# Nonlinear Torsional Vibration Analysis and Control of Semidirect Electromechanical Coupling Transmission System in Shearer

Authors: Song Jiang; Wei Li; Lianchao Sheng; Jiajun Chen; Min Li
Journal: Shock and Vibration
 (2019)

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## Abstract

The nonlinear torsional vibration and instability oscillation caused by nonlinear damping in the shearer electromechanical coupling cutting transmission system in shearer driven by the permanent magnet synchronous motor (PMSM) is investigated in this paper. The electromechanical coupling transmission system in the shearer is equivalent to a concentrated mass model for the purpose of establishing the system dynamic model by the Lagrange–Maxwell equation. Then, the Routh–Hurwitz criterion is used to determine the torsional vibration critical point and stability region for the Hopf bifurcation for the cutting transmission system. According to the Routh–Hurwitz stability criterion, the Hopf bifurcation type and the effect of the supercritical Hopf bifurcation in the torsional vibration of the cutting transmission system are analyzed. Furthermore, based on the washout filter, the Hopf bifurcation controller is designed for suppressing the transmission system’s large vibration amplitude and unstable oscillation. In addition, the influences of the linear gain and nonlinear gain on the bifurcation point and the limit cycle amplitude are discussed. Finally, the numerical simulation results indicate the effectiveness of the designed controller. The research achievements can provide a theoretical basis for design or optimize the cutting transmission system of high-reliability shearer driven by PMSM.


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## Body

## 1. Introduction

In recent years, with the continuous development of coal mining mechanization, it requires higher requirements for the fully mechanized mining equipment in high efficiency and high reliability [1, 2]. As an integral part of fully mechanized mining equipment, the shearers may easily break down due to the terrible working environment. Currently, the traditional main transmission mode of the cutting transmission system is “three-phase asynchronous motor + three-stage straight tooth retarder + planetary gear reducer + cutting drum,” which has a long transmission chain and many transmission components. The long-chain transmission mode is prone to failure [3–5]. Therefore, a semidirect driving system [6, 7] is proposed in this paper, which consists of PMSM, three-stage straight tooth retarder, and cutting drum, and is shown in Figure 1.Figure 1
Schematic diagram semidirect cutting transmission system.Comparing to the original system, the electromagnetic torque and load torque will greatly increase due to the system reduction ratio that is reduced by removing the planetary gear [8, 9], which is prone to torsional vibration of the cutting transmission system. Therefore, it has a great significance to research the torsional vibration of the cutting transmission system. Fortunately, this multimass elastic torsional system like the shearer cutting section has been researched by many domestic and foreign scholars who mainly study from the perspectives of natural frequency and disturbance. It is found that the nonlinear factors are the main cause of torsional vibration, and furthermore, bifurcation and chaos may occur in the system due to some nonlinear factors [10–12]. On the coupled bending and torsional vibration model of rubbing rotor, with rotational speed as the bifurcation parameter, the bifurcation and chaotic behavior of the rotor system are studied by Al-Bedoor [13]. Liu et al. [14] studied the static bifurcation characteristics and rotor system stability with time-delay nonlinear parameters, and then the vibration was controlled with the delay feedback. In [15], the self-excited oscillations caused by nonlinear factors in the power system are analyzed. And the system Hopf bifurcation motion was researched with nonlinear mode and Floquet theory with different resistance parameters. The mechanical rotor system Hopf bifurcation was researched by Yuan [16], which was frequently encountered in the mechanical structure, and may cause the shafting instability and damage. Saigo et al. [17] studied the suppression of forced vibration of the concentrated mass torsion system by a wave absorption. The results demonstrate that smaller mass and larger spring stiffness of an imaginary system absorb vibration energy better.However, the cutting transmission system in shearer not only includes the motor but also mechanical parts, which belongs to the electromechanical coupling system. Due to the effect of electromechanical coupling, the cutting transmission system will exhibit complex dynamic behavior [18–20]. Therefore, the torsional vibrations of the system cannot be interpreted rationally, neither from electrical nor from mechanical aspect. It is important to establish an accurate model of the electromechanical coupling system to further analyze the torsional vibration dynamic characteristics in the semidirect transmission system of the shearer and control the torsional vibration from the aspect of motor [21]. Gustavsson and Aidanpää [22] established the model of electromagnetic excitation produced by the uniform magnetic field, and the influence of nonlinear magnetic pull on rotor vibration of the hydrogenerator unit was discussed. Szolc et al. [23] researched the dynamic electromechanical coupling effects in machine drive systems driven by asynchronous motors, and the electromagnetic stiffness and damping coefficients of the interaction between motor and mechanical system were determined by means of the analytical method. Ju et al. [24] constructed the global electromechanical-coupling dynamic models for the main transmission system of the scraper conveyor, and the Hopf bifurcation characteristic is analyzed, which was caused by sliding friction. In view of the above mentioned, Hopf bifurcation is a significant factor affecting the stability of the electromechanical coupling transmission system. However, the dynamic analysis of the electromechanical coupling system mainly focus on the induction motor, and the PMSM are rarely considered. It is a remarkable fact that the damping of the semidirect transmission shafting would change with the shafting vibration, the shaft fatigue, the wear, and lubrication of the bearing [25, 26]. Torsional vibrations of the cutting transmission system caused by shaft damping variations would further lead to the shafting destabilization [27, 28].Different from the above references, this paper analyses the effect of Hopf bifurcation produced by variable damping in shearer semidirect transmission system driven by the PMSM. Furthermore, based on the washout filter, the Hopf bifurcation controller is designed for suppressing the transmission system’s unstable oscillation. Finally, this paper is organized as follows. The electromechanical-coupling dynamics modelling for the main transmission system of the shearer driven by the high-torque and low-speed PMSM is provided in Section2. The Hopf bifurcation characteristics analysis for the system torsional vibration and numerical simulation verification is given in Section 3. The washout filter controller for the system torsional vibration is designed, and numerical simulation verification is given in Section 4. Finally, the conclusions are drawn in Section 5.


