Abstract
This work is focused on a rolling mill’s main drive electromechanical coupling system. Firstly, we equip electromechanical coupling system with fractional-order time delay. Secondly, we, respectively, derive the conditions for occurrence of Hopf bifurcation around equilibriums and . It is found that the fractional order and time delay in the system play an important role on the system stability. Finally, numerical simulations are given to verify the analytic results.
1. Introduction
The rolling mill system, as a typical complex nonlinear system, involves many disciplines such as rolling, machines, motors, and automatic control. Due to the complex structure of the rolling mill, the system contains a variety of nonlinear factors, which closely affect the intensity and frequency of torsional vibration during the production process of the rolling mill. Frequent torsional vibrations during rolling mill production have a serious impact on the quality of rolled products. The phenomenon of torsional vibration not only causes vibration marks on the surface of the strip and rolls and reduces the service life of the rolls but also deteriorates the operating environment, shortens the fatigue life of parts, and ultimately threatens the safety of the rolling mill. Therefore, being able to propose effective measures to suppress the torsional vibration of the rolling mill [1–10] has become the key to solving the torsional vibration of the main drive of the rolling mill. At the same time, this issue has also attracted scholars’ attention.
Rotating machinery systems contain various nonlinear factors, which lead to complex dynamic behaviors such as Hopf bifurcation and chaos of the system (see [11–13]). The researchers [14–16] abstracted the main drive system of the rolling mill as a two-mass or multimass relative rotation system and studied the bifurcation and chaos of the torsional vibration of the system. In our previous work [17], we designed a nonlinear controller to control the Hopf bifurcation in the main drive delayed system of the rolling mill.
In 2015, Liu et al. [16] deduced a nonlinear electromechanical coupling system by using the dissipation Lagrange equation, and they introduced a time delay feedback to control the dynamic behaviors of the system. The model takes the following form: where is the moment of inertia, and are the angle of rotation and rotational speed, respectively, is the torsional stiffness of drive shaft, is the shafting damping coefficient, and is the electromagnetic damping coefficient (the meaning of above parameters can refer to Ref. [16]).
By applying appropriate transformations, system (1) is equivalent to system (2): where . At present, the stability and the Hopf bifurcation for system (2) were investigated in [16]. Ding et al. [18] focused on Hopf and Hopf-pitchfork bifurcations for system (2) by applying the multiple time scale method and the normal forms.
For electromechanical coupling system, the unstable bifurcation may cause a destructive vibration. Therefore, finding more efficient measures to control vibration is vital to system (2). In the existing literature, the electromechanical coupling systems are used ordinary differential equations or delay differential equations to investigate through the integer-order mathematical modeling. However, few studies have applied fractional order to system (2). At present, in the process of studying some real world problems, it is found that the problems can be better described by the fractional-order models [19–21] than the integer-order models. Therefore, fractional-order systems may be the basis for many future studies. In [22–25], when introducing fractional orders to existing systems, the usual method is to directly replace the integer order with the fractional order. Motivated by the aforementioned works, we follow the same processing method and try to introduce the fractional order into system (2) and explore the effect of fractional order on the damping term of the system (1); the new model is described by the following fractional-order differential equations where represents the Caputo fractional derivative and is the fractional and derivative order of damping, which may represent the nonlocal displacement effects of dissipation of energy (internal friction). and are time delays. The main contributions of this paper are as follows: (1)A fractional-order nonlinear electromechanical coupling system with delay is proposed(2)The conditions of Hopf bifurcation of system (3) around equilibrium are given(3)Some simulations are implemented to corroborate the obtained conclusions
The layout of this work is organized as follows: In Section 2, the basic definition and the required theorem are introduced. In Section 3, stability analysis of fractional-order nonlinear electromechanical coupling system with delay is obtained. In Section 4, some simulations are carried out to verify the theoretical results. Conclusions are given in Section 5.
2. Preliminaries
Before starting the analysis and establishing the results, we firstly introduce the basic definition and the required theorem.
Definition 1 [26]. The Caputo fractional derivative operator of order is defined as where is an integer, .
