## 2 Theoretical Model of the Cycloid Disc Loading  
This paper studies a classic single-stage cycloid speed reducer with two cycloid discs shown in Figure 1. The paper is orientated in cycloid disc dynamic analyses and elements that interact with it during the cycloid speed reducer operation. A cycloid speed reducer is a very complex system from the aspect of its geometry as well as from the aspect of its kinematic and dynamic structure. In order to perform the analysis, it is necessary to know the loads that act on the cycloid disc as the main element of the cycloid speed reducer. Figure 2 shows the forces acting on the cycloid disc during meshing. They are the following [1,2]:  
Figure 1: 3D model of a classic single-stage cycloid speed reducer: 1—input shaft; 2 and 2′—cycloid discs; 3 and 3′—output rollers; 4—exiting mechanism, and 5 (5′)—ring gear rollers. * \(F_{\rm E}\)--eccentric cam force (the force where the vertical component \(F_{\rm EV}\) generates the drive torque \(T_{1}\) on the cycloid disc due to its eccentric rotation),
* \(F_{\rm N}\)--normal force at the current contact point of the cycloid disc tooth and the stationary ring gear roller (normal force),
* \(F_{\rm K}\)--normal force at the current contact point of the output roller and the opening in the cycloid disc (output force). The torques due to which these forces occur are:  
* \(T_{1}\)--drive torque of the cycloid disc,
* \(T_{2}\)--torque on the ring gear,
* \(T_{3}\)--output torque of the cycloid disc. From Figure 2, it is evident that this is a statically indeterminate system. The only known force is the vertical component of the eccentric cam force \(F_{\rm EV}\). [PERSON] defined the first procedure for calculation of the forces acting on the cycloid disc in his book _Planetary Gear Train_ in 1966 [1]. Both [PERSON] and [PERSON] [2], as pioneers in this field, analyzed the ideal case of the cycloid disc meshing with other elements of the cycloid speed reducer. In this case, there are no clearances between the teeth of the cycloid disc and the rollers of the stationary ring gear, hence the assumption that half of the teeth of the cycloid disc transfer the load simultaneously. This is obviously not the case in real operating conditions. In reality, clearances exist for many reasons: as the result of manufacturing faults, to facilitate assembly and disassembly for maintenance purposes, and to enable adequate lubrication, among other reasons. [PERSON]'s procedure for calculating the normal force occurring in the contact of the cycloid disc and the stationary ring gear roller is described in great detail in the literature [1]. In this paper, only the basic expressions are presented in order to help understand the calculation of the normal force in experimental and numerical analyses. The normal force on the \(i\)-th roller is calculated based on Equation (1):  
\[F_{\rm Ni}=F_{\rm Nmax}\cdot\frac{l_{\rm i}}{r_{1}} \tag{1}\]  
where \(F_{\rm Nmax}\)--is the maximum value of the normal force, \(l_{\rm i}\)--is the normal distance from the center of the cycloid disc to the corresponding normal force (Figure 2), \(r_{1}\)--is the radius of the stationary circle [1]. Figure 2: Forces on the cycloid disc [35]. Since the output torque of the cycloid disc \(T_{3}\) is given by Equation (2):  
\[T_{3}=F_{\text{Nmax}}\cdot r_{1}\cdot z_{2}\cdot\frac{\sum\limits_{i}l_{1}^{2}}{ r_{1}^{2}\cdot z_{2}} \tag{2}\]  
Having solved Equation (2), the maximum value of the normal force can be calculated as in Equation (3):  
\[F_{\text{Nmax}}=\frac{4\cdot T_{3}}{r_{1}\cdot z_{2}} \tag{3}\]  
where \(z_{2}\) is the number of the rollers of the stationary ring gear. The size of the clearances between the ring gear and the cycloid disc is directly related to the number of the ring gear rollers meshing with the cycloid disc. The larger the clearances, the smaller the number of the meshed teeth of the cycloid disc and the rollers of the ring gear. In this paper, three types of meshing were analyzed: triple-tooth, double-tooth, and single-tooth meshing.