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Modeling of Self-Vibratory Drilling Head-Spindle System for Predictions of Bearings Lifespan

Modeling of Self-Vibratory Drilling Head-Spindle System for Predictions of Bearings Lifespan Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 606087, 10 pages doi:10.1155/2011/606087 Research Article Modeling of Self-Vibratory Drilling Head-Spindle System for Predictions of Bearings Lifespan 1 2 2 3 F. Forestier, V. Gagnol, P. Ray, and H. Paris LaMI Laboratory, Clermont Universit´e, UBP, EA 3867, BP2235, 03101 Montlucon, France LaMI Laboratory, Clermont Universit´e, IFMA, EA 3867, BP10448, 63000 Clermont-Ferrand, France G-SCOP Laboratory, G-SCOP, 46 Avenue F´elix Viallet, 38031 Grenoble, France Correspondence should be addressed to F. Forestier, fabien.forestier@moniut.univ-bpclermont.fr Received 13 January 2011; Accepted 7 June 2011 Academic Editor: Atma Sahu Copyright © 2011 F. Forestier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The machining of deep holes is limited due to inadequate chip evacuation, which induces tool breakage. To limit this drawback, retreat cycles and lubrication are used. An alternative response to the evacuation problem is based on high-speed vibratory drilling. A specific tool holder induces axial self-maintained vibration of the drill, which enables the chips to be split. The chips are thus of a small size and can be evacuated. To anticipate the potential risk of decreased spindle lifespan associated with these vibrations, a model of the behavior of the system (spindle—self-vibrating drilling head—tool) is elaborated. In order to assess the dynamic behavior of the system, this study develops a rotor-based finite element model, integrated with the modelling of component interfaces. The current results indicate that the simulations are consistent with the experimental measurements. The influence of spindle speed and feed rate on bearing lifespan is highlighted. 1. Introduction The SVDH is composed of an axial vibrating system, con- sisting of the SVDH vibrating subsystem, mounted on a spe- High-speed vibratory drilling allows chips to be split thanks cific HSK 63 taper, and called the SVDH body. The SVDH- to self-maintained vibration during cutting [1]. When these vibrating subsystem is composed of the vibrating parts of vibrations have a magnitude greater than the advance per the drill holder. The SVDH body guides the axial vibrating tooth, the drill continuously enters and exits the material, subsystem through a ball retainer and a classic HSK63 taper which allows the fragmentation of the chips, as shown by connexion with the spindle head. The self-excited vibrations Tichkiewitchetal. [2]. The chips are small and can thus be must be tuned and controlled in order to have a magnitude removed easily without retreat cycles or lubrication. High- greater than the advance per tooth. Mathematical models of speed vibratory drilling enables productivity to increase by a the SVDH dynamics appearing in most of previous works factor of three compared to traditional techniques. Currently, lead to a one-dimensional linear or nonlinear model in the high-speed vibratory drilling is the only process capable of axial vibration direction governed by the mass, damping, producing deep holes using high-speed machining centers, and stiffness of the considered system [3]. In these studies, without lubricants, and with high productivity [3]. damping, which plays an important role in SVDH dynamic However, the excitation generated by the cutting process behaviour, is estimated but not experimentally identified. can also be a source of damage to the self-vibrating drilling Moreover, the spindle behaviour and the interfaces between head and the spindle. So the industrialization of high-speed the spindle and the SVDH are assumed to be rigid and have vibratory drilling requires the effects of vibration on the not been taken into account. However, many works have machine, and more particularly on the spindle, to be predict- showed that tool point dynamics can be significantly influ- ed. For this purpose, a model of the dynamic behavior of enced by spindle dynamics, as well as the spindle-holder— the system (self-vibrating drilling head spindle) is elaborated tool interfaces. This creates a demand for predictive knowl- from a realistic assembly of the components, in order to edge models that are capable of investigating the influence of predict spindle bearing lifespan. cutting conditions on high-speed spindle-SVDH system. 2 Advances in Acoustics and Vibration Many authors have investigated the dynamic behaviour is proposed. This finite element model takes into account of machine tool spindle-bearing systems, both analytically rotor dynamics effects, bearing stiffness and the real behavior and experimentally. They show that spindle dynamics are in- of interfaces between different system components. The sim- fluenced by a large number of factors, including holder char- ulation allows achieving the best possible cutting parameters acteristics [4], spindle shaft geometry and drawbar force without damaging the bearings of the spindle. [5, 6], and the stiffness and damping provided by the bear- In the second section, the spindle-SVDH rotor dynamics ings [7]. Most of these factors are independent of spindle model is presented. A special rotor-beam element is imple- speed, contrary to bearing properties [8, 9] and the spindle mented. The rolling bearing stiffness matrices are calculated rotor dynamics, which change according to spindle speed. around a static function point on the basis of Lim and Such rotating systems have been successfully modelled Singh’s [20] formulation and then integrated into the global through rotor-dynamics studies [10, 11]. In previous works finite element model. The identification of contact dynamics [12, 13], a dynamic high-speed spindle-bearing system mod- in tool-SVDH-spindle assemblies is carried out using the el based on rotor dynamics prediction was presented. Ele- RC method on the basis of experimental substructure char- ment kinematics were formulated in a corotational coordi- acterisation. The identified models are then integrated into nate frame and enabled a special rotor beam element to be the global spindle-SVDH-tool model. Finally, numerical and developed. Model results showed that spindle dynamics are experimental tool tip FRF, in radial and axial directions, is influenced by the gyroscopic coupling and the spin softening compared in order to validate the global assembled model. of the rotating shaft due to high rotation speeds. The literature on the modelling and analysis of spindles Section 3 is dedicated to the definition of optimal cutting shows that the tool tip FRF is also greatly influenced by the conditions with respect to industrial objectives. The studied industrial context requires the maximum material cutting contact dynamics of the spindle-holder-tool interfaces. The flexibility of the afore-mentioned interfaces can dominate rate and a rational use of the tool-SVDH-spindle set in order to guarantee adequate rolling bearing lifespan. As a result, the dynamics of the spindle. The tool tip FRF is usually ob- tained using experimental measurements, which require sig- a recommendation for the use of a spindle-SVDH-tool set nificant testing time to take into account the large number which respects the defined zone of interest, combining reli ability and productivity constraints, is proposed. of spindle, holder, and tool combinations. Thus, semi-ana- lytical approaches have been proposed to minimise experi- Finally, conclusions are presented. mental approaches. Erturk et al. [14] use a receptance cou- 2. Model Building pling (RC) and structural modification method to connect the spindle shaft and the tool holder. Schmitz et al. [15] The vibratory drilling system is composed of a SVDH body model the