# Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears
## 2 Influence of Backlash on Time-Varying Mesh Stiffness  
### Mesh Stiffness Model with Backlash  
Available mesh stiffness models do not consider the influence of backlash. Precision tooth profile equations play an important role in the analysis method to calculate the mesh stiffness. According to the generating method of gear cutting, the tooth profile equations consist of two parts: the involute and transition curve, as shown in Figure 1.  
The subscripts \(i\) and \(t\) in Figure 1 denote the involute curve and transition curve. Figure 1: The principle of the generating method: (**a**) involute; (**b**) transition curve. According to the potential energy method, the potential energy deposited in the gear set can be divided into Hertzian energy \(U_{h}\), bending energy \(U_{b}\), shear energy \(U_{s}\), axial compressive energy \(U_{a}\), and fillet foundation energy \(U_{f}\). The relationship between the potential energy and corresponding stiffness can be expressed as follows:  
\[U_{h} = \frac{{{F_{n}}^{2}}}{{2K_{h}}},\ U_{b} = \frac{{{F_{n}}^{2}}}{{2K_{b}}},\ U_{s} = \frac{{{F_{n}}^{2}}}{{2K_{s}}},\ U_{a} = \frac{{{F_{n}}^{2}}}{{2K_{a}}},U_{f} = \frac{{{F_{n}}^{2}}}{{2K_{f}}} \tag{7}\]  
where \(K_{h}\), \(K_{b}\), \(K_{s}\), \(K_{a}\), and \(K_{f}\) denote the Hertzian contact stiffness, bending stiffness, shear stiffness, axial compressive stiffness, and fillet foundation stiffness, respectively. The stiffness contribution of the gear body is omitted [31]. Then, the total potential energy \(U\) for a pair of teeth can be achieved by summing the energy components, i.e.,  
\[U = \frac{{F}^{2}}{{2K}} = U_{h}\ +\ \sum\limits_{i\ =\ 1}^{2}\left(U_{bi}\ +\ U_{si}\ +\ U_{ai}\ +\ U_{fi}\right) \tag{8}\]  
Then, the single-tooth-pair mesh stiffness for a pair of spur gears with a contact ratio between 1 and 2 can be expressed as follows:  
\[K = 1/\left(\sum\limits_{i\ =\ 1}^{2}\left(\frac{1}{{K_{bi}}}\ +\ \frac{1}{{K_{si}}}\ +\ \frac{1}{{K_{si}}}\ +\ \frac{1}{{K_{fi}}}\right)\ +\ \frac{1}{{K_{h}}}\right) \tag{9}\]  
where the subscripts \(i\ =\ 1,\ 2\) represent the gear and the pinion. According to Reference [32], the Hertzian contact stiffness \(K_{h}\) can be expressed as  
\[K_{h} = \frac{\pi EL}{4(1\ -\ v^{2})} \tag{10}\]  
where \(E\) is [PERSON]'s modulus; \(L\) is the tooth width; and \(v\) is the [PERSON]'s ratio. The fillet foundation stiffness \(K_{f}\) can be calculated according to [13]  
\[K_{f} = \frac{1}{\frac{{sin^{2}\delta_{\varphi}}}{{EL}}\left\{{L^{*}}{ \left(\frac{{u_{f}}}{{S_{f}}}\right)^{2}}\ +\ M^{*}{\left(\frac{{u_{f}}}{{S_{f}}}\right)}\ +\ P^{*}{\left(1\ +\ Q^{*}/\tan^{2}\delta_{\varphi}\right)}\right\}} \tag{11}\]  
where \(\delta_{\varphi}\ =\ \tan(s(\varphi))\) is the angle between normal force and axis \(x\) in Figure 2. The parameters \(L^{*},M^{*},\ P^{*},\ Q^{*}\) can be calculated according to the following equation:  
\[X_{i}^{*}\left(h_{fi},\theta_{f}\right) = \frac{A_{i}}{\theta_{f}^{2}}\ +\ B_{i}h_{fi}^{2}\ +\ \frac{{\rm{C}}_{i}h_{fi}}{\theta_{f}}\ +\ D_{i}/\theta_{f}\ +\ E_{i}h_{fi}\ +\ F_{i} \tag{12}\]  
where \(h_{fi}\ =\ r_{f}/r_{int}\), \(r_{int}\) and \(\theta_{f}\) are described in Figure 2, and \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(E_{i}\), \(F_{i}\) are listed in Table 1.