# Bifurcation Analysis of a Fractional-Order Delayed Rolling Mill’s Main Drive Electromechanical Coupling System

Authors: Rui Zhang; Jinbin Wang; Lifeng Ma
Journal: Advances in Mathematical Physics
 (2021)

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## 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 equilibriumsE00,0,0,0 and E1x1∗,0,x3∗,0. It is found that the fractional order α and time delay τ in the system play an important role on the system stability. Finally, numerical simulations are given to verify the analytic results.


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## Body

## 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:
(1)J1φ¨1+Kφ1−φ2+Cφ̇1−φ̇2+Ceφ̇1−Keφ1=k0+k2φ12+k3φ13J2φ¨2−Kφ1−φ2−Cφ̇1−φ̇2=TL+gclφ2t−τ1+gc2φ23t−τ2,where Jii=1,2 is the moment of inertia, φii=1,2 and φ̇ii=1,2 are the angle of rotation and rotational speed, respectively, K is the torsional stiffness of drive shaft, C is the shafting damping coefficient, and Ce 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):
(2)ẋ1=x2,ẋ2=−ω12x1−b1x2+a1x3+d1x4+l1x12+l2x13,ẋ3=x4,ẋ4=a2x1+d2x2−ω22x3−b2x4+β1x3t−τ1+β2x33t−τ2,where ω12=K−Ke/J1b1=C+Ce/J1,a1=K/J1,d1=C/J1,l1=k2/J1,l2=k3/J1,ω22=K/J2,b2=C/J2,a2=K/J2,d2=C/J2,β1=gc1/J2,β2=gc2/J2. 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
(3)Dαx1=x2,Dαx2=−ω12x1−b1x2+a1x3+d1x4+l1x12+l2x13,Dαx3=x4,Dαx4=a2x1+d2x2−ω22x3−b2x4+β1x3t−τ1+β2x33t−τ2,where Dα represents the Caputo fractional derivative and α is the fractional and derivative order of damping, which may represent the nonlocal displacement effects of dissipation of energy (internal friction). τ1 and τ2 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 conclusionsThe layout of this work is organized as follows: In Section2, 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 orderαα>0 is defined as
(4)D0αfx=dnfxdxn,α=n∈N+,1Γn−α∫0xx−tn−α−1fntdt,0<n−1≤α≤n,where n is an integer, x>0.Theorem 2 [19, 27] (the conditions of stability).
Consider the following fractional-order system:(5)D0αxt=ft,x0=x0,where 0<α<1. The equilibrium points of the above system are the solutions of fx=0. An equilibrium point is locally asymptotically stable if all eigenvalues λi of the Jacobian matrix J=∂f/∂x evaluated at the equilibrium point satisfy argλi>απ/2. Further, if argλi=απ/2, then the system undergoes a Hopf bifurcation.Theorem 3 [28, 29].
Consider the followingn-dimensional fractional-order system with delay:
(6)D0αxjt=fjx1t,x2t,⋯,xnt,τ,j=1,2,⋯,n,where 0<α<1 and time delay τ≥0. System (6) undergoes a Hopf bifurcation at the equilibrium point x∗=x1∗,x2∗,⋯,xn∗ when τ=τ0 if the following conditions are satisfied:
(i)
All the eigenvaluesλjj=1,2,⋯,n of the coefficient matrix A of the linearized system of (7) with τ=0 satisfy argλi>απ/2(ii)
The characteristic equation of the linearized system of (7) has a pair of purely imaginary roots ±ω0 when τ=τ0(iii)
Redsτ/dτω=ω0,τ=τ0≠0


