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Stability-based spindle speed control during flexible workpiece high-speed milling
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International Journal of Machine Tools & Manufacture 48 (2008) 184–194 www.elsevier.com/locate/ijmactool
Stability-based spindle speed control during flexible workpiece high-speed milling I. Man˜e´, V. Gagnol, B.C. Bouzgarrou, P. Ray LaMI, IFMA/UBP-Campus des Ce´zeaux, BP 265 63175 Aubiere Cedex, France Received 24 May 2007; received in revised form 9 August 2007; accepted 16 August 2007 Available online 30 August 2007
Abstract High-speed machining (HSM) is a technology used to increase productivity and reduce production costs. The prediction of stable cutting regions represents an important issue for the machining process, which may otherwise give rise to spindle, cutter and part damage. In this paper, the dynamic interaction of a spindle-tool set and a thin-walled workpiece is analysed by a finite element approach for the purpose of stability prediction. The gyroscopic moment of the spindle rotor and the speed-dependent bearing stiffness are taken into account in the spindle-tool set finite element model and induce speed-dependent dynamic behaviour. A dedicated thin-walled workpiece is designed whose dynamic behaviour interacts with the spindle-tool set. During the machining of this flexible workpiece, chatter vibration occurs at some stages of machining, depending on the cutting conditions and also on the tool position along the machined thin wall. By coupling the dynamic behaviour of the machine and the workpiece, respectively, dependent on the spindle speed and the relative position of both the systems, an accurate stability lobes diagram is elaborated. Finally, the proposed approach indicates that spindle speed regulation is a necessary constraint to guarantee optimum stability during machining of thin-walled structures. r 2007 Elsevier Ltd. All rights reserved. Keywords: Spindle-bearing system; Rotor-dynamics prediction; Chatter vibration; Thin-walled structure; Stability lobes; Spindle speed control
1. Introduction High-speed machining (HSM) is widely used in the aeronautical and automotive domains and enables increased productivity and reduced production costs. These two sectors require competitive productivity, in terms of time, number of pieces and accuracy, which HSM can deliver. During workpiece machining in these sectors, large quantities of material are removed in high-removal-rate conditions, with the risk of instability in the process. This unstable phenomenon, known as chatter, is quite critical since it produces bad surface quality and may lead to spindle, cutter and part damage. Several studies have been conducted since Taylor first identified and described chatter in 1907, for example, by Corresponding author. Tel.: +33 4 73 28 80 19; fax: +33 4 73 28 80 85.
E-mail address: [email protected] (V. Gagnol). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.08.018
Tobias and Fishwick [1], Tlusty and Polacek [2], Merritt [3] and Altintas and Budak [4]. All these authors present a fundamental understanding of regenerative chatter as a feedback mechanism for the growth of self-excited vibrations due to variations in chip thickness and cutting force and subsequent tool vibration. These studies have led to graphic charts, commonly referred to as stability lobe diagrams (SLDs) showing the stability information as a function of chip thickness and spindle speed. Stability studies have two important research approaches. On the one hand, many authors have studied it through machine behaviour, assuming a rigid workpiece. The tool tip transfer function is elaborated through models or experimental approaches. Somehow, most of the previous models made the assumption that spindle-tool set dynamics do not change over the full spindle speed range. This assumption needs to be reconsidered in HSM, where gyroscopic moments and centrifugal forces on both bearings and
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spindle shaft induce spindle speed-dependent dynamics changes. For accurate dynamics prediction, spindle speeddependent dynamics must be evaluated. Faassen et al. [5] and Schmitz et al. [6] proposed considering such dependencies on the basis of experimental transfer function identification at different spindle speeds. Their method uses impulse hammer excitation and contactfree (capacitive probe) response measurement of a rotating tool at different discrete spindle speeds. Experimental results revealed speed-dependent variations in spindle dynamics and hence in the stability limit. However, many conventional testing techniques prove unsuccessful at determining spindle behaviour during high-speed rotation. Indeed, in an experimental modal analysis of a high-speed rotating shaft, the location of the hammer impact point and the sensor measurement point is not properly defined. Other authors have investigated the dynamic behaviour of machine tool spindle-bearing systems through modelling. They show that spindle dynamics are influenced by a large number of factors, including holder characteristics [7], spindle shaft geometry and drawbar force, [8] and the stiffness and damping provided by the bearings [9–11]. Most of these factors are independent of spindle speed, contrary to bearing stiffness and damping, which change according to preload and spindle speed. Jorgensen and Shin [9] showed that the dynamic characteristics of the spindle system vary with speed-dependent changes in bearing stiffness, affecting chatter stability. Wang and Chang [10] performed a finite element model (FEM) analysis on a two-bearing spindle using a Rayleigh beam, without taking high-speed effects into account. Results show the importance of internal bearing contact angle variation on the vibration modes. Nelson [11] presented in 1980 a rotor bearing formulation based on Timoshenko beam theory which included shear deformation. His formulation has been reused for the purpose of stability prediction [12] or to integrate thermo-mechanical considerations [13] or also to validate experimental approaches like the contactless dynamic spindle testing (CDST) equipment of Rantatalo et al. [14]. In previous works [15,16], we have presented a dynamic high-speed spindlebearing system model based on rotor-dynamics prediction. Element kinematics is formulated in a co-rotational coordinate frame and allows a special rotor beam element to be developed. Model results showed that dynamic effects due to high rotational speed, such as gyroscopic coupling and spin softening, have a significant influence on the spindle behaviour. By integrating the modelled speeddependent spindle transfer function into the chatter vibration stability approach of Altintas and Budak [4], a new dynamic SLD was predicted. On the other hand, other research has been carried out through workpiece behaviour, assuming a rigid spindletool set. This assumption is relevant in specific industrial applications like very thin-walled aeronautical structures or turbine blades. Workpiece vibration is strongly dominant in comparison with spindle-tool set vibration. Thevenot
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et al. [17] analysed stability during the machining of a thinwalled structure and proposed a 3D lobes diagram with an additional axis characterized by the tool position along the workpiece. These approaches have recently been extended by coupling the behaviour of the workpiece and the spindletool set [18–20]. The dynamic interaction of spindle-tool set and the workpiece during machining is representative of numerous industrial applications. Chatter comes from both the machine and workpiece vibration due to the similar dynamic behaviour of both the systems. Bravo et al. [19] proposed considering such applications and suggested a method for obtaining a 3D stability lobe to cover all the intermediate machining stages. Le Lan et al. [20] presented a similar approach by combining the FEM representation of a flexible automotive workpiece and the spindle-tool set. Both the approaches used the milling cutting force model formulated by Altintas and are based on the restrictive assumption that spindle-tool set dynamics do not change over the full spindle speed range. This paper describes a method for predicting the stability of a milling process when both the machine and the workpiece vibration are in interaction. The proposed approach takes into account the dynamic interaction of a rotating spindle-tool set and a flexible workpiece by combining their respective transfer functions. The effects of the rotor gyroscopic moment and speed-dependent bearing stiffness on the dynamics of the system are taken into account. A thin-walled workpiece representative of a typical industrial application is modelled by FEM to vibrate simultaneously with the spindle-tool structure. A specific SLD is then elaborated by scanning the dynamic properties of both the workpiece and the rotating spindletool set throughout the machining process. Then investigations are carried out to optimize the spindle speed according to industrial constraints. Section 2 describes the modelling approach to milling interaction. The model consists of a sub-section which describes the spindle-tool set finite element rotor model and a sub-section which describes the flexible workpiece dynamics. The dynamic interaction of both the models is then presented. Results take into account the speed dependency of the spindle model and the tool-position dependency of the machined thin-walled workpiece. Section 3 is dedicated to milling stability prediction. Chatter is dependent on the relative vibration between the two mechanical systems that are in contact during milling. The spindle model predicts that both the speed-dependent gyroscopic moment of the rotor and the speed-dependent bearing stiffness affect the stability limit in every discrete relative tool/workpiece position. A specific SLD is elaborated which reflects the respective dynamic changes that both the spindle and the workpiece system undergo during the milling process. Finally, Section 4 presents spindle speed control in order to guarantee optimum stable condition throughout the machining of the flexible workpiece. The regulation of
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HSM spindle speed to a specific predicted rotating speed will lead to an appreciable gain in productivity under stable machining conditions. 2. Model building Fig. 2. The finite element model of the spindle-bearing system.