## 2. Electromechanical Coupling Torsional Vibration Modelling

The semidirect cutting transmission system of the shearer driven by PMSM mainly consists of PMSM, three-stage gear, shafting, cutting drum, and so on. There are both resilient elements with small quality and inertia with large mass. Therefore, it is a multimass spring damping system with electromechanical coupling. To analyze the coupling dynamic characteristic of the electrical parameters and mechanical parameters, the PMSM and the cutting drum are regarded as the power source and load system, respectively, and ignore the effect of the reducer gear clearance. As a protective shaft for PMSM, the elastic torque shaft is more vulnerable to damage due to the relief groove and the low torsional strength. However, all other components have the feature of large mass and the small elasticity. Hence, the effect of the elastic torque shaft is only considered, and the other shaft and gear are equivalent to the PMSM and cutting drum. Then, the transmission system is equivalent to the two-mass spring torsional vibration system with resilient elements (PMSM and cutting drum) and inertia element (elastic torque shaft), as shown in Figure2. To simplify the analysis, take the following assumptions: (1) neglecting the iron core saturation effect in the PMSM; (2) no consideration of eddy current and hysteresis loss in the PMSM; and (3) the motor stator winding current is a three-phase symmetric sine current.Figure 2
Electromechanical coupling torsional vibration physical model of cutting transmission shafting.In Figure2, J1 is the PMSM inertia; J2 is the sum of the moment of the cutting drum and the transformation of the gear and other elements inertia; ia, ib, and ic are the three-phase stator current of the PMSM, respectively; K and C are the torsional stiffness and damping coefficient of the elastic torque shaft; Tm and TL are the electromagnetic torque of the PMSM and load torque of cutting drum; and θ1 and θ2 are the mechanical angle of the PMSM shaft and cutting drum shaft.Two mass systems are modelled by the Lagrange equation [29]. The kinetic energy of the shearer cutting transmission shafting mainly includes the PMSM rotational kinetic energy and the cutting drum rotational kinetic energy, which can be expressed as follows:(1)E=E1+E2=12J1θ12+12J2θ22.The system potential energy can be represented as(2)V=12Kθ1−θ22.As a result of the downhole environment, water is sprayed to reduce the dust concentration, so the cutting environment is wet. In the process of cutting coal, system will produce nonlinear sliding friction forces due to relative displacement of cutting drum and coal seam. In view of the linear damping and nonlinear stick, slip frictional force is studied [26], which can be shown as follows:(3)F1c=−Cθ˙1−θ˙2−C′θ˙1−θ˙23,F2c=−Cθ˙2−θ˙1−C′θ˙2−θ˙13.The nonconservative generalized force of the electromagnetic system is electromagnetic torque and damping force. And the nonconservative generalized force of the mechanical part is the external load and damping force, and it are expressed as(4)Qi=∑j=12Fjj∂θj∂qi,i=1,2,where ∂qii=1,2 is the generalized coordinate. Substituting equation (3) into equation (4) yields the generalized torque:(5)Q1=Tm−Cθ˙1−θ˙2−C′θ˙1−θ˙23,Q2=−TL−Cθ˙2−θ˙1−C′θ˙2−θ˙13.The system Lagrange–Maxwell equation is(6)ddt∂L∂q˙i−∂L∂q˙i+∂F∂q˙i=Qi.For the mechanical system, the dynamic equation of the PMSM can be obtained by taking the rotation angleθ1 of the PMSM as the generalized coordinates (q1) and substituting equations (1)∼(5) into equation (6):(7)J1θ¨1+Kθ1−θ2+Cθ˙1−θ˙2+C′θ˙1−θ˙23=Te.Similarly, the dynamic equation of the cutting drum is(8)J2θ¨2−Kθ1−θ2−Cθ˙1−θ˙2−C′θ˙1−θ˙23=−TL.The mathematical model of PMSM inabc three-phase stationary coordinate system can be obtained [30] as follows:(9)ua=Rsia+ddtLsia+ψfa,ub=Rsib+ddtLsib+ψfb,uc=Rsic+ddtLsic+ψfc.Based on the electromagnetic field theory, electromagnetic torqueTe is equal to the deflection of magnetic energy storage on the mechanical angular displacement θ1. ψa,ψb,andψc are the magnetic chain of three-phase winding. ψfa,ψfb,andψfc are the magnetic chain produced by a permanent magnet on the three-phase winding, and p is the number of pole pairs:(10)Te=12p∂∂θ1iaψa+ibψb+icψc=12p∂∂θ1Lsia+ψfa+Lsib+ψfb+Lsic+ψfc.By introducing the Clark transform and Park transform, the stator current of the PMSM can be transformed from the three-phase stationary coordinate system (abc) to two-phase rotating coordinate system (dq). Equation (9) can be converted to(11)uduq=23sinθesinθe−2π3sinθe+2π3cosθecosθe−2π3cosθe+2π3uaubuc=Rdid+Lddiddt−pω1LqiqRqiq+Lqdiqdt+ω1Ldid+ω1ψf,where p is the number of pole pairs, Rd=Rq=R, id and iq are the direct and quadrature axis current in the dq coordinate system, Ld and Lq are the direct and quadrature axis inductor in the dq coordinate system and Ld=Lq in stick-type PMSM, ψf is the magnetic chain of the permanent magnets, and θe is the electric mechanical angular displacement.Based on the electromagnetic field theory, in the two-phase rotating coordinate system (dq), the electromagnetic torqueTe is(12)Te=32piqidLd−Lq+ψf=32pψfiq.When the load torque and the electromagnetic torque are balanced, the mechanical angular velocity of the rotor is constant; therefore,θ1=∫Ωdt+φ1, θ2=∫Ωdt+φ2, θ1−θ2=φ1−φ2, and J1φ1+J2φ2=0 [31], so(13)θ1−θ2=φ1−φ2=J1+J2J2φ1,ω1=θ˙1=Ω+φ˙1=Ω+J2J1+J2θ˙1−θ˙2.Finally, substituting equation (13) into equation (11) and combing with equations (7), (8), and (12), the electromechanical coupling torsional vibration model of the cutting transmission shafting is(14)I˙d=−RLdId+LqLdIqΩ+J2J1+J2θ˙1−θ˙2p+udLd,I˙q=−RLqIq−LdLqIdΩ+J2J1+J2θ˙1−θ˙2p−ψfLqΩ+J2J1+J2θ˙1−θ˙2+uqLqp,θ¨1+KJ1θ1−θ2+CJ1θ˙1−θ˙2+C′J1θ˙1−θ˙23=−3p2J1ψfIq,θ¨2−KJ2θ1−θ2−CJ2θ˙1−θ˙2−C′J2θ˙1−θ˙23=−TLJ2.In equation (13), Id,Iq,θ1,andθ2 are variable items which will change with time. This system will exhibit complex dynamical behavior due to the variation of the damping C.