## 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 E00,0,0,0 and a nonzero equilibrium point E1x1∗,0,x3∗,0. The zero equilibrium point E1x1∗,0,x3∗,0 satisfies the following equations:
(7)x2=0,−ω12x1+a1x3+l1x12+l2x13=0,x4=0,a2x1−ω22x3+β1x3+β2x33=0.Then, by a direct computation, we can derive thatx1 satisfies the following equation:
(8)m1x18+m2x17+m3x16+m4x15+m5x14+m6x13+m7x12+m8x1+m9=0.The detailed expression of parameters (mi,i=1⋯9) 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 x1∗>0. Noticing the fact that x3=1/a1ω12x1−l1x12−l2x13, we can calculate x3∗. Furthermore, if we suppose x3∗>0, then system (3) has a nonzero equilibrium point E1x1∗,0,x3∗,0.Next, we, respectively, investigate the stability of system (3) around equilibrium points E00,0,0,0 and E1x1∗,0,x3∗,0. In what follows, we divide the problem into two cases:Case 1.
Stability ofE0=0,0,0,0 with τ1≥0,τ2=0.
The characteristic equation of linear system of system (3) is
(9)s4α+e1s3α+e2s2α+e3sα+e4+f1s2α+f2sα+f3e−τ1s=0,where e1=b1+b2, e2=b1b2+ω12+ω22−d1d2, e3=b2ω12+b1ω22−d1a2−d2a1, e4=ω12ω22−a1a2, f1=−β1, f2=−b1β1, and f3=−β1ω12.For simplicity, Equation (9) is equivalent to
(10)Ps+e−τ1sQs=0,where Ps=s4α+e1s3α+e2s2α+e3sα+e4 and Qs=f1s2α+f2sα+f3.Whenτ1=0, the characteristic Equation (10) at E0=0,0,0,0 becomes
(11)Fλ=λ4+h1λ3+h2λ2+h3λ+h4,where λ=sα, h1=e1, h2=e2+f1, h3=e3+f2, and h4=e4+f3. Applying Routh-Hurwitz criterion for fractional-order differential equations [30], one can deduce that fractional-order system (11) is asymptotically stable when τ1=0 if and only if
(12)H1:hi>0i=1,2,3,4,h1h2−h3>0,h1h2h3−h4h12−h32=0.Whenτ1≠0, multiplying esτ1 on both sides of Equation (10), then one has
(13)Pseτ1s+Qs=0.By replacings=iω=cosπ/2+isinπ/2ω into Equation (13) and separating into its real and imaginary parts, we can obtain
(14)p1ωcosωτ1−p2ωsinωτ1+q1ω=0,p2ωcosωτ1+p1ωsinωτ1+q2ω=0,where Piω=p1ω+ip2ω and Qiω=q1ω+iq2ω. Then, by solving Equation (14), we can get the following
(15)cosωτ1=−p1ωq1ω−p2ωq2ωp12ω+p22ω≐H1ω,sinωτ1=−p1ωq2ω+p2ωq1ωp12ω+p22ω≐H2ω,where p1ω=ω4αcos2πα+e1cos3πα/2ω3α+e2ω2αcosπα+e3cosπα/2ωα+e4, p2ω=ω4αsin2πα+e1sin3πα/2ω3α+e2ω2αsinπα+e3sinπα/2ωα, q1ω=f1ω2αcosπα+f2cosπα/2ωα+f3, and q2ω=f1ω2αsinπα+f2sinπα/2ωα.Bycos2θ+sin2θ=1, then one has
(16)H1ω2+H2ω2=1.Without loss of generality, we further suppose that Equation (16) has at least a positive root. By following from Equation (15), we can obtain
(17)τk1=1ωarccos−p1ωq1ω−p2ωq2ωp12ω+p22ω+2π,k=0,1,2,⋯.For the convenience of further analysis, we define the bifurcation point(18)τ10=minτk1,k=0,1,2⋯,.By differentiating Equation (10) with respect to τ, we have
(19)P′sdsdτ+Q′sdsdτe−sτ+Qs−τdsdτ−se−sτ=0,dsdτ=sQse−sτP′s+Q′s−τQse−sτ=AsBs,and it implies
(20)Redsdτω=ω0,τ=τ10=A1B1+A2B2B12+B22.The detailed calculations can be found in AppendixA.Theorem 4.
SupposingA1B1+A2B2/B12+B22≠0. For system (3), the following results can be obtained.
(i)
If (H1), then the equilibriumE0,0,0,0 is locally asymptotically stable when τ∈0,τ10(ii)
System (3) undergoes a Hopf bifurcation at E0,0,0,0 when τ=τ10Case 2.
Stability ofE1x1∗,0,x3∗,0 with τ1=τ2=τ.
LetX1=x1−x1∗, X2=x2, X3=x3−x3∗, and X4=x4, then we can obtain linearized system:
(21)DαX1=X2,DαX2=n1X1−b1X2+a1X3+d1X4,DαX3=X4,DαX4=a2X1+d2X2−ω22X3−b2X4+n2X3t−τ,where n1=−ω12+2l1x1∗+3l2x1∗2 and n2=β1+3x3∗2β2.
The characteristic system (21) becomes
(22)s4α+ε1s3α+ε2s2α+ε3sα+ε4+ζ1s2α+ζ2sα+ζ3e−τs=0,
whereε1=b1+b2, ε2=b1b2−n1+ω22−d1d2, ε3=−b2n1+b1ω22−d1a2−d2a1, ε4=−n1ω22−a1a2, ζ1=−n2, ζ2=−b1n2, and ζ3=n2n1.Remark 5.
Since Case2 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.
SupposingC1D1+C2D2/D12+D22≠0. For system (3), the following results can be obtained.
(i)
The equilibrium pointE1x1∗,0,x3∗,0 is locally asymptotically stable when τ∈0,τ0(ii)
System (3) undergoes a Hopf bifurcation at E1x1∗,0,x3∗,0 when τ=τ0