spindle holder experimentally and couple it with clamped to the spindle by a standard HSK63A tool-holder in- an analytical model of the overhang portion of the tool. terface. A SVDH-vibrating subsystem is jointed to the SVDH Recent approaches consider distributed springs and dampers body with a specific spring and axially guided by a ball retain- between the tool and the holder along the interface contact. er. Finally, a long drill is held in the SVDH-vibrating sub- [15, 16]. Contact stiffness and damping values alter the frequencies and peak values respectively of dominant tool system with a standard ER25 collet chuck. The spindle has four angular bearings in overall back-to-back configuration tip vibration modes. The fast and accurate identification of contact dynamics in spindle-tool assemblies has become an (Figure 1). important issue. Ren and Beards [17], Schmitz et al. [15], The spindle-SVDH-tool finite element model is restrict- and Movahhedy and Gerami [18] treat identification as a ed to the rotating system composed of the spindle shaft, nonlinear optimisation problem. Movahhedy and Gerami the SVDH and the drill. An experimental modal identifica- [18] uses a genetic algorithm to find the global minimum of tion procedure was carried out on the different spindle sub- the optimisation function structures and showed that spindle behaviour can be re- ⎧ ⎫  stricted to rotating structure behaviour [12]. The interfaces ⎨ ⎬ Re g − g p m represented by the HSK63 taper, spring and ball retainer, and g = ,(1) ⎩ ⎭ collet chuck were taken into account in the model. The CNC Im g − g p m milling machine structure was assumed to be infinitely rigid where g and g are, respectively, predicted and measured compared to the other parts of the system. p m receptances of the assembly. Ozsahin et al. [19]present The numerical model of the spindle SVDH tool is based an original identification procedure based on experimental on the integration of the rotating system’s finite element measurements. In their work, the elastic RC equations allow model (FEM), the rolling bearing model, and the interface the stiffness and damping parameters of the spindle-tool as- model. Figure 2 summarises the various stages of model semblies to be obtained in closed-form expressions. In order development. The system substructures were modelled to predict the drill dynamics and the adequate cutting con- through rotor-dynamics formulations. A readjustment pro- ditions that lead to controlled self-excited vibrations, an cedure was carried out on undefined FEM material proper- accurate comprehensive dynamic model of the cutting proc- ties in order to fit model results to experimental ones. The re- ess and spindle-SVDH dynamics is required. ceptance coupling method was used to identify the dynamic In this paper, a hybrid model based on numerical and ex- parameters of the system’s interfaces. Once each structural perimental approaches of the dynamic behavior of the system subsystem model was validated, the identified interface Advances in Acoustics and Vibration 3 SVDH-vibrating subsystem Drill Spindle shaft SVDH-body Housing Collet chuks Spring and HSK 63 taper Bearings ball retainer Drill Collet chuks Drill holder Spindle shaft Spring and HSK 63 taper ball retainer Bearing Interface Figure 1: The spindle-SVDH-tool system and its finite element model. behaviour parameters were integrated to obtain the assem- The set of differential equations can be written as bled global model. M q q ¨ + C q , q˙ + D q˙ + Kq = F(t),(3) N N N N N N 2.1. Modelling of Structural Subsystems. The spindle SVDH where M is the mass matrix, and C matrix contains the rota- tool system is composed of four structural subsystems: the tional dynamics effects. q and F(t) are the nodal displace- drill, the SVDH-vibrating subsystem, the SVDH body, and ment and force vectors. An accelerating rotor gives rise to the spindle. The motion of the rotating structure is consid- previous time-variant equations (3), but treatment of the ered as the superposition of rigid and elastic body displace- rotor using a pseudoconstant speed approach can still be de- ments. Dynamic equations were obtained using Lagrange scribed by means of linear time-invariant models and is valid formulation associated with a finite element method. Due to in many cases (4): the size of the rotor sections, shear deformations had to be taken into account. Then, the rotating substructure was M q ¨ + (2ΩG + D)q˙ + K − Ω N q = F(t),(4) 0 N N N derived using Timoshenko beam theory. The relevant shape where M is the constant part of matrix M,and G and N functions were cubic in order to avoid shear locking. A spe- 0 proceed from the decomposition of matrix C. cial three-dimensional rotor-beam element with two nodes and six degrees of freedom per node was developed in 2.2. Modelling Angular Contact Ball Bearings. The rotating the corotational reference frame. The damping model used system is supported by four (two front and two rear) hybrid draws on Rayleigh viscous equivalent damping, which makes angular contact bearings. The rolling bearing stiffness matri- it possible to regard the damping matrix D as a linear ces were calculated using in-house software developed on the combination of the mass matrix M and the spindle rigidity basis of Lim and Singh’s [20] formulation. The bearing stiff- matrix K ness model represents the load-displacement relation com- bined with the Hertzian contact stress principle and was D = aM + bK,(2) calculated around a static function point characterised by where a and b are damping coefficients. the bearing preload: δ. Based on Rantatalo’s prediction [9], 4 Advances in Acoustics and Vibration Structural subsystem model and interface model Subsystem validation Interface behavior identification Numerical FRFs Experimental FRFs Receptance coupling method Comparison Readjustment algorithm Frequency and magnitude readjustment Validated subsystem model + + Assembled global model Figure 2: Development of the numerical model. 0 0 Outer ring 0 5000 10000 15000 20000 25000 30000 Inner ring Spindle speed (rpm) (a) Angular contact ball bearing Hertzian and (b) Bearing stiffness variation centrifugal force Figure 3: Bearing stiffness variation depending on spindle speed [12]. the initially calculated bearing stiffness is spindle speed de- 2.3. Structural Systems Dynamic Readjustment. While de- pendent because of the gyroscopic and centrifugal force tailed knowledge of the spindle-SVDH-tool system is in gen- F , which acts on each ball (Figure 3(a)). As the speed eral not available in a manufacturing environment, models increased, the load conditions between the balls and the need to be readjusted in order to fit experimental results. rings in the bearing changed because of the centrifugal force The readjustment parameters are the Young modulus: E, (Figure 3(a)). Then, speed-dependent bearing stiffness was the damping coefficients: (a, b), and the rolling bearing pre- integrated into the global spindle FEM and influenced the load: δ. The readjustment procedure is proposed to tune natural frequencies of the spindle-SVDH-tool unit under the previous variable in order to fit the model results to consideration. the experimental frequency response function (FRF). These Relative stiffness variation (%) Advances in Acoustics and Vibration 5 Substructure AB: F F F F F F a1, Ext→A a2, H→A h1, A→H h2, B→H b1, H→B b2, Ext→B a1 a2 b1 b2 x = AB F + AB F , h1 h2 ab1 11 ab1,Ext→ AB 12 ab2,Ext→ AB (8) x x x x x x a1 a2 h1 h2 b1 b2 x = AB F + AB F . ab2 21 ab1,Ext→ AB 22 ab2,Ext→ AB Substructure A Interface H Substructure B Interface: F F ab1, Ext→AB ab2, Ext→AB x − x = HF , h2 h1 h2,B → H ab1 ab2 (9) x x ab1 ab2 F =−F . h2,B → H h1,A→ H Assembled structure AB Compatibility conditions: Figure 4: Receptance coupling notation. x = x , a1 ab1 x = x , b2 ab2 parameters are readjusted by minimizing the gap between the