Investigation on Innovative Dynamic Cutting Force Modelling in Micro-milling and Its Experimental Validation
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Abstract
In this paper, an innovative cutting force modelling concept is presented by modelling cutting forces against micro-cutting processes such as micro-milling, ultraprecision turning and abrasive micromachining, and also taking account of micro-cutting dynamics. The modelling represents the underlying micro-cutting mechanics and physics in micro-milling in an innovative multi-scale manner, i.e. the specific cutting force at the unit length, unit area and unit volume by considering the size effect, cutting fracture energy, the material modulus, and the cutting heat and temperature partition. A novel instantaneous chip thickness algorithm is introduced to analyse the real chip thickness by taking account of the effects of the micro-tool geometry change brought up by the tool run-out and further contribute to the force model through a numerical iterative algorithm. The measured cutting forces are compensated by a Kalman filter to achieve the accurate cutting forces. This is further utilized to calibrate the model coefficients using least square method. The cutting force modelling is evaluated and validated through well-designed micro-milling trials, which can be used for optimizing the cutting process and tool cutting performance in particular.
Keywords
Cutting force modelling Micro-milling Instantaneous chip thickness Multi-scale modelling Micro-cutting mechanics Specific cutting forceList of symbols
- A_{p}
Ploughing area between the tool and workpiece (μm^{2})
- e
Energy consumption on unit volume (J/mm^{3})
- f
Feed rate (µm/min)
- f_{t}
Feed per tooth (μm)
- f′
Feed rate (µm/s)
- F_{lx}(t, k), F_{ly}(t, k)
Cutting force at unit length (N)
- F_{sx}(t, k), F_{sy}(t, k)
Cutting force on unit area (N)
- F_{x}(t, k), F_{y}(t, k)
Orthogonal cutting force and thrust force (N)
- h(t, k)
Actual chip thickness (μm)
- h(θ_{i}(z))
Uncut chip thickness (μm)
- k
Tooth number of micro-mill /
- K_{rp}, K_{tp}
Ploughing coefficients in radial and tangential directions (N/mm^{2})
- K_{re}, K_{te}
Edge constants in radial and tangential directions (N/mm^{2})
- K_{rc}, K_{tc}
Cutting force coefficients in radial and tangential directions (N/mm^{2})
- l
Total length of cutting flute engaged with material (μm)
- N
Number of tool teeth /
- r
Run-out (μm)
- R_{k}
Actual tool radius (μm)
- s
Cross section of the chip generated in micro-milling (μm^{2})
- t
Time point of milling process (s)
- t_{1}, t_{2}
Time delay between two adjacent cutting tooth acting at the same position on the workpiece (s)
- V
Volume of material being removed in micro-milling (μm^{3})
- Z
Axial height of flute (μm)
- α
Run-out location angle of actual tool centre (rad)
- β
Helix angle of cutting tools (°)
- γ_{c}
Change of rake angle (°)
- w
Angular velocity (rad/s)
- θ
Position angle of kth tooth (°)
- θ_{p}
Position angle of (k − 1)th tooth (°)
- \( \theta_{\text{en}}^{k} \), \( \theta_{\text{ex}}^{k} \)
Actual tool entry and exit angles of the kth tool tooth (°)
- θ_{en}
Engage entry angle of cutting flute (°)
- θ_{ex}
Disengage exit angle of cutting flute (°)
1 Introduction
Cutting force modelling and analysis, as an important process indicator in micro-milling, can collectively reflect the various cutting process phenomena and dynamics such as size effect, chip formation, energy consumption and cutting heat partition, and the machining instability and chatter. It can also be correlated with the tool cutting performance particularly with the tool wear and tool life. Therefore, cutting force is seen as a key formulation to optimize the cutting process variables and tool geometries in micro-cutting processes. A number of cutting force models are proposed and studies for micro-cutting processes [1, 2, 3, 4, 5, 6, 7, 8], which falls in four categories, i.e. analytical modelling [9, 10, 11, 12], numerical modelling [7, 8, 13], empirical modelling [14, 15, 16, 17] and hybrid modelling (combining the strengths of previous three modelling approaches). Davoudinejad et al. [18] proposed a 3D FEM for studying the cutting forces in full slot end milling process. The simulation and experimental results indicated that the effect of the run-out phenomenon was visible in the cutting force results and different cutting action could be significantly observed for the two teeth. He also investigated the influence of the worn tool affected by built-up edge (BUE) on micro-end milling process performance via FEM, which demonstrates that the predicted micro-milling cutting forces resulted affected by BUE with different teeth engagements [19]. Zhang et al. [20] proposed an analytical model by considering the minimum chip thickness effect, tool run-out (axial offset and tilt offset) and trochoidal trajectory to determine the cutting forces. These were seen as the most related influence