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Dynamic modeling of machine tool spindle bearing system and model based diagnosis of bearing fault caused by collision
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Procedia CIRP 00 (2018) 000–000
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Procedia CIRP 00 (2017) 000–000 Procedia CIRP 77 (2018) 614–617 www.elsevier.com/locate/procedia
8th CIRP Conference on High Performance Cutting (HPC 2018)
Dynamic modeling of machine tool spindle system and model 28th CIRP Design Conference, May 2018,bearing Nantes, France based diagnosis of bearing fault caused by collision A new methodology to analyze the functional and physical architecture of 1* Songtao Hongrui Caooriented , Xuefengproduct Chen2, Linkai Niu1 identification existing products for Xi an1, assembly family 22
1 1School of Mechanical Engineering, Xi’an Jiaotong University, Xi'an 710049, China State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi'an 710049, China
Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat
* Corresponding author. Tel.: +86-29-82663689; fax: +86-29-82663689. E-mail address: [email protected] École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France
* Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Abstract
This paper presents a dynamic modeling approach of machine tool spindle bearing system supported by both angular contact ball bearing and Abstract floating displacement bearing with the consideration of bearing defect. The dynamic model of angular contact ball bearing and floating displacement bearing are developed by discrete element method based on the geometrical relationship between different bearing components and In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of Hertz contact theory. The spindle shaft is modeled by finite element method with Timoshenko beam theory. With the developed dynamic model, agile and reconfigurable production systems emerged to cope with various products and product families. To design and optimize production the dynamic response of the machine tool spindle bearing system with or without bearing defect can all be simulated. Base on the developed systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to spindleabearing model,on thethe dynamic characteristics a vertically installed tool spindle with defects analyze productsystem or onedynamic product family physicalvibration level. Different productoffamilies, however, maymachine differ largely in terms of multiple the number and on inner race is investigated. The proposed model is validated on a machine tool spindle with bearing defects caused by collision in the laboratory. nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production © 2018AThe Authors. Published by Elsevier Ltd. This is an open access CC BY-NC-ND license system. new methodology is proposed to analyze existing products in article view ofunder theirthe functional and physical architecture. The aim is to cluster © 2018 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/) these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review underBased responsibility of Flow the International Scientificstructure Committee of products the 8th CIRP Conference on Highsubassemblies Performance are Cutting (HPC and assembly on under Datum Chain, the physical of the is analyzed. Functional Selectionsystems. and peer-review responsibility of the International Scientific Committee of the 8th CIRP Conference on Highidentified, Performance 2018). aCutting functional analysis (HPC 2018). is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative Keywords: tool; Spindle system; model; Bearing fault; example of Machine a nail-clipper is usedbearing to explain theDynamic proposed methodology. AnCollision; industrial case study on two product families of steering columns of thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.
1. Introduction
stability prediction and design optimization of the machine tool spindle. Later, Li et al. [6] developed a spindle bearing system Keywords: Assembly; Design method; Family identification The spindle bearing system is the core component of the model based on DeMul’s quasi-static bearing model and FE machine tool. An in-depth understanding of the dynamics of spindle shaft model, with the consideration of the thermothe machine tool spindle bearing system is of significant mechanical effect of the spindle system. A noticeable research 1.influence Introduction of product rangeby and characteristics manufactured and/ora on making full use of the machine tool and realizing hasthe been conducted Cao and Altintas [7]. They proposed assembled in this for system. In this context, the main challenge in high performance cutting, and conducting effective condition general method the modeling of spindle-bearing systems Due to and thediagnosis fast development modelling andimproved analysis Jones is nowbearing not only to cope with single monitoring application [1,in2]. the domain of based on the model and Timoshenko communication and an have ongoing of digitization and products, a limited product Jiang rangeetoral.existing product Numerous studies beentrend conducted on dynamic beam theory. Additionally, [8], Hentati et al.families, [9] and digitalization, areoffacing important but to be ablealso to analyze and to compare products to define modeling andmanufacturing characteristic enterprises investigation rolling bearing Li also et al. [10] presented significant investigation on challenges in The today’s market environments: new product families. can bebearing observed that classical existing rotor systems. dynamic model development aof continuing the rolling dynamic modelling ofItspindle system. tendency towards reduction of product development and product regrouped in function or features. bearing-rotor systems and spindle bearing system oftimes machine Mostfamilies of the are current dynamic modelsofofclients spindle bearing shortened