The system under study is composed of two subsystems: the spindle-tool set and the flexible workpiece. In this section both the models are described, as are coupling details. The dynamic behaviours of both the mechanical subsystems, represented, respectively, by the spindle-tool set and the workpiece, interact simultaneously due to the milling cutting forces. Dynamic interactions of the two systems are investigated. Assumptions are made for the finish milling of a thinwalled workpiece. The machining technique employed is downmilling, with the restrictions that the tool and the workpiece are always in contact with each other and that the radial depth of cut is small. During the milling operation, the flexibility of both the subsystems is taken into account in the direction perpendicular to the tool trajectory (Fig. 1). 2.1. Spindle model The spindle model is restricted to the rotating entity. An experimental modal identification procedure was carried out on the different spindle entities [21] and showed that spindle behaviour can be restricted to rotating entity behaviour. The rotating entity is composed of five subentities, namely the shaft, rotor, tool, tool holder and tool holder mount (Fig. 2). It is supported by five (three front and two rear) hybrid angular contact bearings. The following studies focus on the rotating entity dynamics. The motion of the rotating entity is considered as the superposition of rigid and elastic body displacements.
Dynamic equations are obtained using Lagrange formulation associated with a finite element method. Due to the size of the rotor sections, shear deformations must be taken into account. Then the rotating entity is derived using Timoshenko beam theory. The relevant shape functions are cubic in order to avoid shear-locking [22]. A special 3D rotor beam element with two nodes and six degrees of freedom per node is developed. As in a classic finite element procedure, dynamic rotor equations are obtained by assembling element matrices. The rotor entity model is assembled with the rolling bearing model. The interfaces between the different sub-entities which compose the rotating entity are assumed to be rigid. The damping model used draws on Rayleigh viscous equivalent damping, which makes it possible to regard the damping matrix D as a linear combination of the mass matrix M, the spindle rigidity matrix Kspindle and the rolling bearing rigidity matrix Kbearing. The set of differential equations can be written as MðqN Þ€qN þ ðCðqN ; q_ N Þ þ DÞ_qN þ KqN ¼ FðtÞ,
(1)
where M(qN) and K are the mass and global stiffness matrices, CðqN ; q_ N Þ matrix contains the rotational dynamics effect, D is the viscous equivalent damping matrix and qN and F(t) are the nodal displacement and force vectors. An accelerating rotor gives rise to time-variant matrices, since M(qN) and CðqN ; q_ N Þ are time-dependent matrices. The treatment of rotor dynamics with a pseudo-constant rotational speed approach establishes that CðqN ; q_ N Þ_qN ¼ 2OG_qN O2 NqN . G and N proceed from the decomposition of matrix C and are, respectively, representative of the gyroscopic and spin softening effects. O is the rotor’s angular velocity. Hence, considering the assumption of pseudo-constant rotational speed, rotor dynamics are described by means of linear time-invariant matrices: M0 q€ N þ ð2OG þ DÞ_qN þ ðK O2 NÞqN ¼ FðtÞ,
Fig. 1. A schematic representation of the milling interaction.
(2)
where M0 is the constant part of the matrix M. In a high-speed spindle the bearing axial preload is kept essentially constant during thermally generated changes to spindle dimensions by spring or hydraulic arrangements. The rolling bearing stiffness matrices are calculated around a static function point characterized by the bearing preload [9,16]. Indeed, stiffness matrices, when the spindle is nonrotating, depend on the geometry and the preload. Based on Rantatalo’s prediction [14], the initially calculated bearing stiffness is spindle speed dependent because of
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the gyroscopic and centrifugal effects induced in the rotating balls (Fig. 3). Then, speed-dependent bearing stiffness is integrated into the global spindle FEM. 2.1.1. Experimental characterization and model readjustment While the detailed knowledge of the spindle is in general not available in a manufacturing environment, the model can be readjusted in order to fit experimental results. The material properties of the rotor and the tool holder mount, such as the Young modulus E, the shear modulus G and the volumic mass r, cannot be precisely known, due to their geometrical and physical complexity. The parameter allows rigidity variations caused by the rear bearings floating mount to be taken into account. The modelling of the damping to represent the dissipative properties of the
Fig. 3. The changes in the bearing stiffness depending on the spindle speed.
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whole set of spindle components also represents a real problem. Hence, a readjustment procedure based on a two-step algorithm is proposed to tune the previous variable in order to fit the model results to experimental ones (Fig. 4). These parameters are readjusted by minimizing the gap between the measured and the modelled tool tip node FRF, respectively, Hexp(o) and Hnum(o), for a non-rotating spindle [15,16], using an optimization Matlabr routine and a least squares-type objective function defined as ¼
o2 X
jH num ðoÞ H exp ðoÞj2 .
o¼o1
The readjustment procedure, applied to the spindle-tool set configuration shown in Fig. 2, was carried out to perform the best modal correlation on a bandwith of 0–3500 Hz. Indeed, the interaction between the spindle bending modes and the flexible workpiece bending modes inside this bandwith (Fig. 9) plays a dominant role in the definition of the frequency range of interest introduced in the stability prediction section. Experimental determination of the dynamic characteristics of the spindle is carried out by an impact hammer test using a Bru¨el and Kjaer acquisition rack controlled by a PC equipped with B&K Pulse software. Finally the spindle database is updated to contain the geometrical and the final readjusted material parameters. 2.1.2. Changes in the spindle dynamics according to the rotation speed The numerical model can be used to investigate highspeed rotational effects on the spindle dynamic behaviour. Natural frequencies are determined as a function of the
Fig. 4. An algorithm of numeric model readjustment.
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Fig. 5. The 3D speed-dependent frequency response function.
rotation speed. A loop on both frequency and spindle speed ranges allows us to generate a Campbell diagram (Fig. 5). This figure illustrates the results of the model’s tool tip node modal analyses with respect to spindle rotation speed. Dynamic effects due to high rotational speed and elastic deformations, such as gyroscopic coupling and spin softening, have a significant influence on the spindle behaviour. In a non-rotating spindle, two orthogonal bending shapes are associated with the same bending frequency, due to the axial symmetry of the rotor. When the spindle rotates, the frequencies associated with these two bending shapes move apart, one above and one below the rest value, as a function of the spindle speed. These two different solutions are called forward and backward whirls in rotor-dynamics literature [23]. 2.2. Workpiece model In order to have a workpiece representative of aeronautical milling problems, the study was focused on a typical thin-walled part. The workpiece is made from AS7 aluminium alloy, with physical properties: E ¼ 70 GN/m2 and density ¼ 2700 kg/m3. The thin wall was downmilled in finishing with axial depth of cut of ap, a radial immersion of 0.1 mm and a feedrate of 0.1 mm/tooth. The workpiece dimensions (Fig. 1) are chosen in order that both spindletool and workpiece systems have similar dynamic behaviour and hence vibrate simultaneously. Thus, satisfying the points announced previously, the part chosen for the study is a rectangular prism of 50 10 100 mm with a thin wall of 5 mm thickness and 30 mm height. That geometry embodies a part that can be
Fig. 6. The workpiece FEM.
found easily in the industry, as, for example, an inner pocket wall. During the finish machining of the part, dynamic properties evolve according to the tool position [19,20]. In order to predict this evolution, a workpiece FEM is established. The tool path is discretized over the whole workpiece and a modal analysis is carried out. At every position the FRF is established. The FEM (Fig. 6) is composed of parallelepiped elements with eight nodes and three degrees of freedom per node. The limit conditions are established by fixing the nodes of the workpiece base. This configuration allows the thin wall to vibrate freely as under real machining conditions.
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Fig. 7. The workpiece frequency response function depending on the tool position.
