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ABSTRACT
In the present work, a mechanistic model of cutting forces is developed with a novel approach to arrive at the cutting edge geometry as well as the cutting mechanics. The geometry of cutting elements derived and verified using a virtual tool generated in CAD environment is considered. The cutting and edge force coefficients at every discrete point on the cutting edge of micro-ball end mill are established in a novel way from the basic metal cutting principles and fundamental properties of materials, considering edge radius and material strengthening effects. Further, measured edge radius is used in the model. Full slot micro-ball end milling experiments are conducted on a high-precision high-speed machining center using a 0.4 mm diameter tungsten carbide tool and cutting forces are measured using a high-sensitive piezo-electric dynamometer. It is established that the predicted as well as experimental cutting forces are higher at very low uncut chip thickness in comparison with the cutting edge radius in micro-ball end milling also. Amplitudes of cutting forces and instantaneous values with incremental rotation of the tool are compared with predicted values over two revolutions for validation of proposed model.
Nomenclature
| dFpj | = | Cutting forces in radial (p = r), tangential (p = t) and axial (p = a) direction, N |
| dFqj, | = | Cutting forces (q = x, y, z) in Cartesian co-ordinate system, N |
| j | = | Tooth index |
| ψ | = | Angular position of a point on cutting edge, radian |
| κ | = | Axial immersion angle, radian |
| Krc, Ktc, Kac | = | Cutting force coefficients in radial, tangential and axial direction, N/mm2 |
| Kre, Kte, Kae | = | Edge force coefficients in radial, tangential and axial direction, N/mm |
| tc | = | Uncut chip thickness, mm |
| db | = | Uncut chip width, mm |
| f | = | Feed rate, mm/min |
| N | = | Rotational speed of micro-ball end mill, rpm |
| nf | = | No of flutes |
| ϕp | = | Tooth spacing or pitch angle, radian |
| δ | = | Lag angle, radian |
| δmax | = | Maximum lag angle, radian |
| θ | = | Tool rotation angle, radian |
| αc | = | Global helix angle or helix angle on cylindrical portion, radian |
| α | = | Local helix angle or helix angle on ball portion, radian |
| da | = | Axial immersion depth, mm |
| z | = | Position of cutting edge element along Z-axis from ball tip, mm |
| R(z) | = | Radius of the disc containing elemental cutting edge about Z-axis, mm |
| r | = | Radius of ball end mill, mm |
| ψl, ψu | = | Upper and lower integration limits, radian |
| τ | = | Shear strength of workpiece material, MPa |
| ϕn | = | Normal shear angle, radian |
| βn | = | Normal friction angle, radian |
| γn | = | Normal rake angle, radian |
| λ | = | Inclination angle, radian |
| η | = | Chip flow angle, radian |
| γr | = | Radial rake angle, radian |
| re | = | Cutting edge radius, mm |
| γef | = | Effective rake angle, radian |
| τy | = | Shear yield strength of workpiece, MPa |
| po | = | Normal pressure at tool tip, MPa |
| ρ | = | Exponential constant for pressure distribution |
| µ | = | Sliding friction coefficient |
| θo | = | Stagnation angle, radian |
| Cm | = | Machining constant |
| σre | = | Reference flow stress, MPa |
| A | = | Johnson–Cook (J–C) material strength coefficient for yield strength, MPa |
| B | = | J-C material strength coefficient for hardening modulus, MPa |
| ϵ | = | Equivalent plastic strain |
| n | = | J–C hardening coefficient |
| C1 | = | J–C material strength coefficient for strain rate sensitivity |
| ἐ | = | Equivalent plastic strain rate, s−1 |
| ἐ0 | = | Reference plastic strain rate, s−1 ( =1 s−1) |
| T | = | Working temperature, °C |
| Tr | = | Reference (room) temperature, °C |
| Tm | = | Melting temperature, °C |
| m | = | J–C thermal softening coefficient |
| K | = | Proportion of primary shear zone |
| ht | = | Primary shear zone thickness, mm |
| Vc | = | Cutting velocity, mm/s |
| l | = | Material length scale, mm |
| σ | = | Flow stress, MPa |
| ηsg | = | Effective strain gradient, mm−1 |
| αm | = | Empirical constant |
| G | = | Shear modulus, MPa |
| b | = | Magnitude of burger vector, mm |
| FqM | = | Predicted cutting forces in feed (q = x) and transverse (q = y) directions, N |
| FqE | = | Experimentally obtained cutting forces in feed (q = x) and transverse (q = y) directions, N |
| Fqa | = | Amplitude of cutting forces in (q = x) and transverse (q = y) directions, N |