# 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 Fitting Method  
In many studies, time-varying mesh stiffness is fitted in the form of the Fourier series expansion [33; 34]. Despite being fitted by a high-level Fourier series, the difference between time-varying mesh stiffness and the fitting one is still very obvious in the alternate mesh area for single and double teeth. Here, the single-tooth-pair mesh stiffness, rather than the time-varying mesh stiffness, will be expressed in the form of the Fourier series expansion:  
\[k(t) = k_{0}\ +\ \sum_{j\ =\ 1}^{\gamma_{1}}\Big{(}k_{aj}\cos\Big{(}j\frac{w_{ k}}{\varepsilon T}t\Big{)}\ +\ k_{bj}\sin\Big{(}j\frac{w_{k}}{\varepsilon T}t\Big{)}\Big{)} \tag{21}\]  
where \(k_{0}\) is the mean value of the single-tooth-pair mesh stiffness. \(k_{aj}\), \(k_{bj}\) are the amplitudes of the \(j\)th order harmonic. \(w_{f}\) is the stiffness fitting frequency when the mesh period is \(T\ =\ 1/\varepsilon\).  
Similarly, the stiffness coefficient can also be expressed in the form of Fourier series expansion:  
\[\eta(t)\ =\ \eta_{0}+\sum_{j\ =\ 1}^{\gamma_{2}}\Big{(}\eta_{aj}\cos\Big{(}j \frac{w_{k}}{\varepsilon T}t\Big{)}+\eta_{bj}\sin\Big{(}j\frac{w_{k}}{ \varepsilon T}t\Big{)}\Big{)} \tag{22}\]  
where \(\eta_{0}\) is the mean value of the stiffness coefficient\(\eta_{aj}\), \(\eta_{bj}\) are the amplitudes of the \(j\)th order harmonic. \(w_{\rho}\) is the coefficient fitting frequency when the mesh period is \(T\ =\ 1/\varepsilon\).  
For the example gear pair, it can be found that the fitting effect is very well when \(\gamma_{1}\ =\ 1\), \(\gamma_{2}\ =\ 3\). The corresponding parameters are listed in Table 3.  
Moreover, \(\eta_{0}\), \(\eta_{aj}\), and \(\eta_{bj}\) can be further expressed as functions of backlash:  
\[\eta_{0} = -311.4\ \times\ b\ +\ 1 \tag{23a}\] \[\eta_{a1} = 159.4\ \times\ b\ -\ 1.359\ \times\ 10^{-4}\] (23b) \[\eta_{b1} = 276\ \times\ b\ -\ 2.365\ \times\ 10^{-4}\] (23c) \[\eta_{a2} = 41.07\ \times\ b\ -\ 4.85\ \times\ 10^{-5}\] (23d) \[\eta_{b2} = -71.13\ \times\ b\ +\ 8.464\ \times\ 10^{-5}\] (23e) \[\eta_{a3} = -12.88\ \times\ b\ +\ 2.284\ \times\ 10^{-5}\] (23f) \[\eta_{b3} = -3.448\ \times\ 10^{-12}\ \times\ b\ +\ 4.033\ \times\ 10^{-16} \tag{23g}\]  
The fitting results are shown in Figure 6a:  
\begin{table}
\begin{tabular}{c c c c c} \hline \(k_{0}\) (N/m) & \(k_{a1}\) (N/m) & \(k_{b1}\) (N/m) & \(w_{k}\) & \(w_{\eta}\) \\ \hline
1.487 \(\times\ 10^{8}\) & 4.749 \(\times\ 10^{7}\) & 1.226 \(\times\ 10^{8}\) & 2.403 & 2.094 \\ \hline \end{tabular}
\end{table}
Table 3: The fitting parameters for the example gear pair.  
According to Figure 6b, the relative error is less than 0.1%, which means the Fourier fitting has high precision.  
Then, the time-varying mesh stiffness can be obtained by defining two pairs of teeth in mesh. In the meshing process of a gear set with a contact ratio between 1 and 2, as shown in Figure 7, the meshing teeth pair with the contact point located between point A and point C is defined as pair #1, and the meshing teeth pair with the contact point located between point C and point D is defined as pair #2.  
The piecewise stiffness functions can be expressed as  
\[\begin{array}{rcl}K_{1}(t)&=&\big{(}k_{0}\ +\ k_{a1}\cos\big{(}\frac{w_{b}}{ kT}mod(t,T)\big{)}\ +\ k_{b1}\sin\big{(}\frac{w_{b}}{ kT}mod(t,T)\big{)}\big{)}(\eta_{0}\\ &&\ +\ \sum\limits_{j=1}^{72}\Big{(}\eta_{aj}\cos\Big{(}j\frac{w_{b}}{