measured and the modelled tool tip node FRF for nonro- x = x , h1 a2 tating components, using an optimisation routine and a least squares type objective function defined as x = x , h2 b1 (10) max 2 F = F , a2,H → A h1,A→ H r = H (ω, (E, a, b, δ)) − H (ω) ,(5) num exp ω=ω min F = F , b1,H → B h2,B → H where H (ω,(E, a, b, δ)) and H (ω) are, respectively, the num exp F = F , a1,Ext→ A ab1,Ext→ AB numerical and experimental FRF. F = F . b2,Ext→ B ab2,Ext→ AB 2.4. Modelling and Identification of Spindle-SVDH-Tool Inter- By integrating the compatibility and interface equations ((9) faces. The dynamic behaviour of the interfaces represented and (10)) into (6)to(8), system (11) is obtained by the HSK63 taper, spring and ball retainer, and collet chuck was taken into account. The identification procedure of the interface models was based on the receptance coupling F (A − AB ) + F (A ) ab1,Ext→ AB 11 11 a2,H → A 12 method. +F (−AB ) = 0, ab2,Ext→ AB 12 2.4.1. Receptance Coupling Background. In this section, the F (A ) + F (A + B ) ab1,Ext→ AB 21 a2,H → A 22 11 RC equations are established. At the top of Figure 4, the (11) substructure A and the substructure B are represented, +F (−B ) = 0, ab2,Ext→ AB 12 connected by interface H. At the bottom of Figure 4, the assembled structure is represented. This figure enables the F (−AB ) + F (−B ) ab1,Ext→ AB 21 a2,H → A 21 excitation point and the measurement points used in the +F (B − AB ) = 0, ab2,Ext→ AB 22 22 receptance coupling approach to be located. The notation A (ω) = (x (ω)/F (ω))| ij ai aj,Ext→ A F ak,Ext→ A=0,∀k = / j AB and AB are elaborated from the definition of 11 21 refers to the spatial receptance vector, whose output is the AB (ω) = (x (ω)/F (ω))| , when ij abi ab j,Ext→ AB F abk,Ext→ AB=0,∀k = j translation of point ai and whose input is the force at point F = 0in(11). The receptance equations are ab2,Ext→ AB aj, when all other forces applied to substructure A are obtained ((12)-(13)) zero. The receptance of the assembled structure AB can be ij expressed according to the receptance of substructures A −1 ij AB = A − A [B + A + H] A , (12) 11 11 12 11 22 21 and B and the receptance of interface H. The movement ij −1 equations of each end point of substructures A and B and AB = A [B + A + H] B . (13) 21 21 11 22 21 of structure AB are written as shown in (6)to(8). The behaviour of the interface, assuming zero mass, is given in Similarly, to obtain AB and, AB it is sufficient to impose 22 12 (9), and the compatibility conditions are written in (10). F = 0in(11). Then the standard receptance ab1,Ext→ AB Substructure A: equations are obtained(14) x = A F + A F , a1 11 a1,Ext→ A 12 a2,H → A −1 AB = B − B [B + A + H] B , 22 22 12 11 22 21 (6) (14) x = A F + A F . a2 21 a1,Ext→ A 22 a2,H → A −1 AB = A [B + A + H] B . 12 12 11 22 12 Substructure B: From the receptance of the system substructures A , B , ij ij x = B F + B F , b1 11 b1,H → B 12 b2,Ext→ B associated with the interface model H, the reconstruction of (7) the assembled structure FRF: AB is possible. x = B F + B F . ij b2 21 b1,H → B 22 b2,Ext→ B 6 Advances in Acoustics and Vibration that the AB and AB curves are in good 11-reconstructed 11-direct H: spring and A: SVDH-vibrating B: SVDH body agreement, which enables the spring-damper interface model ball retainer subsystem and the identified values (k = 1.13 × 10 N/m and c = 24 Ns/m) to be validated. 2.4.3. Application to the Collet Chuck Interface. The interface between the drill and the SVDH-vibrating subsystem is a collet chuck joint. Receptance AB is preferred to 21-direct (a) AB in order to facilitate the experimental identifica- 11-direct tion procedure. Moreover, the mass of the accelerometer is k = 0.5 × 10 N/m c = 0 Ns/m not negligible compared to the mass of the drill. In addition, −100 the SVDH-vibrating subsystem is not in the same material configuration when the collet chuck is tightened on a drill as when it is not. For these reasons, receptance AB 21-reconstructed (13) was obtained using numerical receptance for compo- nents A and B. The identification procedure provided collet chuck stiffness and damping factors of, respectively, 14.8 × k = 1.13 × 10 N/m 6 10 N/m and 4 Ns/m. c = 24 Ns/m (identified values) 2.4.4. Application to the HSK 63 Taper Interface. The exper- imental modal analyses carried out on the spindle/SVDH body system in the axial direction allow the HSK 63 interface to be considered as a rigid connection (Figure 6). Indeed, no −200 100 1000 specific mode for the HSK 63 taper interface appears between Frequency (Hz) 0 and 3000 Hz. AB 11-reconstructed AB 2.5. Model Assembly and Experimental Validation. As in a 11-reconstructed AB 11-direct classic finite element procedure, dynamic equations of the overall system, composed of the drill, the SVDH-vibrating (b) subsystems, the SVDH body, and the spindle, were obtained Figure 5: (a) Components (SVDH-body, SVDH vibrating subsys- by assembling element matrices. The spring-damper connec- tem) and interfaces (spring and ball retainer), (b) identification of tion parameters between the drill and the SVDH-vibrating the stiffness: k and the damping factor: c by minimizing the gap subsystems and between the SVDH-vibrating subsystems between AB and AB . 11-reconstructed 11-direct and SVDH body, identified by the receptance coupling meth- od, enabled the rotor-beam models of the components to be assembled. 2.4.2. Application to the SVDH-Vibrating Subsystem/SVDH The spindle-SVDH-tool assembled model was validated Body Interface. The interface between the SVDH-vibrating by comparison between numerical and experimental FRF, as subsystem and the SVDH body is a prismatic joint whose shown in Figure 7. axial stiffness is controlled by a spring. The SVDH-vibrating Figure 7(a) represents experimental and numerical axial subsystem is guided in axial translation through a ball FRF of the assembled system. The 60 Hz and 4700 Hz retainer, which controls the radial stiffness of the interface. modes are, respectively, due to the spring-ball retainer and Figure 5(a) shows these components. The axial dynamic collet chuck interfaces as mentioned in the previous subsys- behaviour of the interface was modelled using a spring tem identification procedure. Some parasitical experimental damper: H = 1/(k + icω). This interface was assumed model bending modes at 193 Hz, 1237 Hz, and 3433 Hz are present to be rigid in the radial direction. in the experimental FRF. The k and c values were determined by minimizing the Figure 7(b) represents experimental and numerical radial following r criteria: FRF of the assembled system. The 193, 1237, and 3433 Hz modes are related to the drill’s bending modes. The 376 Hz ω +ε mode is controlled by the bearings. Additional numerical r = (AB (ω, k, c) − AB (ω)) , 11-reconstructed 11-direct modes are present at 1830 and 3740 Hz. These frequency ω=ω −ε peaks are related to the dynamics behaviour of the rear side (15) of the spindle. They do not appear in the experimental FRF where AB and AB represent, respectively, since the displacements of the rear side of the real spindle are 11-reconstructed 11-direct the RC-constructed FRF obtained by (12) and the measured blocked by the motor. For the axial and radial FRF, a good FRF on the assembled system. ω is the interface mode correspondence between the numerical and experimental pulsation. The optimisation procedure was carried out on a curves enabled the numerical model to be used for further 3 dB bandwidth around ω .In Figure 5(b),itcan be noticed investigations. Magnitude (dB) Advances in Acoustics and Vibration 7 −150 Impact −200 Measure 0 1000 2000 3000 Frequency (Hz) (a) (b) Impact Measure (c) Figure 6: HSK 63 experimental modal analyses and associated axial FRF. measure impact measure impact −70 −70 −90 −100 −100 −130 0 50 100 −150 −150 −200 −200 0 2500 5000 0 2500 5000 193 1237 3433 4700 193 376 1237 3433 Frequency (Hz) Frequency (Hz) Experimental Experimental Numerical Numerical (a) (b) Figure 7: Numerical versus experimental system FRF. Axial FRF(dB) Axial FRF (dB) Radial FRF (dB) 8 Advances in Acoustics and Vibration 0.15 High speed vibratory 0.1 drilling head 0.05 Long drill 4000 8000 12000 16000 Spindle speed (rpm) Dynamometer Regular chips fragmentation No chips fragmentation (a) (b) Figure 8: Experimental setup. 