factors in the force model and validated through experiments. Matsumura and Tamura [21] developed an analytical model to evaluate the effect of the cutter run-out on the cutting force in order to perform the milling operation properly. In addition, the experimental results from Kang et al. [22] presented that the effect of cutting edge radius on cutting forces was significant in the micro-scale milling. However, the modelling approaches and techniques developed so far are almost all focused on obtaining the absolute value of cutting forces. In ultraprecision and micro-cutting processes, the cutting forces are at the 0.1–1 N scale compared to that in conventional machining normally being at 100–1000 N scale. The accurate measurement and analysis of their absolute values are challenging and often not applicable particularly in-process and taking account of the cutting dynamics as normally required in industrial applications. In addition, the cutting process phenomenon above mentioned cannot be appropriately reflected as cutting force down to such a small scale. Moreover, most researches focus on the effects of cutting edge, tool run-out and dynamic modulation in cutting force models, while the effect of feed rate on the time delay and actual chip thickness is always ignored. As a result, micro-cutting force modelling by considering the instantaneous chip thickness and cutting dynamics comprehensively is still less understood. Therefore, dynamic cutting force modelling and understanding the dynamic aspects of the cutting force are becoming essentially important for micro-cutting particularly for developing the relevant in-process algorithms and analytics for smart micro-machining.
This paper presents an innovative dynamic cutting force model by considering instantaneous chip thickness in micro-milling and further investigates the scientific understanding of the relevant micro-cutting mechanics and the process dynamics. A comprehensive cutting force modelling, representing the micro-milling forces at the unit length, unit area and further at the unit volume, is proposed in order to establish scientific understanding of the underlying micro-cutting mechanics and physics in a multi-scale manner. This innovative modelling is expected to be industrial feasible and realistic compared to the existing models, and also to take account of the size effect, chip formation, tool wear mechanism and the cutting temperature partition, etc. in a quantitative analysis manner while with physical engineering meanings. The approach is evaluated and validated through well-designed experimental trials, which will likely help the micro-milling process optimization with the application to industrial micro-manufacturing.
2 Chip Thickness Modelling in Micro-milling
2.1 Analytical Chip Thickness
However, in micro-milling with the scaled-down cutting variables, the function in Eq. (1) is not appropriate or accurate due to the real movement of the cutting tool centre is not taken into account. The tool diameter normally falls less than 1 mm even down to 100 μm, the feed rate usually ranges from 0.1 to 10 μm and the extremely small cutting edge radius cannot be ignored which means the chip thickness calculation in each revolution is even complex in micro-milling process. In addition, dynamic run-out of cutting tool normally ranges from 1 to 5 μm which is extremely tiny in conventional milling. However, the ratio of run-out to tool diameter becomes more significant in micro-milling. Thus, the run-out of tools needs to be considered in order to achieve a precise chip thickness prediction in each revolution. Bao and Tansel [9] proposed an analytical cutting force model which firstly considered the tool run-out effects. Zaman et al. [12] developed a model which determined the theoretical chip area at any specific angular position of the cutting tool tip. Li et al. [23] presented a new algorithm to calculate the actual chip thickness considering the trochoidal trajectory of tool tip with run-out and the minimum chip thickness (MCT) effects.
However, these methods and approaches mentioned above only considered the tool run-out in the machine coordinate system and overlooked the change of cutting tool geometry caused by the run-out of cutting tool. This study proposes a new algorithm to accurately calculate the chip thickness based on the real trajectory of tool tip in the workpiece coordinate system and the actual chip thickness is determined by comparing the nominal chip thickness with MCT at each calculation interval. As a result, this model simulates discontinuous chip formation at particular locations.
However, the tool actually rotates about spindle centre O and the rotational radius of each tool tip has changed that represented by red line in Fig. 1. Given the nominal radius of cutting tool, tool run-out and its location angle, the following expression of actual tool radius based on laws of cosines is obtained.