Experimental modal characterization of the workpiece was carried out in order to adjust the FEM results to the real behaviour of the part. These tests were made through a classic hammer test and mainly enable the frequency damping ratio of the model to be readjusted. In order to integrate the workpiece model to the spindletool set model at the interaction point representative of milling interaction, the dynamic equations of the part have to be established. The assembly procedure is carried out using Matlabs. The workpiece dynamic equations are established using a reduction of the FEM model to a specific master degrees of freedom using the Guyan algorithm. The workpiece is divided into 11 equidistant points, enabling an accurate workpiece dynamic prediction to be made throughout the machining process. The retained MDOF are chosen at the interaction point between the spindle-tool set and the machined workpiece in the radial direction representative of the main vibration direction. In this way, position 1 corresponds to the first point of study and position 11 the last (Fig. 6). The assumption is made that the workpiece FEM has a constant geometry during finishing machining due to the small amount of material removed. Dynamic equations of the reduced workpiece FEM are determined: MW q€ N þ CW q_ N þ KW qN ¼ FðtÞ,
(3)
where MW is the mass matrix, CW the damping matrix and KW the stiffness matrix of the workpiece FEM. 2.2.1. Changes in the workpiece dynamics according to the tool position Assembling all the points mentioned before, the workpiece model is obtained. A harmonic analysis is carried out
Fig. 8. The milling interaction.
and allows the workpiece frequency response function to be established (Fig. 7) at the different interacting points 1–11. The 3D diagram represents the FRF amplitude between the excitation node P and the observing node P as detailed in Fig. 8 for each discrete tool/workpiece position during milling. The node 1 FRF amplitude presents three bending modes in the frequency range of 0–6000 Hz, at, respectively 800, 1500 and 2080 Hz. The 800 and 1500 Hz modes represent, respectively, a simple flexion around the ~ z-axis and a semiflexion, also around the ~ z-axis but symmetrical about the centre of the workpiece. It can be observed that
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both the frequency and the amplitude of the modes depend on the relative tool/workpiece position. In particular, the amplitude of the first mode at 800 Hz increases when the variable position increases, unlike the second mode amplitude at 1500 Hz, which decreases with the position. This kind of difference affects the final coupled system, as is shown in the next section. 2.3. The coupled model at the milling interaction point Once the transfer functions of both the mechanical systems of the spindle-tool set and the workpiece have been analysed, the dynamic interaction between them can be evaluated. The procedure for coupling these two systems is described in this section. Fig. 8 presents the case of a single degree of freedom cutting process. The vibrations of the spindle-tool set and that of the thin-walled piece are considered in direction ~ z perpendicular to the cutting speed, where part flexibility is dominant. During the cutting process, the tool creates a cutting force on the machined workpiece, which then, by the principle of action and reaction, generates a reaction force on the tool with the same magnitude but in the opposite direction. Each subsystem vibration is described by a set of differential equations obtained through a dedicated finite element modelling approach. The projection of these equations onto the ~ z-axis leads to the resulting equations: M W z€W þ C W z_W þ K W zW ¼ F T=W ðtÞ,
(4)
ð5Þ
Eqs. (4) and (5) describe, respectively, the workpiece and the spindle-tool set vibrations. During milling, both the subsystems interact at a contact point modelled by the tool tip node on the spindle model. The FRF of both the subsystems are generated at this interaction point P (Fig. 8), where zW(t) and zT(t) correspond to the small displacements, respectively, of the workpiece and the tool around their equilibrium position. The dynamic chip thickness h is assumed to be the difference between the vibration of the current coupled system, determined by (zT(t)zW(t)), and the vibration of the coupled system at the previous tooth period t determined by (zT(tt) zW(tt)). Consequently, the chip thickness h encountered by the tooth is hðtÞ ¼ h0 ½ðzT ðtÞ zW ðtÞÞ ðzT ðt tÞ zW ðt tÞÞ.
(6)
The cutting forces in the radial ~ z direction in the time domain are modelled by F T=W ðtÞ ¼ K t ap hðtÞ.
hðsÞ ¼ h0 þ ðesT 1Þ½Z T ðsÞ ZW ðsÞ.
(7)
The radial force FT/W is assumed to be proportional to the chip thickness h(t) and the axial depth of cut ap Kt is a constant cutting coefficient.
(8)
Eqs. (4) and (5) are expressed in the Laplace domain by ½M W s2 þ C W s þ K W Z W ðsÞ 1 Z W ðsÞ ¼ F T=W ðsÞ, ¼ FW ðsÞ ½M T s2 þ ð2OGT þ DT Þ s þ ðK T O2 N T Þ Z T ðsÞ 1 ZT ðsÞ ¼ F W=T ðsÞ, ¼ FT ðsÞ
ð9Þ
ð10Þ
with FW(s) and FT(s) the transfer functions at the interacting point P of, respectively, the workpiece and the spindle-tool set. The equation of the closed loop governing dynamic chip thickness leads to hðsÞ 1 . ¼ h0 1 þ K t ap ð1 esT Þ½FW ðsÞ þ FT ðsÞ
(11)
Stability prediction of the coupled system composed of the flexible workpiece and the spindle-tool set requires the analysis of the transfer function resulting from the sum of the two respective subsystem transfer functions. The absolute value of the relative displacement comes from the sum of both subsystem displacements [19]. The resulting FRF of the coupled system at the interacting milling point P is defined as FCPP ¼ FWPP þ FTPP .
M T z€T þ ð2OG T þ DT Þ z_T þ ðK T O2 N T Þ zT ¼ F W=T ðtÞ.
Considering the Laplace transformation ZT(s) ¼ ‘(zT(t)) and ZW(s) ¼ ‘(zW(t)). Eq. (6) becomes
(12)
A graphical representation of the coupled system FRF amplitude is given in Fig. 9, as well as those of the separate workpiece and the spindle-tool set FRF. The coupled system FRF presented in Fig. 9 is determined for a specific relative tool/workpiece position and also at a specific spindle rotation speed of 5000 rpm. The coupled FRF is strongly influenced by both the workpiece and the spindle-tool set FRF. The amplitude of the coupled system FRF presents its maximum value at the respective dominant modes of both the subsystems. The first amplitude peak is clearly influenced by the first spindle-tool set bending mode. The second amplitude peak on the coupling FRF results from the combination of the backward component of the second spindle bending mode and the first workpiece bending mode. The third coupled FRF peak represents the forward component of the second spindle bending mode. The fourth FRF peak is predominantly composed of the second bending mode of the machining part. When the spindle-tool set flexibility is preponderant compared to the workpiece flexibility, the coupled behaviour is mostly influenced by the spindle-tool set dynamics variation, the converse being the case when the part is the most flexible system. Thus, in relation to the interacting spindle/workpiece model presented, the system with the
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Fig. 9. A comparison of the transfer functions of the spindle-tool set, the workpiece and the coupled system.
Fig. 10. The 3D milling interaction FRF amplitude.
most flexibility controls the behaviour of the coupled system. By scanning the different respective positions of the tool along the piece a 3D FRF can be elaborated. Results of such an investigation can be observed in Fig. 10, where 3D vibratory evolution is presented. Fig. 10 is established at a spindle speed of 5000 rpm. FRF amplitude is strongly dependent on the relative position between the tool and the workpiece. A shift can be observed, to the left of mode 2, and to the right of mode 3, depending on the tool/workpiece position. The vibration mode amplitudes are also varied. Some mode amplitudes
are reduced and others are increased. The symmetry of workpiece geometry at the node 6 vertical layer can also be observed in the dynamic behaviour. Such an investigation can also be made for a given tool/ workpiece relative position, depending on the spindle speed variation. Consequently, the dependency of the coupled FRF on both the spindle rotation speed and the relative position between the workpiece and the spindle had to be investigated in order to identify the consequences on stability prediction. This aspect is analysed in the next section.
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3. Stability analysis of the milling system In this section, the dynamic model of the coupled system composed of the spindle-tool set and the workpiece is used for the purpose of the stability analysis. The vibration of the coupled structure due to the cutting forces is predominant in direction ~ z perpendicular to the cutting speed. The milling stability prediction is based on Altintas and Budak’s approach [4]. However, the proposed analysis technique can be used generically with other milling stability approaches.
In Altintas and Budak’s approach [4], the time-variant coefficients of the dynamic milling equations, which depend on the angular orientation of the cutter as it rotates through the cut, are expanded into a Fourier series and then truncated to include only the average component: alim ¼
The dynamic stiffness of the modes induces permissible depths of cut into stability limits theory. When the critical depth of cut of the stability lobe at the cutting spindle speed is maximized, cutting is at most stable. The resulting stability relationships are shown in Eqs. (13)–(14) where alim is the limiting chip width or axial depth of cut in peripheral end milling operations, Kt is the specific cutting energy coefficient, which relates the radial cutting force Fr to the chip area, and n is the spindle speed in rpm. The cutter is assumed to have Z number of teeth with a zero helix angle. The dynamics-based receptance FCPP ðjoÞ of the coupled system at the interacting milling node P in the radial ~ z direction can be deduced from Eq. (12). FCPP ðjoÞ ¼ RCPP ðoÞ þ jI CPP ðoÞ where j2 ¼ 1. RCPP and I CPP are, respectively, the real part and the imaginary part of the coupled system frequency response function. The negative real part RCpp and the growing phase j of the transfer function allows delimiting a frequency range of interest [15,16]. Then, the optimal cutting parameters (alim; n) that maximize the chatter-free material removal rate are determined, through the frequency range of interest, according to Eqs. (13), (14).
