# PREDICTION OF MACHINE TOOL SPINDLE S DYNAMICS BASED ON A THERMO-MECHANICAL MODEL

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2 There are three kinds of equations that form the bearing model: Hertzian force-deflection characteristics of ball-ring contacts in normal direction; force and moment equilibrium on bearing balls, Figure 4; geometrical formulas stating the relation of bearing ring positions, contact angles and ball-ring normal deflection. Figure 2. Solution flow diagram. The closed-loop structure of the analysis was built using ANSYS FEM software with its APDL programmer s capabilities. 3.2 Thermal model The model for simulation of spindle thermal field was build according to theories of heat transfer within solid materials and fluids, Figure 3. The bearing model from [Harris 1991] uses the Newton-Raphson method for computing the internal condition of bearings (contact forces, angles, rigidity) resulting from the external condition ring position calculated by FEM analysis. The result of the bearing model is always the complete stiffness matrix of each bearing in the assembly with diagonal as well as off-diagonal terms. 3.5 Results and outputs of the thermo-mechanical model The results of the thermo-mechanical model are time-dependent and specific for each of the bearings: preloading force; stiffness matrix; relative radial and axial displacement between bearing rings. Values of bearing ring s displacement and stiffness matrix are exported and used as input values for the subsequent mechanical spindle model. 4. Model of spindle mechanics The model of spindle mechanics connecting finite-element and analytical description of the spindle components is used for modeling the spindle mechanics. The schema of the model is shown in Figure 5. Figure 3. Heat transfer model of bearing surrounding. Very important is the formula of heat generation in bearings it depends on speed, preload, viscosity of lubricant, internal bearing design and other parameters. The theoretical formulas presented in [Harris 1991] were adjusted according to experimental results obtained on our test rig. These formulas match well in large range of operational conditions and bearing designs. The character of the problem, performed simulations and experiments shows, that mainly the accuracy of the heat generation characteristic plays the key role in estimation of thermal issues in HS spindles. 3.3 Structural model The FEM analysis of spindle deformations considers the thermal deformations based on the current thermal field resulting from last thermal solution. The deformations caused by external and internal forces are considered as well. 3.4 Bearing model In high speed operation of ball or roller bearings the rolling element centrifugal forces can be significantly large compared to the forces applied to the bearing externally. High speed affects the contact forces and angles, Figure 4, and also the dynamics of the bearing-supported rotor system. Figure 4. Ball loading. Figure 5. Schema of the model of spindle mechanics. Mechanical properties of spindle shaft and housing are described by its stiffness and mass matrices. Properties of other spindle system components (bearings, tool holder spindle interface) are described by corresponding stiffness matrices obtained separately. All matrices describe an undamped system which can be updated by additional damping information. 4.1 Modeling of spindle geometry The hybrid model uses the FEM software Ansys for generation of mass and stiffness matrices of spindle shaft, spindle housing and the tool. The full FEM model with 3D SOLID elements has too many degrees of freedom (DOF) therefore a reduction is necessary. The Guyan static reduction was used for decreasing the number of DOFs and only several nodes representing shaft/housing cross-sections were selected (see figure 6, reduction points are marked by arrows). The reduced mass and stiffness matrices of spindle shaft and housing