2.2 Actual Chip Thickness
3 Innovative Dynamic Cutting Force Modelling in Micro-milling
3.1 Orthogonal Cutting Force Modelling
Due to the cutting edge radius cannot be ignored, the cutting process in micro-milling is clearly divided into two distinct regimes. One is ploughing-dominant regime where material deforms plastically but no chips are formed; the other is shearing-dominant regime where material is removed with chips formation. The material splits and separates under the round cutting edge. As a result, the upper part forms the chip and the lower part flows beneath the tool and forms the machined surface.
The above cutting coefficients are related to the friction and pressure at the interface between tool and workpiece. These coefficients need to be calibrated against the model based on the experimental data. This can be achieved by curve fitting the cutting forces into the model and using least square methods to find the optimal coefficients.
It should be noted that as the chip thickness changes periodically, both ploughing-dominant cutting and shearing-dominant cutting will take place in each revolution. Thus, the cutting forces should be calculated using both Eqs. (15) and (16).
3.2 The Proposed Innovative Approach to Cutting Force Modelling
The afore-described modelling technique [6] can be used to predict the cutting forces. However, previous research and experiments show that cutting force in micro-machining is quite small down to the 0.1–1 N scale in magnitude. The direct usage of absolute cutting force imposes the technical challenge in accurate prediction of the micro-cutting forces particularly in the micro-cutting process. Furthermore, it is essential in developing the cutting force model for bridging the gaps between understanding the micro-cutting mechanics and the process optimization together with cutting performance enhancement. Therefore, an innovative cutting force modelling is proposed to provide quantitative analysis into micro-milling mechanics and the cutting process, which is further presented in detail below.
3.2.1 Cutting Force at Unit Length
The cutting force at the unit length can be used to predict the burrs formation in micro-milling and likely render the foundation for the process optimization particularly in avoiding the burrs formation along the edges of micro-milled surface.
3.2.2 Cutting Force at Unit Area
The cutting force on unit area can be closely linked with the Young’s modulus of the workpiece material, which can provide insightful information on the chip formation and breakage, surface generation and even tool wears at both rake and flank surfaces of cutting tool [25].
3.2.3 Cutting Force at Unit Volume
The energy consumption on unit volume can be used to compute the cutting heat partition in the micro-milling tool, chips and workpiece. Scientific understanding and quantitative determination of the heat partition ratio in micro-milling are essentially important, as it can likely provide further quantitative analysis on the tool wear mechanism and cutting performance in the process.
4 Experimental Evaluation and Validation
4.1 Cutting Trials Design and Experimental Set-up
Cutting process parameters used in micro-milling experiments
Tool type | ϕ0.4 mm diamond tool | ϕ0.4 mm tungsten carbide tool | ϕ1 mm tungsten carbide tool | ϕ1 mm tungsten carbide tool |
---|---|---|---|---|
Number of flutes | 1 straight | 1 straight | 1 helix | 2 helix |
Feed rate (μm/tooth) | 0.2, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0 | 0.2, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0 | 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 8.0, 10.0, 12.0, 14.0 | 0.5,1.0, 2.0, 3.0, 4.0, 6.0, 9.0, 12.0 |
Spindle speed (k, rpm) | 12, 21, 30 | 12, 21, 30 | 12, 15, 18, 21, 24, 27, 30 | 12, 15, 18, 21, 24, 27, 30 |
Depth of cut (μm) | 20 | 20 | 20 | 20 |
4.2 Measurement of the Tool Run-out
The run-out of cutting tool and its position angle is measured at the tool tip. Various factors can contribute to the run-out magnitude such as manufacturing error, alignment error and tool dynamics. Previous research and experiments show that the most significant run-out error is introduced by the alignment error. This mainly consists of the parallel tool offset and tilt error. For most cases in the experiments, the tilt error has extremely small effect on the tool run-out as the small cutting depth adopted in cutting trials. Thus, the tool run-out error measurement is mainly focused on the tool parallel offset and its position angle.