(13)
where K is a time-invariant but immersion-dependent directional cutting coefficient. 60 60 2pf c ¼ T Zðp þ 2kp 2jÞ I C ðoÞ , j ¼ tan1 PP RCPP ðoÞ
n¼ 3.1. Dynamics-based SLD
2pK 1 , K tZ RCPP ðoÞ
with ð14Þ
where f c (Hz) is the chatter vibration frequency, and k is an integer that corresponds to the number of vibration waves left on the workpiece surface. A dynamics-based SLD is elaborated for each relative position of the tool along the workpiece, also taking into account the effective dynamic properties of the spindle at the considered rotation speed. The proposed stability prediction is developed through a two-step algorithm [16]. In the considered study, the half-workpiece is meshed into six nodes, and, with one stability chart per node, this results in six dynamic stability lobes. As a result, a graphic stability map can be elaborated by the combination of the relative position-dependent stability lobes (Fig. 11). The 3D stability diagram represents the limit depth of cut alim depending on the two dimensions represented by the spindle speed and the relative tool/workpiece position Y. The evolution of stable conditions along the machining part can be observed. The dynamic behaviour symmetry of the workpiece observed in Fig. 7 induces a symmetry in the stability lobes prediction. Due to this symmetry, only the stability lobes on nodes 1–6 are
Fig. 11. The 3D stability lobes diagram depending on the spindle speed and the tool–workpiece position.
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Fig. 12. The stability-based spindle speed variation.
represented. Stability prediction on nodes 7–11 can be deduced from node 5 to 1. During workpiece machining, variations occur in the stability peak position and also in the permissible depth of cut without chatter. Position 1 presents three weak stability peaks at the respective spindle speeds of 7600, 10,100 and 14,000 rpm. During machining, these peaks evolve according to the tool/workpiece position. At position 4 it is possible to observe four stability peaks at 7200, 9800, 11,600 and 13,690 rpm. The analysis of such an accurate 3D stability map enables the most adequate machining strategy to be selected in accordance with industrial objectives. The next section is related to the definition of stability-based spindle speed rules during the machining of a flexible part.
4. Stability-based spindle speed variation The evolution of the stability frontier depending on the spindle speed during flexible part machining shows that gains can be achieved through the definition of dynamicsbased variable spindle speed rules. Nowadays, spindle speed follows a constant speed setting from the beginning of milling until the end. This kind of simple instruction does not allow the best machining conditions to be achieved. Indeed, with regard to the stability conditions, the lowest spindle speed will be retained, and thus productivity will be greatly affected. An inspection of the proposed 3D SLD (Fig. 11) makes it possible to extract specific spindle speeds that maximize industrial criteria such as productivity and stability. In our approach, we propose to maximize the material removal rate at stable conditions by varying the spindle speed with respect to the predicted stability frontier. The resultant optimized spindle-drive rules are elaborated by linking the most stable machining points all along the tool path (Fig. 12).
The different spindle speed rules elaborated through the proposed approach are represented in Fig. 12. Four spindle speed rules can be defined by linking the dominant stability peaks of all 11 nodes. Hence, the regulation of spindle speed to these rules, predicted by the model, will guarantee the maximal stability. The black vertical line positioned on node 6 indicates the symmetry line for spindle speed prediction. In fact, different machining strategies can be defined, depending on the optimization criteria. For example, the two spindle speed rules starting near 10,000 rpm allow the maximized depth of cut to be implemented. The rule starting near 14,000 rpm maximizes the spindle speed in stable machining conditions, but with a reduced permissible depth of cut. Hence, the optimized rule can be defined depending on the industrial strategies. An innovative study approach, concerning the development of a new generation of spindle controls, has been highlighted. Some prospects, concerning the feasibility of continuous speed variation according to optimized criteria and with respect to spindle dynamics, need to be developed further.
5. Conclusion In this paper, an integrated spindle–workpiece model is developed by coupling a rotor-dynamics-based spindle FEM model [15,16] and a thin-walled workpiece FEM model-based common industrial parts. Experimental validations of the modelled spindle and the thin-walled workpiece results have been carried out separately. The vibratory interaction between the two systems during milling is studied. The dynamic behaviour of the coupled system is shown to be greatly dependent on two variables, respectively, the spindle–workpiece relative position and the spindle rotation speed.
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The proposed method enables accurate lobes diagrams to be established through the consideration of the relative transfer function of the interacting milling system. The evolution of the stability boundaries according to the spindle rotation speed and the tool–workpiece relative position give rise to a 3D lobes diagram. The stability prediction results are in accordance with the results of Bravo et al. [19] and take into account the additional effects due to spindle speed-dependent dynamics. Finally, by examining the predicted 3D stability map, specific spindle speed rules which optimize the maximum material removal rate without chatter during the machining of the flexible workpiece are elaborated. The spindle speed rules are dependent on the relative spindle-tool/ workpiece position and take into account the speeddependent spindle dynamics. The optimized machining conditions regarding industrial objectives require the spindle speed to be regulated according to the predicted rule. This approach enables design and operating rules to be established for HSM spindles, and can provide help for designers while attempting to eliminate costly experimentbased design trials. Acknowledgements This study was carried out within the framework of the TIMS Research Group, using grants from the Regional Council of Auvergne, the French Ministry of Research, the CNRS and the Cemagref. References [1] S. Tobias, W. Fishwick, A theory of regenerative chatter, The Engineer, February (1958). [2] J. Tlusty, M. Polacek, The stability of machine tools against selfexcited vibrations in machining, in: Proceedings of the ASME International Research in Production Engineering, Pittsburgh, USA, 1963, pp. 465–474. [3] H. Merritt, Theory of self-excited machine tool chatter, Transactions of the ASME Journal of Engineering for Industry 87 (1965) 447–454. [4] Y. Altintas, E. Budak, Analytical prediction of stability lobe in milling, CIRP Annals 44 (1) (1995) 357–362. [5] R.P.H. Faassen, N. van de Wouw, J.A.J. Oosterling, H. Nijmeijer, Prediction of regenerative chatter by modelling and analysis of highspeed milling, International Journal of Machine Tools and Manufacture 43 (2003) 1437–1446. [6] T. Schmitz, J. Ziegert, C. Stanislaus, A method for predicting chatter stability for systems with speed-dependent spindle dynamics, Society
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of Manufacturing Engineers Technical Paper TPO4PUB182, Transactions of the NAMRI/SME 32 (2004) 17–24. J. Agapiou, E. Rivin, C. Xie, Toolholder/spindle interfaces for CNC machine tools, Annals of the CIRP 44 (1) (1995) 357–362. S. Smith, P. Jacobs, J. Halley, The effects of drawbar force on metal removal rate in milling, Annals of the CIRP 48 (1) (1999) 293–296. B.R. Jorgensen, Y.C. Shin, Dynamics of spindle-bearing systems at high speeds including cutting load effects, Journal of Manufacturing Science and Engineering 120 (1998) 387–394. W.R. Wang, C.N. Chang, Dynamic analysis and design of a machine tool spindle-bearing system, Journal of Vibration and Acoustics 116 (1994) 280–285. H.D. Nelson, A finite rotating shaft element using Timoshenko beam theory, Journal of Mechanical Design 102 (1980) 793–803. G.L. Xiong, J.M. Yi, C. Zeng, H.K. Guo, L.X. Li, Study of the gyroscopic effect of the spindle on the stability characteristics of the milling system, Journal of Materials Processing Technology 138 (1–3) (2003) 379–384. C.W. Lin, J.F. Tu, J. Kamman, An integrated thermo-mechanicaldynamic model to characterized motorized machine tool spindles during very high-speed rotation, International Journal of Machine Tools and Manufacture 43 (10) (2003) 1035–1050. M. Rantatalo, J.O. Aidanpa¨a¨, B. Go¨ranssonn, P. Norman, Milling machine spindle analysis using FEM and non-contact spindle excitation and response measurement, International Journal of Machine Tools and Manufacture 47 (7) (2006) 1034–1045. V. Gagnol, C.B. Bouzgarrou, P. Ray, C. Barra, Model-based stability prediction for HSM spindle, International Journal of Machine Tools and Manufacture 47 (7–8) (2007) 1176–1186. V. Gagnol, C.B. Bouzgarrou, P. Ray, C. Barra, Stability-based spindle design optimization, ASME Journal of Manufacturing Science and Engineering 129 (2007) 407–415. V. Thevenot, L. Arnaud, G. Dessein, G. Cazenave-Larroche, Integration of dynamic behavior in stability lobes method: 3D lobes construction and application to thin walled structure milling, International Journal of Advanced Manufacturing Technology 27 (2006) 638–644. L. Masset, J.F. Debongnie, W. Belluco, Chatter maps for process design of powertrain components, in: Proceedings of the Seventh CIRP International Workshop on Modelling of Machining Operations, 2004, pp. 251–258. U. Bravo, O. Altuzarra, L.N. Lopez de Lacalle, J.A. Sanchez, F.J. Campa, Stability limits of milling considering the flexibility of the workpiece and the machine, International Journal of Machine Tools and Manufacture 45 (2005) 1669–1680. J.V. Le Lan, A. Marty, J.F. Debongnie, Providing stability maps for milling operations, International Journal of Machine Tools and Manufacture 47 (9) (2006) 1493–1496. V. Gagnol, C.B. Bouzgarrou, P. Ray, C. Barra, Modelling approach for a high speed machine tool spindle-bearing system, in: CD-ROM Proceedings of the International Design Engineering Technical Conferences, Long Beach, USA, September 24–28, 2005. G. Genta, Vibration of Structures and Machines, Practical Aspects, Springer, Berlin, 1995. M. Lalanne, G. Ferraris, Rotordynamics Prediction in Engineering, second ed., Wiley, New York, 2001.
