## 1 Introduction
### Trochoidal Milling
#### 1.2.4 Cutting Forces in Trochoidal Milling  
The total machining force \(F\) is the action force the tool exerts on the workpiece and at which it is pressed into the machined material. The machined material exerts an equally large and inversely oriented resistive reaction force on the tool. The total machining force is the resultant one of the two components, namely, the active component \(F_{a}\) and the passive component \(F_{p}\), but in principle, it does not act in a simplified manner in the 2D space; it acts in the general direction and can be divided into three mutually perpendicular directions in the 3D space. The feed component \(F_{f}\) acts in the \(x\) axis; the normal component \(F_{fh}\) acts in the \(y\) axis as a normal to the feed component; and the passive component \(F_{p}\) acts in the \(z\) axis (Figure 5). The active force component \(F_{a}\) can be considered the result of the cutting component \(F_{c}\) and its normal \(F_{cn}\), and at the same time, as the result of the feed component \(F_{f}\) and its normal \(F_{fn}\). Machining force components can be measured indirectly; for example, from the power output of the machine electric motor or the torque of the power relative to the rotation axis [7; 30]. In the presented experiments, exact, direct measurements of the components of the total machining force were used using a stationary dynamometer, which measures the rectangular projections of the components of the forces \(F_{f}\), \(F_{fh}\) and \(F_{p}\) in the 3D coordinate system axes \(x\), \(y\) and \(z\). The active component of the force \(F_{a}\) is the cutting force, the largest and most important component of the machining force, acting in the general direction, so it cannot be measured directly by the dynamometer. However, its magnitude can be measured indirectly as the resultant of the feed component \(F_{f}\) (acting in the \(x\) axis direction) and the normal to the feed component \(F_{fh}\) (acting in the \(y\) axis direction). Knowledge of the active force component is a fundamental basis for calculating the stresses of the milling cutting tool and the workpiece chuck; and the energy balance of the machine tool. The component of the feed force \(F_{f}\) acts in the direction of the machine feed; its magnitude is mainly the basis for calculating the strength of the tool. The passive component of the force \(F_{p}\) is the smallest component of the machining power; in the case of trochoidal milling, the active force component is often more significant by orders of magnitude than the passive component, but the passive component has a high information value from the technological point of view. In principle, it pushes the tool out stroke with the workpiece, resulting in vibration during the machining process. The vibration is highly undesirable, primarily because it negatively affects the accuracy of the workpiece dimensions and the quality of the machined surface. Since the passive component is very small in the case of trochoidal machining, the cut is relatively stable.The presented results follow on from the current research of trochoidal milling and focus on the analysis of the course and character of the machining forces in the formation of the trochoid path using CAM software. Trochoidal milling is a dynamic machining method which, thanks to the use of the new CAM control systems, allows one to not only maintain constant cutting conditions (feed and angular speed of the cutting tool), but also the parameters of the  
Figure 5: Scheme of the components for total machining force \(F\): in 2D-plan (**a**) and 3D-side (**b**) views. trochoid, the step, the engagement angle, the mean chip thickness and the surface roughness of the workpiece. For each application and each tool, the cutting conditions and parameters of the trochoid must be optimized. The use of the trochoid grooving method places high demands on machine tools that must be able to provide sufficient acceleration to ensure the prescribed workpiece trajectory and have sufficient power to provide the requested speed.