## 3. Cutting Transmission System Bifurcation Analysis

For the convenience of the calculation, the difference value of two angles (θ1,θ2) is used to describe the strength of the vibration. Therefore, let the third type minus the fourth type in equation (14) merge nonvariable items in the system, defined as follows:(15)x1=Id,x2=Iq,x3=θ1−θ2,x˙3=x4=θ˙1−θ˙2,x˙4=θ¨1−θ¨2,a=−RLd,b=ΩpLqLd,c=pLqLdJ2J1+J2,d=−ΩpLdLq,e=−RLq,f=−pLdLqJ2J1+J2,g=−pψfLqJ2J1+J2,h=−3pψf2J1,m=−KJ1+KJ2,k=−C′J1+C′J2,s=uqLq−ΩpψfLq,n=−CJ1+CJ2,l=udLd,z=TL,where x1 is the quadrature axis current in the dq coordinate system, x2 is the direct axis current in the dq coordinate system, x3 is the difference between the mechanical angle of the PMSM and the angle of the cylinder, and x4 is x3’s derivative of time.Therefore, equation (14) is simplified to(16)x˙=gx,C,x˙1=ax1+bx2+cx2x4+l,x˙2=dx1+ex2+fx1x4+gx4+s,x˙3=x4,x˙4=hx2+mx3+nx4+kx43+z.After the cutting transmission system of the shearer is changed to semidirect drive, the damping will change in the following cases: (1) the intensity of the vibration in the shaft system and (2) the wearing and the lubrication condition of the shaft. In addition, there may be oscillation vibration and Hopf bifurcation with the damping changes in the shafting, so the dampingC is selected as the bifurcation parameter.The equilibrium point of equation (16) can be transferred to the coordinate origin through simple linear transformation. Therefore, the dynamic characteristics of the system balance at the origin are of universal significance without loss of generality. Hence, let l=0,s=0,andz=0, and then equilibrium point can be transferred to the coordinate origin x0=0,0,0,0 for equation (16). The Jacobian matrix and nonlinear terms of the system at x0, respectively, are as follows:(17)Ax0,C=ab00de0g00010hmn,fx0,C=cx2x4fx1x40kx43.According to the Hopf bifurcation theory, the necessary and sufficient conditions for Hopf bifurcation are expressed as follows: there are only one pair of conjugate imaginary roots in the characteristic root ofAx0,C and the other n−2 characteristic roots have negative real parts. The system is unstable at the origin when positive real part exists. While the eigenvalues have all negative real parts, the system is asymptotically stable. In order to simplify the calculation, the Hurwitz criterion is used to determine the existence of the Hopf bifurcation in this paper.The characteristic polynomial ofAx0,C can be expressed as(18)λ4+p1λ3+p2λ2+p3λ1+p4=0,where p1=−n−e−a,p2=ae−m−bd+an−gh+en, and p3=am+em+agh−aen+bdn,p4=bdm−aem.Hurwitz criterion can be shown as(19)piC∗>0,i=1,2,3,4,ΔiC∗>0,i=1,2,3,Δ3C∗=0,dΔ3CdCC=C∗≠0.In equation (19), Δi is the order principal minor determinant of equation (18):(20)Δi=p1p3p5⋯p2i−11p2p4⋯p2i−20p1p3⋯p2i−3⋮⋮⋮⋮⋮001⋯pi,where the possible threshold is derived from Δ3C∗=0. And if the resulting point C∗ satisfies the other requirements of equation (19) at the same time, the Hopf bifurcation occurs at this point. The parameters in the model are selected according to the data in the actual system shearer. Therefore, the main parameters of semidirect cutting transmission system in the form are as follows:(21)J1=1.96kg⋅m2,J2=1.425kg⋅m2,Rd=Rq=R=0.136Ω,Ld=Lq=0.0284H,ψf=1.235v⋅s,Ω=5rad/s,K=4100N⋅m/rad,where Ω is the