applied sciences Article
Investigation of the Machining Stability of a Milling Machine with Hybrid Guideway Systems Jui-Pin Hung 1, *, Wei-Zhu Lin 1,2 , Yong-Jun Chen 1 and Tzou-Lung Luo 2 1 2
*
Graduate Institute of Precision Manufacturing, National Chin-Yi University of Technology, Taichung 41170, Taiwan;
[email protected] (W.-Z.L.);
[email protected] (Y.-J.C.) Intelligent Machine Tool Technology Center, ITRI Central Region Campus, Taichung 54041, Taiwan;
[email protected] Correspondence:
[email protected]; Tel.: +886-4-2392-4505 (ext. 7181); Fax: 886-4-2392-5714
Academic Editor: Chih Jer Lin Received: 29 November 2015; Accepted: 23 February 2016; Published: 8 March 2016
Abstract: This study was aimed to investigate the machining stability of a horizontal milling machine with hybrid guideway systems by finite element method. To this purpose, we first created finite element model of the milling machine with the introduction of the contact stiffness defined at the sliding and rolling interfaces, respectively. Also, the motorized built-in spindle model was created and implemented in the whole machine model. Results of finite element simulations reveal that linear guides with different preloads greatly affect the dynamic responses and machining stability of the horizontal milling machine. The critical cutting depth predicted at the vibration mode associated with the machine tool structure is about 10 mm and 25 mm in the X and Y direction, respectively, while the cutting depth predicted at the vibration mode associated with the spindle structure is about 6.0 mm. Also, the machining stability can be increased when the preload of linear roller guides of the feeding mechanism is changed from lower to higher amount. Keywords: dynamic compliance; machining stability; hybrid guideways
1. Introduction For satisfying multipurpose or specific industrial applications, most of the modern milling machine centers nowadays have been designed and constructed by different modularized main components [1]. In the configuration of the machine tool, the arrangement of linear feeding mechanism plays an important role in determining machining performance because the various linear elements or other joints are weaker links of different structural components [2,3]. Although modularized design concept brings various machine configurations under consideration, evaluation of dynamic performance of the candidates that can satisfy the required performance is a perquisite for fabricating the prototype [4,5]. However, the feeding mechanism of the control axis has been verified to greatly affect the structural dynamic characteristics and machining stability [6]. The joints in the guideway system cannot be treated as a rigid interface; on the contrary, they form weak links between components and greatly affect the overall structural characteristics of the assembled machine [7,8]. As a consequence, the modeling of an interface and the identification of the interfacial characteristics are of importance in the dynamic analysis of a machine tool structure. Currently, the horizontal milling machine with moving column has been constructed to achieve multiple machining functions. The configuration of the feeding mechanism is considered a dominant role for innovative design of such kind of machines. Generally, linear guides with rolling balls are appropriate for light duty milling machines, while linear roller guide moduli or sliding guideway systems were adopted in heavy duty milling machine to undertake more loads in machining. Essentially, each of the various guideway systems possesses different kinematical and dynamic Appl. Sci. 2016, 6, 76; doi:10.3390/app6030076
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properties, which may thus bring the machine tool to demonstrate different dynamic performances. Research studies [9,10] have shown that a rolling guide exhibits different vibration characteristics depending on preloaded amount. Regarding the sliding contact guide, the contact stiffness of a sliding surface is the function of surface finish, pairing of contact material, and interface pressure distribution [11]. The interface pressure was also verified to be an important factor affecting the contact stiffness and the machining performance [12]. The purpose of this study was to analyze the machining stability of a horizontal milling machine with hybrid guideway systems which were assembled with linear roller guides and sliding guides with coated antifriction layers. Currently, the finite element approach has been recognized as an effective tool in modeling the machine tool, which can accurately predict the dynamic behavior of the prototype designs without physically building any parts. This approach of designing prototype machines in software has been termed virtual prototyping [13,14]. To be a realistic model for assessing the frequency responses of the milling machine, a finite element model incorporating the machine frame with a spindle unit was a prerequisite. Using the FE modeling approach, Sulitka et al., [15,16] demonstrated the dynamic compliance of the spindle to vary with the changing of the machine configuration. Various factors, such as the spindle bearing preload or structural geometry, influencing the FRFs at the tool tip were also investigated [17,18]. The optimization of the machine tool design has been studied by incorporating the FE models for full machine analyses with the consideration of machining stability in process [19,20]. Studies of Mousseigne et al., [21,22] found the variation of the tool point dynamic response with the change of milling, which further caused great influences on the stability lobe curves. Their study also suggested that a precise estimation of the natural frequencies is of importance for lessening the uncertainty of the stability lobe curves. The milling machine discussed in this study was driven on hybrid guideway systems which were assembled with a linear roller guide and a sliding guide with coated antifriction layers. In creating the finite element models of the milling machine, apart from mechanical frame structure, the sliding guides and roller guides were also modeled by different interface elements with appropriate interface stiffness. Besides, the motorized built-in spindle model was created and implemented in the whole machine model. As a dynamic evaluation of the machine tool, the vibration modes, dynamic stiffness, and machining stability of the spindle tool system were predicted. The results were used to compare the influence of the use of linear roller guides with different preload. 2. Construction of Milling Machine In this study, a horizontal milling machine was designed, as shown in Figure 1, in which the table was mounted on the machine base and was driven through the X-axis linear feeding mechanism. The vertical column was mounted on the machine base through the Z-axis feeding mechanism. The spindle ram was mounted on the vertical column, which can move along vertical direction through the Y-axis feeding mechanism. To enhance the rigidity of the feeding mechanism, the guideway systems were assembled by linear roller guides and sliding guides, in which the sliding guides were grounded, hardened, and then coated with scrapped Turcite-B liners. The use of the Turcite-B slideways can also increase the damping property of the machining system [23]. All the structural components were fabricated from gray iron. In each feeding mechanism, the driven ball screw (Hiwin-R32-10K4-FSC) has a diameter of 32 mm, a lead pitch of 10 mm, and a basic dynamic load rating C of 2.52 kN [24]. The screw shaft was slightly preloaded to a level of 0.05C so as to lessen the axial backlash. Moreover, two standardized ball-screw support units coded EK15 were used at both ends of the screw shaft to ensure its rigidity. Each feeding system was also equipped with linear roller guide modules (INA-RUS19069), which can be preloaded to different amounts by means of the adjusting the gib in spaces, normally between 10% and 20% of the basic dynamic loading capacity [25]. In addition, a motorized spindle unit was installed on the feeding ram of the spindle.