International Journal of Machine Tools & Manufacture 48 (2008) 184–194 www.elsevier.com/locate/ijmactool
Stability-based spindle speed control during flexible workpiece high-speed milling I. Man˜e´, V. Gagnol, B.C. Bouzgarrou, P. Ray LaMI, IFMA/UBP-Campus des Ce´zeaux, BP 265 63175 Aubiere Cedex, France Received 24 May 2007; received in revised form 9 August 2007; accepted 16 August 2007 Available online 30 August 2007
Abstract High-speed machining (HSM) is a technology used to increase productivity and reduce production costs. The prediction of stable cutting regions represents an important issue for the machining process, which may otherwise give rise to spindle, cutter and part damage. In this paper, the dynamic interaction of a spindle-tool set and a thin-walled workpiece is analysed by a finite element approach for the purpose of stability prediction. The gyroscopic moment of the spindle rotor and the speed-dependent bearing stiffness are taken into account in the spindle-tool set finite element model and induce speed-dependent dynamic behaviour. A dedicated thin-walled workpiece is designed whose dynamic behaviour interacts with the spindle-tool set. During the machining of this flexible workpiece, chatter vibration occurs at some stages of machining, depending on the cutting conditions and also on the tool position along the machined thin wall. By coupling the dynamic behaviour of the machine and the workpiece, respectively, dependent on the spindle speed and the relative position of both the systems, an accurate stability lobes diagram is elaborated. Finally, the proposed approach indicates that spindle speed regulation is a necessary constraint to guarantee optimum stability during machining of thin-walled structures. r 2007 Elsevier Ltd. All rights reserved. Keywords: Spindle-bearing system; Rotor-dynamics prediction; Chatter vibration; Thin-walled structure; Stability lobes; Spindle speed control
1. Introduction High-speed machining (HSM) is widely used in the aeronautical and automotive domains and enables increased productivity and reduced production costs. These two sectors require competitive productivity, in terms of time, number of pieces and accuracy, which HSM can deliver. During workpiece machining in these sectors, large quantities of material are removed in high-removal-rate conditions, with the risk of instability in the process. This unstable phenomenon, known as chatter, is quite critical since it produces bad surface quality and may lead to spindle, cutter and part damage. Several studies have been conducted since Taylor first identified and described chatter in 1907, for example, by Corresponding author. Tel.: +33 4 73 28 80 19; fax: +33 4 73 28 80 85.
E-mail address: [email protected] (V. Gagnol). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.08.018
Tobias and Fishwick [1], Tlusty and Polacek [2], Merritt [3] and Altintas and Budak [4]. All these authors present a fundamental understanding of regenerative chatter as a feedback mechanism for the growth of self-excited vibrations due to variations in chip thickness and cutting force and subsequent tool vibration. These studies have led to graphic charts, commonly referred to as stability lobe diagrams (SLDs) showing the stability information as a function of chip thickness and spindle speed. Stability studies have two important research approaches. On the one hand, many authors have studied it through machine behaviour, assuming a rigid workpiece. The tool tip transfer function is elaborated through models or experimental approaches. Somehow, most of the previous models made the assumption that spindle-tool set dynamics do not change over the full spindle speed range. This assumption needs to be reconsidered in HSM, where gyroscopic moments and centrifugal forces on both bearings and
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spindle shaft induce spindle speed-dependent dynamics changes. For accurate dynamics prediction, spindle speeddependent dynamics must be evaluated. Faassen et al. [5] and Schmitz et al. [6] proposed considering such dependencies on the basis of experimental transfer function identification at different spindle speeds. Their method uses impulse hammer excitation and contactfree (capacitive probe) response measurement of a rotating tool at different discrete spindle speeds. Experimental results revealed speed-dependent variations in spindle dynamics and hence in the stability limit. However, many conventional testing techniques prove unsuccessful at determining spindle behaviour during high-speed rotation. Indeed, in an experimental modal analysis of a high-speed rotating shaft, the location of the hammer impact point and the sensor measurement point is not properly defined. Other authors have investigated the dynamic behaviour of machine tool spindle-bearing systems through modelling. They show that spindle dynamics are influenced by a large number of factors, including holder characteristics [7], spindle shaft geometry and drawbar force, [8] and the stiffness and damping provided by the bearings [9–11]. Most of these factors are independent of spindle speed, contrary to bearing stiffness and damping, which change according to preload and spindle speed. Jorgensen and Shin [9] showed that the dynamic characteristics of the spindle system vary with speed-dependent changes in bearing stiffness, affecting chatter stability. Wang and Chang [10] performed a finite element model (FEM) analysis on a two-bearing spindle using a Rayleigh beam, without taking high-speed effects into account. Results show the importance of internal bearing contact angle variation on the vibration modes. Nelson [11] presented in 1980 a rotor bearing formulation based on Timoshenko beam theory which included shear deformation. His formulation has been reused for the purpose of stability prediction [12] or to integrate thermo-mechanical considerations [13] or also to validate experimental approaches like the contactless dynamic spindle testing (CDST) equipment of Rantatalo et al. [14]. In previous works [15,16], we have presented a dynamic high-speed spindlebearing system model based on rotor-dynamics prediction. Element kinematics is formulated in a co-rotational coordinate frame and allows a special rotor beam element to be developed. Model results showed that dynamic effects due to high rotational speed, such as gyroscopic coupling and spin softening, have a significant influence on the spindle behaviour. By integrating the modelled speeddependent spindle transfer function into the chatter vibration stability approach of Altintas and Budak [4], a new dynamic SLD was predicted. On the other hand, other research has been carried out through workpiece behaviour, assuming a rigid spindletool set. This assumption is relevant in specific industrial applications like very thin-walled aeronautical structures or turbine blades. Workpiece vibration is strongly dominant in comparison with spindle-tool set vibration. Thevenot
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et al. [17] analysed stability during the machining of a thinwalled structure and proposed a 3D lobes diagram with an additional axis characterized by the tool position along the workpiece. These approaches have recently been extended by coupling the behaviour of the workpiece and the spindletool set [18–20]. The dynamic interaction of spindle-tool set and the workpiece during machining is representative of numerous industrial applications. Chatter comes from both the machine and workpiece vibration due to the similar dynamic behaviour of both the systems. Bravo et al. [19] proposed considering such applications and suggested a method for obtaining a 3D stability lobe to cover all the intermediate machining stages. Le Lan et al. [20] presented a similar approach by combining the FEM representation of a flexible automotive workpiece and the spindle-tool set. Both the approaches used the milling cutting force model formulated by Altintas and are based on the restrictive assumption that spindle-tool set dynamics do not change over the full spindle speed range. This paper describes a method for predicting the stability of a milling process when both the machine and the workpiece vibration are in interaction. The proposed approach takes into account the dynamic interaction of a rotating spindle-tool set and a flexible workpiece by combining their respective transfer functions. The effects of the rotor gyroscopic moment and speed-dependent bearing stiffness on the dynamics of the system are taken into account. A thin-walled workpiece representative of a typical industrial application is modelled by FEM to vibrate simultaneously with the spindle-tool structure. A specific SLD is then elaborated by scanning the dynamic properties of both the workpiece and the rotating spindletool set throughout the machining process. Then investigations are carried out to optimize the spindle speed according to industrial constraints. Section 2 describes the modelling approach to milling interaction. The model consists of a sub-section which describes the spindle-tool set finite element rotor model and a sub-section which describes the flexible workpiece dynamics. The dynamic interaction of both the models is then presented. Results take into account the speed dependency of the spindle model and the tool-position dependency of the machined thin-walled workpiece. Section 3 is dedicated to milling stability prediction. Chatter is dependent on the relative vibration between the two mechanical systems that are in contact during milling. The spindle model predicts that both the speed-dependent gyroscopic moment of the rotor and the speed-dependent bearing stiffness affect the stability limit in every discrete relative tool/workpiece position. A specific SLD is elaborated which reflects the respective dynamic changes that both the spindle and the workpiece system undergo during the milling process. Finally, Section 4 presents spindle speed control in order to guarantee optimum stable condition throughout the machining of the flexible workpiece. The regulation of
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HSM spindle speed to a specific predicted rotating speed will lead to an appreciable gain in productivity under stable machining conditions. 2. Model building Fig. 2. The finite element model of the spindle-bearing system.