rated speed of PMSM, J1 is the moment of inertia for PMSM, J2 is the sum of the moment of the cutting drum and the transformation of the gear and other elements inertia, and K is the stiffness coefficient of the elastic torque shaft. R, Ld, Lq, and ψf can be obtained, respectively, by off-line identification of PMSM.By calculation, the couple transmission system of the shearer semidirect drive met equation (19) at C∗=41.4675. In this case, the characteristic root of Ax0,C has one pair of conjugate imaginary roots. The pure virtual root is ω0i, and vector U and W are right and left feature vectors corresponding to the characteristic root Ax0,C∗, respectively. Their relationship can be expressed as(22)UA=ω0iU,AW=ω0iW,UW=1.Through equation (22), the stability index b of the Hopf bifurcation is defined as(23)b=Re−UfxxxWWW∗+2UfxxWA−10fxxWW∗+UfxxW∗×A0−2iω0I−1fxxWW.In equation (23), W∗ is the conjugate plural of W, and A0=Ax0,CC=C∗:(24)fxxxWWW∗=∂∂x∂∂x∂fx,C∂x×WWWC=C∗,x=x0.According to the value ofbC∗, the stability of periodic solutions of Hopf bifurcation in the original system can be determined. If bC∗<0, the bifurcation type of the system is subcritical Hopf bifurcation, and the periodic solution is unstable. On the contrary, bC∗>0, the bifurcation type of the system is supercritical Hopf bifurcation, and the bifurcated periodic solution is stable. The stability index of Hopf bifurcation is bC∗=0.00174>0 through calculation, and therefore, the type of Hopf bifurcation is supercritical Hopf bifurcation.The Runge–Kutta method is used to verify the dynamic characteristics generated by the system Hopf bifurcation pointC∗, and the parameters are brought into equation (16). In Figure 3 of torsional vibration amplitude figure variation with the damping C, the system exhibits different dynamics on both sides of the bifurcation point because of supercritical Hopf bifurcation. When C is greater than the bifurcation point C∗, the running state of the main transmission system is stable and no limit cycle appears, as shown in Figure 4.Figure 3
The torsional vibration amplitude curve with the dampingC.Figure 4
The time history of the system atC=48.WhenC is smaller than the bifurcation point C∗, the system exhibits a stable limit cycle motion. Figure 3 shows that when the C is less than the critical value C∗, the smaller the value of the damping C, the bigger the limit cycle. At the same time, when the damping is greater than the critical damping, the amplitude of the vibration is close to 0. It can be available from the comparison of time-history response of identical initial value in Figure 5, and the amplitude of the limit cycles is obviously larger in the case of smaller damping. Furthermore, when C is fixed, the amplitude of the limit cycle has nothing to do with the initial value, as shown in Figure 6, which shows that the amplitudes are basically the same under two different initial values. From the above analysis, the conclusion of the simulation results is consistent with the theoretical analysis.Figure 5
Phase trajectory contrast diagram of the system with different dampings: (a)C=20; (b) C=10.
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(b)Figure 6
Comparison of amplitudes of the system at different initial values:C=20.
(a)
(b)


