A dualpendulumrotor system widely appears in aeropower plant, mining screening machines, parallel robots, and the like of the other rotation equipment. Unfortunately, the synchronous behavior related to the dualpendulumrotor system is less reported. Based on the special backgrounds, a simplified mechanical model of the dualpendulumrotor system is proposed in the paper, and the intrinsic mechanisms of synchronous phenomenon in the system are further revealed with employing the Poincaré method. The research results show that the spring stiffness, the installation angular of the motor, and rotation direction of the rotors have a large influence on the existence and stability of the synchronization state in the coupling system, and the mass ratios of the system are irrelevant to the synchronous state of the system. It should be noted that to ensure the implementation of the synchronization of the system, the values of the parameters of the system should be far away to the two “critical points”.
National Natural Science Foundation of China51705437Southwest Petroleum University2017QHZ009plan of science and technology of Sichuan province2016RZ00591. Introduction
“Synchronization phenomenon” widely exists in nature and daily life. The socalled synchronization phenomenon refers to selfadjusting different frequencies of oscillating objects to a unified frequency relying on their internal weak couplings [1, 2]. In recent years, the study of synchronization phenomenon is involved in the fields of physics, chemistry, and biology, such as the practical application on complex dynamic network systems [3–5], nonlinear coupled chaotic systems [6, 7], coupled pendulum system [8–12], and rotor system [1, 13–19]. Nevertheless, the latter two systems can be categorized as synchronous problem of mechanics system in detail. In the synchronous study of the pendulum system, antiphase selfsynchronization of clock pendulum is firstly discovered by Huygens in 1665. In the modern computation and analysis, the difference of spring stiffness influenced on the synchronous state is concerned, and the synchronization in inphase and antiphase state for Huygens model is analyzed. Meanwhile, the derivatizations of Huygens model, including two coupled double pendula, coupling pendula under elastic forces, and pendula connected with linear springs, have attracted many scholars’ attention. In the selfsynchronous study of the rotor system, I.I.Blekhman proposed Poincaré method to explore the synchronous mechanism in multirotor systems. Nowadays, the Poincaré method is still widely used in modern engineering design. Based on the foundation of I.I.Blekhman’s method, many scientists proposed the other approaches to study selfsynchronization of the rotor system. Wen Bangchun introduced an averaging method to calculate the synchronization and stability between multiple coupled rotors [13]. Zhao Chunyu [14, 15] gave a revised small parameter method to describe the synchronous process of rotor system, which greatly simplifies the solving process of synchronization problem in rotor system. Sperling presented a twoplane automatic balancing device for equilibration of rigidrotor unbalance, on which the synchronization of the rigid rotors is determined with numerical method. Similarly, Balthazar [16] examined selfsynchronization of four nonideal exciters in nonlinear vibration system via numerical simulations. Djanan. A. A. N [17] explored the system, three motors working on the same plate; the synchronous state depends on the physical characteristics of the motors and the plate.
The objects of the abovementioned researches are mainly synchronization of the coupled pendula or the coupled rotors; however, the synchronization of the pendulumrotor system is less concerned. The extensive application of the pendulumrotor system in aeropower plant, mining screening machines, parallel robots, and the like of the other rotation equipment is important for our industrial production and daily life [20–22]. The aim of this paper is to investigate a system composed of two unbalanced rotors coupled with two pendula in a horizontal plane. But mass of the motor and the rotor, which can be relatively heavy or light, may affect the system motion. Therefore, the proposed problem is different from the dynamics of classical pendulum systems presented in papers [8–11] and classical rotor systems presented in papers [12–20, 23–27]. We consider an ideal source of energy supply to asynchronous motors, which drag the unbalanced rotors. We also investigate selected aspects of the synchronization phenomenon. There are different synchronization states possible in mechanical systems. We will search for a complete synchronization when the rotors perform rotation with the difference direction or for the synchronous state when the phase differences between the rotors are stabilized.