The system under study is composed of two subsystems: the spindle-tool set and the flexible workpiece. In this section both the models are described, as are coupling details. The dynamic behaviours of both the mechanical subsystems, represented, respectively, by the spindle-tool set and the workpiece, interact simultaneously due to the milling cutting forces. Dynamic interactions of the two systems are investigated. Assumptions are made for the finish milling of a thinwalled workpiece. The machining technique employed is downmilling, with the restrictions that the tool and the workpiece are always in contact with each other and that the radial depth of cut is small. During the milling operation, the flexibility of both the subsystems is taken into account in the direction perpendicular to the tool trajectory (Fig. 1). 2.1. Spindle model The spindle model is restricted to the rotating entity. An experimental modal identification procedure was carried out on the different spindle entities [21] and showed that spindle behaviour can be restricted to rotating entity behaviour. The rotating entity is composed of five subentities, namely the shaft, rotor, tool, tool holder and tool holder mount (Fig. 2). It is supported by five (three front and two rear) hybrid angular contact bearings. The following studies focus on the rotating entity dynamics. The motion of the rotating entity is considered as the superposition of rigid and elastic body displacements.
Dynamic equations are obtained using Lagrange formulation associated with a finite element method. Due to the size of the rotor sections, shear deformations must be taken into account. Then the rotating entity is derived using Timoshenko beam theory. The relevant shape functions are cubic in order to avoid shear-locking [22]. A special 3D rotor beam element with two nodes and six degrees of freedom per node is developed. As in a classic finite element procedure, dynamic rotor equations are obtained by assembling element matrices. The rotor entity model is assembled with the rolling bearing model. The interfaces between the different sub-entities which compose the rotating entity are assumed to be rigid. The damping model used draws on Rayleigh viscous equivalent damping, which makes it possible to regard the damping matrix D as a linear combination of the mass matrix M, the spindle rigidity matrix Kspindle and the rolling bearing rigidity matrix Kbearing. The set of differential equations can be written as MðqN Þ€qN þ ðCðqN ; q_ N Þ þ DÞ_qN þ KqN ¼ FðtÞ,
(1)
where M(qN) and K are the mass and global stiffness matrices, CðqN ; q_ N Þ matrix contains the rotational dynamics effect, D is the viscous equivalent damping matrix and qN and F(t) are the nodal displacement and force vectors. An accelerating rotor gives rise to time-variant matrices, since M(qN) and CðqN ; q_ N Þ are time-dependent matrices. The treatment of rotor dynamics with a pseudo-constant rotational speed approach establishes that CðqN ; q_ N Þ_qN ¼ 2OG_qN O2 NqN . G and N proceed from the decomposition of matrix C and are, respectively, representative of the gyroscopic and spin softening effects. O is the rotor’s angular velocity. Hence, considering the assumption of pseudo-constant rotational speed, rotor dynamics are described by means of linear time-invariant matrices: M0 q€ N þ ð2OG þ DÞ_qN þ ðK O2 NÞqN ¼ FðtÞ,
Fig. 1. A schematic representation of the milling interaction.
(2)
where M0 is the constant part of the matrix M. In a high-speed spindle the bearing axial preload is kept essentially constant during thermally generated changes to spindle dimensions by spring or hydraulic arrangements. The rolling bearing stiffness matrices are calculated around a static function point characterized by the bearing preload [9,16]. Indeed, stiffness matrices, when the spindle is nonrotating, depend on the geometry and the preload. Based on Rantatalo’s prediction [14], the initially calculated bearing stiffness is spindle speed dependent because of
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the gyroscopic and centrifugal effects induced in the rotating balls (Fig. 3). Then, speed-dependent bearing stiffness is integrated into the global spindle FEM. 2.1.1. Experimental characterization and model readjustment While the detailed knowledge of the spindle is in general not available in a manufacturing environment, the model can be readjusted in order to fit experimental results. The material properties of the rotor and the tool holder mount, such as the Young modulus E, the shear modulus G and the volumic mass r, cannot be precisely known, due to their geometrical and physical complexity. The parameter allows rigidity variations caused by the rear bearings floating mount to be taken into account. The modelling of the damping to represent the dissipative properties of the
Fig. 3. The changes in the bearing stiffness depending on the spindle speed.
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whole set of spindle components also represents a real problem. Hence, a readjustment procedure based on a two-step algorithm is proposed to tune the previous variable in order to fit the model results to experimental ones (Fig. 4). These parameters are readjusted by minimizing the gap between the measured and the modelled tool tip node FRF, respectively, Hexp(o) and Hnum(o), for a non-rotating spindle [15,16], using an optimization Matlabr routine and a least squares-type objective function defined as ¼
o2 X
jH num ðoÞ H exp ðoÞj2 .
o¼o1
The readjustment procedure, applied to the spindle-tool set configuration shown in Fig. 2, was carried out to perform the best modal correlation on a bandwith of 0–3500 Hz. Indeed, the interaction between the spindle bending modes and the flexible workpiece bending modes inside this bandwith (Fig. 9) plays a dominant role in the definition of the frequency range of interest introduced in the stability prediction section. Experimental determination of the dynamic characteristics of the spindle is carried out by an impact hammer test using a Bru¨el and Kjaer acquisition rack controlled by a PC equipped with B&K Pulse software. Finally the spindle database is updated to contain the geometrical and the final readjusted material parameters. 2.1.2. Changes in the spindle dynamics according to the rotation speed The numerical model can be used to investigate highspeed rotational effects on the spindle dynamic behaviour. Natural frequencies are determined as a function of the
Fig. 4. An algorithm of numeric model readjustment.
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Fig. 5. The 3D speed-dependent frequency response function.
rotation speed. A loop on both frequency and spindle speed ranges allows us to generate a Campbell diagram (Fig. 5). This figure illustrates the results of the model’s tool tip node modal analyses with respect to spindle rotation speed. Dynamic effects due to high rotational speed and elastic deformations, such as gyroscopic coupling and spin softening, have a significant influence on the spindle behaviour. In a non-rotating spindle, two orthogonal bending shapes are associated with the same bending frequency, due to the axial symmetry of the rotor. When the spindle rotates, the frequencies associated with these two bending shapes move apart, one above and one below the rest value, as a function of the spindle speed. These two different solutions are called forward and backward whirls in rotor-dynamics literature [23]. 2.2. Workpiece model In order to have a workpiece representative of aeronautical milling problems, the study was focused on a typical thin-walled part. The workpiece is made from AS7 aluminium alloy, with physical properties: E ¼ 70 GN/m2 and density ¼ 2700 kg/m3. The thin wall was downmilled in finishing with axial depth of cut of ap, a radial immersion of 0.1 mm and a feedrate of 0.1 mm/tooth. The workpiece dimensions (Fig. 1) are chosen in order that both spindletool and workpiece systems have similar dynamic behaviour and hence vibrate simultaneously. Thus, satisfying the points announced previously, the part chosen for the study is a rectangular prism of 50 10 100 mm with a thin wall of 5 mm thickness and 30 mm height. That geometry embodies a part that can be
Fig. 6. The workpiece FEM.
found easily in the industry, as, for example, an inner pocket wall. During the finish machining of the part, dynamic properties evolve according to the tool position [19,20]. In order to predict this evolution, a workpiece FEM is established. The tool path is discretized over the whole workpiece and a modal analysis is carried out. At every position the FRF is established. The FEM (Fig. 6) is composed of parallelepiped elements with eight nodes and three degrees of freedom per node. The limit conditions are established by fixing the nodes of the workpiece base. This configuration allows the thin wall to vibrate freely as under real machining conditions.