product lifecycles. thereetisal.an[4]. increasing However, orientedon product are hardly to find. tools are reviewed by Cao etInal.addition, [3] and Lin Gagnol system areassembly mainly focused ACBBfamilies supported spindles. The demand customization, beingmodel at theofsame time tool in a spindle global On themodeling product family level, products differ by mainly in two et al. [5]ofpresented a dynamic machine dynamic of spindle system supported both ACBB competition withbased competitors over shaft the world. main characteristics: (i) thebearing number(FDB) of components and studied (ii) the bearing system on the FEall spindle model This and atrend, static and floating displacement has not been which is inducing thestiffness development macrocontact to micro type of components (e.g.model mechanical, electrical, electronical). non-linear equivalent model from of angular ball as there is no dynamic of FDB. Additionally, in these markets, in and diminished lot dynamic sizes dueanalysis, to augmenting Classical methodologies considering single products bearing results (ACBB), used for chatter models, bearings are mainly modeled mainly by nonlinear stiffness product varieties (high-volume to low-volume production) [1]. or solitary, already existing product families analyze the To cope with this augmenting variety as well as to be able to product structure on a physical level (components level) which 2212-8271 possible © 2018 The optimization Authors. Publishedpotentials by Elsevier Ltd. open access causes article under the CC BY-NC-ND license an efficient definition and identify in This theis an existing difficulties regarding (http://creativecommons.org/licenses/by-nc-nd/3.0/) production system, it is important to have a precise knowledge comparison of different product families. Addressing this Peer-review of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting (HPC 2018).. 2212-8271 ©under 2018responsibility The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection © and peer-review under responsibility of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting 2212-8271 2017 The Authors. Published by Elsevier B.V. (HPC 2018). Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018. 10.1016/j.procir.2018.08.197
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Songtao Xi et al. / Procedia CIRP 77 (2018) 614–617 Author name / Procedia CIRP 00 (2018) 000–000
coefficients, Jones’ or DeMul’s quasi-static bearing model. However, the effects of the cage and relative slid between different bearing components cannot be included in these kinds of bearing model. Recently, Xi et al. [11] presented a dynamic modeling approach for spindle bearing system supported by both ACBB and FDB. The dynamic models of both ACBB and FDB were developed based on discrete element method (DEM) and Gupta dynamic bearing modeling theory. Full understanding of vibrations behaviour of the spindle bearing system based on dynamic models is of great significance of condition monitoring and diagnosis application of machine tool spindle bearing system. As for the study on the vibration response of the bearing with localized defect, Ahmad et al. [12] investigated the dynamic response and linear stability of rolling element bearing systems based on a nonlinear dynamic mode. In this model, the rolling elements were modeled as nonlinear springs, and the effect of the single point defect of the raceways were model as a series of impulses with corresponding characteristic frequencies. Similar model was also used by Patil et al. [13]. Patel and Tandon et al. [14] studied the vibration response of deep groove ball bearings considering single and multiple defects in races based on a proposed dynamic model with the consideration of ball mass and Jones-Harris bearing model. Additionally, Niu et al. [15] developed a dynamic model of high speed rolling ball bearings with localized surface defects in raceways based on Gupta dynamic bearing model with each bearing component having six degrees of freedom, and investigated the vibration response and passing frequencies of the bearing. In this paper, a dynamic model of machine tool spindle supported by both ACBB and FDB considering localized bearing defects is presented. The dynamic response characteristics of the spindle bearing system with bearing localized defect is investigated combining a practice spindle bearing fault caused by collision.
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based on Gupta dynamic bearing modeling theory. The spindle shaft is modeled using 5DOFs Timoshenko beam theory. In order to study the dynamic vibration response of the system with bearing defects, the bearing fault model used in [15] is adopted in this study. In the proposed dynamic model, bearing inner races are fixed with the spindle at their corresponding installation nodes. The installation directions of the bearings are considered when constructing the coupled dynamic model of the spindle bearing system. As the dynamic model of ACBB has been described in detail in [15, 16], only the dynamic model of FDB is introduce here. The geometrical and interactional relationship of different components of FDB is shown in Fig.2.
Fig. 2 geometrical and interactional relationship between components of FDB
In Fig.2, d denotes the ball diameter; rIRo and rORi are the geometrical dimensions of inner and outer races. u r is the unilateral radial clearance of the FDB, and rORc is the orbit radius of the outer raceway groove curvature center. These rORi f o d , where parameters satisfy rORi rIRo d ur and rORc f o is the groove curvature factor of outer raceway. The frame ( X air , Y air , Z air ) denotes the azimuth-in-inner raceway frame. The ball inner/outer raceway interaction is given in Fig. 3. The ball-inner raceway interaction is much different from the ballouter raceway interaction due to their different structure forms. The ball-outer race interaction is similar to that of ACBB.
