Simulation and experiment on surface topography of complex surface in single point diamond turning based on determined tool path

Abstract

Surface topography is an important element that reflects the machining quality of workpieces and the functional performance of components. To avoid expensive cutting experiments, forecasting the surface topography before actual machining is necessary. Surface topography of diamond turned surfaces is directly affected by generated tool path in ultraprecision single point diamond turning (SPDT). In this study, a novel surface topography modeling method was presented. The planned tool path based on active machining precision control was considered in this method. Simulation and experiment of an umbrella surface diamond turning were conducted to testify the usability of the proposed model. Through comparing the simulation and experimental data of surface topography and machining error, the results showed that the proposed model can be applied to forecast the surface topography and machining error in SPDT using the generated tool path.

Appendix

Since i = 500, j = 500, L_x = L_y = 8mm, d_x = d_y = 0.01mm, based on Eq. (1) the XY coordinates can be calculated as A.1:

$$ \left\{\begin{array}{c}{X}_k=500\times 0.01-\frac{8}{2}=1\\ {}{Y}_k=500\times 0.01-\frac{8}{2}=1\end{array}\right. $$

(A.1)

According to Eq. (2), the polar coordinates of P_k is given by A.2:

$$ \left\{\begin{array}{c}{\rho}_k=\sqrt{1^2+{1}^2}=\sqrt{2}\\ {}{\theta}_k=\arctan \left(\frac{1}{1}\right)=\frac{\pi }{4}\end{array}\right. $$

(A.2)

As presented in Fig. 1, to calculate the Z coordinate of P_k, the CCPs closest to P_k on two adjacent circles of the planned tool path should be determined firstly. Since f = 0.0629 mm/r, on the basis of Eq. (3), the number of entire cycle N_t before point P_k can be obtained by A.3:

$$ {N}_t=\left\lfloor \sqrt{2}/0.0629\right\rfloor =22 $$

(A.3)

In the light of Eq. (4), the angle coordinate θ_kl of point P_k can be indicated as A.4:

$$ {\theta}_{kl}=\frac{\pi }{4}+22\times 2\pi =139.0155 $$

(A.4)

The planned tool path of the umbrella surface workpiece machining is given in Fig. 3. According to the coordinates of P_k (ρ_k, θ_kl) calculated above, the coordinates of CCPs that closest to P_k were determined as P_l1(1.4047, 139.0058, −0.0189) and P_l2(1.4680, 145.2767,   − 0.0216). To calculate the Z coordinate of P_k, it is necessary to ascertain the cutting positions on the planned tool path that enables the tool to cut to point P_k. As shown in Fig. 1, the cutting position may be between two adjacent CCPs, so the cutting sections [P_l1,P_l1 + 1] and [P_l2,P_l2 + 1] should be further confirmed. It can be obtained from Fig. 3 that the coordinates of the other CCPs in the cutting sections [P_l1,P_l1 + 1] and [P_l2,P_l2 + 1] are P_l1 + 1(1.4052,  139.0591,   − 0.0116) and P_l2 + 1(1.4686,  145.3286,   − 0.0140). Because there is a small straight line between the two adjacent CCPs on the planned tool path, the coordinates of the cutting positions P_kl1 and P_kl2 can be calculated by the interpolation method expressed in Eq. (5), as shown in A.5:

$$ \left\{\begin{array}{c}{\theta}_{kl1}=139.0155\\ {}{\rho}_{kl1}=1.4047-\left[\left(1.4047-1.4052\right)\times \frac{139.0058-139.0155}{139.0058-139.0591}\right]=1.3984\\ {}{z}_{kl1}=-0.0189-\left[\left(-0.0189+0.0116\right)\times \frac{139.0058-139.0155}{139.0058-139.0591}\right]=-0.0176\end{array}\right. $$

$$ \left\{\begin{array}{c}{\theta}_{kl2}=139.0155\\ {}{\rho}_{kl2}=1.4680-\left[\left(1.4680-1.4686\right)\times \frac{145.2767-139.0155}{145.2767-145.32861}\right]=1.3984\\ {}{z}_{kl2}=-0.0216-\left[\left(-0.0216+0.0140\right)\times \frac{145.2767-139.0155}{145.2767-145.32861}\right]=-0.0184\end{array}\right. $$

(A.5)

The coordinates of CLPs corresponding to the cutting position can be deduced and calculated as P_kl1 ′ (1.3984, 139.0155, 0.4884) and P_kl2 ′ (1.4619, 139.0155, 0.4876) using the same way. According to Eq. (6), the residual heights z_l1 and z_l2 generated at point P_k when the tool was cutting at those two cutting positions can be calculated by A.6:

$$ {z}_{l1}=0.4884-\sqrt{0.506^2-{\left(\sqrt{2}-1.3984\right)}^2}=-0.0174 $$

$$ {z}_{l2}=0.4876-\sqrt{0.506^2-{\left(\sqrt{2}-1.4619\right)}^2}=-0.0161 $$

(A.6)

From Eq. (7), the minimum value of the two residual heights was z coordinate of the grid point P_k, as given by A.7:

$$ {z}_k=\min \left(-0.0174,-0.0161\right)=-0.0174 $$

(A.7)

After the above calculation, the coordinate of grid point P_k was obtained as (1, 1, −0.0174).