## 4. Cutting Transmission System Vibration Control

The numerical simulation results in Figures3 and 5 show that, the system oscillates and the limit cycle appears when the damping C is smaller than C∗. When the system responds to a smaller limit cycle, it has little harm to the elastic torque shaft. However, when the external conditions cause damping C to be smaller, the amplitude of the limit cycle will increase. At this time, the cutting transmission shaft system of the shearer will have strong vibration, which will cause great harm to the fatigue damage of the elastic torque shaft. In order to control the unstable oscillation of shafting, the washout filter controller is introduced, which is a kind of the extended linear or nonlinear state feedback method.The state equation of the washout filter can be represented as(25)ω˙=α−εω,where ω is the state variables of the washout filter, α is the input variables, ε is the time constant, and α−εω is the output variable. In this paper, from the perspective of implementation, x2 is used as the input variable, and x5 is used as the state variables. In nonlinear systems with Hopf bifurcation, the squared term and the cubic term can affect the stability of Hopf bifurcation. So, nonlinear feedback controller can be shown as w=kmx2−εx5+knx2−εx53; km and kn are the linear gains and nonlinear control gain, respectively. w is added to the second term in equation (16) as the output feedback.So, the controlled system can be shown as(26)x˙1=ax1+bx2+cx2x4,x˙2=dx1+ex2+fx1x4+gx4+w,x˙3=x4,x˙4=hx2+mx3+nx4+kx43,x˙5=x2−εx5.The Jacobian matrix and nonlinear terms of the control system can be represented as(27)Bx0,C=ab000de+km00−εkm000100hmn00100−ε,hx0,C=cx2x4fx1x4+knx2−εx530kx430.Ifε=0.1,km≠0,kn=0,andC=41.4675, the washout filter will not change the position of the equilibrium point. And the characteristic equation of Bx0,C in equation (26) is(28)λ5+p1λ4+p2λ3+p3λ2+p4λ+p5=0,where p1=59.93−km,p2=−55.05km+5972,p3=−5210km+124400,p4=−23800km+8077000,andp5=806400.If all eigenvalues of matrixBx0,C have the negative real part, the controlled system will not occur Hopf bifurcation at C∗, which means Δi>0i=1,2,3,4,km<−0.006597. When ε=0.1, km=−1, and kn=0, the roots of Bx0,C have negative real parts at this time, and the system is asymptotically stable, as shown in Figure 7(b). At this moment, the bifurcation characteristics are controlled at the Hopf bifurcation point C∗, but the other dynamics of the system have not been determined.Figure 7
Comparison of phase trajectory between original system and controlled system at the critical damping point: (a) original system; (b) controlled system.
(a)
(b)According to the Hopf bifurcation theory, when Hopf bifurcation occurs at the equilibrium point, system (26) must satisfy the condition of equation (29). The relationship between the linear control gain km and the bifurcation critical point C∗ can be drawn, which is shown in Figure 8. When km=0, the controlled system has Hopf bifurcation same as the original system at C=41.4675. And if km=−1, the controlled system has the Hopf bifurcation point at C=18.09. Comparing to the original system, the system stability domain increases greatly. Therefore, when the linear gain part of the controller selects the appropriate value, the bifurcation characteristics of the nonlinear system can be effectively controlled and the bifurcation point is transferred to the safe working area:(29)piC∗>0,i=1,2,3,4,5,ΔiC∗>0,i=1,2,3,4,Δ4C∗=0,dΔ4CdCC=C∗≠0.Figure 8
