# Trochoidal Milling Path with Variable Feed. Application to the Machining of a Ti-6Al-4V Part

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## Abstract

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## 1. Introduction

_{e}and a small axial depth of cut a

_{p}, or the removal of a small chip width and a large axial cut, Figure 1.

_{p}as much as possible, due to its limited influence [2] in tool wear and surface quality [3]. The reduction of a

_{e}produces a decrease of the chip width, reducing heat generation and improving tool wear.

_{p}are equal to triple the tool diameter, while a

_{e}is almost 0.06D, which can be considered as finishing conditions, allowing cutting speeds over 90 m/min. The decrease of tool wear facilitates the possibility of using sharper cutting edges (less cutting force). Those cutting edges are micro-rounded in order to raise their robustness, Figure 2.

_{p}× a

_{e}), the number of passes required to complete the machining would be the same. Nevertheless, if the chip width decreases there are new advantages:

- The tooth gap can be smaller for the same feed (the chip width is reduced), so more teeth can be implemented and, for the same feed per tooth, the final feed rises and, as a result, the machining time decreases.
- As a consequence, the mill core has a wider section, being able to support bigger flection and torque forces, Figure 3. This bigger rigidity makes possible to decrease deformation and vibrations, being more suitable for materials with superior cutting requirements [4], such as titanium and nickel alloys.

## 2. Materials and Methods

#### 2.1. Selection of the Radial Cut Depth

_{e}and h are clearly related, as shown in Figure 4, considering the maximum chip thickness and approximating the CB chord to the tangent in B. This approximation is valid because the feed per tooth, f

_{z}, is small in comparison to the mill dimensions. In this case, the feed per tooth, f’

_{z}, on the milling surface, is equal to the feed per tooth, f

_{z}, in the centre of the mill:

- Considering that a
_{e}is small in comparison to the milling tool radius r_{m}. - Avoiding the use of the squared parenthesis. According to the previous aspect, it is not significant compared to the other equation terms.
- Using the milling tool diameter instead of the radius.

_{z}, which is required to determine the feed:

_{z}= f

_{z}′.

_{e}as constant as possible.

_{eff}, which has a much higher value, compared to a

_{e}. Furthermore, the feed per tooth and, therefore, the feed in (7) are different in the centre of the milling tool, which is the point to be programmed in the numeric control (NC) machine, and in the most external point (highest cut radius), as shown in Figure 5.

- For the same feed in the milled part, the programmed feed must be lower in the interior tool path.
- According to Figure 5 and Equations (2) and (8), for the interior tool path, the engagement angle is higher for the same radial depth, a
_{e}, as a_{eff}is larger than a_{e}. - As the value of the feed per tooth is low in comparison to the milling tool dimensions, the arch of fz′ is close to a straight line. Thus, the mean thickness can be approximated to the previous case, considering a
_{eff}instead of a_{e}:$${h}_{m}={f}_{z}{}^{\prime}\sqrt{\frac{{a}_{eff}}{D}}$$ - As a
_{e}is constant, the values of the engagement angle and the effective axial depth are constant. However, they are not constant in trochoidal milling.

- The mean chip depth should not be decreased. For this reason, in this study, to let h
_{m}be constant when a_{eff}varies, the feed per tooth fz′ will be continuously modified.

_{z}was selected according to the tool manufacturer’s technical advice in relation to the straight-line peripheral milling for surface finishing. With that value in Equation (6), h

_{m}can be obtained. This value makes it possible to obtain f

_{z}′ in Equation (10). The instantaneous feed per tooth in the centre of the milling tool can be obtained with (9) for each instantaneous a

_{e}.

_{e}is not constant and, consequently, neither is a

_{eff}. For this reason, the engagement angle θ

_{e}is variable.

_{emax}.

_{eff}.

_{2}HT

_{1}, with $r=b-{r}_{m}$:

_{eff}. To do so, there are two possibilities: using analytic geometry or trigonometry. In both cases, the coordinate origin is placed in O

_{2}.

_{1}and in H, T

_{1}is obtained.

_{2}and H, T

_{2}can be obtained:

_{2}and T

_{2}, intersected with the line which contains T

_{1}and is perpendicular to the previous one:

_{2}can be obtained:

_{1}HT

_{1}, the cosine theorem is applied:

_{1}are determined as follows:

_{1}T

_{2}of the triangle T

_{1}HT

_{2}:

_{1}HT

_{2}:

#### 2.2. NC Program of a Trochoid with Adaptive Feed

_{m}= 12 mm, a

_{emax}= 0.5 mm. The maximum value of the effective radial depth is 1.32 mm, and the corresponding ωt angle is 65.3°.