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Figure 1. Schematics of a horizontal milling machine with moving column and hybrid guideways. (a) Main structure modules of the milling machine, (b) Y-axis guideway system, (c) X and Z-axis Figure 1. Schematics of a horizontal milling machine with moving column and hybrid guideways. (a) guideway system. Main structure modules of the milling machine, (b) Y‐axis guideway system, (c) X and Z‐axis guideway system.
The key factor for accurate construction of the analysis model of a machine tool structure is the The key factor for accurate construction of the analysis model of a machine tool structure is the modeling of the feeding mechanism, which usually consists of various linear components of rolling modeling of the feeding mechanism, which usually consists of various linear components of rolling motion type. These interfaces thestructure structurecharacteristics characteristics in loading damping motion type. These interfaces mostly mostly affect affect the in loading and and damping capacities. Hence the the modeling of of thethe interface greatly the simulation simulationresults results can capacities. Hence modeling interface greatly determines determines whether whether the approach the real characteristics of the system. A method for the simulation of the rolling and sliding can approach the real characteristics of the system. A method for the simulation of the rolling and interfaces is presented in next section as follows. sliding interfaces is presented in next section as follows. 3. Modeling of the Milling Machine 3. Modeling of the Milling Machine 3.1. Modeling of the Joint/Interface 3.1. Modeling of the Joint/Interface
3.1.1.3.1.1. Rolling Interface Rolling Interface According to Hertzian theory, the contact force between a rolling element and the raceways can According to Hertzian theory, the contact force between a rolling element and the raceways can be related to the local deformation at the contact point as the expression (Figure 2): be related to the local deformation at the contact point as the expression (Figure 2): Q K h δ 3/2
Q “ Kh δ3{2
(1)
(1)
where Q denotes the contact force and is the elastic deformation at the contact point. Kh represents the Hertz constant, which is determined by the contact geometry of the ball groove or raceway and where Q denotes the contact force and δ is the elastic deformation at the contact point. Kh represents the material properties of the contacting components. Details are available in the literatures [26,27]. the Hertz constant, which is determined by the contact geometry of the ball groove or raceway and The contact stiffness at the contact point can further be determined from Equation 2.
the material properties of the contacting components. Details are available in the literatures [26,27]. dQ 3 3 2/3 1/3 from Equation (2). 2 The contact stiffness at the contact point K n canfurther K h δ1/be determined Kh Q (2) dδ 2 2 dQ 3 3 2{3 For a roller guide, the relationship between the contact force Q and the local deformation at Kn “ “ Kh δ1{2 “ Kh Q1{3 (2) dδ 2 2 the contact point of the roller to the flat raceway can be described as For a roller guide, the relationship between πL2 Eeq force Q and the local deformation δ at the 2Q the contact δ ln( ) as (3) contact point of the roller to the flat raceway can be described πLEeq Q δ“
πL2 Eeq 2Q lnp q πLEeq Q
Eeq “ 2{pp1 ´ µ2a q{Ea ` p1 ´ µ2b q{Eb q
(3)
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E eq 2 / ((1 μ 2a ) / E a (1 μ b2 ) / E b )
In above equation, Eeq is the relative elastic stiffness between the contacting bodies. L is the length In above equation, E eq is the relative elastic stiffness between the contacting bodies. L is the of the roller. The normal stiffness is thus given as length of the roller. The normal stiffness is thus given as ff « 2 πL 2Eeq 2 ´11 2 (4) K A “ 2 lnp πL Eeqq ´ 2 K A πLEeq ln( Q ) πLEeq (4)
πLEeq
Q
πLEeq
Figure 2. Modeling of the rolling and sliding interfaces. (a) Schematic of angular contact bearing, Figure 2. Modeling of the rolling and sliding interfaces. (a) Schematic of angular contact bearing, (b) (b) Modeling of the contact interface between rolling ball and groove, (c) Schematic of sliding guideway Modeling of the contact interface between rolling ball and groove, (c) Schematic of sliding guideway and the modeling of the antifriction sliding interface. and the modeling of the antifriction sliding interface.