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Fig. 7. The workpiece frequency response function depending on the tool position.
Experimental modal characterization of the workpiece was carried out in order to adjust the FEM results to the real behaviour of the part. These tests were made through a classic hammer test and mainly enable the frequency damping ratio of the model to be readjusted. In order to integrate the workpiece model to the spindletool set model at the interaction point representative of milling interaction, the dynamic equations of the part have to be established. The assembly procedure is carried out using Matlabs. The workpiece dynamic equations are established using a reduction of the FEM model to a specific master degrees of freedom using the Guyan algorithm. The workpiece is divided into 11 equidistant points, enabling an accurate workpiece dynamic prediction to be made throughout the machining process. The retained MDOF are chosen at the interaction point between the spindle-tool set and the machined workpiece in the radial direction representative of the main vibration direction. In this way, position 1 corresponds to the first point of study and position 11 the last (Fig. 6). The assumption is made that the workpiece FEM has a constant geometry during finishing machining due to the small amount of material removed. Dynamic equations of the reduced workpiece FEM are determined: MW q€ N þ CW q_ N þ KW qN ¼ FðtÞ,
(3)
where MW is the mass matrix, CW the damping matrix and KW the stiffness matrix of the workpiece FEM. 2.2.1. Changes in the workpiece dynamics according to the tool position Assembling all the points mentioned before, the workpiece model is obtained. A harmonic analysis is carried out
Fig. 8. The milling interaction.
and allows the workpiece frequency response function to be established (Fig. 7) at the different interacting points 1–11. The 3D diagram represents the FRF amplitude between the excitation node P and the observing node P as detailed in Fig. 8 for each discrete tool/workpiece position during milling. The node 1 FRF amplitude presents three bending modes in the frequency range of 0–6000 Hz, at, respectively 800, 1500 and 2080 Hz. The 800 and 1500 Hz modes represent, respectively, a simple flexion around the ~ z-axis and a semiflexion, also around the ~ z-axis but symmetrical about the centre of the workpiece. It can be observed that
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both the frequency and the amplitude of the modes depend on the relative tool/workpiece position. In particular, the amplitude of the first mode at 800 Hz increases when the variable position increases, unlike the second mode amplitude at 1500 Hz, which decreases with the position. This kind of difference affects the final coupled system, as is shown in the next section. 2.3. The coupled model at the milling interaction point Once the transfer functions of both the mechanical systems of the spindle-tool set and the workpiece have been analysed, the dynamic interaction between them can be evaluated. The procedure for coupling these two systems is described in this section. Fig. 8 presents the case of a single degree of freedom cutting process. The vibrations of the spindle-tool set and that of the thin-walled piece are considered in direction ~ z perpendicular to the cutting speed, where part flexibility is dominant. During the cutting process, the tool creates a cutting force on the machined workpiece, which then, by the principle of action and reaction, generates a reaction force on the tool with the same magnitude but in the opposite direction. Each subsystem vibration is described by a set of differential equations obtained through a dedicated finite element modelling approach. The projection of these equations onto the ~ z-axis leads to the resulting equations: M W z€W þ C W z_W þ K W zW ¼ F T=W ðtÞ,
(4)
ð5Þ
Eqs. (4) and (5) describe, respectively, the workpiece and the spindle-tool set vibrations. During milling, both the subsystems interact at a contact point modelled by the tool tip node on the spindle model. The FRF of both the subsystems are generated at this interaction point P (Fig. 8), where zW(t) and zT(t) correspond to the small displacements, respectively, of the workpiece and the tool around their equilibrium position. The dynamic chip thickness h is assumed to be the difference between the vibration of the current coupled system, determined by (zT(t)zW(t)), and the vibration of the coupled system at the previous tooth period t determined by (zT(tt) zW(tt)). Consequently, the chip thickness h encountered by the tooth is hðtÞ ¼ h0 ½ðzT ðtÞ zW ðtÞÞ ðzT ðt tÞ zW ðt tÞÞ.
(6)
The cutting forces in the radial ~ z direction in the time domain are modelled by F T=W ðtÞ ¼ K t ap hðtÞ.
hðsÞ ¼ h0 þ ðesT 1Þ½Z T ðsÞ ZW ðsÞ.
(7)
The radial force FT/W is assumed to be proportional to the chip thickness h(t) and the axial depth of cut ap Kt is a constant cutting coefficient.
(8)
Eqs. (4) and (5) are expressed in the Laplace domain by ½M W s2 þ C W s þ K W Z W ðsÞ 1 Z W ðsÞ ¼ F T=W ðsÞ, ¼ FW ðsÞ ½M T s2 þ ð2OGT þ DT Þ s þ ðK T O2 N T Þ Z T ðsÞ 1 ZT ðsÞ ¼ F W=T ðsÞ, ¼ FT ðsÞ
ð9Þ
ð10Þ
with FW(s) and FT(s) the transfer functions at the interacting point P of, respectively, the workpiece and the spindle-tool set. The equation of the closed loop governing dynamic chip thickness leads to hðsÞ 1 . ¼ h0 1 þ K t ap ð1 esT Þ½FW ðsÞ þ FT ðsÞ
(11)
Stability prediction of the coupled system composed of the flexible workpiece and the spindle-tool set requires the analysis of the transfer function resulting from the sum of the two respective subsystem transfer functions. The absolute value of the relative displacement comes from the sum of both subsystem displacements [19]. The resulting FRF of the coupled system at the interacting milling point P is defined as FCPP ¼ FWPP þ FTPP .
M T z€T þ ð2OG T þ DT Þ z_T þ ðK T O2 N T Þ zT ¼ F W=T ðtÞ.
Considering the Laplace transformation ZT(s) ¼ ‘(zT(t)) and ZW(s) ¼ ‘(zW(t)). Eq. (6) becomes
(12)
A graphical representation of the coupled system FRF amplitude is given in Fig. 9, as well as those of the separate workpiece and the spindle-tool set FRF. The coupled system FRF presented in Fig. 9 is determined for a specific relative tool/workpiece position and also at a specific spindle rotation speed of 5000 rpm. The coupled FRF is strongly influenced by both the workpiece and the spindle-tool set FRF. The amplitude of the coupled system FRF presents its maximum value at the respective dominant modes of both the subsystems. The first amplitude peak is clearly influenced by the first spindle-tool set bending mode. The second amplitude peak on the coupling FRF results from the combination of the backward component of the second spindle bending mode and the first workpiece bending mode. The third coupled FRF peak represents the forward component of the second spindle bending mode. The fourth FRF peak is predominantly composed of the second bending mode of the machining part. When the spindle-tool set flexibility is preponderant compared to the workpiece flexibility, the coupled behaviour is mostly influenced by the spindle-tool set dynamics variation, the converse being the case when the part is the most flexible system. Thus, in relation to the interacting spindle/workpiece model presented, the system with the
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Fig. 9. A comparison of the transfer functions of the spindle-tool set, the workpiece and the coupled system.
Fig. 10. The 3D milling interaction FRF amplitude.
most flexibility controls the behaviour of the coupled system. By scanning the different respective positions of the tool along the piece a 3D FRF can be elaborated. Results of such an investigation can be observed in Fig. 10, where 3D vibratory evolution is presented. Fig. 10 is established at a spindle speed of 5000 rpm. FRF amplitude is strongly dependent on the relative position between the tool and the workpiece. A shift can be observed, to the left of mode 2, and to the right of mode 3, depending on the tool/workpiece position. The vibration mode amplitudes are also varied. Some mode amplitudes
are reduced and others are increased. The symmetry of workpiece geometry at the node 6 vertical layer can also be observed in the dynamic behaviour. Such an investigation can also be made for a given tool/ workpiece relative position, depending on the spindle speed variation. Consequently, the dependency of the coupled FRF on both the spindle rotation speed and the relative position between the workpiece and the spindle had to be investigated in order to identify the consequences on stability prediction. This aspect is analysed in the next section.