10 in order to give rules of uses to obtain the maximum removal rate with respect to system lifespan. For bearing lifespan calculations, experiments were car- ried out to measure the cutting force for different cutting conditions, using a three-component dynamometer (Kistler dynamometer type 9257B). High-speed vibratory drilling operations, representative of the industrial context, were performed on 35MnV7 steel, with a drill of 116 mm length and of 5 mm diameter. The experimental setup is represented in Figure 8(a). Rotation speed and feed rate were tested, respectively, between 5500 and 15500 rpm and between 0.05 and 0.15 mm/rev. Only the drilling operations that led to regular chips fragmentation were used to predict bearing lifespan, as shown in Figure 8(b). The time-variant value of 20000 hours the radial forces, which depends on the angular orientation of the cutter as it rotates through the cut, was expanded into a Fourier series and then truncated to include only N = 12000 rpm N = 9000 rpm the fundamental frequency. These forces, combined with the f = 0.05 mm/rev f = 0.125 mm/rev numerical FRF of the model, between the excitation node and the bearings nodes, allow the resulting rolling bearings Front bearings Rear bearings solicitation to be determined. The rolling bearings lifespan is 6 3 10 C L = , (16) 60N P where C is the basic dynamic load and P the equivalent dynamic load in Newtons. N is the spindle speed in rpm. The Figure 9: Influence of location and cutting conditions on bearings values obtained are expressed in hours and can be compared lifespan. to an industrial objective of 20000 hours. 3.1. Effect of Cutting Conditions on Bearing Lifespan. First, 3. Bearing Lifespan Predictions the bearings with the shortest lifespan is sought, because it is The industrial context of the proposed paper is to realize the element upon which the lifespan of the entire spindle de- high-speed vibratory drilling operation, with a drill diameter pends. For all cutting conditions, the bearings with the short- of 5 mm, with a drill depth of 100 mm, without retreat cycles est lifespan is always the front bearings. From Figure 9, or lubrication, in a mass production system. which represents the extreme lifespan of the different bear- In this section, the numerical model will be used to ings, the strong influence of the cutting conditions on the predict bearings lifespan for various cutting conditions, in bearings lifespan can be noticed. Industrial recommenda- this context. The calculated bearing lifespan, thanks to exper- tions for spindle lifespan are 20000 hours. The shortest imental data, was compared to industrial recommendations calculated lifespan is 55000 and corresponds to the cutting Bearings lifespan (hours) Feed rate (mm/rev) Advances in Acoustics and Vibration 9 6 −3 ×10 ×10 0 3 0 0 0.13 0.05 0.1 0.13 0.05 0.1 Feed rate (mm/rev) Feed rate (mm/rev) (a) (b) Figure 10: (a) Influence of feed rate on bearings lifespan, for a spindle speed of 12000 rpm, and (b) influence of the feed rate on the inverse of experimental axial cutting force magnitude. higher than 0.125 mm/rev, no stable vibrations of the drill were obtained. The bigger the feed rate, the lower the bear- ings Lifespan. However, even the shortest bearings Lifespan is compatible with industrial standard. Hence, the optimal feed rate of 0.125 mm/rev is retained. Figure 10(b) shows that cutting forces are modified by the feed rate. Similarities between the curves in Figures 10(a) and 10(b) indicate that the bearing lifespan is mainly influenced by the feed rate. Figure 11(b) shows the variations in bearing lifespan de- 0 5000 10000 15000 pending on spindle speed, for a feed rate of 0.05 mm per rev- Spindle speed (rpm) olution. The curve enables the determination of a spindle 2nd mode speed at 8500 rpm which maximizes bearing Lifespan and a 1st mode spindle speed at 11500 rpm which minimizes it. In order to Drill oscillations (a) optimize productivity, as shown on Figure 8(b), the recom- mended cutting spindle speed is 12000 rpm. The curves of Figure 11 illustrate the dynamic effects due to high rotational speed, such as gyroscopic coupling and spin softening, on system behaviour, and hence on bearing lifespan. Figure 11(a) represents the effect of spindle speed on the first two radial modes of the system. The dotted line 5000 10000 indicates the drill oscillation frequency and is plotted from Spindle speed (rpm) experimental data measured during high-speed vibratory Front bearing drilling operations. The critical speed, at 11500 rpm, corre- (b) sponds to the intersection of the radial mode frequencies Figure 11: (a) Campbell diagram. (b) Influence of spindle speed on with the excitation line due to the drill oscillations. bearings lifespan, for a feed rate of 0.05 mm/rev. 4. Conclusions conditions given rise to the best material removal rate (N = In this paper, a comprehensive approach to developing a hy- 12000 rpm, f = 0.125 mm/rev). Thus, a high-speed vibra- brid model of the dynamic behavior of the spindle self-vibra- tory drilling operation is always compatible with industri- tory drilling head—tool system has been proposed. This ap- al standards, even under the best material removal rate cut- proach has resulted in a numerical model enriched with ting conditions. physical data. The various components of the system are Figure 10(a) shows the influence of the feed rate on bear- modelled using a specific beam element, taking into account ings lifespan, for a spindle speed of 12000 rpm. Only feed the gyroscopic effects, centrifugal forces, and shear deflec- rateshigherthan0.05mm/revweretested, becausefor feed tion. The receptance coupling method is used to identify the rates below this limit, the productivity is too low and high- speed vibratory drilling loses its relevance. For feed rates dynamic behavior of the interface. The complete system is Frequency (Hz) Lifespan (hours) Lifespan (hours) −1 Axial cutting forces (1/N) 10 Advances in Acoustics and Vibration then obtained by assembling the beam model of each com- [7] A. Erturk, H. N. Ozguven, and E. Budak, “Effect analysis of bearing and interface dynamics on tool point FRF for chatter ponent using spring-damper elements. 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Budak, “Analytical modeling F: force vector of spindle-tool dynamics on machine tools using Timoshenko E:Youngmodulus beam model and receptance coupling for the prediction of δ: rolling bearing preload tool point FRF,” International Journal of Machine Tools and N: spindle speed in rpm Manufacture, vol. 46, no. 15, pp. 1901–1912, 2006. L : bearing lifespan [15] T. L. Schmitz, K. Powell, D. Won, G. S. Duncan, W. G. C: basic dynamic load Sawyer, and J. C. Ziegert, “Shrink fit tool holder connection P: equivalent dynamic bearing load. stiffness/damping modeling for frequency response prediction in milling,” International Journal of Machine Tools and Manu- facture, vol. 47, no. 9, pp. 1368–1380, 2007. References [16] H. Ahmadian and M. Nourmohammadi, “Tool point dynam- [1]A.M.Gouskov,S.A.Voronov,and S. A. 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Modeling of Self-Vibratory Drilling Head-Spindle System for Predictions of Bearings Lifespan