To reduce the complexity in model creation and mesh generation of the motion components, To reduce the complexity in model creation and mesh generation of the motion components, such such as the ball bearings in the spindle unit, the contact configuration between rolling ball and the as the ball bearings in the spindle unit, the contact configuration between rolling ball and the raceway raceway is modeled as a two‐point contact mode. The outer and inner raceways were respectively is modeled as a two-point contact mode. The outer and inner raceways were respectively simplified simplified as a part of the spindle shaft and housing in geometry. The outer and inner raceways are as a part of the spindle shaft and housing in geometry. The outer and inner raceways are directly directly connected using a series of spring elements by neglecting the rolling elements, as shown in connected using a series of spring elements by neglecting the rolling elements, as shown in Figure 2. Figure 2. 3.1.2. Sliding Interface 3.1.2. Sliding Interface The contact properties of the elastic interface layer such as Turcite-B in the sliding guide system can be modeled by the following exponential relationship [11]. The contact properties of the elastic interface layer such as Turcite‐B in the sliding guide system can be modeled by the following exponential relationship [11]. δn “ Cn Pnmm (5) δ n C n Pn (5) 2 ), C is coefficient of normal where isnormal deformation deformation (mm), (mm), PPnn isis normal normal pressure pressure (N/mm (N/mm2), Cnn is coefficient of normal Where δnn isnormal contact flexibility, and m is coefficient of non-linearity of deformations. The contact stiffness in the contact flexibility, and m is coefficient of non‐linearity of deformations. The contact stiffness in the normal direction can be obtained as normal direction can be obtained as
dPn 1 1 m m KKn “ d Pn “ 1 pp1n´ n n dδ C dδnn Cnnm m
(6) (6)
For modeling sliding guides in a finite element model, the spring layer model was employed to For modeling sliding guides in a finite element model, the spring layer model was employed to simulate the contact characteristics of the antifriction liners between the guide rail and sliding block. simulate the contact characteristics of the antifriction liners between the guide rail and sliding block. The antifriction layer was replaced by interface elements distributed over this interface (Figure 2), The antifriction layer was replaced by interface elements distributed over this interface (Figure 2), which was described by the stiffness matrix, which was described by the stiffness matrix,
[ rK K ee s] “ rNs N T rTs T T rDs N rTs D TdA dA rNs A ż
T
A
T
(7) (7)
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where [N] and [T] is the interpolation function and coordinate transformation matrix, respectively, and [D] is the contact stiffness matrix of the interface layer described as [D] = diag[Kn, Ks]. Kn and Ks is where [N] and [T] is the interpolation function and coordinate transformation matrix, respectively, and the contact stiffness in normal and tangential directions of the contact interface. [D] is the contact stiffness matrix of the interface layer described as [D] = diag[Kn , Ks ]. Kn and Ks is the contact stiffness in normal and tangential directions of the contact interface. 3.2. Modeling of Spindle Unit 3.2. Modeling Spindle Unit Figure 3 ofshows the motorized spindle installed in the milling machine, in which the spindle shaft Figure is supported in housing two pairs of bearings 7009C at front and rear ends in shaft DT 3 shows the motorizedby spindle installed in the milling machine, in which the spindle arrangement, respectively, which were preloaded at medium amount of 290 N with an axial rigidity is supported in housing by two pairs of bearings 7009C at front and rear ends in DT arrangement, of 66 N/μm. In addition, a cutter with the diameter of 20 mm and length of 60 mm was clamped on respectively, which were preloaded at medium amount of 290 N with an axial rigidity of 66 N/µm. the spindle tool holder (BT30). To investigate the dynamic behavior of the spindle, a solid model of In addition, a cutter with the diameter of 20 mm and length of 60 mm was clamped on the spindle tool the spindle bearing system including the rotating shaft and the spindle housing was prepared and holder (BT30). To investigate the dynamic behavior of the spindle, a solid model of the spindle bearing meshed with tetrahedral element (see the Figure 3). housing The built in prepared motor was as tetrahedral a cylinder system including the rotating shaft and spindle was andsimplified meshed with having equivalent weight as the original unit. Regarding the supporting bearings, the outer element (see Figure 3). The built in motor was simplified as a cylinder having equivalent weight asand the inner rings were respectively simplified as a part of the spindle shaft and housing in geometry. A original unit. Regarding the supporting bearings, the outer and inner rings were respectively simplified series of spring elements were employed to directly connect the inner and outer rings, modeling the as a part of the spindle shaft and housing in geometry. A series of spring elements were employed contact state between rolling ball and groove. For finite element analysis, all the metal components to directly connect the inner and outer rings, modeling the contact state between rolling ball and have the following material properties of carbon steel: elastic modulus E = 200 GPa, Poisson′s ratio groove. For finite element analysis, all the metal components have the following material properties 3 = 0.3, and density = 7800 Kg/m of carbon steel: elastic modulus . The overall contact stiffness of each bearing was calculated as 340 E = 200 GPa, Poisson1 s ratio µ = 0.3, and density ρ = 7800 Kg/m3 . N/μm. The stiffness of each bearing is was distributed on spring The elements circumferentially The overall contact stiffness of each bearing calculated as the 340 N/µm. stiffness of each bearing surrounding the spindle shaft created in the model. The tool holder the with cutter was modeled as a is distributed on the spring elements circumferentially surrounding spindle shaft created in the solid cylinder and assumed to be firmly connected with the spindle nose, which was considered as a model. The tool holder with cutter was modeled as a solid cylinder and assumed to be firmly connected part of the spindle shaft. with the spindle nose, which was considered as a part of the spindle shaft.
Figure 3. Solid model and finite element model of spindle unit. Figure 3. Solid model and finite element model of spindle unit.
3.3. Finite Element Model of Milling Machine 3.3. Finite Element Model of Milling Machine Figure presents finite element of machinestructure. milling machinestructure. Each component structural Figure 44 presents thethe finite element modelmodel of milling Each structural component of was the meshed system was ten‐node elements, tetrahedral elements, a total of 124,648 of the system withmeshed with ten-node tetrahedral with a total ofwith 124,648 elements and elements and 224,175 nodes. The components of the feeding mechanism, such linear as ball screw/nut, 224,175 nodes. The components of the feeding mechanism, such as ball screw/nut, roller guides, linear roller guides, and sliding guideways were included in the machine model. For linear roller and sliding guideways were included in the machine model. For linear roller guides, the overall guides, the overall structural stiffness K n in normal direction was 500 and 850 N/μm for normal and structural stiffness Kn in normal direction was 500 and 850 N/µm for normal and medium preloaded medium preloaded roller guides. Regarding to the sliding guideway with Turcite‐B antifriction layer, roller guides. Regarding to the sliding guideway with Turcite-B antifriction layer, the contact stiffness the stiffness measured as area 750 kN/μm unit in meter [23]. The overall wascontact measured as 750was kN/µm per unit in meterper [23]. Thearea overall contact stiffness at ball contact groove stiffness at ball groove was estimated as 1.62 kN/μm according to the specifications of the ball screw was estimated as 1.62 kN/µm according to the specifications of the ball screw [24]. In order to obtain a [24]. order to obtain whole machine, analysis model of a milling system machine, the spindle‐bearing system wholeIn analysis model of aa milling the spindle-bearing created in the above section was created in the inabove section was asincorporated in the shown analysis, in Figure Also, in incorporated the spindle ram, shown in Figure 4. spindle Also, in ram, finiteas element the4. materials finite element analysis, the materials used for structural components are made of gray cast iron with used for structural components are made of gray cast iron with an elastic modulus E = 660 GPa, 3 . The materials of an elastic modulus E = 660 GPa, Poisson′s ratio = 0.3, and density = 7200 Kg/m . The materials of Poisson1 s ratio µ = 0.3, and density ρ = 7200 Kg/m linear rolling3components have 1 3 .The vibration linear rolling components have an elastic modulus E = 210 GPa, Poisson′s ratio = 0.3, and density an elastic modulus E = 210 GPa, Poisson s ratio µ = 0.3, and density ρ = 7800 Kg/m 3 = 7800 Kg/m mode shapes.The vibration mode shapes associated with the frequencies of the milling system were associated with the frequencies of the milling system were obtained by implementing obtained by implementing the modal analysis into the finite element computation. the modal analysis into the finite element computation.
