## 5 The Meshing Stiffness of the Trochoid Roller Pinion Rack
### Definition of Meshing Stiffness of Trochoidal Roller Pinion Rack  
The meshing stiffness of gear pair is in the normal direction of the line of action. The line of action of the gear pair is a straight line and the pressure angle does not change during the transmission process. However, the pressure angle of the trochoid roller pinion rack changes with time as shown in Figure 19. The interval 1 is a meshing period, the interval 2 is the double-tooth meshing interval, and the interval 3 is the single-tooth meshing interval. It can be seen that the pressure angle of the meshing point of the two roller pins in the double-tooth meshing interval is quite different. This meshing characteristic is different from the fixed pressure angle of the gear. In the traditional gear transmission, the meshing stiffness of the double-tooth meshing area is the direct superposition of the meshing stiffness of the two meshing points, but it cannot be fully applied to the trochoidal roller pinion rack.  
Figure 18: The influence law of \(c_{y}\) on the line of action.  
Since the output of the trochoid roller pinion rack is a circular motion, it is more practical to convert the meshing stiffness into the torsional stiffness on the roller pinion. The meshing stiffness \(k_{rp}\) of the trochoid roller pinion rack is defined as the torque value required for the roller pinion to produce 1 rad torsional deformation, which is the deformation of the tangential direction of the meshing point converted to the deformation on the roller pinion. The pressure angle of the two meshing points in the double-tooth meshing area of the trochoid roller pinion rack is quite different, and the normal direction of the meshing point is not the same as the tangential direction of the roller pinion where the meshing point is located. Therefore, it is necessary to convert the meshing stiffness at the two meshing points during calculation. The normal meshing stiffness \(k\) of the meshing point is calculated by the energy method, and then it is first converted into the tangential meshing stiffness \(k_{t}\) of the roller pinion, and then \(k_{t}\) is converted into the torsional meshing stiffness \(k_{rp}\) of the roller pinion. The conversion method between the stiffnesses is described as following.  
As shown in Figure 20, \(D_{i}\) represents the \(i\)-th (\(i\) = 1,2) meshing point, \(O\) is the pitch node of the rack, \(D_{i}O\) represents the force direction at the meshing point \(D_{i}\), and \(l_{i}\) represents the tangent direction of the meshing point \(D_{i}\) on the roller pinion direction, and \(\theta_{i}\) represents the angle between the force direction and the tangent direction of the meshing point \(D_{i}\), \(R_{i}\) represents the distance between the meshing point \(D_{i}\) and the \(O^{\prime}\) which is the center of the roller pinion. Using the energy method, the meshing stiffness \(k_{i}\) of \(D_{i}\) can be directly calculated, the normal meshing stiffness \(k_{i}\) is in the \(D_{i}O\) direction. The meshing stiffness of the two meshing points is superimposed as the system normal meshing stiffness \(k\). The calculation method of the tangential meshing stiffness \(k_{t}\) of the trochoid roller pinion rack is shown in Equation (7), and the calculation method of the meshing stiffness \(k_{rp}\) is shown in Equation (8).  
\[k_{t}=k\cos(\theta_{i}) \tag{7}\]  
\[k_{rp}=k_{t}R_{i}^{2} \tag{8}\]  
Figure 19: Time-varying curve of pressure angle of the trochoidal roller pinion rack.  
Figure 20: Schematic diagram of each parameter used for meshing stiffness calculation.