# Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears
## 3 Influence of Backlash on Dynamics
### Numerical Results and Discussion  
The dynamic Equation (29) can be numerically integrated by using the fourth order Runge-Kutta method. The simulation runs for 20,000 periods and only the data of the last 50 periods are plotted to guarantee the data relates to the steady state conditions. Here, a system with \(T_{p}~{}=~{}500\) N\(\cdot\)m, \(n_{p}~{}=~{}2500\) rpm, \(\zeta~{}=~{}0.11\), \(e(t)~{}=~{}0\) is established.  
Figure 8: The non-linear dynamic model.  
Figure 9 shows the dynamic response of the system as \(b\ =\ b_{min}\), the response period of the time history chart equals the excitation period, the phase diagram presents a closed non-circle and non-elliptic curve, and the discrete points are located at the frequencies of \(if_{m}(i=1,2,\cdots)\) in the FFT spectrogram, which means that the system exhibits a non-harmonic-single-periodic motion, but the periodicity is not strict. Partially enlarged views of the graphics showed that the phase diagram had 50 curves, the resulting trace in the Poincare map was concentrated in 50 points, and the enlarged spectrum had obvious perturbation peaks. That is to say, the plotted 50 periods were not strictly repetitive, and the situation was the same as the simulation runs for more periods. Thus, we were certain that the system exhibited a non-strictly non-harmonic-single-periodic vibration.  
In practical engineering applications, the value range of the backlash is \(b_{min}\ \leq\ b\ \leq\ 2b_{min}\). Here, the research range of the backlash was expanded to \(3b_{min}\) to conduct a better study on the influence of backlash on the dynamics. Figure 10 shows the system response in terms of the dynamic transmission error for various backlash values. In the case of \(b\ =\ 1.25b_{min}\), the dynamic motion is non-harmonic-single-periodic as a whole. Partially enlarged views indicated that the system presented a low-frequency vibration, as shown in Figure 10a. The phase diagram had 16 curves, the resulting trace in the Poincare map was concentrated in 16 points, and the enlarged spectrum had peaks at the points of \(if_{m}/16\). As a result, the system mainly exhibited non-harmonic-single-periodic vibration and also exhibited 16T-periodic low-frequency vibration with a small amplitude.  
As the backlash increased to \(1.5b_{min}\), the motion was similar to that of \(b\ =\ b_{min}\) where no significant period of low-frequency vibration was presented. As the backlash further increased, the low-frequency vibrations were 7T-periodic in the case of \(b\ =\ 1.75b_{min}\) and 30T-periodic in the case of \(b\ =\ 2b_{min}\), respectively. According to Figure 10e-h, it was clear that the low-frequency vibrations were 8T-periodic, 16T-periodic, 11T-periodic, and 7T-periodic, respectively.  
The dynamic transmission error is a key parameter to evaluate the gear system's vibration and noise. Available studies have often been concerned with the vibration whose minimum response frequency equals that of the excitation frequency, and those small-amplitude and low-frequency motions have received less attention. According to Figures 9 and 10, it is quite clear that the backlash had a great influence on the small-amplitude and low-frequency vibration, which can be regarded as secondary vibration and can guide us to understanding the mechanism of some special noise.  
Figure 9: The system response in terms of the dynamic transmission error for \(b\ =\ b_{min}\).