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Figure 4. Finite element models of the milling machine. Figure 4. Finite element models of the milling machine.
Figure 4. Finite element models of the milling machine. The harmonic analysis was performed to measure the frequency response at the end of the The harmonic analysis was performed to measure the frequency response at the end of the cuter. cuter. In the finite element governing equation for harmonic analysis, the damping matrix was In the finite governing equation for harmonic analysis, the damping matrix The element harmonic analysis was performed to measure the frequency response at the was end assumed of the to assumed to be proportional to the structural stiffness matrix [K] according to the relationship [C] = cuter. In the element governing equation harmonic the damping was value be proportional tofinite the structural stiffness matrix [K]for according toanalysis, the relationship [C] = matrix β[K]. The [K]. The value β mr, representing the structural damping factor, is calculated from 2 mr/wr, where mr βmris the modal damping ratio for spindle dominant vibration mode, about 2.5%. , assumed to be proportional to the structural stiffness matrix [K] according to the relationship [C] = representing the structural damping factor, is calculated from 2 ξmr /wr , where ξmr is the modal [K]. The value β mr, representing the structural damping factor, is calculated from 2 mr/wr, where mr damping ratio for spindle dominant vibration mode, about 2.5%.
is the modal damping ratio for spindle dominant vibration mode, about 2.5%. 4. Results and Discussions 4. Results and Discussions 4. Results and Discussions
4.1. Frequency Response Function of Spindle Unit 4.1. Frequency Response Function of Spindle Unit 4.1. Frequency Response Function of Spindle Unit Figure 5 shows the main vibration modes of the spindle unit and the first four modal Figure 5 shows the main vibration modes of the spindle unit and the first four modal frequencies frequencies are 1046, 1567, 2972, and 3855 Hz respectively. The harmonic analysis was performed to Figure 5 shows the main vibration modes of the spindle unit and the first four modal areassess the frequency response of spindle under unit force at the tool tip. For validation of the finite 1046, 1567, 2972, and 3855 Hz respectively. The harmonic analysis was performed to assess the frequencies are 1046, 1567, 2972, and 3855 Hz respectively. The harmonic analysis was performed to frequency response of spindle under unit force at the tool tip. For validation of the finite element element model, the dynamic characteristic of the spindle was measured by conducting the vibration assess the frequency response of spindle under unit force at the tool tip. For validation of the finite model, the dynamic characteristic of the spindle was measured by conducting vibration test on the test on the spindle unit. A cutter with the diameter of 20 mm and length the of 60 mm was firmly element model, the dynamic characteristic of the spindle was measured by conducting the vibration spindle unit. A cutter with the diameter of 20 mm and length of 60 mm was firmly clamped on clamped on the spindle tool holder, with an overhang length of 30 mm. test on the spindle unit. A cutter with the diameter of 20 mm and length of 60 mm was firmly the clamped on the spindle tool holder, with an overhang length of 30 mm. spindle tool holder, with an overhang length of 30 mm.
Figure 5. Fundamental modal shapes of spindle. Figure 5. Fundamental modal shapes of spindle. Figure 5. Fundamental modal shapes of spindle.
In testing, the accelerometer was mounted on the cutter to measure the vibration signal excited In testing, the accelerometer was mounted on the cutter to measure the vibration signal excited In testing, thehammer accelerometer wasThe mounted on the cutter to measure the vibration signal excited the impact at tool tool tip. tip. was then extracted from the by by the impact hammer at The frequency frequency response response function function was then extracted from the by recorded FFT spectrum. As shown in Figure 6, the spindle shaft vibrates greatly at a frequency of the impact hammer at tool tip. The frequency response function was then extracted from recorded FFT spectrum. As shown in Figure 6, the spindle shaft vibrates greatly at a frequency of the recorded FFT spectrum. As shown in Figure 6, the spindle shaft vibrates greatly at a frequency of
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1043 Hz, but has less vibration at 1454 Hz. The spindle unit was found to become more compliant 1043 Hz, but has less vibration at 1454 Hz. The spindle unit was found to become more compliant when the first bending mode was excited. The maximum compliance predicted by finite element model when the first bending mode was excited. The maximum compliance predicted by finite element Appl. Sci. 2016, 6, 76 7 of 13 and measured from physical unit is 0.85 µm/N at 1048 Hz and 0.73 µm/N at 1046 Hz, respectively. model and measured from physical unit is 0.85 μm/N at 1048 Hz and 0.73 μm/N at 1046 Hz, 1043 Hz, but has less vibration at 1454 Hz. The spindle unit was found to become more compliant It is obvious that the spindle model has a dynamic behavior comparable to that measured in the respectively. It is obvious that the spindle model has a dynamic behavior comparable to that when the first bending mode was excited. The maximum compliance predicted by finite element physical spindle. measured in the physical spindle.