2. Dynamic modeling of spindle bearing system The dynamic model of the machine tool spindle bearing system supported both by ACBB and FDB is show in Fig.1. As shown in Fig.1, the FDB is often used to support the spindle shaft combining with the ACBBs. ACBBs are used to support the spindle shaft and provide the axial preload, and the FDB is designed to compensate the axial thermal expansion of the shaft to achieve a maximum rotating speed.
Fig. 1. The dynamic model of spindle bearing system supported by both ACBBs and FDB [11].
In the proposed spindle bearing system dynamic model, the dynamic model of ACBB and FDB is developed using DEM
Fig. 3 The ball-inner/outer raceway interaction
The contact loads between different bearing components are calculated based on Hertzian contact theory. The elastic contact deform between ball-inner raceway interaction can be easily calculated in the azimuth-in-inner raceway frame by bir rIRo 1 2 d rbirair3 , where rbirair3 is the 3rd component of rbirair . The normal contact force between ball and inner raceway is calculated by K bir bir bir 0 Qbir bir 0 0 There is some things should be noted when calculating the ball-inner race Hertz contact stiffness coefficient Kbir which is related to the curvature and curvature sum of two contacting bodies, due to their special contact form. The ball-inner/outer race contact form is shown Fig. 4. As the inner raceway of FDB
Songtao Xi et al. / Procedia CIRP 77 (2018) 614–617 Author name / Procedia CIRP 00 (2018) 000–000
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is a cylinder surface, the radius of inner raceway in plane 1 rⅡ1 , as shown in Fig. 4. Thus, the curvature of inner raceway in plane 1 equals 1/∞，i.e., 0 in FDB ball-inner race contact, and that should be noticed when calculating Kbir .
Fig. 4 Ball-inner/outer geometrical contact.
It should also be made attention when calculating the radius of the ball-inner raceway contact deformed pressured surface. According to [16], it is defined by j 2 f j d (2 f j 1) , where j=i or o, which denotes the inner or outer raceway. As for ballinner race contact, as rⅡ1 , the groove curvature factor of inner raceway fi . As a result, the radius of the ball-inner raceway contact deformed pressured surface should be i 2 fi d (2 fi 1) f d , which relates to the ball-inner i
raceway relative slip velocity calculation at the contact point. As for the bearing defect model, the model used in [15] is adopted in this study. The definition of the defect is shown in Fig. 5. In this model, the raceway defects of the bearing change the contact condition and elastic contact deformation between the ball and raceways, and result in an additionally contact load between bearing components and finally reflect on the dynamic response of the system.
Fig. 5 The bearing defect model.
The coupling restriction between the dynamic bearing models and spindle shaft model is constructed through the forces and responses restriction at bearing installation nodes through bearing inner races. First, the interaction forces and moments between bearing components are calculated by the developed dynamic bearing models, and the forces acted on the inner races are transmitted to the FE shaft model at corresponding installation nodes. As inner races are rigidly connected with the shaft at corresponding nodes, they own the same responses. Thus, the calculated responses of the shaft are in turn transmitted to the bearing inner races for the interaction forces calculation in the next step. More details about the spindle system dynamic modeling approach can refer to [11]. 3. The dynamic model application The investigated machine tool spindle is shown in Fig. 6, which is a Kessler spindle supported both ACBBs and FDB,
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and vertically installed. Two FAG HC 71914 ACBBs are installed at the front end, and one FAG 1011 FDB is installed at the rear end of the shaft, just as shown in Fig.1. After once collision between the tool and workpiece caused by maloperation, the vibration of the spindle increases and together with abnormal noise. After overhauling and changing bearings, the vibration of the spindle returns normal condition. It was found that there are three tiny pits on the inner raceway of the 2nd ACBB caused by the collision. The pits are so small that it can be difficultly detected by naked eyes, and need to be detected with the help of electron microscope. The three pits appear adjacently with an equal angle 11.25°, which is equal to the spacing angle of the balls. The pits are about 100-200μm wide and 25-50μm deep.
Fig. 6 Pictures of the spindle and bearing defects.