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Copyright © 2011 F. Forestier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 606087, 10 pages doi:10.1155/2011/606087 Research Article Modeling of Self-Vibratory Drilling Head-Spindle System for Predictions of Bearings Lifespan 1 2 2 3 F. Forestier, V. Gagnol, P. Ray, and H. Paris LaMI Laboratory, Clermont Universit´e, UBP, EA 3867, BP2235, 03101 Montlucon, France LaMI Laboratory, Clermont Universit´e, IFMA, EA 3867, BP10448, 63000 Clermont-Ferrand, France G-SCOP Laboratory, G-SCOP, 46 Avenue F´elix Viallet, 38031 Grenoble, France Correspondence should be addressed to F. Forestier, fabien.forestier@moniut.univ-bpclermont.fr Received 13 January 2011; Accepted 7 June 2011 Academic Editor: Atma Sahu Copyright © 2011 F. Forestier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The machining of deep holes is limited due to inadequate chip evacuation, which induces tool breakage. To limit this drawback, retreat cycles and lubrication are used. An alternative response to the evacuation problem is based on high-speed vibratory drilling. A specific tool holder induces axial self-maintained vibration of the drill, which enables the chips to be split. The chips are thus of a small size and can be evacuated. To anticipate the potential risk of decreased spindle lifespan associated with these vibrations, a model of the behavior of the system (spindle—self-vibrating drilling head—tool) is elaborated. In order to assess the dynamic behavior of the system, this study develops a rotor-based finite element model, integrated with the modelling of component interfaces. The current results indicate that the simulations are consistent with the experimental measurements. The influence of spindle speed and feed rate on bearing lifespan is highlighted. 1. Introduction The SVDH is composed of an axial vibrating system, con- sisting of the SVDH vibrating subsystem, mounted on a spe- High-speed vibratory drilling allows chips to be split thanks cific HSK 63 taper, and called the SVDH body. The SVDH- to self-maintained vibration during cutting [1]. When these vibrating subsystem is composed of the vibrating parts of vibrations have a magnitude greater than the advance per the drill holder. The SVDH body guides the axial vibrating tooth, the drill continuously enters and exits the material, subsystem through a ball retainer and a classic HSK63 taper which allows the fragmentation of the chips, as shown by connexion with the spindle head. The self-excited vibrations Tichkiewitchetal. [2]. The chips are small and can thus be must be tuned and controlled in order to have a magnitude removed easily without retreat cycles or lubrication. High- greater than the advance per tooth. Mathematical models of speed vibratory drilling enables productivity to increase by a the SVDH dynamics appearing in most of previous works factor of three compared to traditional techniques. Currently, lead to a one-dimensional linear or nonlinear model in the high-speed vibratory drilling is the only process capable of axial vibration direction governed by the mass, damping, producing deep holes using high-speed machining centers, and stiffness of the considered system [3]. In these studies, without lubricants, and with high productivity [3]. damping, which plays an important role in SVDH dynamic However, the excitation generated by the cutting process behaviour, is estimated but not experimentally identified. can also be a source of damage to the self-vibrating drilling Moreover, the spindle behaviour and the interfaces between head and the spindle. So the industrialization of high-speed the spindle and the SVDH are assumed to be rigid and have vibratory drilling requires the effects of vibration on the not been taken into account. However, many works have machine, and more particularly on the spindle, to be predict- showed that tool point dynamics can be significantly influ- ed. For this purpose, a model of the dynamic behavior of enced by spindle dynamics, as well as the spindle-holder— the system (self-vibrating drilling head spindle) is elaborated tool interfaces. This creates a demand for predictive knowl- from a realistic assembly of the components, in order to edge models that are capable of investigating the influence of predict spindle bearing lifespan. cutting conditions on high-speed spindle-SVDH system. 2 Advances in Acoustics and Vibration Many authors have investigated the dynamic behaviour is proposed. This finite element model takes into account of machine tool spindle-bearing systems, both analytically rotor dynamics effects, bearing stiffness and the real behavior and experimentally. They show that spindle dynamics are in- of interfaces between different system components. The sim- fluenced by a large number of factors, including holder char- ulation allows achieving the best possible cutting parameters acteristics [4], spindle shaft geometry and drawbar force without damaging the bearings of the spindle. [5, 6], and the stiffness and damping provided by the bear- In the second section, the spindle-SVDH rotor dynamics ings [7]. Most of these factors are independent of spindle model is presented. A special rotor-beam element is imple- speed, contrary to bearing properties [8, 9] and the spindle mented. The rolling bearing stiffness matrices are calculated rotor dynamics, which change according to spindle speed. around a static function point on the basis of Lim and Such rotating systems have been successfully modelled Singh’s [20] formulation and then integrated into the global through rotor-dynamics studies [10, 11]. In previous works finite element model. The identification of contact dynamics [12, 13], a dynamic high-speed spindle-bearing system mod- in tool-SVDH-spindle assemblies is carried out using the el based on rotor dynamics prediction was presented. Ele- RC method on the basis of experimental substructure char- ment kinematics were formulated in a corotational coordi- acterisation. The identified models are then integrated into nate frame and enabled a special rotor beam element to be the global spindle-SVDH-tool model. Finally, numerical and developed. Model results showed that spindle dynamics are experimental tool tip FRF, in radial and axial directions, is influenced by the gyroscopic coupling and the spin softening compared in order to validate the global assembled model. of the rotating shaft due to high rotation speeds. The literature on the modelling and analysis of spindles Section 3 is dedicated to the definition of optimal cutting shows that the tool tip FRF is also greatly influenced by the conditions with respect to industrial objectives. The studied industrial context requires the maximum material cutting contact dynamics of the spindle-holder-tool interfaces. The flexibility of the afore-mentioned interfaces can dominate rate and a rational use of the tool-SVDH-spindle set in order to guarantee adequate rolling bearing lifespan. As a result, the dynamics of the spindle. The tool tip FRF is usually ob- tained using experimental measurements, which require sig- a recommendation for the use of a spindle-SVDH-tool set nificant testing time to take into account the large number which respects the defined zone of interest, combining reli ability and productivity constraints, is proposed. of spindle, holder, and tool combinations. Thus, semi-ana- lytical approaches have been proposed to minimise experi- Finally, conclusions are presented. mental approaches. Erturk et al. [14] use a receptance cou- 2. Model Building pling (RC) and structural modification method to connect the spindle shaft and the tool holder. Schmitz et al. [15] The vibratory drilling system is composed of a SVDH body model the spindle holder experimentally and couple it with clamped to the spindle by a standard HSK63A tool-holder in- an analytical model of the overhang portion of the tool. terface. A SVDH-vibrating subsystem is jointed to the SVDH Recent approaches consider distributed springs and dampers body with a specific spring and axially guided by a ball retain- between the tool and the holder along the interface contact. er. Finally, a