Theorem 2 [19, 27] (the conditions of stability). Consider the following fractional-order system: where . The equilibrium points of the above system are the solutions of . An equilibrium point is locally asymptotically stable if all eigenvalues of the Jacobian matrix evaluated at the equilibrium point satisfy . Further, if , then the system undergoes a Hopf bifurcation.
Theorem 3 [28, 29]. Consider the following -dimensional fractional-order system with delay: where and time delay . System (6) undergoes a Hopf bifurcation at the equilibrium point when if the following conditions are satisfied: (i)All the eigenvalues of the coefficient matrix of the linearized system of (7) with satisfy (ii)The characteristic equation of the linearized system of (7) has a pair of purely imaginary roots when (iii)
3. Stability and Hopf Bifurcation
Herein, the conditions of Hopf bifurcation of system (3) around feasible equilibrium points are mainly investigated. Noticing that system (3) has a zero equilibrium point and a nonzero equilibrium point . The zero equilibrium point satisfies the following equations:
Then, by a direct computation, we can derive that satisfies the following equation:
The detailed expression of parameters () in Equation (8) can be found in Appendix A.
In order to meet the actual problem, we suppose that Equation (8) has a positive root . Noticing the fact that , we can calculate . Furthermore, if we suppose , then system (3) has a nonzero equilibrium point .
Next, we, respectively, investigate the stability of system (3) around equilibrium points and . In what follows, we divide the problem into two cases:
Case 1. Stability of with .
The characteristic equation of linear system of system (3) is
where , , , , , , and .
For simplicity, Equation (9) is equivalent to where and .
When , the characteristic Equation (10) at becomes where , , , , and . Applying Routh-Hurwitz criterion for fractional-order differential equations [30], one can deduce that fractional-order system (11) is asymptotically stable when if and only if
When , multiplying on both sides of Equation (10), then one has
By replacing into Equation (13) and separating into its real and imaginary parts, we can obtain where and . Then, by solving Equation (14), we can get the following where , , , and .
By , then one has
Without loss of generality, we further suppose that Equation (16) has at least a positive root. By following from Equation (15), we can obtain
For the convenience of further analysis, we define the bifurcation point
By differentiating Equation (10) with respect to , we have and it implies
The detailed calculations can be found in Appendix A.
Theorem 4. Supposing . For system (3), the following results can be obtained. (i)If (H1), then the equilibrium is locally asymptotically stable when (ii)System (3) undergoes a Hopf bifurcation at when
Case 2. Stability of with .
Let , , , and , then we can obtain linearized system:
where and .
The characteristic system (21) becomes
where , , , , , , and .
Remark 5. Since Case 2 is quite similar to those for Case 1, we put the detailed mathematical derivation in Appendix B. Here, we only give the corresponding conclusions.
Theorem 6. Supposing . For system (3), the following results can be obtained. (i)The equilibrium point is locally asymptotically stable when (ii)System (3) undergoes a Hopf bifurcation at when
4. Numerical Simulation
In this section, we use the Adama-Bashforth-Moulton predictor-corrector method [31, 32] and show some numerical simulations to illustrate the analysis results in the previous sections. By choosing or as bifurcation parameters, we mainly analyse the Hopf bifurcation of system (3).
Case 1. To facilitate numerical simulations, the parameters take the values based on Ref. [16], as follows:
, , , , , , , , , , and .
By performing some calculations based on previous analysis, we can obtain . Figure 1 shows that bifurcation diagrams of system (3) for the parameters with , , and . The above parameters satisfy the conditions of Theorem 4; then, it demonstrates that is locally asymptotically stable when and fractional-order , as shown in Figures 2 and 3. Besides, as increases, the Hopf bifurcation will occur at the expense of system’s (3) stability. Figures 4 and 5 show that the solution of system (3) is unstable when . The phase diagram depicts that the system undergoes a stable limit cycle around zero equilibrium point , as shown in Figure 5.
When , although the system (3) is asymptotically stable with different values of the fractional order, the time required for the system to reach the stable state becomes long as the fractional-order value decreases, as shown in Figure 6 (can be found in Appendix C). It demonstrates that not only the time delay has an impact on the stability of the system, but the fractional order also plays an important role on the system stability.