PRedicted FRF
0.8 0.7 0.6 1.0 0.5 0.9 0.4 0.8 0.3 0.7 0.2 0.6 0.1 0.5 0.0 0.4
Measured FRF
Frequency response function at tool end PRedicted FRF Measured FRF
0
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Frequency, (Hz)
(a) 0
500
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0.6 0.4 0.2 1.0 0.0 0.8 -0.2 0.6 -0.4 0.4 -0.6 0.2 -0.8 0.0 -1.0 -0.2 0 -0.4 -0.6 -0.8 -1.0 0
FRF, Real (um/N) FRF, Real (um/N)
FRF, Amplitude, (um/N) FRF, Amplitude, (um/N)
model and measured from physical unit is 0.85 μm/N at 1048 Hz and 0.73 μm/N at 1046 Hz, Frequency response function at tool end to that respectively. Frequency It is obvious that at tool the endspindle model has a dynamic behavior comparable response function 1.0 1.0 measured in the physical spindle. 0.9 0.8 Predicted FRF
Measured FRF
Frequency response function at tool end
Predicted FRF Measured FRF
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Frequency, (Hz)
(b) 500
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Frequency,of (Hz) Figure 6. Comparisons the frequency response functions measured Frequency, from the spindle unit and (Hz)
Figure 6. Comparisons of the frequency response functions measured from the spindle unit and predicted by the spindle model. (a) Amplitude of the FRFs, (b) Real part of the FRFs. (b)the FRFs. predicted by the spindle(a) model. (a) Amplitude of the FRFs, (b) Real part of Figure 6. Comparisons of the frequency response functions measured from the spindle unit and 4.2. Natural Vibration Modes of Milling Machine predicted by the spindle model. (a) Amplitude of the FRFs, (b) Real part of the FRFs. 4.2. Natural Vibration Modes of Milling Machine
Results of modal analysis indicate that the fundamental modes are associated with the vibration
of the spindle ram and vertical column as illustrated in are Figure 7. The first mode is the Results of modal analysis indicate that structure, the fundamental modes associated with the vibration 4.2. Natural Vibration Modes of Milling Machine rolling motion of vertical column of the Z axis at 34 Hz, which is dominated by the feeding of the spindle ram and vertical column structure, as illustrated in Figure 7. The first mode is the rolling Results of modal analysis indicate that the fundamental modes are associated with the vibration mechanism of Z axis. The second mode is the twisting vibration of the vertical column and spindle motion of vertical the Zcolumn axis at structure, 34 Hz, which is dominated by 7. the feeding mechanism of the spindle column ram and of vertical as illustrated in Figure The first mode is the of ram about Y‐axis at 63 Hz. The third and fourth modes at 98 and 119 Hz are yawing (twisting) Z axis. The second mode is the twisting vibration of the vertical column and spindle ram about Y-axis rolling motion of vertical column of the Z axis at 34 Hz, which is dominated by the feeding vibration of spindle ram about the Z‐axis. It is noticed that in the second and fourth modes the at 63 mechanism of Z axis. The second mode is the twisting vibration of the vertical column and spindle Hz. The third and fourth modes at 98 and 119 Hz are yawing (twisting) vibration of spindle ram spindle nose shows a greater displacement, which can affect the machining ability of the machine. In Y‐axis 63 Hz. that The in third fourth 98 and the 119 spindle Hz are yawing (twisting) aboutram theabout Z-axis. It is at noticed the and second andmodes fourthat modes nose shows a greater addition, these modes are affected by the feeding mechanisms of the vertical column and spindle vibration of spindle ram about the Z‐axis. It ability is noticed that machine. in the second and fourth modes the displacement, which can affect the machining of the In addition, these modes ram. The other high frequency modes are associated with the bending vibration of the spindle shaft. are spindle nose shows a greater displacement, which can affect the machining ability of the machine. In affected by the feeding mechanisms of the vertical column and spindle ram. The other high frequency addition, these modes are affected by the feeding mechanisms of the vertical column and spindle modes are associated with the bending vibration of the spindle shaft. ram. The other high frequency modes are associated with the bending vibration of the spindle shaft.
Figure 7. Fundamental vibration modes of the milling machine.
Figure 7. Fundamental vibration modes of the milling machine. Figure 7. Fundamental vibration modes of the milling machine.
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Appl. Sci. 2016, 6, 76 4.3. Frequency Response Functions of the Milling Machine
4.3. Frequency Response Functions of the Milling Machine
Figure 8 illustrates the predicted frequency response at the tool end of the milling machine in Figure 8 illustrates the predicted frequency response at the tool end of the milling machine in X X and Y directions, respectively, which are expressed in terms of dynamic compliance in magnitude as a and Y directions, respectively, which are expressed in terms of dynamic compliance in magnitude as function of the frequency. As observed in the figure, the spindle of the milling machine behaves similar a function of the frequency. As observed in the figure, the spindle of above the milling behaviors in X and Y axis, especially, at the higher frequency mode 1000machine Hz. Thebehaves maximum similar behaviors in X and Y axis, especially, at the higher frequency mode above 1000 Hz. The dynamic compliance occurs at the frequency of 1290 Hz, which is higher than the measured natural maximum dynamic compliance occurs at the frequency of 1290 Hz, which is higher than the frequency of the spindle unit. This is because of the constrained of the spindle housing on the spindle measured natural frequency of the spindle unit. This is because of the constrained of the spindle ram. The maximum compliance in X and Y direction is 1.42 µm/N (1285 Hz) and 1.49 µm/N (1290 Hz). housing on the spindle ram. The maximum compliance in X and Y direction is 1.42 m/N (1285 Hz) The mode inducing the peak vibration of the cutter is mainly associated with the bending vibration of and 1.49 m/N (1290 Hz). The mode inducing the peak vibration of the cutter is mainly associated the spindle shaft. The other apparent lower frequency below 200 Hz mainly with the bending vibration of the modes spindle occurring shaft. The at other apparent modes occurring at are lower caused by the vibration of the spindle ram and movable column. frequency below 200 Hz are mainly caused by the vibration of the spindle ram and movable column. Frequency reponse functions at tool end 1.40
FRFs, Real part [um/N]
FRFs, Amplitude [um/N]
1.60 1.20 1.00
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0.80
FRF in Y-axis
0.60 0.40 0.20 0.00 0
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800 1000 1200 1400 1600 1800 2000
Frequency reponse functions at tool end
1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80
FRF in X-axis FRF in Y-axis
0
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Frequency, (Hz)
(a)
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(b)
Figure 8. Comparisons of the predicted frequency response functions at tool end of milling machine
Figure 8. Comparisons of the predicted frequency response functions at tool end of milling machine modelin X and Y directions. (a) Amplitude of the FRFs, (b) Real part of the FRFs. modelin X and Y directions. (a) Amplitude of the FRFs, (b) Real part of the FRFs.