long drill is held in the SVDH-vibrating sub- [15, 16]. Contact stiffness and damping values alter the frequencies and peak values respectively of dominant tool system with a standard ER25 collet chuck. The spindle has four angular bearings in overall back-to-back configuration tip vibration modes. The fast and accurate identification of contact dynamics in spindle-tool assemblies has become an (Figure 1). important issue. Ren and Beards [17], Schmitz et al. [15], The spindle-SVDH-tool finite element model is restrict- and Movahhedy and Gerami [18] treat identification as a ed to the rotating system composed of the spindle shaft, nonlinear optimisation problem. Movahhedy and Gerami the SVDH and the drill. An experimental modal identifica- [18] uses a genetic algorithm to find the global minimum of tion procedure was carried out on the different spindle sub- the optimisation function structures and showed that spindle behaviour can be re- ⎧ ⎫  stricted to rotating structure behaviour [12]. The interfaces ⎨ ⎬ Re g − g p m represented by the HSK63 taper, spring and ball retainer, and g = ,(1) ⎩ ⎭ collet chuck were taken into account in the model. The CNC Im g − g p m milling machine structure was assumed to be infinitely rigid where g and g are, respectively, predicted and measured compared to the other parts of the system. p m receptances of the assembly. Ozsahin et al. [19]present The numerical model of the spindle SVDH tool is based an original identification procedure based on experimental on the integration of the rotating system’s finite element measurements. In their work, the elastic RC equations allow model (FEM), the rolling bearing model, and the interface the stiffness and damping parameters of the spindle-tool as- model. Figure 2 summarises the various stages of model semblies to be obtained in closed-form expressions. In order development. The system substructures were modelled to predict the drill dynamics and the adequate cutting con- through rotor-dynamics formulations. A readjustment pro- ditions that lead to controlled self-excited vibrations, an cedure was carried out on undefined FEM material proper- accurate comprehensive dynamic model of the cutting proc- ties in order to fit model results to experimental ones. The re- ess and spindle-SVDH dynamics is required. ceptance coupling method was used to identify the dynamic In this paper, a hybrid model based on numerical and ex- parameters of the system’s interfaces. Once each structural perimental approaches of the dynamic behavior of the system subsystem model was validated, the identified interface Advances in Acoustics and Vibration 3 SVDH-vibrating subsystem Drill Spindle shaft SVDH-body Housing Collet chuks Spring and HSK 63 taper Bearings ball retainer Drill Collet chuks Drill holder Spindle shaft Spring and HSK 63 taper ball retainer Bearing Interface Figure 1: The spindle-SVDH-tool system and its finite element model. behaviour parameters were integrated to obtain the assem- The set of differential equations can be written as bled global model. M q q ¨ + C q , q˙ + D q˙ + Kq = F(t),(3) N N N N N N 2.1. Modelling of Structural Subsystems. The spindle SVDH where M is the mass matrix, and C matrix contains the rota- tool system is composed of four structural subsystems: the tional dynamics effects. q and F(t) are the nodal displace- drill, the SVDH-vibrating subsystem, the SVDH body, and ment and force vectors. An accelerating rotor gives rise to the spindle. The motion of the rotating structure is consid- previous time-variant equations (3), but treatment of the ered as the superposition of rigid and elastic body displace- rotor using a pseudoconstant speed approach can still be de- ments. Dynamic equations were obtained using Lagrange scribed by means of linear time-invariant models and is valid formulation associated with a finite element method. Due to in many cases (4): the size of the rotor sections, shear deformations had to be taken into account. Then, the rotating substructure was M q ¨ + (2ΩG + D)q˙ + K − Ω N q = F(t),(4) 0 N N N derived using Timoshenko beam theory. The relevant shape where M is the constant part of matrix M,and G and N functions were cubic in order to avoid shear locking. A spe- 0 proceed from the decomposition of matrix C. cial three-dimensional rotor-beam element with two nodes and six degrees of freedom per node was developed in 2.2. Modelling Angular Contact Ball Bearings. The rotating the corotational reference frame. The damping model used system is supported by four (two front and two rear) hybrid draws on Rayleigh viscous equivalent damping, which makes angular contact bearings. The rolling bearing stiffness matri- it possible to regard the damping matrix D as a linear ces were calculated using in-house software developed on the combination of the mass matrix M and the spindle rigidity basis of Lim and Singh’s [20] formulation. The bearing stiff- matrix K ness model represents the load-displacement relation com- bined with the Hertzian contact stress principle and was D = aM + bK,(2) calculated around a static function point characterised by where a and b are damping coefficients. the bearing preload: δ. Based on Rantatalo’s prediction [9], 4 Advances in Acoustics and Vibration Structural subsystem model and interface model Subsystem validation Interface behavior identification Numerical FRFs Experimental FRFs Receptance coupling method Comparison Readjustment algorithm Frequency and magnitude readjustment Validated subsystem model + + Assembled global model Figure 2: Development of the numerical model. 0 0 Outer ring 0 5000 10000 15000 20000 25000 30000 Inner ring Spindle speed (rpm) (a) Angular contact ball bearing Hertzian and (b) Bearing stiffness variation centrifugal force Figure 3: Bearing stiffness variation depending on spindle speed [12]. the initially calculated bearing stiffness is spindle speed de- 2.3. Structural Systems Dynamic Readjustment. While de- pendent because of the gyroscopic and centrifugal force tailed knowledge of the spindle-SVDH-tool system is in gen- F , which acts on each ball (Figure 3(a)). As the speed eral not available in a manufacturing environment, models increased, the load conditions between the balls and the need to be readjusted in order to fit experimental results. rings in the bearing changed because of the centrifugal force The readjustment parameters are the Young modulus: E, (Figure 3(a)). Then, speed-dependent bearing stiffness was the damping coefficients: (a, b), and the rolling bearing pre- integrated into the global spindle FEM and influenced the load: δ. The readjustment procedure is proposed to tune natural frequencies of the spindle-SVDH-tool unit under the previous variable in order to fit the model results to consideration. the experimental frequency response function (FRF). These Relative stiffness variation (%) Advances in Acoustics and Vibration 5 Substructure AB: F F F F F F a1, Ext→A a2, H→A h1, A→H h2, B→H b1, H→B b2, Ext→B a1 a2 b1 b2 x = AB F + AB F , h1 h2 ab1 11 ab1,Ext→ AB 12 ab2,Ext→ AB (8) x x x x x x a1 a2 h1 h2 b1 b2 x = AB F + AB F . ab2 21 ab1,Ext→ AB 22 ab2,Ext→ AB Substructure A Interface H Substructure B Interface: F F ab1, Ext→AB ab2, Ext→AB x − x = HF , h2 h1 h2,B → H ab1 ab2 (9) x x ab1 ab2 F =−F . h2,B → H h1,A→ H Assembled structure AB Compatibility conditions: Figure 4: Receptance coupling notation. x = x , a1 ab1 x = x , b2 ab2 parameters are readjusted by minimizing the gap between the measured and the modelled tool tip node FRF for nonro- x = x , h1 a2 tating components, using an optimisation routine and a least squares type objective function defined as x = x , h2 b1 (10) max 2 F = F , a2,H → A h1,A→ H r = H (ω, (E, a, b, δ)) − H (ω) ,(5) num exp ω=ω min F = F , b1,H → B h2,B → H where H (ω,(E, a, b, δ)) and H (ω) are, respectively, the num exp F = F , a1,Ext→ A ab1,Ext→ AB numerical and experimental