4.3 Cutting Force Model Calibration and Validation
4.3.1 Parameters Calibration
Cutting force model coefficients for different cutting regimes
Material | Cutting regime | K_{rc} (N/mm^{2}) | K_{re} (N/mm) | K_{tc} (N/mm^{2}) | K_{te} (N/mm) |
2476 | 5.44 | 1808 | 6.05 | ||
Aluminium 6082-T6 | Ploughing regime | K_{rp} (N/mm^{3}) | K_{tp} (N/mm^{3}) | ||
1150 | 1480 |
4.3.2 Model Validation
4.4 Model Interpretation of the Machining Process
It can be found that the magnitude of specific cutting force increases along with feed rate. The fluctuations are due to material impurity, tool vibration and different chip formation mechanism. It should be noted that the magnitude at proximity of 180° is higher than that at 0°, which is shown in both plots. This is probably due to the reason of intact chips being removed by the tool at 180°, while the tool just engages with material at 0° and doesn’t remove any material until chips are formed.
As for straight flute cutting tools, the cutting force on unit area and energy consumption on unit volume of material are the same from the point of mathematical computation. Figure 11 also indicates the stress exerted on the tool, at around 180°, the stress rises up dramatically from 50 to 500 GPa. Comparing with the major properties of tungsten carbide tool with shear modulus of 274 GPa, tensile strength of 344 MPa and compression strength of 2.7 GPa, it can be indicated that although the force magnitude is small, the stresses that the tool is exposed to are extremely large. This can be used to explain the tool wear in micro-milling process. The constituent elements of tool are gradually removed in every revolution.
5 Conclusions
- 1.
An innovative cutting force modelling approach is presented for computing the instantaneous chip thickness while taking account of the micro-milling tool run-out. It is found that the tool run-out has significant effects on the actual tool radius and cutting geometry in micro-milling. Furthermore, the MCT is also included in the associated new algorithm supported by computational procedures.
- 2.
The cutting force model for the micro-milling process is developed in a multi-scale manner, including the cutting force at the unit length, specific cutting force at the unit area, and further cutting force or cutting energy consumption in the unit volume of materials being machined. This model is of significance in likely leading to better scientific understanding of micro-milling mechanics, which will help render the in-depth analysis of the sharp edge quality (burrs and voids) in micro-milling, chip formation, surface generation and micro-cutting heat and temperature partition.
- 3.
Experimental cutting trials are conducted to validate the modelling approach and the cutting force model. The method to determine the tool run-out is introduced experimentally. For accurate measurement of cutting forces, a Kalman filter is developed and applied to compensate the distortion in cutting force signals due to the dynamic transmission characteristics of the tooling-workpiece system. Based on the compensated cutting forces, the cutting force model is calibrated and proved with the good agreement with experimental trials.
- 4.
The cutting force modelling presented can be applied to scientifically interpret and better understand the underlying micro-cutting mechanics in the micro-milling process particularly linking to burrs-free machining, chip formation and surface generation, and the tool cutting performance and tool wear. Furthermore, the modelling and the associated analytics can likely be used to gauge the cutting process in real time and thus able to simulate the dynamic surface generation by combining a virtual micro-milling system and digital twin in the future.
Notes
Acknowledgements
The authors would like to thank the Korean Institute of Materials and Machinery (KIMM) for partial funding support of this research (Grant No. R31063).