4.4. Machining Stability
4.4. Machining Stability
To get insight into the machining performance of the milling machine at design stage, the
To get insight into was the assessed machining performance of the milling machine at design stage, machining stability based on frequency response functions of the tool tip. The the machining stability was assessed based on frequency response functions of the tool tip. The machining machining stability of the milling machine was predicted using the analytical model developed in study In their machine approach, was the predicted time‐varying force coefficient of model the dynamic milling stability of [28]. the milling using the analytical developed inprocess study [28]. model was approximated by Fourier‐series components. The stability relationship between the was In their approach, the time-varying force coefficient of the dynamic milling process model chatter‐free axial cutting depths (Z min) and the spindle speed (n) in end‐mill operation as follows. approximated by Fourier-series components. The stability relationship between the chatter-free axial The speed‐dependent transfer function H(jw) representing the ratio of the Fourier transform of cutting depths (Zmin ) and the spindle speed (n) in end-mill operation as follows. the displacement at the tool tip over the dynamic cutting force can be expressed as H(jw) = Re(w) + The speed-dependent transfer function H(jw) representing the ratio of the Fourier transform of the jIm(w), where Re and Im are, respectively, the real and imaginary parts of the transfer function of the displacement at the tool tip over the dynamic cutting force can be expressed as H(jw) = Re (w) + jIm (w), spindle tool tip. The limit cutting depth Z min for stable machining at spindle speed n is defined as where Re and Im are, respectively, the real and imaginary parts of the transfer function of the spindle 2πRe I tool tip. The limit cutting depth Zmin forZstable speed n is defined as (8) (1 mat)2spindle machining min
n
Zmin 60ω c N (2kπ )
NKt
Re
´2πRe Im 2 “ p1 ` q Re , NKt
(8)
(9)
60ω1c , k lobes(0,1, 2...) 2 tan ` ( Iφq n“π Np2kπ m / Re) (9) ´1 pI {R q k “ lobesp0, 1, 2...q φ “ π ´ 2tan In the above equation, Kt is the cutting resistance coefficients in the tangential direction to the m e cutter, N is the number of cutter teeth, and k is the lobe number. The machining stability was
In the above equation, Kt is the cutting resistance coefficients in the tangential direction to the calculated based on the vibration modes that cause the tool to deform greatly. Therefore, according cutter, N is the number of cutter teeth, and k is the lobe number. The machining stability was calculated to the tool end FRFs, the apparent peak vibrations at lower and high frequency were selected for basedprediction of the machining stability. The analysis of machining stability was used to evaluate how on the vibration modes that cause the tool to deform greatly. Therefore, according to the tool end FRFs, the apparent peak vibrations lower and high frequency were selected for prediction of the the linear guide preload will affect at the dynamic behavior and machining. Considering that the machining stability. The analysis of machining stability was used to evaluate how the linear guide yawing and twisting modes cause the tool to deform greatly and the two modes occur aligning in X and Y directions, therefore, the machining stability in X and Y directions are calculated, respectively. preload will affect the dynamic behavior and machining. Considering that the yawing and twisting modes cause the tool to deform greatly and the two modes occur aligning in X and Y directions, therefore, the machining stability in X and Y directions are calculated, respectively. In this way, the
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coupled effect of the dominant vibration modes was ignored when calculating the stability lobe curves. In this way, the coupled effect of the dominant vibration modes was ignored when calculating the Appl. Sci. 2016, 6, 76 A two-tooth carbide cutter was employed to machine the stock material of Al7075 alloys and9 of 13 titanium stability lobe curves. A two‐tooth carbide cutter was employed to machine the stock material of alloysIn this way, the coupled effect of the dominant vibration modes was ignored when calculating the Ti-6Al-4V alloys at high alloys and low speed machining. The low cutting resistance coefficients were Al7075 alloys and titanium Ti‐6Al‐4V alloys at high and speed machining. The cutting 2 of for Al7075 alloys and K = 2000 N/mm2 for Ti6Al4V alloys [28,29]. calibrated as K = 796 N/mm stability lobe curves. A two‐tooth carbide cutter was employed to machine the stock material of 2 t t resistance coefficients were calibrated as Kt = 796 N/mm of for Al7075 alloys and Kt = 2000 N/mm2 Al7075 alloys and titanium alloys Ti‐6Al‐4V alloys at low speed machining. The cutting Figure 9 presents the stability lobes in the X and Yhigh and directions predicted at the vibration mode of for Ti6Al4V alloys [28,29]. 2 of for Al7075 alloys and Kt = 2000 N/mm2 resistance coefficients were calibrated as K t = 796 N/mm Figure 9 presents the stability lobes in the X and Y directions predicted at the vibration mode of machine tool structure, respectively. The limited depth is 4.2 mm in the X-direction and 10.1 mm in the for Ti6Al4V alloys [28,29]. machine tool structure, respectively. The limited depth is 4.2 mm in the X‐direction and 10.1 mm in Y-direction, respectively. Comparison of the results of stability analysis indicates that the machine tool the Figure 9 presents the stability lobes in the X and Y directions predicted at the vibration mode of Y‐direction, Comparison of the stability analysis indicates the has a different limitedrespectively. depth for stable machining in results X and of Y directions when the cutter that is operated machine tool structure, respectively. The limited depth is 4.2 mm in the X‐direction and 10.1 mm in machine tool has a different limited depth for stable machining in X and Y directions when the under lower speed. As found from the predicted tool end FRFs of milling machine, the results of the Y‐direction, respectively. Comparison of the results of stability analysis indicates that the cutter is operated under lower speed. As found from the predicted tool end FRFs of milling machine, stability analysis