2. Dynamic Model
A simplified dualpendulumrotor system is shown in Figure 1. This system consists of a rigid oscillating body of mass m0 [Kg] elastically supported via a linear damping spring with stiffness kx [N/m] and damping fx [Ns/m]. Unbalanced rotor i actuated by an asynchronous motor is modelled by a point mass mi [Kg] (for i=1,2) and attached at the end of a massless rod of length r [m]; the rotation angle of the rotors is defined by φi∈S1 (for i=1,2) in [rad]. It should be noted that all the motors are installed rigid vibrating body by linear torsion springs with stiffness kφN/rad and damping fφ[N·m/rad/s]. Thus, small oscillating should exist in the two motors when the motors are operated in steady state; the oscillating angle is denoted by φj∈S1 (for j=3,4) in [rad]. Therefore, the oscillating characteristics of the motors are just like a pendulum or pendula with length l and lumped mass (equal to the mass of the motors) mj [Kg] (for j=3,4). The angular installation of the motors is denoted by β∈S1 in [rad]; x represents oscillation of centroid of the system in xdirection, respectively. σ is the rotation direction of the rotors; l [m] is distance between pivot of the motor and centroid O of the system.
The dualpendulumrotor system.
In rotating coordinate o′x″y″, the rotating centers of the rotors Φ1″, Φ2″ can be expressed as(1)Φ1″=σrcosφ1lcosφ3+βrsinφ1+lsinφ3+β,Φ2″=rcosφ2+lcosφ4+βrsinφ2+lsinφ4+β.
And the rotating center of the motors Φ3″ and Φ4″ is represented as(2)Φ3″=lcosβ+φ3lsinβ+φ3,Φ4″=lcosβ+φ4lsinβ+φ4.
In fixed coordinate oxy, coordinates Φ1,Φ2,Φ3,Φ4 can be obtained by(3)Φi=Φ0+Φi″.i=1,2,3,4
where Φ0 is displacement vector of the vibrating base, Φ0=x,0T.
According to kinetic theory, kinetic energy T of the dualpendulumrotor system should be written as follows:(4)T=12m0x˙+12∑i=14miΦ˙iTΦ˙i+12∑i=12Jiφ˙i2.
J1 and J2 are the rotational inertia of the rotors, and symbols •˙ and •¨ denote d•/dt and d2•/dt2, respectively.
In addition, potential energy V of the dualpendulumrotor system can be expressed by(5)V=12kxx2+12kφφ32+12kφφ42
Moreover, dissipated energy D of the system can be obtained by(6)D=12fxx˙2+12f1φ˙12+12f2φ˙22+12f3φ˙32+12f4φ˙42
According to the Lagrange equation(7)ddt∂T∂q˙∂TV∂q+∂D∂q=Qi,
the dynamic equation of the dualpendulumrotor system can be derived. In the system, coordinate matrix q=x,φ1,φ2,φ3,φ4T is considered as the generalized coordinate matrix; meanwhile, the generalized force matrix of the system is assumed as(8)QxQφ1Qφ2Qφ3Qφ4T=0Me1Re1Me2Re200T
Substituting (4), (5), (6), and (8) into (7) and, meanwhile, considering m1,m2≪m0,∑k=04mk=M. As J1 and J2 are sums of rotational inertia of motor’s rotor about spin axis and that of eccentric lump, compared with m1r2 and m1r2, the rotational inertia of motor’s rotor about spin axis is too small to be neglected; i.e., J1≈m1r2,J2≈m2r2, and the dynamic equation of the dualpendulumrotor system is written by (9a)J1φ¨1=Me1Re1+σm1rsinφ1x¨m1rlφ¨3cosβ+σφ1+m1rlφ˙32sinβ+σφ1(9b)J2φ¨2+J2φ¨2=Me2Re2+m2rsinφ2x¨m2rlφ¨4cosβφ2+m2rlφ˙42sinβφ2(9c)Mx¨+fxx˙+kxx=σm1rφ¨1sinφ1+φ˙12cosφ1+m2rφ¨2sinφ2+φ˙22cosφ2m1+m3lφ¨3sinβ+φ˙32cosβ+m2+m4lφ¨4sinβ+φ˙42cosβ(9d)m1l2+m3l2φ¨3+fφφ˙3+kφφ3=m1+m3lsinβx¨m1lrφ¨1cosβ+σφ1σφ˙12sinβ+σφ1(9e)m2l2+m4l2φ¨4+fφφ˙4+kφφ4=m2+m4lsinβx¨m2lrφ¨2cosβφ2+φ˙22sinβφ2
