FE simulation, analytical prediction, and experimentation of cutting force in longitudinal vibration-assisted milling (LVAM) during Ti-6Al-4 V cutting

Abstract

In longitudinal vibration-assisted milling (LVAM), the tool flute is retracted from the bottom of the machined surface along the axial/longitudinal direction. Nevertheless, there is still a limitation in the knowledge of LVAM during Ti-6Al-4 V cutting regarding the cutting forces pattern in analysis using finite element (FE) simulations and analytical prediction comparatively. This study developed an FE model of the cutting force using ABAQUS and a mathematical model of the cutting force using MATLAB. The FE model was developed based on the Johnson–Cook plastic deformation and the Johnson–Cook damage model. The analytical prediction was developed based on the fluctuated undeformed axial chip thickness and the influence of the longitudinal vibration on the undeformed chip thickness. According to this comparative study, the analytical prediction of the cutting force coded in MATLAB has remarkable similarity with the FE simulation model cutting force. The experimental cutting forces were also carried out to validate both the FE simulation model and the mathematical model. According to the comparison results in the LVAM, the FE simulation model has higher quantitative error than that in the mathematical model, with the highest error in the mathematical model being about 7.146% and the highest error in the numerical model being about 9.913%. In addition, a comparative study between LVAM and CM in terms of the experimental cutting temperature, machined surface morphology, and chip morphology was also carried out. In evidence, LVAM provides better machining performance than CM.

Appendix

 

Fig. 28

The undeformed axial chip thickness $z\left(\phi \right)$ in LVAM

According to Eq. 10, the undeformed axial chip thickness $z\left(\phi \right)$ can be derived from Eq. 16. Therefore, $z\left({\phi }_{i}\right)= \frac{1}{-{\Gamma }_{\beta }}\left(-{\phi }_{i}+i{\phi }_{p}+{\phi }_{st}\right).$

CASE 1:

In case 1, the undeformed axial chip thickness $z\left(\phi \right)$ is clearly defined by ${z}_{1}$ according to Fig. 28. Therefore, Eq. 32 is the same as Eq. 17.

$${z}_{1}=z-{z}_{v}= \frac{1}{-{\Gamma }_{\beta }}\left(-{\phi }_{i}+i{\phi }_{p}+{\phi }_{st}\right)- {z}_{v}$$

(32)

CASE 2:

In case 2, the undeformed axial chip thickness $z\left(\phi \right)$ is defined by ${z}_{2}$ according to Fig. 28. Therefore, Eq. 18 in this article can be derived as follows:

$${z}_{2}={A}_{d}-z-{z}_{v}={A}_{d}- \frac{1}{-{\Gamma }_{\beta }}\left(-{\phi }_{i}+i{\phi }_{p}+{\phi }_{st}\right)- {z}_{v}$$

(33)

where, ${{\phi }_{ex}-\phi }_{st}={\Gamma }_{\beta }\cdot {A}_{d}$, Eq. 33 becomes Eq. 36. And it is like Eq. 18 in this article.

$${z}_{2}={A}_{d}- \frac{1}{-{\Gamma }_{\beta }}\left(-{\phi }_{i}+i{\phi }_{p}+{\phi }_{ex}-{\Gamma }_{\beta }\cdot {A}_{d}\right)- {z}_{v}$$

(34)

$${z}_{2}={A}_{d}-(\frac{{\phi }_{i}}{{\Gamma }_{\beta }}-\frac{i{\phi }_{p}}{{\Gamma }_{\beta }}-\frac{{\phi }_{ex}}{{\Gamma }_{\beta }}+{A}_{d})- {z}_{v}$$

(35)

$${z}_{2}= \frac{1}{{\Gamma }_{\beta }}\left(-{\phi }_{i}+i{\phi }_{p}+{\phi }_{ex}\right)- {z}_{v}$$

(36)

where, ${z}_{v}\left(t\right)= \frac{{a}_{z}}{2}\cdot \mathrm{sin}\left(2\pi f.t\right)+\frac{{a}_{z}}{2}$. $t= \frac{{\phi }_{i \bullet 60}}{2\pi {\omega }_{N}}$ by converting the time to immersion angle (${\phi }_{i})$ which is in radian.

$${z}_{v}\left({\phi }_{i}\right)= \frac{{a}_{z}}{2}\cdot \mathrm{sin}\left(f\cdot \left(\frac{{\phi }_{i}\cdot 60}{{\omega }_{N}}\right)\right)+\frac{{a}_{z}}{2}$$

(37)