## 1 Introduction
### Trochoidal Milling
#### 1.2.2 Analysis of Trochoidal Milling  
The term trochoidal milling is based on the mathematical definition which defines the trochoidal path as a curve created by point fixed to a circle rolling along a line. In the machining application, trochoidal milling is referred to as machining where the tool does not move linearly, and the cutting angle is constant. Our research is focused on the first case where the trochoid axis is primarily linear, as discussed below. A jump in technological development allowed for the precise setting of the minimum angle of cut of the cutting tool, which significantly reduces tool vibration. Reduction of the engagement angle is particularly advantageous in the side milling of grooves and pockets where the position of the milling cutting tool relative to the workpiece is easily changeable, and also when machining very hard materials, where relatively large frictional forces are produced, resulting in considerable warming. The centre of the tool moves directionally in the desired direction, with the milling machine rotating; a combination of both types of motion forms a curve, a cyclic spiral with a conventionally fixed name--trochoid (Figure 1) [11]. The milling cutting tool takes the material off in relatively thin layers, but at a relatively fast feed rate; therefore, it is relatively less loaded with considerable performance. The effect of small radial cutting forces increases the stability of the cut, i.e., there is a substantial decrease in pushing the tool away from to the workpiece as a response, and in the vibration. While increasing the cutting speed and reducing chip thickness, the heat generated at the cutting point is reduced, so it is possible to increase the cutting depth as needed. These positively changed milling conditions ensure that the tool and the machine are more wear-resistant, exhibiting a much longer service life, and the machining process thus becomes very effective and desirable in many new applications [5; 19]. The essential knowledge, especially on theoretical issues of trochoidal machining, was already applied in the last century. The technical term \"trochoidal milling\" is now commonly used conventionally, despite certain deviations of the real movement of the milling cutting tool compared to mathematically predicted trochoidal trajectories [23] manifesting themselves mainly as smaller spaces at the beginning of the cut and more significant gaps at the end of the cut (Figure 2). In Figure 2, the outer edges of the trochoidal trajectories can be compared in smaller and larger feeds, and the deviation between the two courses is recorded. In real milling, it is possible to get closer to the continuously smooth outer edge of the cut by increasing the rotary speed of the milling cutting tool, reducing its feed rate, taking a very thin layer of material and applying the knowledge of the dynamics of the machining process [20]. In the analysis of the trochoidal milling model [23], cutting forces were predicted for peripheral milling of circular corner profiles with the main focus on the geometry of the tool and the workpiece, the  
Figure 1: Geometric representation of the tool path [11], where: \(R_{T}\) is the radius of the milling cutting tool; \(R_{P}\) is the radius of the milling cutting circle. Figure 2: Geometric representation of the trochoid and circular tool paths, where: \(R_{T}\) is the radius of the milling cutting tool; \(R_{P}\) is the radius of the trochoid circle. position of the tool relative to the workpiece and the material properties of the workpiece. In cooperation with analytical and numerical solutions, chip formation was developed for precise dynamic modelling in circular finishing milling, taking into account tooth movement and uneven chip thickness [24; 25]. The dynamic model of circular finish milling was also determined by analysing time changes of input and output stroke angles in various machining zones [6; 26], thereby laying the foundation for further tool stability analysis with the trochoid motion of the cutting tool.