A dynamic model of the spindle shown in Fig. 6 is developed based on the proposed dynamic modeling approach with consideration of the bearing defects in 2nd ACBB. The inner race defects parameters in simulation case is given in Table 1. The comparison between simulated and experimental measured dynamic responses of the spindle at the spindle housing is shown in Fig. 7. Table 1 The inner race defects parameters Number Angle Width 1 168.75° 100μm 2 180° 200μm 3 191.25° 100μm
Depth 25μm 50μm 25μm
From the time domain waveform of both the experimental measured and simulated vibration signal, it can be found that there is an obvious amplitude modulation phenomenon of the vibration signals, and the modulation frequency is the 2nd order harmonic of the spindle rotating frequency, as the period of the modulation component is the half of the spindle rotating period. This phenomenon is mainly due to two reasons. Firstly, the spindle is vertically installed without additional radial load applied on the spindle when it is idle. Thus, all the balls in the bearing are in an equal load condition due to the axial spindle preload. Secondly, the defects are appeared on the inner race, orientation of the defects as well as the impact load between the ball and defects change with the rotation of the spindle and inner race, as illustrated in Fig. 8. As a result, the amplitude and direction of the impact vibration change periodically with the rotation of the spindle. The change of the impact vibration directions of both the simulated and experimental signal in one spindle shaft revolution can be illustrated by selected points P1P4 shown in Fig. 7, and their enlarged figures shown in Fig. 9. In each spindle revolution, the amplitude of signal is modulated and changes twice, and the impact direction changes once. Additionally, the contact loads between one considered ball and inner/outer races are simulated and shown in Fig. 10. From Fig. 10, it can be seen that the size of defects has significant effect on the contact load between balls and races.
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amplitude modulation phenomenon of the vibration signal corresponding to the 2nd order spindle rotating frequency is carefully investigated. Results show that the proposed dynamic model can accurately simulate the dynamic response and interaction forces of the spindle bearing system with localized defects. Additionally, it can provide an effective tool for the vibration mechanism investigation and condition monitoring and diagnosis of the spindle bearing system. Acknowledgements Fig.7 Comparison between the simulation and experiment. (a) and (c) are the simulated and experimental time domain waveform; (b) and (d) are the corresponding envelope spectrums.
The authors would like to acknowledge the support of the National Natural Science Foundation of China (No. 51575423 11772244), the Natural Science Foundation of Shaanxi (No. 2017JM5120), and the Fundamental Research Funds for the Central University. References
Fig. 8 Illustration of the vibration modulation.
Fig. 9 The impact direction illustration of P1-P4.
Fig. 10 Ball-inner/outer race contact load.
4. Conclusions In this study, a dynamic model of machine tool spindle system supported by both ACBB and FDB with the consideration of bearing defects is presented and investigated. The dynamic models of ACBB and FDB are developed using DEM method and Gupta dynamic bearing modeling theory. The spindle shaft is modeled based on the Timoshenko beam theory. The developed spindle system modeling approach is validated on an actual vertically installed machine tool spindle with multiple defects on inner race caused by collision. The
[1] E. Abele, Y. Altintas, C. Brecher, Machine tool spindle units, CIRP AnnalsManufacturing Technology, 59 (2010) 781-802. [2] H. Cao, X. Zhang, X. Chen, The concept and progress of intelligent spindles: a review, International Journal of Machine Tools and Manufacture, 112 (2017) 21-52. [3] H. Cao, L. Niu, S. Xi, X. Chen, Mechanical model development of rolling bearing-rotor systems: A review, Mechanical Systems and Signal Processing, 102 (2018) 37-58. [4] C.-W. Lin, Y.-K. Lin, C.-H. Chu, Dynamic models and design of spindlebearing systems of machine tools: A review, International Journal of Precision Engineering and Manufacturing, 14 (2013) 513-521. [5] V. Gagnol, B.C. Bouzgarrou, P. Ray, C. Barra, Model-based chatter stability prediction for high-speed spindles, Int J Mach Tool Manu, 47 (2007) 1176-1186. [6] H. Li, Y.C. Shin, Analysis of bearing configuration effects on high speed spindles using an integrated dynamic thermo-mechanical spindle model, International Journal of Machine Tools and Manufacture, 44 (2004) 347-364. [7] Y. Cao, Y. Altintas, A general method for the modeling of spindle-bearing systems, Journal of Mechanical Design, 126 (2004) 1089-1104. [8] S. Jiang, S. Zheng, A modeling approach for analysis and improvement of spindle-drawbar-bearing assembly dynamics, International Journal of Machine Tools and Manufacture, 50 (2010) 131-142. [9] T. Hentati, M. Barkallah, S. Bouaziz, M. Haddar, Dynamic modeling of spindle-rolling