## 5 The Meshing Stiffness of the Trochoid Roller Pinion Rack
### Calculation of the Stiffness of Each Component  
The energy method is used to calculate the normal meshing stiffness of the trochoid roller pinion rack [21]. The hertzian contact potential energy \(U_{h}\), bending potential energy \(U_{b}\), shear potential energy \(U_{s}\), and axial compression potential energy \(U_{a}\) during the meshing process are mainly considered. The relationship between these potential energy and stiffness is shown in Equation (9) [21]. \[U_{i}=\frac{F^{2}}{2k_{i}} \tag{9}\]  
where \(F\) is the static normal meshing force at the meshing point, \(U_{i}\) (\(i=h\), \(b\), \(s\), \(a\)) is the potential energy existing in meshing, and \(k_{i}\)(\(i=h\), \(b\), \(s\), \(a\)) is the stiffness corresponding to each potential energy. The total energy stored in a pair of meshing pairs is shown in Equation (10) [21]. In the Equation (10), subscripts 1 and 2 represent rack and roller pinion respectively. Equation (11) [21] is used to calculate the normal stiffness at the meshing point. Since the rack of the roller pinion is engaged alternately with single and double tooth, the meshing stiffness in the normal direction of time variation in the whole meshing is Equation (12). \[ \begin{array}{l}U=U_{h}+U_{b1}+U_{s1}+U_{a1}+U_{f1}+U_{b2}+U_{s2}+U_{d2}+U_ {f2}\\ =\frac{F^{2}}{2}\Big{(}\frac{1}{k_{h}}+\frac{1}{k_{h1}}+\frac{1}{k_{a1}}+ \frac{1}{k_{a1}}+\frac{1}{k_{f1}}+\frac{1}{k_{h2}}+\frac{1}{k_{a2}}+\frac{1}{ k_{a2}}+\frac{1}{k_{f2}}\Big{)}\end{array}  \tag{10}\]  
\[k=\frac{1}{\frac{1}{k_{h}}+\frac{1}{k_{h1}}+\frac{1}{k_{a1}}+\frac{1}{k_{a1}}+ \frac{1}{k_{a1}}+\frac{1}{k_{f1}}+\frac{1}{k_{a2}}+\frac{1}{k_{a2}}+\frac{1}{ k_{a2}}+\frac{1}{k_{f2}}} \tag{11}\]  
\[k=\sum_{i=1}^{2}\frac{1}{\frac{1}{k_{h}}+\frac{1}{k_{b1,i}}+\frac{1}{k_{a1,i}} +\frac{1}{k_{a1,i}}+\frac{1}{k_{f1,i}}+\frac{1}{k_{h2,i}}+\frac{1}{k_{a2,i}}+ \frac{1}{k_{a2,i}}+\frac{1}{k_{f2,i}}} \tag{12}\]  
where \(i\) (\(i=1\), \(2\)) is the \(i\)th engagement pair. There is a formula relationship between energy and each component stiffness. The force exerted on a certain instant of the rack in the meshing process is shown in Figure 21, where \(\alpha\) is the meshing pressure angle. Bending energy, shear energy and axial compression energy can be obtained according to the material mechanics method, and can be expressed as Equations (13)-(15) [21] respectively. \[U_{b}=\int_{0}^{d}\frac{M^{2}}{2 EI_{x}}\mathrm{d}s \tag{13}\]  
Figure 21: Force diagram of rack. \[U_{s}=\int_{0}^{d}\frac{1.2{F_{b}}^{2}}{2 GA_{x}}\mathrm{d}s \tag{14}\] \[U_{a}=\int_{0}^{d}\frac{{F_{a}}^{2}}{2 EA_{x}}\mathrm{d}s \tag{15}\]  
where, \(F_{b}=F\mathrm{cos}\alpha\), \(F_{a}=F\mathrm{sin}\alpha\), \(M=F_{b}x-F_{a}h\), \(d\) is the height between the force point and the tooth root which is shown in Figure 20.  
The bending deformation flexibility of the rack in the meshing process can be expressed as Equation (16) [21]. \[\delta_{b}=\int_{0}^{d}\frac{\left(s\cos\alpha-h\sin\alpha\right)^{2}}{EI_{x}} \mathrm{d}s \tag{16}\]  
The shear deformation flexibility of the rack is expressed in Equation (17) [21]. \[\delta_{s}=\int_{0}^{d}\frac{1.2\cos^{2}\alpha}{GA_{x}}\mathrm{d}s \tag{17}\]  
The axial compression deformation flexibility of the rack is expressed in Equation (18) [21]. \[\delta_{a}=\int_{0}^{d}\frac{\sin^{2}\alpha}{EA_{x}}\mathrm{d}s \tag{18}\]  
The contact deformation flexibility of the rack is expressed in Equation (19) [21]. \[k=\sum_{i=1}^{2}\frac{1}{\frac{1}{k_{h}}+\frac{1}{k_{b1,i}}+\frac{1}{k_{a1,i}} +\frac{1}{k_{a1,i}}+\frac{1}{k_{f1,i}}+\frac{1}{k_{h2,i}}+\frac{1}{k_{a2,i}}+ \frac{1}{k_{a2,i}}+\frac{1}{k_{f2,i}}} \tag{12}\]0  
where, \(E\) is the elastic modulus of material, \(G\) is the shear modulus of material, \(G=E/2(1+\
u)\), \(I_{x}\) is the moment of inertia of the section at the meshing point \(M\), \(I_{x}=2/3{h_{x}}^{3}L\), \(A_{x}\) is the cross section area at the meshing point \(M\), \(A_{x}=2{h_{x}}L\), \(\
u\) is the poisson's ratio of material, \(L\) is the tooth width of rack.