4 Table 2. Spindle eigenvalues changing with the thermo-mechanical condition of bearings. Bending Spindle running time at 6000 rpm frequency: 0 hour 0,5 hour 2 hours 8 hours 1. freq. 153 Hz 194 Hz 212 Hz 219 Hz 2. freq. 313 Hz 346 Hz 353 Hz 356 Hz 3. freq. 401 Hz 473 Hz 500 Hz 511 Hz 4. freq. 933 Hz 1088 Hz 1157 Hz 1185 Hz Figure 8. Schematic chart of the front part of the spindle temperature field in the spindle. Figure 9. Preload force of all bearings. Figure 10. Radial stiffness of all bearings. asing preload forces cause the increase of bearing stiffness. It can be seen, that the stiffness of basic mechanical model and the stiffness resulting from the thermo-mechanical simulation differs nearly +300% after an eight-hour run. Figure 11. Influence of centrifugal and thermal effects on bearing stiffness. 5.4 Results of the model of spindle dynamics The results of the thermo-mechanical model were used for computation of spindle dynamic properties; resultant eigenfrequencies are show in Table 2 and Figure 12. Figure 12. Spindle eigenvalues changing with the thermo-mechanical condition of bearings. 6. Conclusions The results show that the thermal field that develops during spindle operation has significant effect on condition of spindle bearings, their properties and resultant spindle dynamics. The effect is very important mainly when position preload of bearings (rigid spacers) is used. The proposed method of building the thermo-mechanical-dynamical models enables to predict spindle properties and to optimize their structure at the design stage. The main task for the future is to connect the mentioned sequential models into one user-friendly tool and to provide complex experimental verification in large range of spindle designs and operational conditions. Acknowledgement This research has been supported by the 1M0507 grant of the Ministry of Education of the Czech Republic and by the SKF company. References [Altintas 2005] Altintas, Y., Cao, Y., Virtual design and optimization of machine tool spindles. In: Annals of the CIRP. 2005, vol. 54, p [Cao 2004] Cao, Y., Altintas, Y., A general method for the modeling of spindle-bearing system. Journal of Mechanical Desin. ASME 2004, vol. 126, p , ISSN: [Harris 1991] Harris, T. A., Rolling Bearings Analysis. Third Edition. New York: John Wiley & Sons [Holkup 2006] Holkup, T., Holý, S., Complex modelling of spindle rolling bearings. Journal of Machine Engineering Eficiency Development of Manufacturing Machines, vol. 6, no.3, 2006, p ISSN [Jones 1960] Jones, A.B., A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions. Journal of Basic Design, Transactions of the ASME. 1960, s [Kolar 2007] Kolar, P., Holkup, T., Modeling of a Machine Tools Spindle using a Hybrid Model. In: Proceedings of 3rd International Conference Virtual Design and Automation. Poznań: Faculty of Mechanical Engineering and Management, Politechnika Poznańska, CD-ROM, paper No. S05_06_01_kolar. 8p

5 [Li 2003] Li, H., Shin, Y. C., Analysis of bearing configuration effects on high speed spindles using an integrated dynamic thermo-mechanical spindle model. International Journal of Machine Tools and Manufacture. 2003, vol. 44, p [Lin 2003] LIN, C. W., et al, An integrated thermo-mechanical-dynamical model to characterize motorized machine tool spindles during very high speed rotation. International Journal of Machine Tools and Manufacture. 2003, vol. 43, p [Li 2004a] Li, H., Shin, Y. C., Integrated dynamic thermo-mechanical modeling of high speed spindles, part 1: model development. Journal of Manufacturing Science and Engineering, Transactions of the ASME. 2004, vol. 126, p [Li 2004b] Li, H., Shin, Y. C., Integrated dynamic thermo-mechanical modeling of high speed spindles, part 2: solution procedure and validation. Journal of Manufacturing Science and Engineering, Transactions of the ASME. 2004, vol. 126, p [Rantatalo 2007] Rantalalo, M., Aidanpää, J. O., Göransson, B., Norman, P., Milling machine spindle analysis using FEM and non-contact spindle excitation and response measurement. International Journal of Machine Tools and Manufacture. 2007, vol. 47, p Contacts Dipl. Ing. Petr Kolar, PhD. Research Center for Manufacturing Technology Faculty of Mechanical Engineering, CTU in Prague Horska 3, Praha 2, Czech Republic tel.: , fax: PREDICTION OF MACHINE TOOL SPINDLE S DYNAMICS BASED ON A THERMO-MECHANICAL MODEL March /171

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