FRF. F = F . b2,Ext→ B ab2,Ext→ AB 2.4. Modelling and Identification of Spindle-SVDH-Tool Inter- By integrating the compatibility and interface equations ((9) faces. The dynamic behaviour of the interfaces represented and (10)) into (6)to(8), system (11) is obtained by the HSK63 taper, spring and ball retainer, and collet chuck was taken into account. The identification procedure of the interface models was based on the receptance coupling F (A − AB ) + F (A ) ab1,Ext→ AB 11 11 a2,H → A 12 method. +F (−AB ) = 0, ab2,Ext→ AB 12 2.4.1. Receptance Coupling Background. In this section, the F (A ) + F (A + B ) ab1,Ext→ AB 21 a2,H → A 22 11 RC equations are established. At the top of Figure 4, the (11) substructure A and the substructure B are represented, +F (−B ) = 0, ab2,Ext→ AB 12 connected by interface H. At the bottom of Figure 4, the assembled structure is represented. This figure enables the F (−AB ) + F (−B ) ab1,Ext→ AB 21 a2,H → A 21 excitation point and the measurement points used in the +F (B − AB ) = 0, ab2,Ext→ AB 22 22 receptance coupling approach to be located. The notation A (ω) = (x (ω)/F (ω))| ij ai aj,Ext→ A F ak,Ext→ A=0,∀k = / j AB and AB are elaborated from the definition of 11 21 refers to the spatial receptance vector, whose output is the AB (ω) = (x (ω)/F (ω))| , when ij abi ab j,Ext→ AB F abk,Ext→ AB=0,∀k = j translation of point ai and whose input is the force at point F = 0in(11). The receptance equations are ab2,Ext→ AB aj, when all other forces applied to substructure A are obtained ((12)-(13)) zero. The receptance of the assembled structure AB can be ij expressed according to the receptance of substructures A −1 ij AB = A − A [B + A + H] A , (12) 11 11 12 11 22 21 and B and the receptance of interface H. The movement ij −1 equations of each end point of substructures A and B and AB = A [B + A + H] B . (13) 21 21 11 22 21 of structure AB are written as shown in (6)to(8). The behaviour of the interface, assuming zero mass, is given in Similarly, to obtain AB and, AB it is sufficient to impose 22 12 (9), and the compatibility conditions are written in (10). F = 0in(11). Then the standard receptance ab1,Ext→ AB Substructure A: equations are obtained(14) x = A F + A F , a1 11 a1,Ext→ A 12 a2,H → A −1 AB = B − B [B + A + H] B , 22 22 12 11 22 21 (6) (14) x = A F + A F . a2 21 a1,Ext→ A 22 a2,H → A −1 AB = A [B + A + H] B . 12 12 11 22 12 Substructure B: From the receptance of the system substructures A , B , ij ij x = B F + B F , b1 11 b1,H → B 12 b2,Ext→ B associated with the interface model H, the reconstruction of (7) the assembled structure FRF: AB is possible. x = B F + B F . ij b2 21 b1,H → B 22 b2,Ext→ B 6 Advances in Acoustics and Vibration that the AB and AB curves are in good 11-reconstructed 11-direct H: spring and A: SVDH-vibrating B: SVDH body agreement, which enables the spring-damper interface model ball retainer subsystem and the identified values (k = 1.13 × 10 N/m and c = 24 Ns/m) to be validated. 2.4.3. Application to the Collet Chuck Interface. The interface between the drill and the SVDH-vibrating subsystem is a collet chuck joint. Receptance AB is preferred to 21-direct (a) AB in order to facilitate the experimental identifica- 11-direct tion procedure. Moreover, the mass of the accelerometer is k = 0.5 × 10 N/m c = 0 Ns/m not negligible compared to the mass of the drill. In addition, −100 the SVDH-vibrating subsystem is not in the same material configuration when the collet chuck is tightened on a drill as when it is not. For these reasons, receptance AB 21-reconstructed (13) was obtained using numerical receptance for compo- nents A and B. The identification procedure provided collet chuck stiffness and damping factors of, respectively, 14.8 × k = 1.13 × 10 N/m 6 10 N/m and 4 Ns/m. c = 24 Ns/m (identified values) 2.4.4. Application to the HSK 63 Taper Interface. The exper- imental modal analyses carried out on the spindle/SVDH body system in the axial direction allow the HSK 63 interface to be considered as a rigid connection (Figure 6). Indeed, no −200 100 1000 specific mode for the HSK 63 taper interface appears between Frequency (Hz) 0 and 3000 Hz. AB 11-reconstructed AB 2.5. Model Assembly and Experimental Validation. As in a 11-reconstructed AB 11-direct classic finite element procedure, dynamic equations of the overall system, composed of the drill, the SVDH-vibrating (b) subsystems, the SVDH body, and the spindle, were obtained Figure 5: (a) Components (SVDH-body, SVDH vibrating subsys- by assembling element matrices. The spring-damper connec- tem) and interfaces (spring and ball retainer), (b) identification of tion parameters between the drill and the SVDH-vibrating the stiffness: k and the damping factor: c by minimizing the gap subsystems and between the SVDH-vibrating subsystems between AB and AB . 11-reconstructed 11-direct and SVDH body, identified by the receptance coupling meth- od, enabled the rotor-beam models of the components to be assembled. 2.4.2. Application to the SVDH-Vibrating Subsystem/SVDH The spindle-SVDH-tool assembled model was validated Body Interface. The interface between the SVDH-vibrating by comparison between numerical and experimental FRF, as subsystem and the SVDH body is a prismatic joint whose shown in Figure 7. axial stiffness is controlled by a spring. The SVDH-vibrating Figure 7(a) represents experimental and numerical axial subsystem is guided in axial translation through a ball FRF of the assembled system. The 60 Hz and 4700 Hz retainer, which controls the radial stiffness of the interface. modes are, respectively, due to the spring-ball retainer and Figure 5(a) shows these components. The axial dynamic collet chuck interfaces as mentioned in the previous subsys- behaviour of the interface was modelled using a spring tem identification procedure. Some parasitical experimental damper: H = 1/(k + icω). This interface was assumed model bending modes at 193 Hz, 1237 Hz, and 3433 Hz are present to be rigid in the radial direction. in the experimental FRF. The k and c values were determined by minimizing the Figure 7(b) represents experimental and numerical radial following r criteria: FRF of the assembled system. The 193, 1237, and 3433 Hz modes are related to the drill’s bending modes. The 376 Hz ω +ε mode is controlled by the bearings. Additional numerical r = (AB (ω, k, c) − AB (ω)) , 11-reconstructed 11-direct modes are present at 1830 and 3740 Hz. These frequency ω=ω −ε peaks are related to the dynamics behaviour of the rear side (15) of the spindle. They do not appear in the experimental FRF where AB and AB represent, respectively, since the displacements of the rear side of the real spindle are 11-reconstructed 11-direct the RC-constructed FRF obtained by (12) and the measured blocked by the motor. For the axial and radial FRF, a good FRF on the assembled system. ω is the interface mode correspondence between the numerical and experimental pulsation. The optimisation procedure was carried out on a curves enabled the numerical model to be used for further 3 dB bandwidth around ω .In Figure 5(b),itcan be noticed investigations. Magnitude (dB) Advances in Acoustics and Vibration 7 −150 Impact −200 Measure 0 1000 2000 3000 Frequency (Hz) (a) (b) Impact Measure (c) Figure 6: HSK 63 experimental modal analyses and associated axial FRF. measure impact measure impact −70 −70 −90 −100 −100 −130 0 50 100 −150 −150 −200 −200 0 2500 5000 0 2500 5000 193 1237 3433 4700 193 376 1237 3433 Frequency (Hz) Frequency (Hz) Experimental Experimental Numerical Numerical (a) (b) Figure 7: Numerical versus experimental system FRF. Axial FRF(dB) Axial FRF (dB) Radial FRF (dB) 8 Advances in Acoustics and Vibration 0.15 High speed vibratory 0.1 drilling head 0.05 Long drill 4000 8000 12000 16000 Spindle speed (rpm) Dynamometer Regular chips fragmentation No chips fragmentation (a) (b) Figure 8: Experimental setup. 