_{z}, the number of teeth z, the cutting speed V

_{c}and the milling tool diameter D

_{m}. The rotation frequency of the milling tool is:

_{max}.

_{f}, which provides a different v for each φ.

- The points that define a step of the trochoidal path were obtained using (36).
- ω was obtained from (30), using the feed f
_{z}for peripheral milling. - Introducing the previous values of Δφ and ω in (37), Δt is obtained.
- With (33), v is obtained.
- Using (34), the coordinates of the trochoidal arc (0 to π) were obtained, with Δφ.
- For each of the previous coordinates (x, y), the effective chip width was obtained (27).
- The mean value of chip width was obtained from (6), for the peripheral milling, with a
_{e}= a_{emax}. - With (10), the value of f’
_{z}can be found. - As the cut is interior, Equation (11) makes it possible to find the corrected f
_{z}for each point. When a_{effmax}is reached, f_{z}is at the minimum, being maximum in the trochoid limits (Figure 11). - Finally, rotation (38) and translation are applied to the points obtained in the previous step. This step is described below.

- The cost of a cylinder (∅ = 183 mm) of Ti-6Al-4V was significantly lower than a rectangular plate. In fact, the local provider had a leftover, which made it more affordable.

_{i}, y

_{i}) belongs to the arch of one of the trochoidal steps (Figure 15). A rotation is applied to this point:

_{s}, y

_{s}), where the tangent was obtained.

_{s}, y

_{s}) are:

_{emax}) gives the number of the generated trochoidal arches. Several expressions can be used to define the length of the spiral arch. Applying differential calculus:

_{m}, obtained with (43).

#### 2.3. Practical Development

- The substitution of angle φ
_{e}(46). - The substitution of the length increment Δl
_{m}by the value of a_{emax}divided by the value obtained from (48).

## 3. Results

_{effma}

_{x}and to decrease the value or φ

_{aeffmax}.

_{eff}and the milling displacement angle with increments of 5°.

_{eff}more stable, being necessary to optimize its parameters [30,31]. However, the circular trochoidal path was tested, with the objective of studying the feed per tooth variation to maintain the chip thickness constant.

_{e}= 0.5 mm and p = 35 mm. With these conditions, the spiral path was milled several times with three similar machines (Figure 17).

^{TM}).

## 4. Discussion

_{eff}stays more stable, although its value is higher, decreasing the feed per tooth. In contrast, the 0° elliptical path starts with a high value of a

_{eff}, making it necessary to decrease the feed per tooth, although it could be increased at the exit.

_{eff}is not high, obtaining its maximum value before tracing the centre of the semitrochoid. The feed per tooth has a high value at the entry and decreases until a

_{effmax}and, after it, its value increases to the exit. From the exit to the following entry, the feed value can be significantly increased.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

García-Hernández, C.; Garde-Barace, J.-J.; Valdivia-Sánchez, J.-J.; Ubieto-Artur, P.; Bueno-Pérez, J.-A.; Cano-Álvarez, B.; Alcázar-Sánchez, M.-Á.; Valdivia-Calvo, F.; Ponz-Cuenca, R.; Huertas-Talón, J.-L.;
et al. Trochoidal Milling Path with Variable Feed. Application to the Machining of a Ti-6Al-4V Part. *Mathematics* **2021**, *9*, 2701.
https://doi.org/10.3390/math9212701

**AMA Style**

García-Hernández C, Garde-Barace J-J, Valdivia-Sánchez J-J, Ubieto-Artur P, Bueno-Pérez J-A, Cano-Álvarez B, Alcázar-Sánchez M-Á, Valdivia-Calvo F, Ponz-Cuenca R, Huertas-Talón J-L,
et al. Trochoidal Milling Path with Variable Feed. Application to the Machining of a Ti-6Al-4V Part. *Mathematics*. 2021; 9(21):2701.
https://doi.org/10.3390/math9212701

**Chicago/Turabian Style**

García-Hernández, César, Juan-José Garde-Barace, Juan-Jesús Valdivia-Sánchez, Pedro Ubieto-Artur, José-Antonio Bueno-Pérez, Basilio Cano-Álvarez, Miguel-Ángel Alcázar-Sánchez, Francisco Valdivia-Calvo, Rubén Ponz-Cuenca, José-Luis Huertas-Talón,
and et al. 2021. "Trochoidal Milling Path with Variable Feed. Application to the Machining of a Ti-6Al-4V Part" *Mathematics* 9, no. 21: 2701.
https://doi.org/10.3390/math9212701