bearings systems in peripheral milling operations, Journal of Vibroengineering, 18 (2016). [10] Y. Li, X. Chen, P. Zhang, J. Zhou, Dynamics modeling and modal experimental study of high speed motorized spindle, Journal of Mechanical Science and Technology, 31 (2017) 1049-1056. [11] S. Xi, H. Cao, X. Chen, L. Niu, A dynamic modeling approach for spindle bearing system supported by both angular contact ball bearing and floating displacement bearing, Journal of Manufacturing Science and Engineering, (2017). [12] A. Rafsanjani, S. Abbasion, A. Farshidianfar, H. Moeenfard, Nonlinear dynamic modeling of surface defects in rolling element bearing systems, Journal of Sound and Vibration, 319 (2009) 1150-1174. [13] M. Patil, J. Mathew, P. Rajendrakumar, S. Desai, A theoretical model to predict the effect of localized defect on vibrations associated with ball bearing, International Journal of Mechanical Sciences, 52 (2010) 1193-1201. [14] V. Patel, N. Tandon, R. Pandey, A dynamic model for vibration studies of deep groove ball bearings considering single and multiple defects in races, Journal of Tribology, 132 (2010) 041101. [15] L. Niu, H. Cao, Z. He, Y. Li, Dynamic modeling and vibration response simulation for high speed rolling ball bearings with localized surface defects in raceways, Journal of Manufacturing Science and Engineering, 136 (2014) 041015. [16] P.K. Gupta, Advanced dynamics of rolling elements, Springer Science & Business Media, 2012.
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Procedia CIRP 00 (2018) 000–000
www.elsevier.com/locate/procedia
Procedia CIRP 00 (2017) 000–000 Procedia CIRP 77 (2018) 614–617 www.elsevier.com/locate/procedia
8th CIRP Conference on High Performance Cutting (HPC 2018)
Dynamic modeling of machine tool spindle system and model 28th CIRP Design Conference, May 2018,bearing Nantes, France based diagnosis of bearing fault caused by collision A new methodology to analyze the functional and physical architecture of 1* Songtao Hongrui Caooriented , Xuefengproduct Chen2, Linkai Niu1 identification existing products for Xi an1, assembly family 22
1 1School of Mechanical Engineering, Xi’an Jiaotong University, Xi'an 710049, China State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi'an 710049, China
Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat
* Corresponding author. Tel.: +86-29-82663689; fax: +86-29-82663689. E-mail address: [email protected] École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France
* Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Abstract
This paper presents a dynamic modeling approach of machine tool spindle bearing system supported by both angular contact ball bearing and Abstract floating displacement bearing with the consideration of bearing defect. The dynamic model of angular contact ball bearing and floating displacement bearing are developed by discrete element method based on the geometrical relationship between different bearing components and In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of Hertz contact theory. The spindle shaft is modeled by finite element method with Timoshenko beam theory. With the developed dynamic model, agile and reconfigurable production systems emerged to cope with various products and product families. To design and optimize production the dynamic response of the machine tool spindle bearing system with or without bearing defect can all be simulated. Base on the developed systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to spindleabearing model,on thethe dynamic characteristics a vertically installed tool spindle with defects analyze productsystem or onedynamic product family physicalvibration level. Different productoffamilies, however, maymachine differ largely in terms of multiple the number and on inner race is investigated. The proposed model is validated on a machine tool spindle with bearing defects caused by collision in the laboratory. nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production © 2018AThe Authors. Published by Elsevier Ltd. This is an open access CC BY-NC-ND license system. new methodology is proposed to analyze existing products in article view ofunder theirthe functional and physical architecture. The aim is to cluster © 2018 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/) these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review underBased responsibility of Flow the International Scientificstructure Committee of products the 8th CIRP Conference on Highsubassemblies Performance are Cutting (HPC and assembly on under Datum Chain, the physical of the is analyzed. Functional Selectionsystems. and peer-review responsibility of the International Scientific Committee of the 8th CIRP Conference on Highidentified, Performance 2018). aCutting functional analysis (HPC 2018). is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative Keywords: tool; Spindle system; model; Bearing fault; example of Machine a nail-clipper is usedbearing to explain theDynamic proposed methodology. AnCollision; industrial case study on two product families of steering columns of thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.
