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Full life-cycle cutting force prediction in ball helical milling based on oblique cutting analysis

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Abstract

In order to predict the change of cutting force in ball helical milling of titanium alloy, considering the kinematics of helical milling and the geometric model of a ball end cutter, the cutting process is divided into three stages, cutting-in, steady, and cutting-out stages, and the shape and volume of undeformed chip in different stages are described. Then, according to the oblique cutting theory and the principle of maximum shear stress, the constraint relationships among the parameters such as friction angle, shear angle, and shear stress in the micro-element are established. Thus, based on the position parameters at different stages, a three-direction full life-cycle cutting force model is presented, and the cutting force coefficients are identified and characterized as Weibull functions of instantaneous undeformed chip thickness. Finally, the predicted cutting forces are compared with the experimental values. The results show that the average and maximum errors are less than 19% which verify the accuracy of the model.

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Abbreviations

\({n}_{\text{rot}}\text{(rpm)}\) :

Rotation speed

\({n}_{\text{orb}}\text{(rpm)}\) :

Orbital revolution speed

\({v}_{\text{t}}\text{, }{v}_{\text{a}}\text{(m/min)}\) :

Tangential and axial feed speeds

\({f}_{\text{t}}\text{, }{f}_{\text{a}}\text{(mm/t)}\) :

Tangential and axial feed rates per tooth

\({R}_{\text{h}}\text{,}{R}_{\text{t}}\text{(mm)}\) :

Radii of hole and tool

N :

Tooth number of the tool

\(\gamma ({\text{radian}})\) :

Helix angle of the tool trajectory

\({a}_{\text{o}}\text{(mm/orb)}\) :

Axial depth of cut per orbital revolution

\(i\text{ (radian)}\) :

Helix angle of the tool

z(mm):

Axial height of point P

\({z}_{1}\text{,}{z}_{2}\text{(mm)}\) :

Axial depths of cut at time t and t-torb

\({R}_{\text{s}}(z)({\text{mm}})\) :

Radial distance of point P

\(\kappa (z)({\text{radian}})\) :

Axial immersion angle of point P

\(\psi (z)({\text{radian}})\) :

Radial lag angle of point P

\({\xi }_{\text{j}}(z),{\theta }_{\text{j}}(z)({\text{radian}})\) :

Radial position angles of point P in \(o-xyz\text{ and }a-{x}_{\text{a}}{y}_{\text{a}}{z}_{\text{a}}\)

\(t({\text{s}})\) :

Tool movement time

\(e({\text{mm}})\) :

Eccentricity

\(\zeta ({\text{radian}})\) :

Tool orbital revolution angle

\({\zeta }_{0}({\text{radian}})\) :

Initial angle of tool orbital revolution angle

\({F}_{\text{X}}\text{,}{F}_{\text{Y}}\text{,}{F}_{\text{Z}}\text{(N)}\) :

Cutting forces in the x, y, z directions

\({H}_{\text{t}}\text{(mm)}\) :

Workpiece thickness

\({\text{d}}S({\text{mm}})\) :

Discrete cutting arc length

\({\text{d}}b({\text{mm}})\) :

Projection length of the micro-element along the cutting speed direction

\({h}_{\text{j}}({\text{mm}})\) :

Instantaneous undeformed chip thickness

\({F}_{\text{d}}(t)({\text{N}})\) :

Indentation force

\(\Delta S({\text{mm}}^{2})\) :

Cross-section area of the material to be cut

\({K}_{\text{db}}{\text{(N/mm}}^{2}\text{)}\) :

Indentation force coefficient

\({K}_{\text{tc}},{K}_{\text{rc}},{K}_{\text{ac}}({\text{N/mm}}^{2})\) :

Shear force coefficients in tangential,radial and axial directions

\({K}_{\text{te}},{K}_{\text{re}},{K}_{\text{ae}}(\text{N/mm})\) :

Edge force coefficients in tangential,radial and axial directions

\(\alpha ,{\alpha }_{\text{n}}({\text{radian}})\) :

Rake and normal rake angles

\({\beta }_{\text{i}},{\beta }_{\text{n}}\text{(radian)}\) :

Friction and normal friction angles

\({\varphi }_{\text{i}},{\varphi }_{\text{n}}({\text{radian}})\) :

Shear and normal shear angles

\(\tau ({\text{MPa}})\) :

Shear stress

\({S}_{\text{s}}{\text{(mm}}^{2}\text{)}\) :

Shear area

\(\eta ({\text{radian}})\) :

Chip flow angle

\({r}_{\text{e}}({\text{mm}}^{-3})\) :

Edge radius of the tool

\({\text{d}}{F}_{\text{j,z}}\text{,d}{F}_{\text{s,j,z}}\text{,d}{F}_{\text{f, j, z}}\text{,d}{F}_{\text{n, j, z}}\text{(N)}\) :

Resultant,shear,friction, and normal micro forces applied at the chip by the flute \({j}_{\text{th}}\) at an axial depth of z

\({\text{d}}{F}_{\text{t,j,z}},{\text{d}}{F}_{\text{r,j,z}},{\text{d}}{F}_{\text{a,j,z}}\text{(N)}\) :

Tangential,radial and axial micro forces for the flute j at an axial depth of z

\({\text{d}}{F}_{\text{tn,j,z}},{\text{d}}{F}_{\text{rn,j,z}},{\text{d}}{F}_{\text{an,j,z}}\text{(N)}\) :

Projections of d \({F}_{\text{t,j,z}},{\text{d}}{F}_{\text{r,j,z}},\) and \({\text{d}}{F}_{\text{a,j,z}}\) on normal plane

\({A}_{1}\text{,}{A}_{2}\text{,}{A}_{3}\text{,}T\) :

Transformation matrixs

\({\phi }_{\text{i}}\text{(radian)}\) :

Angle between \({\text{d}}{F}_{\text{j,z}}\) and the normal plane

\({\phi }_{\text{n}}\text{(radian)}\) :

Angle between the projection of \({\text{d}}{F}_{\text{j,z}}\) on the normal plane and the cutting plane

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Funding

This project has received funding from the Natural Science Foundation of Hebei Province, China (Grant No. E2020501014).

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  1. School of Control Engineering, Northeastern University at Qinhuangdao, Hebei Province, Qinhuangdao, 066004, China

    Zhitong Zhou & Haiyan Wang

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  1. Zhitong Zhou
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Zhou, Z., Wang, H. Full life-cycle cutting force prediction in ball helical milling based on oblique cutting analysis. Int J Adv Manuf Technol 124, 1623–1638 (2023). https://doi.org/10.1007/s00170-022-10402-0

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