alsoa indicate thedepth dynamic characteristic ofin the milling machinewhen is greater machine tool has different that limited for stable machining X and Y directions the in the results of stability analysis also indicate that the dynamic characteristic of the milling machine is the Ycutter is operated under lower speed. As found from the predicted tool end FRFs of milling machine, direction than in the X direction, which further bring the higher limited depth in Y direction greater in the Y direction than in the X direction, which further bring the higher limited depth in Y compared with the X direction. This result gives a suggestion for the improvement of the structure of the results of stability analysis also indicate that the dynamic characteristic of the milling machine is direction compared with the X direction. This result gives a suggestion for the improvement of the the milling machine. greater in the Y direction than in the X direction, which further bring the higher limited depth in Y structure of the milling machine. direction compared with the X direction. This result gives a suggestion for the improvement of the Stability lobes diagram in Y-direction Stability lobes diagram in X-direction structure of the milling machine. 16 9
Stability lobes diagram in X-direction
10 8 97
1412
64 53 3
Stability lobes diagram in Y-direction
1614
86 75
42
Cutting depth, (mm) Cutting depth, (mm)
Cutting depth, (mm) Cutting depth, (mm)
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Figure 9. Predicted stability lobe diagrams in X and Y directions based on the vibration mode associated
(a) lobe diagrams in X and Y directions based on the(b) Figure 9. Predicted stability vibration mode associated with the machine tool structure. (a) Stability lobes in X direction, (b) Stability lobes in Y direction. with Figure 9. Predicted stability lobe diagrams in X and Y directions based on the vibration mode associated the machine tool structure. (a) Stability lobes in X direction, (b) Stability lobes in Y direction. with the machine tool structure. (a) Stability lobes in X direction, (b) Stability lobes in Y direction. On the other hand, the predicted stability lobes diagrams based of the vibration mode of
spindle structure are illustrated in Figure 10. As can be seen, the limited depth for high speed stable On the other hand, the predicted stability lobes diagrams based of the vibration mode of spindle the other hand, the predicted stability lobes diagrams based of the vibration mode of machining in X direction is about 6.02 mm, almost equal to that in Y direction. This is an expected structure On are illustrated in Figure 10. As can be seen, the limited depth for high speed stable machining spindle structure are illustrated in Figure 10. As can be seen, the limited depth for high speed stable phenomenon since this mm, stability is calculated based on direction. the vibration associated phenomenon with the in X direction is about 6.02 almost equal to that in Y Thismode is an expected machining in X direction is about 6.02 mm, almost equal to that in Y direction. This is an expected bending motion of the spindle shaft. sincephenomenon this stability since is calculated based on the vibration mode associated with the bending motion of the this stability is calculated based on the vibration mode associated with the spindle shaft. Stability lobes diagram in X-direction- spindle mode Stability lobes diagram in Y-direction spindle mode bending motion of the spindle shaft. 10
10 Stability lobes diagram in Y-direction spindle mode
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C u ttin g d e p th , (m m ) C u ttin g d e p th , (m m )
C utting de pth, (m m ) C utting de pth, (m m )
Stability lobes diagram in X-direction- spindle mode
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Figure 10. Predicted stability lobe diagrams in X and Y directions based on the vibration mode associated (a) (b) with the spindle structure. (a) Stability lobes in X direction, (b) Stability lobes in Y direction. Figure 10. Predicted stability lobe diagrams in X and Y directions based on the vibration mode associated Figure 10. Predicted stability lobe diagrams in X and Y directions based on the vibration mode with the spindle structure. (a) Stability lobes in X direction, (b) Stability lobes in Y direction. In order to clarify the effect of linear guide preload, the dynamic response of the milling
associated with the spindle structure. (a) Stability lobes in X direction, (b) Stability lobes in Y direction. machine was also assessed from harmonic analysis in which the contact stiffness of the linear guide In order to clarify the effect of linear guide preload, the dynamic response of the milling modulus was assumed as 850 N/μm, a higher value corresponding to medium preload. As shown in machine was also assessed from harmonic analysis in which the contact stiffness of the linear guide In order to clarify the effect of linear guide preload, the dynamic response of the milling machine Figure 11, the dynamic compliances at lower frequency are affected to vary with the change of the modulus was assumed as 850 N/μm, a higher value corresponding to medium preload. As shown in was also assessed from harmonic analysis in which the contact stiffness of the linear guide modulus Figure 11, the dynamic was assumed as 850 N/µm,compliances at lower frequency a higher value correspondingare affected to vary with the change of the to medium preload. As shown in Figure 11,
the dynamic compliances at lower frequency are affected to vary with the change of the preload of
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linear guides on the X and Y axis. The maximum compliance in X direction is 0.58 and 0.61 µm/N preload of linear guides on the X and Y axis. The maximum compliance in X direction is 0.58 and preload of linear guides on the X and Y axis. The maximum compliance in X direction is 0.58 and for the0.61 linear set at medium and low preload; while the maximum compliance in Y direction is 0.61 guide m/N for the linear guide set at medium and low preload; while the maximum compliance in Y m/N for the linear guide set at medium and low preload; while the maximum compliance in Y 0.325 direction is 0.325 µm/N (167 Hz)and 0.352 µm/N (131 Hz). direction is 0.325 m/N (167 Hz) and 0.352 m/N (131 Hz). m/N (167 Hz) and 0.352 m/N (131 Hz).