It can be seen that the terms of φ˙32 and φ˙42, nonlinear terms owing to the small periodic vibration in synchronous state, are included in (9c). The terms related to φ˙32 and φ˙42 can be rewritten as Poincaré style with small parameter μ. Meanwhile, damping coefficient of this system is very small, and so the terms with damping coefficient can be rearranged as Poincaré style. In synchronous state, velocity of the motors is fluctuated slightly and periodically; hence, the acceleration of the rotors is also considered as a small term. Therefore, introducing small parameter μ, the Poincaré style of (9a), (9b),(9c), (9d), (9e) is given as follows [27]:(11a)J1φ¨1=μΦ1(11b)J2φ¨2=μΦ2(11c)Mx¨+kxx=σm1rφ¨1sinφ1+φ˙12cosφ1+m2rφ¨2sinφ2+φ˙22cosφ2m1+m3lφ¨3sinβ+m2+m4lφ¨4sinβμfxx˙+m1+m3lφ˙32cosβm2+m4lφ˙42cosβ(11d)m1l2+m3l2φ¨3+kφφ3=m1+m3lsinβx¨m1lrφ¨1cosβ+σφ1μfφφ˙3σm1lrφ˙12sinβ+σφ1(11e)m2l2+m4l2φ¨4+kφφ4=m2+m4lsinβx¨m2lrφ¨2cosφ2βμfφφ˙4+m2lrφ˙22sinφ2β
where (12a)μΦ1=Me1Re1+σm1rsinφ1x¨m1rlφ¨3cosβ+σφ1+m1rlφ˙32sinβ+σφ1(12b)μΦ2=Me2Re2+m2rsinφ2x¨m2rlφ¨4cosβφ2+m2rlφ˙42sinβφ2
Based on Zhao’s method [15], the electromagnetic torque of the asynchronous motor is linearized at stable point φ˙i=ωs when the two motors are synchronously operated; i.e.,(13)Mei=npLmi2US02Lsi2ωmRriωsnpωmi=1,2
where Lmi is coefficient of mutual asynchronous in the ith motor; Lsi is the ith coefficient of stator inductance in the ith motor; np is polepairs of the motors; ωm is synchronous velocity of the motors; Rri is rotor resistance in the ith motor; US0 is amplitude of stator voltage.
3. Approximate Solution
From formulas (9a) to (9e), it can be seen that the vibration characteristics of the system are coupled related to DOFs φ1,φ2,x,φ3, and φ4. Neglecting the terms of small parameters, introduce the dimensionless parameter as follows:(14)rl=rl,mM=rm,mim=ηii=1,2,3,…,4,ωx=kxM,ωφ=kφm1l2+m3l2,ω~φ=kφ/m1l2+m3l2,f1t=rmrlση1cosφ1+η2cosφ2,f2t=η1rlη1+η3sinφ1+σβ,f3t=η2rlη2+η4sinφ2β.
In addition, velocities φ˙1 and φ˙2 of the two motors approached to a constant are represented by ωm in the steady state; therefore, the accelerations of the two motors should be equal to zero; i.e., φ¨1=0 and φ¨2=0. In this case, (11c) (11e) can be simplified as the dimensionless style:(15a)x¨+ωx2x=ωm2lf1tη1+η3rmlsinβφ¨3+η2+η4rmlsinβφ¨4,(15b)φ¨3+ωφ2φ3=sinβlx¨+ωm2f2t,(15c)φ¨4+ω~φ2φ4=sinβlx¨+ωm2f3t.
For simplifying the solution procedure, consider that the natural frequencies of the torsion spring are quasiidentical; thus, parameter ω~φ should be replaced by ωφ.
Applying Laplace transformation in (16a), (16b), (16c), the coupling terms can be decoupled. Therefore, we have (16a)s2X+ωx2X=ωm2lF1sη1+η3rmls2sinβϕ3+η2+η4rmls2sinβϕ4(16b)s2ϕ3+ωφ2ϕ3=s2sinβlX+ωm2F2s(16c)s2ϕ4+ωφ2ϕ4=s2sinβlX+ωm2F3s
From formulas (16a)(16c), the approximate solutions of the system in complex domain can be given by(17a)X=ωm2lG1sF1sη1+η3ωm2rmls2sinβG1sF2s+η2+η4ωm2rmls2sinβG1sF3s(17b)ϕ3=s2sinβωm2G1sF1s+ωm2G2s+ωm2η1+η3rms4sin2βG1sG2sF2sωm2η2+η4rms4sin2βG1sG2sF3s(17c)ϕ4=ωm2s2sinβG1sF1sωm2η1+η3rms4sin2βG1sG2sF2s+ωm2G2s+ωm2η2+η4rms4sin2βG1sG2sF3s
where G1s=s2+ωx2s2+ωφ2η1+η3+η2+η4rms4sin2β,G2s=s2+ωφ2.
