# Obtaining Various Shapes of Machined Surface Using a Tool with a Multi-Insert Cutting Edge
## 2 Methods and Materials  
### Universal Form Tool with a Multi-Insert Profile  
The design of the universal form tool presented in this paper is based on the concept of variable tool configuration, which is achieved by changing the cutting-edge elements and their geometry [39; 42; 43]. Multi-edge tools are constructed with the ability to change their geometrical characteristics and this feature distinguishes them from conventional fixed geometry tools. This concept is referred to as geometric and kinematic flexibility (GKF) of the tool [42]. [PERSON]'s solution is an example of implementation [40; 41].  
New tool designs that incorporate the GKF concept could potentially replace many current designs available on the market. The GKF tool concept is based on the assumption that the outline of an object consists of simple single profiles. When those profiles are combined in different configurations, new surface profiles can be generated. Examples of the surface division of axisymmetric and planar objects into single profiles are presented in Figure 1a,b, respectively.  
In traditional forming tools, the entire profile is a single tool, limiting its use to a single profile (see Figure 1c). This principle applies to both axisymmetric and planar surfaces.  
The cutting edges in the concept of the universal profile tool are assembled from severalsing multi-inserts, to create the desired profile, as indicated in Figure 1d. The modular design concept of the cutting edge can be processed by the same set of single inserts for the machining of a wide class of surface contours. For example, assuming a single application of the single insert with a different profile number, \(N\), of different machined surface profiles can be determined as the variances without repetition:  
\[N=\frac{n!}{(n-i)!} \tag{1}\]  
where \(n\) is the number of multi-inserts, and \(i\) is the number of surfaces.  
Table 1 shows the number of different machined surface made from single inserts based on Equation (1). Calculation results indicate the possibility of creating many different surface profiles using a limited number of edges as shown in Table 1.  
In the case of form tools for turning, the cutting edges should have one rake surface and in the case of milling tools, shaped edges should be arranged on the circumference of the tool, to avoid tool vibration and shock during simultaneous operation of all cutting edges.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & & \\ \hline \multirow{4}{*}{**Number of Surfaces, \(i\)**} & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \cline{2-9}  & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \cline{2-9}  & 2 & - & 2 & 6 & 12 & 20 & 30 & 42 & 56 \\ \cline{2-9}  & 3 & - & - & 6 & 24 & 60 & 120 & 210 & 336 \\ \cline{2-9}  & 4 & - & - & - & 24 & 120 & 360 & 840 & 1680 \\ \cline{2-9}  & 5 & - & - & - & - & 120 & 720 & 2520 & 6720 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Number \(N\) of different profile of machined surface calculated using Equation (1).  
Figure 1: An example of (**a**) an axisymmetric contoured surface, (**b**) a planar contoured surface, (**c**) a single edge profile, (**d**) a profile combining from another multi-insert.