## 3 Quasi-Static Force Analysis
### Force and Moment Equilibrium  
During the operation of the cycloid gear reducer, force generation is confined to specific interactions between the cycloid gear and a limited set of engaged rollers and output pins, and the discriminant of this range is shown in Equation (11), where \(\varphi_{in}\) is the input angle and, usually, each angle has one-half of the number of rollers and output pins in this range.  
\[\varphi_{in}\leq\theta\leq\varphi_{in}+\pi \tag{11}\]  
Figure 3 depicts a simplified depiction of the forces operating on the cycloid gear, where \(S_{f}\) is a fixed coordinate system, \(F_{p,i}\) represents the roller-generated force of \(i\) on the cycloidal gear, and \(L_{p,i}\) stands for the force arm length of the roller force of \(i\). \(F_{o,j}\) is the output plunger pin force of \(j\) on the cycloid gear and \(L_{o,j}\) is the output plunger pin force of \(j\). \(F_{b}\) is the input crank bearing force on the cycloid gear, \(\varepsilon\) stands for the angle formed by the input crank bearing force and eccentricity's direction, \(r_{w}\) is the radius of the output pin center pitch circle, and \(r_{p}\) represents the pitch circle's radius around the roller center.  
Figure 2: The schematic diagram of contact between two cylinders with parallel central axes.  
Upon achieving force balance on the cycloid gear, the force and moment are balanced in the \(X\)-axis and \(Y\)-axis:  
\[\sum F_{x}=\sum F_{p,i}\cos\theta_{p,i}+\sum F_{o,j}\cos(\varphi_{in}+\pi)+F_{p} \cos(\varphi_{in}+\varepsilon)=0 \tag{12}\]  
\[\sum F_{y}=\sum F_{p,i}\sin\theta_{p,i}+\sum F_{o,j}\sin(\varphi_{in}+\pi)+F_{p} \sin(\varphi_{in}+\varepsilon)=0 \tag{13}\]  
\[\sum M_{O_{c}}=\sum F_{o,j}L_{o,j}-\sum F_{p,i}L_{p,j}=0 \tag{14}\]  
where \(L_{p,i}\) and \(L_{o,j}\) can be obtained from Equations (15) and (16), in which \(\beta_{p}\) is the angle between the two rollers, \(\beta_{o}\) is the angle between the two output pins, and \(\varphi_{out}\) is the output angle.  
\[L_{p,i}=\frac{\left|x_{O_{c}}\tan\theta_{p,i}-y_{O_{c}}+\left[\tau_{p}\sin(i \beta_{p})-r_{p}\cos(i\beta_{p})\tan\theta_{p,i}\right]\right|}{\sqrt{\tan^{2} \bigl{(}\theta_{p,i}\bigr{)}+\left(-1\right)^{2}}} \tag{15}\]  
\[L_{o,j}=\frac{\left|x_{O_{c}}\tan(\varphi_{in}+\pi)-y_{O_{c}}+\left[r_{w}\sin (j\beta_{o}-\varphi_{out})-r_{w}\cos(j\beta_{o}-\varphi_{out})\tan(\varphi_{in }+\pi)\right]\right|}{\sqrt{\tan^{2}(\varphi_{in}+\pi)+\left(-1\right)^{2}}} \tag{16}\]  
The contact state between the roller and the cycloid gear is determined by Equation (17), where \(\triangle\varphi_{out}\) is the change in output angle when the cycloid gear reaches force balance and \(\varphi_{bl,i}\) is the value of the roller backlash of the roller of \(i\) as shown in Figure 4.  
\[f_{c}=\frac{\left|\triangle\varphi_{out}-\varphi_{bl,i}\right|+\left(\triangle \varphi_{out}-\varphi_{bl,i}\right)}{2} \tag{17}\]  
The following two possible cases exist:  
Case (a). \(\triangle\varphi_{out}>\varphi_{bl,i}\), \(f_{c}=\triangle\varphi_{out}-\varphi_{bl,i}\)  
In this case, the change in the output angle adequately compensates for roller back-lash, facilitating contact between the cycloid gear and the roller, while the calculated function value is subsequently expressed as the indentation depth.  
Case (b). \(\triangle\varphi_{out}<\varphi_{bl,i}\), \(f_{c}=0\)  
In this case, roller backlash remains, preventing contact between the cycloidal gear and the roller.  
Figure 3: The mechanics model of force and torque balance.  
The number of uncertainties can be reduced to \(\bigtriangleup\)\(\varphi_{out}\), \(c_{o}\), \(F_{b}\), and \(\varepsilon\), which can be calculated with the help of a Matlab programming solution.