1. Introduction
stability prediction and design optimization of the machine tool spindle. Later, Li et al. [6] developed a spindle bearing system Keywords: Assembly; Design method; Family identification The spindle bearing system is the core component of the model based on DeMul’s quasi-static bearing model and FE machine tool. An in-depth understanding of the dynamics of spindle shaft model, with the consideration of the thermothe machine tool spindle bearing system is of significant mechanical effect of the spindle system. A noticeable research 1.influence Introduction of product rangeby and characteristics manufactured and/ora on making full use of the machine tool and realizing hasthe been conducted Cao and Altintas [7]. They proposed assembled in this for system. In this context, the main challenge in high performance cutting, and conducting effective condition general method the modeling of spindle-bearing systems Due to and thediagnosis fast development modelling andimproved analysis Jones is nowbearing not only to cope with single monitoring application [1,in2]. the domain of based on the model and Timoshenko communication and an have ongoing of digitization and products, a limited product Jiang rangeetoral.existing product Numerous studies beentrend conducted on dynamic beam theory. Additionally, [8], Hentati et al.families, [9] and digitalization, areoffacing important but to be ablealso to analyze and to compare products to define modeling andmanufacturing characteristic enterprises investigation rolling bearing Li also et al. [10] presented significant investigation on challenges in The today’s market environments: new product families. can bebearing observed that classical existing rotor systems. dynamic model development aof continuing the rolling dynamic modelling ofItspindle system. tendency towards reduction of product development and product regrouped in function or features. bearing-rotor systems and spindle bearing system oftimes machine Mostfamilies of the are current dynamic modelsofofclients spindle bearing shortened product lifecycles. thereetisal.an[4]. increasing However, orientedon product are hardly to find. tools are reviewed by Cao etInal.addition, [3] and Lin Gagnol system areassembly mainly focused ACBBfamilies supported spindles. The demand customization, beingmodel at theofsame time tool in a spindle global On themodeling product family level, products differ by mainly in two et al. [5]ofpresented a dynamic machine dynamic of spindle system supported both ACBB competition withbased competitors over shaft the world. main characteristics: (i) thebearing number(FDB) of components and studied (ii) the bearing system on the FEall spindle model This and atrend, static and floating displacement has not been which is inducing thestiffness development macrocontact to micro type of components (e.g.model mechanical, electrical, electronical). non-linear equivalent model from of angular ball as there is no dynamic of FDB. Additionally, in these markets, in and diminished lot dynamic sizes dueanalysis, to augmenting Classical methodologies considering single products bearing results (ACBB), used for chatter models, bearings are mainly modeled mainly by nonlinear stiffness product varieties (high-volume to low-volume production) [1]. or solitary, already existing product families analyze the To cope with this augmenting variety as well as to be able to product structure on a physical level (components level) which 2212-8271 possible © 2018 The optimization Authors. Publishedpotentials by Elsevier Ltd. open access causes article under the CC BY-NC-ND license an efficient definition and identify in This theis an existing difficulties regarding (http://creativecommons.org/licenses/by-nc-nd/3.0/) production system, it is important to have a precise knowledge comparison of different product families. Addressing this Peer-review of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting (HPC 2018).. 2212-8271 ©under 2018responsibility The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection © and peer-review under responsibility of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting 2212-8271 2017 The Authors. Published by Elsevier B.V. (HPC 2018). Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018. 10.1016/j.procir.2018.08.197
2
Songtao Xi et al. / Procedia CIRP 77 (2018) 614–617 Author name / Procedia CIRP 00 (2018) 000–000
coefficients, Jones’ or DeMul’s quasi-static bearing model. However, the effects of the cage and relative slid between different bearing components cannot be included in these kinds of bearing model. Recently, Xi et al. [11] presented a dynamic modeling approach for spindle bearing system supported by both ACBB and FDB. The dynamic models of both ACBB and FDB were developed based on discrete element method (DEM) and Gupta dynamic bearing modeling theory. Full understanding of vibrations behaviour of the spindle bearing system based on dynamic models is of great significance of condition monitoring and diagnosis application of machine tool spindle bearing system. As for the study on the vibration response of the bearing with localized defect, Ahmad et al. [12] investigated the dynamic response and linear stability of rolling element bearing systems based on a nonlinear dynamic mode. In this model, the rolling elements were modeled as nonlinear springs, and the effect of the single point defect of the raceways were model as a series of impulses with corresponding characteristic frequencies. Similar model was also used by Patil et al. [13]. Patel and Tandon et al. [14] studied the vibration response of deep groove ball bearings considering single and multiple defects in races based on a proposed dynamic model with the consideration of ball mass and Jones-Harris bearing model. Additionally, Niu et al. [15] developed a dynamic model of high speed rolling ball bearings with localized surface defects in raceways based on Gupta dynamic bearing model with each bearing component having six degrees of freedom, and investigated the vibration response and passing frequencies of the bearing. In this paper, a dynamic model of machine tool spindle supported by both ACBB and FDB considering localized bearing defects is presented. The dynamic response characteristics of the spindle bearing system with bearing localized defect is investigated combining a practice spindle bearing fault caused by collision.
