**4. Research on Bifurcation Behavior**

The load of the HAGC system will be affected by different forces during the working process. The influence factors are diverse and complex. The research object of this article will focus on some nonlinear factors such as the excitation force, elastic force and damping force. The selected nonlinear forces are important influence factors. The effects of the above feature parameters on bifurcation behavior are explored.

The actual physical parameters of the 650 4/6-roll cold rolling mill from the "National Engineering Research Center for Equipment and Technology of Cold Strip Rolling" are employed in the following research. Some unmeasured parameters are empirical estimates. The parameters are shown in Table 1. The photos of the 650 4/6-roll cold rolling mill are displayed in Figure 5.


**Table 1.** Parameters of the numerical experiment.

(**a**) (**b**)

**Figure 5.** Photos of the 650 4/6-roll cold rolling mill.

The influence of nonlinear elastic force can be reflected by the nonlinear stiffness coefficient α. Hence, the effect of α on bifurcation characteristics is firstly researched. According to Equation (11) derived in the previous section, the bifurcation diagram when α changes is revealed in Figure 6. The jump phenomenon of the vibration amplitude will gradually be enhanced with the increase in α. The degree of jump phenomenon is especially severe in the resonance region. However, when it is far away from the resonance region, the degree of jump will be reduced. The primary reason for the above result is that the change of the nonlinear stiffness coefficient α affects the natural frequency ω<sup>0</sup> of the system. As α increases, the natural frequency of the system increases. The change of natural frequency results in the resonance phenomenon when natural frequency couples with external excitation frequency ω. However, the resonance phenomenon can cause the increase in system instability. Therefore, if α is reasonably adjusted to effectively avoid the resonance region, the stability of the system will be facilitated.

**Figure 6.** Bifurcation diagram with the variation of α.

Additionally, the nonlinear elastic force can be influenced by the liquid column height *L*<sup>1</sup> of control cavity. So, the effect of *L*<sup>1</sup> on bifurcation characteristics was further studied. The bifurcation diagram with the variation of *L*<sup>1</sup> is displayed in Figure 7. When *L*<sup>1</sup> is close to the two ends of hydraulic cylinder, the jump phenomenon of the vibration amplitude is more serious. When *L*<sup>1</sup> is away from the ends of the hydraulic cylinder, the degree of amplitude jump is relatively reduced. The bifurcation phenomenon near the middle position (130 mm) is more complex. The foremost reason is that the stiffness of the hydraulic spring is related to the piston position of the hydraulic cylinder. When the piston is in the middle position, the liquid compressibility is most affected. At this time, the hydraulic spring stiffness is small and the natural frequency of the system is low. Therefore, it shows poor stability in the system. Hence, properly controlling the size of *L*<sup>1</sup> is conducive to the stability of the system.

**Figure 7.** Bifurcation diagram with the variation of *L*1.

The effect of nonlinear damping force can be reflected by the nonlinear damping coefficient β. Hence, the influence of β on bifurcation characteristics was analyzed. The bifurcation diagram with the change of β is illustrated in Figure 8. As can be observed, the jump phenomenon of the vibration amplitude will gradually decrease with the increase in β, and the vibration amplitude is effectively suppressed. However, when the value of β exceeds a certain threshold, the suppression effect for vibration amplitude is no longer obvious, and the jump phenomenon of vibration amplitude still exists. The main reason is that the appropriate β can narrow the frequency band of resonance and decrease the unstable area, which is beneficial to the stability of the system.

**Figure 8.** Bifurcation diagram with the variation of β.

Since the influence of nonlinear excitation force can be reflected by the external excitation amplitude *F*, the effect of *F* on bifurcation characteristics was also investigated. The bifurcation diagram with the variation of *F* is demonstrated in Figure 9. It will be observed that the jump phenomenon of vibration amplitude will gradually strengthen with the increase in *F*. Furthermore, the degree of jump will gradually increase. The main reason is that the increase in *F* can widen the frequency band of resonance and augment the instability of the system.

**Figure 9.** Bifurcation diagram with the variation of *F*.

#### **5. Conclusions**

On the basis of the theory of nonlinear dynamics, the bifurcation characteristics of load vertical vibration of the HAGC system were researched. The effects of some important parameters on bifurcation characteristics were emphatically explored. Through in-depth research, some conclusions are drawn:

(1) The bifurcation curves in each subregion have their own topological structure. With the change of the open fold parameters, the topological structure changes at the transition set. Moreover, the system has different vibration behaviors in diverse subregions, and will exhibit different bifurcation behaviors under various parameter combinations. Therefore, by analyzing the bifurcation characteristics of the system, the parameter region that causes the system to be unstable can be determined.

(2) With the increase in the nonlinear stiffness coefficient α, the jump phenomenon of the vibration amplitude will gradually be enhanced. Especially, the degree of jump phenomenon is severe in the resonance region. However, the degree of jump will be reduced when it is far away from the resonance region. Therefore, if α is reasonably adjusted to effectively avoid the resonance region, the stability of the system will be facilitated.

(3) The jump phenomenon of the vibration amplitude is more serious when the liquid column height *L*<sup>1</sup> is close to the two ends of hydraulic cylinder. The degree of amplitude jump is relatively reduced when *L*<sup>1</sup> is located in the middle section of the hydraulic cylinder. However, the bifurcation phenomenon near the middle position is more complex. Hence, properly controlling the size of *L*<sup>1</sup> is conducive to the stability of the system.

(4) With the increase in the nonlinear damping coefficient β, the jump phenomenon of the vibration amplitude will gradually decrease, and the vibration amplitude is effectively suppressed. However, the suppression effect for vibration amplitude is no longer obvious when the value of β exceeds a certain threshold, and the jump phenomenon of the vibration amplitude still exists. The appropriate β can narrow the frequency band of resonance and decrease the unstable area, which is beneficial to the stability of the system.

(5) With the increase in the external excitation amplitude *F*, the jump phenomenon of the vibration amplitude will gradually strengthen. Moreover, the degree of jump will gradually increase, which is not conducive to the stability of the system.

The acquired results provide a theoretical basis for vibration traceability and suppression of the HAGC system. This research can provide an important basis for the further study on nonlinear dynamic behaviors of the HAGC system.

**Author Contributions:** Conceptualization, Y.Z. and W.J.; Methodology, S.T.; Investigation, Y.Z. and S.T.; Writing—Original Draft Preparation, Y.Z.; Writing—Review & Editing, X.Y. and Y.L.; Supervision, C.W.

**Funding:** This research was funded by National Natural Science Foundation of China (No. 51805214, 51875498), China Postdoctoral Science Foundation (No. 2019M651722), Natural Science Foundation of Hebei Province (No. E2018203339, E2017203129), Nature Science Foundation for Excellent Young Scholars of Jiangsu Province (No. BK20190101), Open Foundation of National Research Center of Pumps, Jiangsu University (No. NRCP201604) and Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (No. GZKF-201820).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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