## 2 Nonlinear Dynamic Modeling of Spur Gear Systems
### Influence of Stiffness Excitation  
#### 3.2.1 Analysis of Time-Varying Meshing Stiffness  
As an important internal excitation of the gear system, the time-varying meshing stiffness has a vital influence on the dynamic characteristics. The time-varying stiffness mentioned here means the dimensionless meshing stiffness, \(K_{1}\), which indicates the ratio of the amplitude of the gear meshing stiffness to the mean value of the gear meshing stiffness.  
Figure 14: Bifurcation diagram with respect to \(\omega\) (\(T_{c}=27\) Nm).  
Figure 12: \(\omega=1.93\); (**a**) phase diagram; (**b**) Poincaré map; (**c**) time-domain diagram.  
Figure 13: \(\omega=2.66\); (**a**) phase diagram; (**b**) Poincaré map; (**c**) time-domain diagram.  
Let the load torque \(T_{c}=5.4\) Nm, the rotational speed be 1500 rpm, the gear transmission error \(e_{0}=0.03\) mm, the damping coefficient \(\xi=0.12\), and the supporting stiffness \(k_{bxy}\) be 1.5 times the average meshing stiffness. As shown in Figures 15-20, the system's bifurcation diagram, phase diagrams, Poincare maps, and time-domain diagrams are obtained when the meshing stiffness takes different values.  
As shown in Figure 15, the bifurcation diagram of the displacement along the LOA is obtained with an increase in \(K_{1}\). When \(K_{1}\) is in (0, 0.131), the system is in a stable periodic motion state. When \(K_{1}\) falls into (0.16, 0.3), the quasiperiod-doubling motion gradually turns into a stable period-doubling motion. Subsequently, the stable period-doubling motion maintains until \(K_{1}\) approaches 0.3. When \(K_{1}\) falls into (0.3, 0.337), the period-doubling motion becomes a period-quadrupling motion through period-doubling bifurcation. Afterward, the period-quadrupling motion becomes a period-octuping motion through period-doubling bifurcation and finally evolves into a chaotic motion. When \(K_{1}\) is in (0.34, 0.45), the system motion is chaotic. On the whole, with an increase in meshing stiffness, the system gradually enters a chaotic state from a single stable period through multiple period-doubling bifurcations, and the system gradually changes from a stable state to an unstable state.  
The representative meshing stiffness was selected for the analysis of the dynamic characteristics. When \(K_{1}\) = 0, as shown in Figure 16, the phase trajectory is in a single closed curve, and the Poincare map has only one isolated dot, which means the system is in a periodic motion. When \(K_{1}\) = 0.16, as shown in Figure 17, the Poincare map exhibits several dots gathering into two places, which means the system falls into a quasiperiod-doubling motion. Similarly, as shown in Figures 18 and 19, when \(K_{1}\) = 0.25 and \(K_{1}\) = 0.33, the system is in a period-doubling motion state and a period-quadrupling motion state, respectively. When \(K_{1}\) = 0.45, it can be seen in Figure 20 that the Poincare map is characterized by discrete dots, which means the system is in a chaotic state.  
In conclusion, with an increase in meshing stiffness, the system's dynamic response gradually evolves from a stable periodic motion to a chaotic motion, thereby deteriorating transmission performance. Decreasing the amplitude of the meshing stiffness tends to obtain a stable periodic motion and good stability. As a result, with the help of a reasonable contact ratio of the gears, both the bifurcation points and the chaotic motion can be effectively evaded, thereby improving system stability.