10 in order to give rules of uses to obtain the maximum removal rate with respect to system lifespan. For bearing lifespan calculations, experiments were car- ried out to measure the cutting force for different cutting conditions, using a three-component dynamometer (Kistler dynamometer type 9257B). High-speed vibratory drilling operations, representative of the industrial context, were performed on 35MnV7 steel, with a drill of 116 mm length and of 5 mm diameter. The experimental setup is represented in Figure 8(a). Rotation speed and feed rate were tested, respectively, between 5500 and 15500 rpm and between 0.05 and 0.15 mm/rev. Only the drilling operations that led to regular chips fragmentation were used to predict bearing lifespan, as shown in Figure 8(b). The time-variant value of 20000 hours the radial forces, which depends on the angular orientation of the cutter as it rotates through the cut, was expanded into a Fourier series and then truncated to include only N = 12000 rpm N = 9000 rpm the fundamental frequency. These forces, combined with the f = 0.05 mm/rev f = 0.125 mm/rev numerical FRF of the model, between the excitation node and the bearings nodes, allow the resulting rolling bearings Front bearings Rear bearings solicitation to be determined. The rolling bearings lifespan is 6 3 10 C L = , (16) 60N P where C is the basic dynamic load and P the equivalent dynamic load in Newtons. N is the spindle speed in rpm. The Figure 9: Influence of location and cutting conditions on bearings values obtained are expressed in hours and can be compared lifespan. to an industrial objective of 20000 hours. 3.1. Effect of Cutting Conditions on Bearing Lifespan. First, 3. Bearing Lifespan Predictions the bearings with the shortest lifespan is sought, because it is The industrial context of the proposed paper is to realize the element upon which the lifespan of the entire spindle de- high-speed vibratory drilling operation, with a drill diameter pends. For all cutting conditions, the bearings with the short- of 5 mm, with a drill depth of 100 mm, without retreat cycles est lifespan is always the front bearings. From Figure 9, or lubrication, in a mass production system. which represents the extreme lifespan of the different bear- In this section, the numerical model will be used to ings, the strong influence of the cutting conditions on the predict bearings lifespan for various cutting conditions, in bearings lifespan can be noticed. Industrial recommenda- this context. The calculated bearing lifespan, thanks to exper- tions for spindle lifespan are 20000 hours. The shortest imental data, was compared to industrial recommendations calculated lifespan is 55000 and corresponds to the cutting Bearings lifespan (hours) Feed rate (mm/rev) Advances in Acoustics and Vibration 9 6 −3 ×10 ×10 0 3 0 0 0.13 0.05 0.1 0.13 0.05 0.1 Feed rate (mm/rev) Feed rate (mm/rev) (a) (b) Figure 10: (a) Influence of feed rate on bearings lifespan, for a spindle speed of 12000 rpm, and (b) influence of the feed rate on the inverse of experimental axial cutting force magnitude. higher than 0.125 mm/rev, no stable vibrations of the drill were obtained. The bigger the feed rate, the lower the bear- ings Lifespan. However, even the shortest bearings Lifespan is compatible with industrial standard. Hence, the optimal feed rate of 0.125 mm/rev is retained. Figure 10(b) shows that cutting forces are modified by the feed rate. Similarities between the curves in Figures 10(a) and 10(b) indicate that the bearing lifespan is mainly influenced by the feed rate. Figure 11(b) shows the variations in bearing lifespan de- 0 5000 10000 15000 pending on spindle speed, for a feed rate of 0.05 mm per rev- Spindle speed (rpm) olution. The curve enables the determination of a spindle 2nd mode speed at 8500 rpm which maximizes bearing Lifespan and a 1st mode spindle speed at 11500 rpm which minimizes it. In order to Drill oscillations (a) optimize productivity, as shown on Figure 8(b), the recom- mended cutting spindle speed is 12000 rpm. The curves of Figure 11 illustrate the dynamic effects due to high rotational speed, such as gyroscopic coupling and spin softening, on system behaviour, and hence on bearing lifespan. Figure 11(a) represents the effect of spindle speed on the first two radial modes of the system. The dotted line 5000 10000 indicates the drill oscillation frequency and is plotted from Spindle speed (rpm) experimental data measured during high-speed vibratory Front bearing drilling operations. The critical speed, at 11500 rpm, corre- (b) sponds to the intersection of the radial mode frequencies Figure 11: (a) Campbell diagram. (b) Influence of spindle speed on with the excitation line due to the drill oscillations. bearings lifespan, for a feed rate of 0.05 mm/rev. 4. Conclusions conditions given rise to the best material removal rate (N = In this paper, a comprehensive approach to developing a hy- 12000 rpm, f = 0.125 mm/rev). Thus, a high-speed vibra- brid model of the dynamic behavior of the spindle self-vibra- tory drilling operation is always compatible with industri- tory drilling head—tool system has been proposed. This ap- al standards, even under the best material removal rate cut- proach has resulted in a numerical model enriched with ting conditions. physical data. The various components of the system are Figure 10(a) shows the influence of the feed rate on bear- modelled using a specific beam element, taking into account ings lifespan, for a spindle speed of 12000 rpm. Only feed the gyroscopic effects, centrifugal forces, and shear deflec- rateshigherthan0.05mm/revweretested, becausefor feed tion. The receptance coupling method is used to identify the rates below this limit, the productivity is too low and high- speed vibratory drilling loses its relevance. For feed rates dynamic behavior of the interface. The complete system is Frequency (Hz) Lifespan (hours) Lifespan (hours) −1 Axial cutting forces (1/N) 10 Advances in Acoustics and Vibration then obtained by assembling the beam model of each com- [7] A. Erturk, H. N. Ozguven, and E. Budak, “Effect analysis of bearing and interface dynamics on tool point FRF for chatter ponent using spring-damper elements. Finally, the assem- stability in machine tools by using a new analytical model bled model is validated by comparisons between numeri- for spindle-tool assemblies,” International Journal of Machine cal and experimental FRFs. The model is used to predict the Tools and Manufacture, vol. 47, no. 1, pp. 23–32, 2007. influence of a high-speed vibratory drilling operation on the [8] B. R. Jorgensen and Y. C. Shin, “Dynamics of spindle-bearing bearing lifespan. The predictions of bearing lifespan are used systems at high speeds including cutting load effects,” Journal to give rules of uses of the high-speed vibratory drilling head. of Manufacturing Science and Engineering, vol. 120, no. 2, pp. 387–394, 1998. Nomenclature [9] M. Rantatalo, J. O. Aidanpaa, B. Goransson, and P. 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Budak, “Analytical modeling F: force vector of spindle-tool dynamics on machine tools using Timoshenko E:Youngmodulus beam model and receptance coupling for the prediction of δ: rolling bearing preload tool point FRF,” International Journal of Machine Tools and N: spindle speed in rpm Manufacture, vol. 46, no. 15, pp. 1901–1912, 2006. L : bearing lifespan [15] T. L. Schmitz, K. Powell, D. Won, G. S. Duncan, W. G. C: basic dynamic load Sawyer, and J. C. Ziegert, “Shrink fit tool holder connection P: equivalent dynamic bearing load. stiffness/damping modeling for frequency response prediction in milling,” International Journal of Machine Tools and Manu- facture, vol. 47, no. 9, pp. 1368–1380, 2007. References [16] H. Ahmadian and M. Nourmohammadi, “Tool point dynam- [1]A.M.Gouskov,S.A.Voronov,and S. A. 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