615
based on Gupta dynamic bearing modeling theory. The spindle shaft is modeled using 5DOFs Timoshenko beam theory. In order to study the dynamic vibration response of the system with bearing defects, the bearing fault model used in [15] is adopted in this study. In the proposed dynamic model, bearing inner races are fixed with the spindle at their corresponding installation nodes. The installation directions of the bearings are considered when constructing the coupled dynamic model of the spindle bearing system. As the dynamic model of ACBB has been described in detail in [15, 16], only the dynamic model of FDB is introduce here. The geometrical and interactional relationship of different components of FDB is shown in Fig.2.
Fig. 2 geometrical and interactional relationship between components of FDB
In Fig.2, d denotes the ball diameter; rIRo and rORi are the geometrical dimensions of inner and outer races. u r is the unilateral radial clearance of the FDB, and rORc is the orbit radius of the outer raceway groove curvature center. These rORi f o d , where parameters satisfy rORi rIRo d ur and rORc f o is the groove curvature factor of outer raceway. The frame ( X air , Y air , Z air ) denotes the azimuth-in-inner raceway frame. The ball inner/outer raceway interaction is given in Fig. 3. The ball-inner raceway interaction is much different from the ballouter raceway interaction due to their different structure forms. The ball-outer race interaction is similar to that of ACBB.
2. Dynamic modeling of spindle bearing system The dynamic model of the machine tool spindle bearing system supported both by ACBB and FDB is show in Fig.1. As shown in Fig.1, the FDB is often used to support the spindle shaft combining with the ACBBs. ACBBs are used to support the spindle shaft and provide the axial preload, and the FDB is designed to compensate the axial thermal expansion of the shaft to achieve a maximum rotating speed.
Fig. 1. The dynamic model of spindle bearing system supported by both ACBBs and FDB [11].
In the proposed spindle bearing system dynamic model, the dynamic model of ACBB and FDB is developed using DEM
Fig. 3 The ball-inner/outer raceway interaction
The contact loads between different bearing components are calculated based on Hertzian contact theory. The elastic contact deform between ball-inner raceway interaction can be easily calculated in the azimuth-in-inner raceway frame by bir rIRo 1 2 d rbirair3 , where rbirair3 is the 3rd component of rbirair . The normal contact force between ball and inner raceway is calculated by K bir bir bir 0 Qbir bir 0 0 There is some things should be noted when calculating the ball-inner race Hertz contact stiffness coefficient Kbir which is related to the curvature and curvature sum of two contacting bodies, due to their special contact form. The ball-inner/outer race contact form is shown Fig. 4. As the inner raceway of FDB
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is a cylinder surface, the radius of inner raceway in plane 1 rⅡ1 , as shown in Fig. 4. Thus, the curvature of inner raceway in plane 1 equals 1/∞，i.e., 0 in FDB ball-inner race contact, and that should be noticed when calculating Kbir .
Fig. 4 Ball-inner/outer geometrical contact.
It should also be made attention when calculating the radius of the ball-inner raceway contact deformed pressured surface. According to [16], it is defined by j 2 f j d (2 f j 1) , where j=i or o, which denotes the inner or outer raceway. As for ballinner race contact, as rⅡ1 , the groove curvature factor of inner raceway fi . As a result, the radius of the ball-inner raceway contact deformed pressured surface should be i 2 fi d (2 fi 1) f d , which relates to the ball-inner i
raceway relative slip velocity calculation at the contact point. As for the bearing defect model, the model used in [15] is adopted in this study. The definition of the defect is shown in Fig. 5. In this model, the raceway defects of the bearing change the contact condition and elastic contact deformation between the ball and raceways, and result in an additionally contact load between bearing components and finally reflect on the dynamic response of the system.
Fig. 5 The bearing defect model.