Case 2. Fixed , , , , , , , , , , , and . By some calculations, we can obtain , , and . Furthermore, we can get nonzero equilibrium of the system (3). The above parameters satisfy the conditions of Theorem 6; then, it demonstrate that is locally asymptotically stable when and fractional-order , as shown in Figures 7 and 8. Besides, as increases, the Hopf bifurcation will occur at the expense of system’s (3) stability. Figures 9 and 10 show that the solution of system (3) is unstable when . The phase diagram depicts that the system undergoes a stable limit cycle around nonzero equilibrium point , as shown in Figure 10.
(a)
(b)
(c)
(d)
When the time delay is small (), the stability of the system can remain stable as the value of the fractional order gradually increases. However, when the time delay becomes big (), the nonzero equilibrium point of the system (3) changes from stable to unstable as the value of fractional order increases; finally, a periodic solution is generated, as shown in Figure 11 (can be found in Appendix C). It shows that the fractional order also plays an important role on the stability of the system.
(a)
(b)
5. Discussions
In Refs. [22–25], when introducing fractional orders to existing systems, the usual method is to directly replace the integer order with the fractional order. Accordingly, we follow the same processing method to introduce the fractional order into system (2). However, such an approach may ignore the relevant mechanics [33] of the actual system. The coupled systems [33] and damping models [34–36] using fractional derivatives had been successfully applied to many dynamic systems in mechanical engineering. Inspired by the above literatures, we introduce fractional order in the coupling term and the damping term in system (2); then, we have where , , , and denote the Grünwald-Letnikov fractional derivative.
Based on the additivity of the fractional-order derivative operator, we reduce the dimensionality of Equation (23), and Equation (23) then becomes
Let , , , , , , , , , then Equation (13) can be rewritten as where , , and denotes the Grünwald-Letnikov fractional derivative. It is clear to find that system (25) can be seen as a more general form of system (3), which better fits the actual model. Therefore, further analysis of the dynamics of system (25) will be our future work.
6. Conclusions
Bifurcation analysis can answer how do the behavior of the system change as the parameters change. In this work, we propose a fractional-order nonlinear electromechanical coupling system with delay. By choosing time delay as a bifurcation parameter, we, respectively, derive the conditions for occurrence of Hopf bifurcation around zero equilibrium and nonzero equilibrium . It is demonstrated that fractional-order and time delay have an important influence on electromechanical coupling system. By applying Adams-Bashforth-Moulton method, we implement some simulations to corroborate our analysis results.
For electromechanical coupling system, the unstable bifurcation may cause a destructive vibration. Our analysis results demonstrate that a combination of fractional order and time delay in the system have an important effect on the dynamic properties of the system. From the results of the numerical simulation, selecting the fractional order and and the feedback parameters can realize the switching of the dynamic properties of the system from a zero equilibrium point to a nonzero equilibrium point. Here, represents the ratio of the torsional stiffness of drive shaft to the moment of inertia (see [16]). Thus, our obtained results may have some implications for reducing the vibration of electromechanical coupled rolling mill systems.
Besides, it was found that the fractal derivative is the mathematical approximation of the fractional derivative in engineering applications, and the fractal derivative can describe the randomness of nonlinear systems (see [37–41]). Therefore, fractional-order systems may be the basis for many future studies. It will be our future work to study the randomness of rolling mill vibration by using fractional-order (or fractal derivative) mathematical modeling.
Appendix
A. The Expression of Parameters in Equation (8) and Equation (20)
B. Theoretical Derivation of Stability and Hopf Bifurcation of System (21)
For the convenience of further analysis, we denote and . Then, Equation (22) is equivalent to
Multiplying on both sides of Equation (B.1), then we get
By replacing into Equation (B.2) and separating into its real and imaginary parts, we can obtain where and . Then, by solving Equation (B.3), one obtains where , , , and .
From , then
Without loss of generality, we further suppose that Equation (B.5) has at least a positive root. By following from Equation (B.4), we can obtain
Furthermore, we define the bifurcation point .
By differentiating Equation (B.1) with respect to , we have
Then, it implies where
C. Waveform Diagrams of System (3) at and
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
This work is supported by the Fund for Shanxi 1331 Project Key Subjects Construction.