases of DP7, while the stiffness from earings, [K B ], remains constant. Therefore, upon determining spindle shaft diameters, the influences of earing stiffness must e taken into account. DP8 means the total length of the spindle shaft in Fig. 3. The effects of DP8 on the are presented in Fig. 11. From the figures, we can see that the system natural IV. ANALYSIS OF THE DPS EFFECTS In the following discussion, the maximum and minimum values of first mode and second mode and their corresponding ratios of DPs are identified from the figures in Section III for all eight DPs with respect to and. Dividing the ratios of NF1 and NF2 y the corresponding ratios of DPs and taking their asolute values, we can otain an index similar to elasticity measurements. The consequent indices are demonstrated with radar charts to facilitate relative importance comparison of the DPs. Fig. 12 and Fig. 13 show the results for the first and second modes respectively. From these two figures, it can e discovered that the corresponding ratios exceed 1 include DP4 (@ ratio 1.1) and DP8 (@ ratio 0.9 and 1.1) on oth modes,and DP5 (@ ratio 0.9 and 1.1) only on second mode. From the aove analysis, we can find that if the system NF1 and NF2 are determined as the design targets, the important design parameters are spacing 2091

Fig. 13. The radar charts for the second mode natural frequency etween the front and rear earing sets (DP4) and spacing etween middle line of the two earing sets and the free end of cutter (DP5), and length of the spindle shaft (DP8), which confirms the results of [6, 7]. On the contrary, the uncritical design parameters include earing initial preload (DP1), spacing etween the earings of rear earing sets (DP3) and diameters of the spindle shaft (DP7). V. CONCLUSION In this paper, we have developed a design flow chart to represent the overall spindle design prolems. Based on this flow chart, eight design parameters are identified. A design sensitivity analysis of these eight design parameters is then conducted ased on an integrated finite element method model to investigate their influence on the of the spindle system. Based on the results, we present the following conclusions: 1. The vital design parameters to the system dynamics include spacing etween the front and rear earing sets, spacing etween middle line of the two earing sets and the free end of cutter, and length of the spindle shaft. 2. The uncritical design parameters to the system dynamics include earing initial preload, spacing etween the earings of rear earing sets, and diameters of the spindle shaft. ACKNOWLEDGMENT This research is supported y the NSC of R.O.C under the grant No. NSC-94-2212-E-159-003. REFERENCES [1] C. Salomon, Verfahren zur eareitung von metallen oder ei eareitung durch schneidende werkzeuge von sich ähnlich verhaltenden werkstoffen, Patent, 1931, deutsches Patent Nr. 523594, April. [2] J. Tu, M. Corless, and J. Jeppsson, Roust control of high speed end milling with unknown process parameter and cnc delay, Journal of Aerospace Engineering, vol. 1, pp. 1 9, 2004. [3] F. Koenigserger and J. Tlusty, Machine Tool Structures. Pergamon Press, 1970, vol. 1. [4] J. Tlusty, Dynamics of high-speed milling, Journal of Engineering for Industry, Transactions of the ASME, vol. 108, pp. 59 67, 1986. [5] Y. Altintas and E. Budak, Analytical prediction of staility loes in milling, CIRP Annual, vol. 44, pp. 357 362, 1995. [6] K. Al-shareef and J. Brandon, On the quasi-static design of machine tool spindles, Preceedings of the Institution of Mechanical Engineers, vol. 204, pp. 91 104, 1990. [7], On the effects of variations in the design parameters on the dynamic performance of machine tool spindle-earing systems, International Journal of Machine Tools and Manufacture, vol. 30, no. 3, pp. 431 445, 1990. [8] Y. Kang, Y.-P. Chang, J.-W. Tsai, S.-C. Chen, and L.- K. Yang, Integrated cae strategies for the design of machine tool spindle-earing systems, Finite Elements in Analysis and Design, vol. 37, pp. 485 511, 2001. [9] H. Li and Y. C. Shin, Analysis of earing configuration effects on high speed spindles using an integrated dynamic thermo-mechanical spindle model, International Journal of Machine Tools and Manufacture, vol. 44, pp. 347 364, 2004. [10] O. Maeda, Y. Cao, and Y. Altintas, Expert spindle design system, International Journal of Machine Tools and Manufacture, vol. 45, pp. 537 548, 2005. [11] C.-W. Lin, J. Tu, and J. Kamman, An integrated thermo-mechanical-dynamic model to characterize motorized machine tool spindles during very high speed rotation, International journal of machine tools and manufacture, vol. 43, pp. 1035 1050, 2003. [12] B. Bossmanns and J. Tu, Thermal model for high speed motorized spindles, International Journal of Machine Tools and Manufacture, vol. 39, no. 9, pp. 1345 1366, 1999. [13], A power flow model for high speed motorized spindles - heat generation characterization, Journal of Manufacturing Science and Engineering, Transactions of the ASME, vol. 123, no. 3, pp. 494 505, 2001. [14] F. Wardle, S. Lacey, and S. Poon, Dynamic and static characteristics of a wide speed range machine tool spindle, Precision Engineering, vol. 5, no. 4, pp. 175 183, 1983. [15] J. Stein and J. Tu, A state-space model for monitoring thermally induced preload in anti-friction spindle earings of high-speed machine tools, Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 116, pp. 372 386, 1994. [16] J. Tu, Thermoelastic instaility monitoring for preventing spindle earing seizure, Triology Transactions, vol. 38, no. 1, pp. 201 207, 1995. 2092