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3. Stability analysis of the milling system In this section, the dynamic model of the coupled system composed of the spindle-tool set and the workpiece is used for the purpose of the stability analysis. The vibration of the coupled structure due to the cutting forces is predominant in direction ~ z perpendicular to the cutting speed. The milling stability prediction is based on Altintas and Budak’s approach [4]. However, the proposed analysis technique can be used generically with other milling stability approaches.
In Altintas and Budak’s approach [4], the time-variant coefficients of the dynamic milling equations, which depend on the angular orientation of the cutter as it rotates through the cut, are expanded into a Fourier series and then truncated to include only the average component: alim ¼
The dynamic stiffness of the modes induces permissible depths of cut into stability limits theory. When the critical depth of cut of the stability lobe at the cutting spindle speed is maximized, cutting is at most stable. The resulting stability relationships are shown in Eqs. (13)–(14) where alim is the limiting chip width or axial depth of cut in peripheral end milling operations, Kt is the specific cutting energy coefficient, which relates the radial cutting force Fr to the chip area, and n is the spindle speed in rpm. The cutter is assumed to have Z number of teeth with a zero helix angle. The dynamics-based receptance FCPP ðjoÞ of the coupled system at the interacting milling node P in the radial ~ z direction can be deduced from Eq. (12). FCPP ðjoÞ ¼ RCPP ðoÞ þ jI CPP ðoÞ where j2 ¼ 1. RCPP and I CPP are, respectively, the real part and the imaginary part of the coupled system frequency response function. The negative real part RCpp and the growing phase j of the transfer function allows delimiting a frequency range of interest [15,16]. Then, the optimal cutting parameters (alim; n) that maximize the chatter-free material removal rate are determined, through the frequency range of interest, according to Eqs. (13), (14).
(13)
where K is a time-invariant but immersion-dependent directional cutting coefficient. 60 60 2pf c ¼ T Zðp þ 2kp 2jÞ I C ðoÞ , j ¼ tan1 PP RCPP ðoÞ
n¼ 3.1. Dynamics-based SLD
2pK 1 , K tZ RCPP ðoÞ
with ð14Þ
where f c (Hz) is the chatter vibration frequency, and k is an integer that corresponds to the number of vibration waves left on the workpiece surface. A dynamics-based SLD is elaborated for each relative position of the tool along the workpiece, also taking into account the effective dynamic properties of the spindle at the considered rotation speed. The proposed stability prediction is developed through a two-step algorithm [16]. In the considered study, the half-workpiece is meshed into six nodes, and, with one stability chart per node, this results in six dynamic stability lobes. As a result, a graphic stability map can be elaborated by the combination of the relative position-dependent stability lobes (Fig. 11). The 3D stability diagram represents the limit depth of cut alim depending on the two dimensions represented by the spindle speed and the relative tool/workpiece position Y. The evolution of stable conditions along the machining part can be observed. The dynamic behaviour symmetry of the workpiece observed in Fig. 7 induces a symmetry in the stability lobes prediction. Due to this symmetry, only the stability lobes on nodes 1–6 are
Fig. 11. The 3D stability lobes diagram depending on the spindle speed and the tool–workpiece position.
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Fig. 12. The stability-based spindle speed variation.
represented. Stability prediction on nodes 7–11 can be deduced from node 5 to 1. During workpiece machining, variations occur in the stability peak position and also in the permissible depth of cut without chatter. Position 1 presents three weak stability peaks at the respective spindle speeds of 7600, 10,100 and 14,000 rpm. During machining, these peaks evolve according to the tool/workpiece position. At position 4 it is possible to observe four stability peaks at 7200, 9800, 11,600 and 13,690 rpm. The analysis of such an accurate 3D stability map enables the most adequate machining strategy to be selected in accordance with industrial objectives. The next section is related to the definition of stability-based spindle speed rules during the machining of a flexible part.
4. Stability-based spindle speed variation The evolution of the stability frontier depending on the spindle speed during flexible part machining shows that gains can be achieved through the definition of dynamicsbased variable spindle speed rules. Nowadays, spindle speed follows a constant speed setting from the beginning of milling until the end. This kind of simple instruction does not allow the best machining conditions to be achieved. Indeed, with regard to the stability conditions, the lowest spindle speed will be retained, and thus productivity will be greatly affected. An inspection of the proposed 3D SLD (Fig. 11) makes it possible to extract specific spindle speeds that maximize industrial criteria such as productivity and stability. In our approach, we propose to maximize the material removal rate at stable conditions by varying the spindle speed with respect to the predicted stability frontier. The resultant optimized spindle-drive rules are elaborated by linking the most stable machining points all along the tool path (Fig. 12).
The different spindle speed rules elaborated through the proposed approach are represented in Fig. 12. Four spindle speed rules can be defined by linking the dominant stability peaks of all 11 nodes. Hence, the regulation of spindle speed to these rules, predicted by the model, will guarantee the maximal stability. The black vertical line positioned on node 6 indicates the symmetry line for spindle speed prediction. In fact, different machining strategies can be defined, depending on the optimization criteria. For example, the two spindle speed rules starting near 10,000 rpm allow the maximized depth of cut to be implemented. The rule starting near 14,000 rpm maximizes the spindle speed in stable machining conditions, but with a reduced permissible depth of cut. Hence, the optimized rule can be defined depending on the industrial strategies. An innovative study approach, concerning the development of a new generation of spindle controls, has been highlighted. Some prospects, concerning the feasibility of continuous speed variation according to optimized criteria and with respect to spindle dynamics, need to be developed further.
5. Conclusion In this paper, an integrated spindle–workpiece model is developed by coupling a rotor-dynamics-based spindle FEM model [15,16] and a thin-walled workpiece FEM model-based common industrial parts. Experimental validations of the modelled spindle and the thin-walled workpiece results have been carried out separately. The vibratory interaction between the two systems during milling is studied. The dynamic behaviour of the coupled system is shown to be greatly dependent on two variables, respectively, the spindle–workpiece relative position and the spindle rotation speed.
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The proposed method enables accurate lobes diagrams to be established through the consideration of the relative transfer function of the interacting milling system. The evolution of the stability boundaries according to the spindle rotation speed and the tool–workpiece relative position give rise to a 3D lobes diagram. The stability prediction results are in accordance with the results of Bravo et al. [19] and take into account the additional effects due to spindle speed-dependent dynamics. Finally, by examining the predicted 3D stability map, specific spindle speed rules which optimize the maximum material removal rate without chatter during the machining of the flexible workpiece are elaborated. The spindle speed rules are dependent on the relative spindle-tool/ workpiece position and take into account the speeddependent spindle dynamics. The optimized machining conditions regarding industrial objectives require the spindle speed to be regulated according to the predicted rule. This approach enables design and operating rules to be established for HSM spindles, and can provide help for designers while attempting to eliminate costly experimentbased design trials. Acknowledgements This study was carried out within the framework of the TIMS Research Group, using grants from the Regional Council of Auvergne, the French Ministry of Research, the CNRS and the Cemagref. References [1] S. Tobias, W. Fishwick, A theory of regenerative chatter, The Engineer, February (1958). [2] J. Tlusty, M. Polacek, The stability of machine tools against selfexcited vibrations in machining, in: Proceedings of the ASME International Research in Production Engineering, Pittsburgh, USA, 1963, pp. 465–474. [3] H. Merritt, Theory of self-excited machine tool chatter, Transactions of the ASME Journal of Engineering for Industry 87 (1965) 447–454. [4] Y. Altintas, E. Budak, Analytical prediction of stability lobe in milling, CIRP Annals 44 (1) (1995) 357–362. [5] R.P.H. Faassen, N. van de Wouw, J.A.J. Oosterling, H. Nijmeijer, Prediction of regenerative chatter by modelling and analysis of highspeed milling, International Journal of Machine Tools and Manufacture 43 (2003) 1437–1446. [6] T. Schmitz, J. Ziegert, C. Stanislaus, A method for predicting chatter stability for systems with speed-dependent spindle dynamics, Society
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