The curve diagram of the bifurcation critical valueC∗ with the linear gain km.Takeε=0.1,km=−1,andkn≠0, and the Hopf bifurcation of the controlled system occurs at C=18.09 through calculation. The Jacobian matrix Bx0,C has a pair of pure imaginary roots λ1,2=±50.5163i, whose corresponding eigenvectors can be expressed as(30)φ=φ¯=0−5.2989−2.6000i0.0362+0.0178i0.6615+0.0432i0.0008+0.0003i.Since it is more complicated to calculate the three-order normal form of the controlled system through the central manifold theorem, the direct calculation method of normal form in [32] can be used. This method only needs to calculate the pure virtual root of the controlled system and its corresponding eigenvector, and then the normal form of the controlled system can be obtained through the nonlinear transformation. The linear transformation is introduced here:(31)X=φu+φ¯u¯+∑j+k≥2Hjkuju¯k.The Hopf bifurcation specification of the controlled system can be shown as(32)u˙=50.5163iu+Eu2u¯.The nonlinear transformation is introduced to the nonlinear terms, and then the polynomial form ofu and u¯ as follows:(33)fX=∑j+k≥2Fjkuju¯k,where F21 is coefficient vector corresponding to u2u¯.(34)F21=F21,1F21,20F21,40,F21,1=cφ2H11,4+φ¯2H20,4+φ4H11,2+φ¯4H20,2=0.0022+0.0015i,F21,2=fφ1H11,4+φ¯1H20,4+φ4H11,1+φ¯4H20,1+knφ2−0.1φ52φ¯2−0.1φ¯5=kn0.4751+0.0009404i+0.001923−0.002212i,F21,4=kφ42φ¯4=1.1735×10−4+2.5612×10−4i.In the above equation,(35)H11=−A−1F11,H20=2×50.5163iI−A−1F20,F11=000φ2φ¯4+φ¯2ϖ40T,F20=000φ2φ¯40T.Finally, the parametersE is(36)E=0.004265+0.4751kn+0.0009404kn−0.0004616i.So normal form of the controlled system can be expressed as(37)u˙=50.5163iu+0.004265+0.4751kn+0.0009404kn−0.0004616iu2u¯.According to the Hopf bifurcation theory, the controlled system will have a supercritical bifurcation atC=18.09 when ReE<0andkn<−0.009. The bifurcation type of the system stays the same by calculating. At the same time, subcritical Hopf bifurcation will occur at kn>−0.009, as shown in Figure 9. It can be seen from Figure 9(b) that the time-history response is stable when the initial values are close to the equilibrium point. When initial values are far from the equilibrium point, the time-history response increases with time in Figure 9(a). It is necessary to avoid the generation of subcritical bifurcation in the actual situation.Figure 9
Comparison of amplitudes at different initial values of the controlled system:C=18.09,km=−1,andkn=0.01.
(a)
(b)The amplitude of the limit cycle can be effectively controlled through the selection of the value ofkm and kn reasonably. Figure 10 shows that the amplitude of the controlled system is obviously lower than that of the original system without subcritical Hopf bifurcation (Figure 10(b) is the time-history diagram of the controlled system at km=−1andkn=−1), and the significant effect is achieved after addition of the control system. In contrast to the two controlled systems in Figure 11 (km=−1andkn=−0.1 in Figure 11(a); km=−1andkn=−10 in Figure 11(b)), if the case of km is fixed, the smaller the kn is, the smaller the amplitude of the limit cycle is.Figure 10
The amplitude contrast diagram of the original and controlled systems atC=10.
(a)
(b)Figure 11
Amplitude comparison of nonlinear gain of controlled systems at different time scales: (a)kn=−0.1; (b) kn=−10.
(a)
(b)The bifurcation point will vary withkm, and the bifurcation type will change with kn. At the same time, it is necessary to calculate the eigenvalue and eigenvector of the system to determine the type of bifurcation, but the eigenvalue and eigenvector are difficult to calculate due to the change of km. So it is hard to calculate the type of bifurcation for the system through the theoretical method when km and kn change simultaneously. However according to the simulation, the subcritical Hopf bifurcation only occurs when kn is near 0. Figure 12 shows the amplitude variation of km and kn change at the same time when kn is far less than 0. It can be seen from Figure 12 that the vibration amplitude decreases with the reduction of linear gain km and nonlinear gain kn.Figure 12
The vibration amplitude variation diagram ofkm and kn change at the same time.