The coupling restriction between the dynamic bearing models and spindle shaft model is constructed through the forces and responses restriction at bearing installation nodes through bearing inner races. First, the interaction forces and moments between bearing components are calculated by the developed dynamic bearing models, and the forces acted on the inner races are transmitted to the FE shaft model at corresponding installation nodes. As inner races are rigidly connected with the shaft at corresponding nodes, they own the same responses. Thus, the calculated responses of the shaft are in turn transmitted to the bearing inner races for the interaction forces calculation in the next step. More details about the spindle system dynamic modeling approach can refer to [11]. 3. The dynamic model application The investigated machine tool spindle is shown in Fig. 6, which is a Kessler spindle supported both ACBBs and FDB,
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and vertically installed. Two FAG HC 71914 ACBBs are installed at the front end, and one FAG 1011 FDB is installed at the rear end of the shaft, just as shown in Fig.1. After once collision between the tool and workpiece caused by maloperation, the vibration of the spindle increases and together with abnormal noise. After overhauling and changing bearings, the vibration of the spindle returns normal condition. It was found that there are three tiny pits on the inner raceway of the 2nd ACBB caused by the collision. The pits are so small that it can be difficultly detected by naked eyes, and need to be detected with the help of electron microscope. The three pits appear adjacently with an equal angle 11.25°, which is equal to the spacing angle of the balls. The pits are about 100-200μm wide and 25-50μm deep.
Fig. 6 Pictures of the spindle and bearing defects.
A dynamic model of the spindle shown in Fig. 6 is developed based on the proposed dynamic modeling approach with consideration of the bearing defects in 2nd ACBB. The inner race defects parameters in simulation case is given in Table 1. The comparison between simulated and experimental measured dynamic responses of the spindle at the spindle housing is shown in Fig. 7. Table 1 The inner race defects parameters Number Angle Width 1 168.75° 100μm 2 180° 200μm 3 191.25° 100μm
Depth 25μm 50μm 25μm
From the time domain waveform of both the experimental measured and simulated vibration signal, it can be found that there is an obvious amplitude modulation phenomenon of the vibration signals, and the modulation frequency is the 2nd order harmonic of the spindle rotating frequency, as the period of the modulation component is the half of the spindle rotating period. This phenomenon is mainly due to two reasons. Firstly, the spindle is vertically installed without additional radial load applied on the spindle when it is idle. Thus, all the balls in the bearing are in an equal load condition due to the axial spindle preload. Secondly, the defects are appeared on the inner race, orientation of the defects as well as the impact load between the ball and defects change with the rotation of the spindle and inner race, as illustrated in Fig. 8. As a result, the amplitude and direction of the impact vibration change periodically with the rotation of the spindle. The change of the impact vibration directions of both the simulated and experimental signal in one spindle shaft revolution can be illustrated by selected points P1P4 shown in Fig. 7, and their enlarged figures shown in Fig. 9. In each spindle revolution, the amplitude of signal is modulated and changes twice, and the impact direction changes once. Additionally, the contact loads between one considered ball and inner/outer races are simulated and shown in Fig. 10. From Fig. 10, it can be seen that the size of defects has significant effect on the contact load between balls and races.
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amplitude modulation phenomenon of the vibration signal corresponding to the 2nd order spindle rotating frequency is carefully investigated. Results show that the proposed dynamic model can accurately simulate the dynamic response and interaction forces of the spindle bearing system with localized defects. Additionally, it can provide an effective tool for the vibration mechanism investigation and condition monitoring and diagnosis of the spindle bearing system. Acknowledgements Fig.7 Comparison between the simulation and experiment. (a) and (c) are the simulated and experimental time domain waveform; (b) and (d) are the corresponding envelope spectrums.
The authors would like to acknowledge the support of the National Natural Science Foundation of China (No. 51575423 11772244), the Natural Science Foundation of Shaanxi (No. 2017JM5120), and the Fundamental Research Funds for the Central University. References
Fig. 8 Illustration of the vibration modulation.
Fig. 9 The impact direction illustration of P1-P4.
Fig. 10 Ball-inner/outer race contact load.
4. Conclusions In this study, a dynamic model of machine tool spindle system supported by both ACBB and FDB with the consideration of bearing defects is presented and investigated. The dynamic models of ACBB and FDB are developed using DEM method and Gupta dynamic bearing modeling theory. The spindle shaft is modeled based on the Timoshenko beam theory. The developed spindle system modeling approach is validated on an actual vertically installed machine tool spindle with multiple defects on inner race caused by collision. The
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