## 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 τ1 or τ2 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:
a1=6, a2=4.5, b1=2, b2=0.5, d1=1, d2=0.5, ω12=7, ω22=5, l1=0.5, l2=−0.1, β1=0.8 and β2=0.4.
By performing some calculations based on previous analysis, we can obtainτ10=5.1257. Figure 1 shows that bifurcation diagrams of system (3) for the parameters a2∈2,4.5 with α=0.95, τ1=0, and h=2−6. The above parameters satisfy the conditions of Theorem 4; then, it demonstrates that E00,0,0,0 is locally asymptotically stable when τ1=4.8<τ10 and fractional-order α=0.95, as shown in Figures 2 and 3. Besides, as τ1 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 τ1=5.13>τ10. The phase diagram depicts that the system undergoes a stable limit cycle around zero equilibrium point E00,0,0,0, as shown in Figure 5.
Whenτ1=1, 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.
Fixedα=0.98, a1=6, a2=5, b1=2, b2=0.5, d1=1, d2=0.5, ω12=7, ω22=5, l1=0.5, l2=−0.1, β1=1 and β2=−0.04. By some calculations, we can obtain x1∗=3.6196, x3∗=3.9215, and τ0=1.0750. Furthermore, we can get nonzero equilibrium E13.6196,0,3.9215,0 of the system (3). The above parameters satisfy the conditions of Theorem 6; then, it demonstrate that E13.6196,0,3.9215,0 is locally asymptotically stable when τ=1<τ0 and fractional-order α=0.98, as shown in Figures 7 and 8. Besides, as τ 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 τ=1.2>τ0. The phase diagram depicts that the system undergoes a stable limit cycle around nonzero equilibrium point E13.6196,0,3.9215,0, as shown in Figure 10.Figure 1
Bifurcation diagrams of system (3) for the parameters a2∈2,4.5 with α=0.95 and h=2−6.
(a)(b)(c)(d)Figure 2
The system (3) at E00,0,0,0 is locally asymptotically stable, with α=0.95, τ1=4.8<τ10, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 3
Phase portraits of system (3) at E00,0,0,0 with α=0.95, τ1=4.8<τ10, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 4
The system (3) at E00,0,0,0 is unstable, with α=0.95, τ1=5.13>τ10, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 5
Phase portraits of system (3) at E00,0,0,0 with α=0.95, τ1=5.13>τ10, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 6
Waveform diagrams of system (3) at E00,0,0,0 with τ=1, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 7
The system (3) at E13.6196,0,3.9215,0 is locally asymptotically stable, with α=0.98, τ=1<τ0, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 8
Phase portraits of system (3) at E13.6196,0,3.9215,0 with α=0.98, τ=1<τ0, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 9
The system (3) at E13.6196,0,3.9215,0 has a stable periodic solution, with α=0.98, τ=1.2>τ0, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.Figure 10
Phase portraits of system (3) at E13.6196,0,3.9215,0 with α=0.98, τ=1.2>τ0, h=0.05, and x10,x20,x30,x40=0.15,0.15,0.2,0.2.When the time delay is small (τ=1), the stability of the system can remain stable as the value of the fractional order gradually increases. However, when the time delay becomes big (τ=4), 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.Figure 11
Waveform diagrams of system (3) at E13.6196,0,3.9215,0 with h=0.05 and x10,x20,x30,x40=0.15,0.15,0.2,0.2. (a) τ=1; (b) τ=4.
(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 Cφ̇1−φ̇2 and the damping term Ceφ̇1 in system (2); then, we have
(23)J1φ¨1+Kφ1−φ2+CDαφ1−C′Dβφ2+CeDαφ1−Keφ1=k0+k2φ12+k3φ13,J2φ¨2−Kφ1−φ2−CDαφ1+C′Dβφ2=TL+gclφ2t−τ1+gc2φ23t−τ2,where α,β∈0,1, α≠β, Dα, and Dβ 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
(24)Dαφ1=φ3,D2−αφ3=φ¨1=1J1−Kφ1−φ2−Cφ3+C′φ4−Ceφ3+Keφ1+k0+k2φ12+k3φ13,Dβφ2=φ4,D2−βφ4=φ¨2=1J2Kφ1−φ2+Cφ3−C′φ4+TL+gc1φ2t−τ1+gc2φ23t−τ2.Letα1=α, α2=2−α, α3=β, α4=2−β, φ1=x1, φ3=x2, φ2=x3, φ4=x4, ω12=K−Ke/J1b1=C+Ce/J1,a1=K/J1,d1′=C′/J1,l1=k2/J1,l2=k3/J1,ω22=K/J2,b2′=C′/J2,a2=K/J2,d2=C/J2,β1=gc1/J2,β2=gc2/J2, then Equation (13) can be rewritten as
(25)Dα1x1=x2,Dα2x2=−ω12x1−b1x2+a1x3+d1′x4+l1x12+l2x13,Dα3x3=x4,Dα4x4=a2x1+d2x2−ω22x3−b2′x4+β1x3t−τ1+β2x33t−τ2,where αii=1,3∈0,1, αii=2,4∈1,2, and Dαii=1,2,3,4 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 delayτ as a bifurcation parameter, we, respectively, derive the conditions for occurrence of Hopf bifurcation around zero equilibrium E00,0,0,0 and nonzero equilibrium E1x1∗,0,x3∗,0. It is demonstrated that fractional-order α and time delay τ 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 orderα and time delay τ 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 α and a2 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, a2 represents the ratio of the torsional stiffness of drive shaft K to the moment of inertia J2 (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.



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