## 5. Conclusion

In this paper, the Hopf bifurcation caused by the change of the system damping was studied concerning the shearer transmission shafting semidirect driven by the PMSM. The electromechanical coupling dynamic model was established by the Lagrange–Maxwell equation. The dynamic bifurcation behavior of the system transmission shafting was analyzed through theoretical analysis and numerical simulation. Based on the washout filter, the torsional vibration bifurcation of the shafting was effectively controlled. The research work of this article mainly includes the following:(1)
Due to the harsh environment of coal cutting, there is the supercritical and subcritical Hopf bifurcation with the diversification of the damping in the cutting transmission system. When theC is less than the critical value C∗, the smaller the value of the damping C, the bigger the limit cycle in supercritical Hopf bifurcation. Simultaneously, the amplitude of the limit cycle has nothing to do with the initial value.(2)
Based on the normative theory, the Hopf bifurcation controller of the shearer cutting transmission shaft was designed by the washout filter. The results show that the system stability domain increases with the linear gain of the controller and the generation of the subcritical Hopf bifurcation can be inhibited by the cube nonlinear parts.(3)
For supercritical Hopf bifurcation, selecting the linear and nonlinear gain reasonably can effectively control the torsional vibration and the shafting torsional vibration. And the limit cycle amplitude decreases with the nonlinear gain of the controller when the linear gain is zero. At the same time, whenkn and km are varied simultaneously, the limit cycle amplitude decreases with the reduction of linear gain km and nonlinear gain kn in supercritical Hopf bifurcation.All these results are of important theoretical significance to reduce the torsional vibration caused by nonlinear factors in the shearer cutting transmission system and provide a guarantee for the safe and reliable application of the low-speed and large-torque PMSM on the shearer.



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