## 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.

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## Data availability

The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

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## Funding

This work is supported by Key R&D Projects of the Ministry of Science and Technology of China (Grant Nos. 2017YFA0701200 and 2018YFB1107600), National Natural Science Foundation of China (Grant No. 51775237), and Graduate Innovation Fund of Jilin University (Grant No. 101832020CX122).

## Author information

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### Contributions

P.N., S.J., J.Z., and J.L. contributed to the conception of the study; P.N., S.J., and J.L. studied the surface topography prediction model; P.N., J.Z, and J.L. performed a surface topography simulation of an umbra surface to verify the proposed model; H.D. and J.L. performed the machining experiment and were in charge of data management; P.N., S.J., and J.L. analyzed the experimental results and concluded the whole work; P.N. wrote the original draft; S.J. reviewed and edited the manuscript; J.Z. supervised all works; The supporting projects for this study were administered by P.N., S.J., J.Z., and H.D.

### Corresponding authors

## Ethics declarations

### Ethics approval

The research did not involve human participants or animals.

### Consent to participate

The research did not involve human participants or animals.

### Consent to publish

We would like to submit the enclosed manuscript entitled “Simulation and experiment on surface topography of complex surface in single point diamond turning based on determined tool path” to *The International Journal of Advanced Manufacturing Technology* to be considered for publication. No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. We would like to declare on behalf of my coauthors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.

### Competing interests

The authors declare no conflict of interest.

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## Appendix

### 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:

According to Eq. (2), the polar coordinates of *P*_{k} is given by 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:

In the light of Eq. (4), the angle coordinate *θ*_{kl} of point *P*_{k} can be indicated as 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:

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:

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:

After the above calculation, the coordinate of grid point *P*_{k} was obtained as (1, 1, −0.0174).

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### Cite this article

Ning, P., Zhao, J., Ji, S. *et al.* Simulation and experiment on surface topography of complex surface in single point diamond turning based on determined tool path.
*Int J Adv Manuf Technol* **113, **2555–2562 (2021). https://doi.org/10.1007/s00170-021-06671-w

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### Keywords

- Surface topography simulation
- Tool path
- Slow tool servo (STS)
- Single point diamond turning (SPDT)
- Machining error