Thread milling errors

The study shows that thread milling (including internal threads) is one of the most forward-looking technologies that offers max thread cutting performance. The reasons are the shortest cut-in path equal to the thread depth, and the full thread profile machining time equal to the period of one revolution of the workpiece. The paper highlights that thread milling is an efficient, general-purpose machining process. In this study, we solved the thread profile error estimation problem. Thread milling with parallel workpiece and cutter axis was investigated with a milling simulation model. The optimal thread milling tool parameters for internal thread machining were found. The method efficiency was proven by measuring undercuts and overcuts in pipe thread milling. For instance, the thread error (overcuts/undercuts measured flatwise to the thread profile) diametral compensation does not exceed 0.0268 mm while the pipe thread tolerance G1.25`` is 0.36 mm.


Introduction
Thread milling (including internal threads) is one of the most forward-looking technologies that offer max thread cutting performance [1...7].
Standard CNC thread milling cycles can be used. It is preferable to develop an NC program with CCS (an application that estimates the cutting mode and selects the machining strategy.) The study objective is reducing efforts, milling time and cost through investigating the thread milling process and measuring the undercuts and overcuts in special buttress threading on thin-walled pipes for selecting the most suitable cutter.
We used simulation to estimate the undercuts and overcuts.

Research method
The simulation and under-(over-)cuts calculation were made to achieve the aim. Other researchers tried to solve this problem analytically [8]. The solution was two-dimensional, while shaping is a three-dimensional process. It was noted in [10]: "The difference between the 3D part model and the 3D reference model is the undercut (overcut) volume." In this paper, we identified errors of radiator nipple pipe thread milling.
The final stage of the simulation is creating a 3D model of the thread milling errors. The overcuts were generated by subtracting the resulting part model from the reference model (figure 1); for the undercuts, the reference model was subtracted from the resulting part 3D model ( To find the max value of the overcuts or undercuts (the max distance between the resulting thread surface and reference one) we used the "Surface deviation" tool in the CCS software.
With the surface deviation results we traced the max deviation point, then cut a section at this point, and estimated the errors. Then the simulation model was used in a series of experiments to find the most suitable cutter in terms of minimizing the shape errors. Afterwards, we generated a fitting criterion function.
The simulation objective was minimizing the thread geometry errors. The geometric errors have two components: thread profile overcuts and undercuts. In real applications, one error component is sometimes emphasized. We considered all the components. The objective function is the sum of two components: The function arguments are the cutter radius u R and the number of teeth z. Hereinafter, we will denote the variables as follows: where  is the    f domain,  is the max acceptable thread geometry error.
We used a numerical method for finding the multi-valuable function extremum to find the max value of the two-argument function    f .
A solution to the problem is generating a sequence   ..., , , k , function values are a decreasing convergent sequence.
These sequence generating methods are called 'descent methods' [10]: where k P is the descending vector from In this case, it is impossible to estimate the partial derivatives (there is no analytical relation), so we replaced them with finite differences [3]: The function value is decreased every time the distance increment increases, that is: ( As a result, the iterative process meeting the condition (3) is as follows. Two initial approximations are specified: Table 1.
The distance increment 1  is found. It is constant for subsequent iterations. Eq. (2) is estimated, and the condition (3) is checked at every iteration.
If the condition is met, the same distance increment is used for the following iterations. Otherwise, the increment is decreased until the condition is met. The process stops when the following condition is met: ( The key milling variable is the engagement angle between the cutter and the blank  . Refer to Table 2 for the results. The diagrams based on the table are shown in figure 3.    Table 3 for the initial data used to find the    f error function maximum value. The lateral undercut is 0.29 mm. That is, such cutter dimensions produce unacceptable machining results. Refer to figure 4 for the part model machined as described above.

Discussion
As can be seen from the simulation results presented in In this case, the thread profile has linearity errors. For the simulation, we measured these errors flatwise to the nominal thread profile. Then we found a factor to recalculate the overcut values measured flatwise into the actual thread profile error values. These errors are recalculated by the mean diameter as follows: Refer to figure 5 for the relation between the root faceting, the cutter radius, and the number of teeth. With this relation, one can assess the impact of the cutter radius and the number of teeth on the root facet formation, so the process planner can choose the optimal tool parameters on an ad hoc basis.

Conclusion
The 13 th iteration simulation results (the cutter diameter is 100 mm and the number of teeth is 32) are the best for the pipe thread  25 1, G milling. What we found is: milling simplifies the manufacturing process. Moreover, it requires a fewer number of tools, the machining time is reduced, while the manufacturing becomes more flexible and efficient.
With the presented thread milling simulation, the optimal cutter parameters for the specific thread milling can be found.
With a larger cutter radius, more teeth can be fitted, and it minimizes the errors (to 0.0268 mm if the number of teeth is 32.) The error is within the average diameter tolerance equal to 0.36 mm.
Thread milling is even more efficient for cutting oppositely directed threads like radiator nipples.