(a)
(b)
(a) (b) Figure 11. Comparisons of the tool end FRFs predicted for milling machine with different preloaded Figure 11. Comparisons of the tool end FRFs predicted for milling machine with different preloaded linear guides mounted on spindle head and movable column. (a) FRFs in X direction, (b) FRFs in Y linearFigure 11. Comparisons of the tool end FRFs predicted for milling machine with different preloaded guides mounted on spindle head and movable column. (a) FRFs in X direction, (b) FRFs in direction. linear guides mounted on spindle head and movable column. (a) FRFs in X direction, (b) FRFs in Y Y direction. direction. The influence of the linear guide preload on the machining stability can be observed from
Figure 12, which presents the stability lobes in the X and Y directions. This also compares the
The influence of the linear guide preload stability can observed from The influence of the linear guide preload on on the the machining machining stability can be be observed from difference of the limited cutting depth when the linear guides are preloaded at different amounts. FigureFigure 12, which presents thethe stability lobes Y directions. directions.This This also compares 12, which presents stability lobes inin the the X X and and Y also compares the the The limited cutting depth for stable machining in X‐direction is 4.2 and 5.6 mm for linear guides difference of the limited cutting depth when the linear guides are preloaded at different amounts. difference of the limited cutting depth when the linear guides are preloaded at different amounts. with low and medium preload, respectively and the limited depth in Y‐direction is 10.1 and 10.6 The limited cutting depth stable machining in X‐direction is 4.2 and 5.6 mm for linear guides mm, cutting respectively. Comparison of the results in from stability analysis indicates that for the linear milling guides The limited depth for for stable machining X-direction is 4.2 and 5.6 mm with low and medium preload, respectively and the limited depth in Y‐direction is 10.1 and 10.6 10.6 mm, machine has a different limited cutting depth for stable machining in X and Y directions, which are with low and medium preload, respectively and the limited depth in Y-direction is 10.1 and mm, respectively. Comparison of the results from stability analysis indicates that the milling also affected to change with the preload of the linear guides in spindle head. As found, the milling respectively. Comparison of the results from stability analysis indicates that the milling machine has a machine with high preloaded linear guides has a larger cutting depth as compared to that with machine has a different limited cutting depth for stable machining in X and Y directions, which are differentlower preloaded guides. limited cutting depth for stable machining in X and Y directions, which are also affected to also affected to change with the preload of the linear guides in spindle head. As found, the milling change with the preload of the linear guides in spindle head.cutting As found, the machine machine with high preloaded linear guides has a larger depth as milling compared to that with with high Stability lobes diagram in X-direction Stability lobes diagram in Y-direction preloaded linear guides has a larger cutting depth as compared to that with lower preloaded guides. 10 lower preloaded guides. 8 Cutting depth, (mm)
10
Stability lobes diagram in X-direction
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Figure 12. Comparisons of the stability lobe diagram predicted at the vibration mode associated with machine tool structure with different preloaded guideways. (a) Stability lobes in X direction, (b) Stability lobes in Y direction.
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5. Conclusions This study employed the finite element modeling technology to investigate the machining stability of a horizontal milling machine with a movable column on hybrid guideway mechanism. Currently, a number of studies based on finite element methods have been employed to analyze the dynamic behavior of a machine tool with different configurations. As mentioned above, a realistic model should be created to reflect the various characteristics of different components such as the linear guide system and spindle bearing system. In this study, a full FE model of the milling machine with a hybrid guide system was created in this way. Apart from the machine frame, the linear components such as linear roller guides were also simulated by solid elements and interface elements at the sliding/rolling interfaces. These interfaces were respectively characterized by appropriate contact properties, which were determined based on associated mathematical models according to their preloaded status. A solid spindle-bearing model, which was experimentally verified by means of physical spindle, was also incorporated into the machine frame model. The full FE model is different from the rigid-flexible coupling models in which reduced FE model of the main modules were coupled at the connection joints with spring elements. The coupled FE model provides a relevant approximation of the dynamic properties and less vibration modes because of the reduced degree of freedoms in each flexible body. Whereas the full FE machine model can offer accurate visualizations on the dynamic behaviors of the spindle system and machine structure. According to the finite element simulation, the fundamental vibration modes relating to the spindle module are dominated by the feeding mechanisms of the movable column and the spindle ram. The frequency response of the milling tool shows two apparent characteristics, including the lower modes dominated by the yawing vibration of the spindle ram coupled with the twisting vibration of the movable vertical column, and the higher modes associated with the bending of the spindle shaft. The machining stability based on the machine structure mode allows the cutter to operate with high axial depth under lower spindle speed, while the machining stability based on spindle structure mode is valid for a high speed machine with less cutting depth. In addition, in this case the horizontal spindle head with low preloaded linear guides show a lower cutting depth for stable machining than that with higher preloaded guides. As a conclusion, the realizations on the dynamic behaviors of a milling machine based on the proposed FE model can provide valuable design improvements of the machine structure or guideway system of the machine tool. Acknowledgments: We gratefully acknowledge the support for this work provided by National Science Council in Taiwan through project number MOST103-2221-E-167-004. Author Contributions: Jui-Pin Hung contributed to the organization of the research work as well as the analysis of the machine system and manuscript preparation. Wei-Zhu Linand Yong-Jun Chen contributed to the design and analysis of the machine structure. Tzou-Lung Luo conducted the experimental work of the spindle systems. Conflicts of Interest: The authors declare no conflict of interest.
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