References
- 1.Vogler MP, DeVor RE, Kapoor SG (2003) Microstructure-level force prediction model for micro-milling of multi-phase materials. Trans J Manuf Sci Eng 125(2):202–209CrossRefGoogle Scholar
- 2.Vogler MP, Kapoor SG, DeVor RE (2005) On the modeling and analysis of machining performance in micro-endmilling, part II: cutting force prediction. Trans ASME J Manuf Sci Eng 126(4):695–705CrossRefGoogle Scholar
- 3.Jun MB, DeVor RE, Kapoor SG (2006) Investigation of the dynamics of microend milling—part II: model validation and interpretation. Trans J Manuf Sci Eng 128(4):901–912CrossRefGoogle Scholar
- 4.Cheng K, Huo D (eds) (2013) Micro cutting: fundamentals and applications. Wiley, ChichesterGoogle Scholar
- 5.Bissacco G, Hansen HN, Slunsky J (2008) Modelling the cutting edge radius size effect for force prediction in micro milling. CIRP Ann Manuf Technol 57(1):113–116CrossRefGoogle Scholar
- 6.Malekian M, Park SS, Jun MB (2009) Modeling of dynamic micro-milling cutting forces. Int J Mach Tools Manuf 49(7):586–598CrossRefGoogle Scholar
- 7.Afazov S, Ratchev S, Segal J (2010) Modelling and simulation of micro-milling cutting forces. J Mater Process Technol 210(15):2154–2162CrossRefGoogle Scholar
- 8.Jin X, Altintas Y (2012) Prediction of micro-milling forces with finite element method. J Mater Process Technol 212(3):542–552CrossRefGoogle Scholar
- 9.Bao WY, Tansel IN (2000) Modeling micro-end-milling operations, part I: analytical cutting force model. Int J Mach Tools Manuf 40(15):2155–2173CrossRefGoogle Scholar
- 10.Liu X, DeVor R, Kapoor S (2006) An analytical model for the prediction of minimum chip thickness in micromachining. Trans J Manuf Sci Eng 128(2):474–481CrossRefGoogle Scholar
- 11.Wang JJ, Zheng C (2002) An analytical force model with shearing and ploughing mechanisms for end milling. Int J Mach Tools Manuf 42(7):761–771CrossRefGoogle Scholar
- 12.Zaman M, Kumar AS, Rahman M, Sreeram S (2006) A three-dimensional analytical cutting force model for micro end milling operation. Int J Mach Tools Manuf 46(3):353–366CrossRefGoogle Scholar
- 13.Özel T, Altan T (2000) Process simulation using finite element method—prediction of cutting forces, tool stresses and temperatures in high-speed flat end milling. Int J Mach Tools Manuf 40(5):713–738CrossRefGoogle Scholar
- 14.Huo D, Cheng K (2010) An experimental investigation on micro milling of OFHC copper using tungsten carbide, CVD and single crystal diamond micro tools. Proc IMechE Part B J Eng Manuf 224(B6):995–1003CrossRefGoogle Scholar
- 15.Kang I, Kim J, Kim J, Kang M, Seo Y (2007) A mechanistic model of cutting force in the micro end milling process. J Mater Process Technol 187:250–255CrossRefGoogle Scholar
- 16.Park S, Malekian M (2009) Mechanistic modeling and accurate measurement of micro end milling forces. CIRP Ann Manuf Technol 58(1):49–52CrossRefGoogle Scholar
- 17.Pérez H, Vizán A, Hernandez J, Guzmán M (2007) Estimation of cutting forces in micromilling through the determination of specific cutting pressures. J Mater Process Technol 190(1):18–22CrossRefGoogle Scholar
- 18.Davoudinejad A, Tosello G, Parenti P, Annoni M (2017) 3D finite element simulation of micro end-milling by considering the effect of tool run-out. Micromachines 8(6):187CrossRefGoogle Scholar
- 19.Davoudinejad A, Tosello G, Annoni M (2017) Influence of the worn tool affected by built-up edge (BUE) on micro end-milling process performance: a 3D finite element modeling investigation. Int J Precis Eng 18(10):1321–1332CrossRefGoogle Scholar
- 20.Zhang X, Yu T, Wang W (2018) Prediction of cutting forces and instantaneous tool deflection in micro end milling by considering tool run-out. Int J Mech Sci 136:124–133CrossRefGoogle Scholar
- 21.Matsumura T, Tamura S (2017) Cutting force model in milling with cutter runout. Procedia CIRP 58:566–571CrossRefGoogle Scholar
- 22.Kang I, Kim J, Seo Y (2011) Investigation of cutting force behaviour considering the effect of cutting edge radius in the micro-scale milling of AISI 1045 steel. Proc Inst Mech Eng Part B J Eng Manuf 225(2):163–171CrossRefGoogle Scholar
- 23.Li C, Lai X, Li H, Ni J (2007) Modeling of three-dimensional cutting forces in micro-end-milling. J Micromech Microeng 17(4):671–678CrossRefGoogle Scholar
- 24.Jiao F (2015) Investigation on micro-cutting mechanics with application to micro-milling. Ph.D. thesis, Brunel University LondonGoogle Scholar
- 25.Niu Z, Jiao F, Cheng K (2018) An innovative investigation on chip formation mechanisms in micro-milling using natural diamond and tungsten carbide tools. J Manuf Process 31(1):382–394CrossRefGoogle Scholar
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