Then, the numerators and denominators of (17a), (17b), (17c) in the right are divided by divisor ωφ4ωx2; moreover, introduce frequency ratios(18)nx=ωmωx,nφ=ωmωφ
into (17a)(17c). In this case, the coefficients of spring stiffness kx and kφ are converted into the frequency ratios nx and nφ in light of (14), respectively.
Finally, through applying the inverse Laplace transformation to (18), the approximate solutions of the system in time domain can be obtained by(19a)x=μ11lf1t+μ12η1+η3lrmf2t+μ13η2+η4lrmf3t,(19b)φ3=μ21f1t+μ22η1+η3rmf2t+μ23η2+η4rmf3t,(19c)φ4=μ31f1t+μ32η1+η3rmf2t+μ33η2+η4rmf3t,
In light of functions f1t,f2t, and f3t in (14), the approximate solutions of the system can be rearranged as the following:(21a)x=σμ11η1rrmcosφ1+μ11η2rrmcosφ2+μ12η1rrmsinφ1+σβ+μ13η2rrmsinφ2β,(21b)φ3=σμ21η1rmrlcosφ1+μ21η2rmrlcosφ2+μ22η1rlrmsinφ1+σβ+μ23η2rlrmsinφ2β,(21c)φ4=μ31rmrlση1cosφ1+η2cosφ2+μ32η1rlrmsinφ1+σβ+μ33η2rlrmsinφ2β.
4. Synchronization
Based on the approximate solutions above, the synchronization and stability of the system can be determined with Poincaré method [20, 25–27]. In addition, the phase angle of the rotors can be assumed by (22)φ1=ωmt+α1φ2=ωmt+α2
Substituting (22) into (21a), (21b), (21c), the approximate solutions of the system can be written by (23a)x=μ11rrmση1cosωmt+α1+η2cosωmt+α2+μ12rrmη1sinωmt+α1+σβ+μ13rrmη2sinωmt+α2β,(23b)φ3=μ21rmrlση1cosωmt+α1+η2cosωmt+α2+μ22rlrmη1sinωmt+α1+σβ+μ23rlrmη2sinωmt+α2β,(23c)φ4=μ31rmrlση1cosωmt+α1+η2cosωmt+α2+μ32η1rlrmsinωmt+α1+σβ+μ33η2rlrmsinωmt+α2β.
As the values of the phase difference of the rotors may be different when the asynchronous motors are operated in the steady state, the phase difference between the rotors is noted by(24)α=α1α2
The second derivatives of x,φ3, and φ4 with respect to time t can be obtained. Substituting x¨,φ¨3, and φ¨4 into (12a) and (12b), and then integrating and averaging these equations related to t over period T, respectively, we can obtain (25)P1=1T∫0TμΦ1dt=Me1Re1+12η1η2mr2ωm2rmσμ11sinα+σμ12cosα+β+μ21cosα+σβμ23sinα+σ+1β(26)P2=1T∫0TμΦ2dt=Me2Re2+12η1η2mr2ωm2rmσμ11sinα+σμ12cosα+β+μ21cosα+σβμ23sinα+σ+1β
P1 and P2 represent the averaging residual torque of the first and second motor in single period T, respectively. As we all know, when the motors are operated in synchronous state, μ is a small parameter approached to zero. Therefore, the residual torque in the motors is approximated to zero; i.e., (27)P1≈P2≈0
Substituting (27) into (25) and (26), the balanced torques of the motor are written by (28a)Me1Re1=12η1η2mr2ωm2rmσμ11sinασμ12cosα+βμ21cosα+σβ+μ23sinα+σ+1β(28b)Me2Re2=12η1η2mr2ωm2rmσμ11sinα+σμ12cosα+β+μ21cosα+σβμ23sinα+σ+1β
Mei,Reii=1,2 represent average electromagnetic and friction torque in the motors over period T, respectively. When the two rotors are operated synchronously, the vibrating system transmits electromagnetic torque between the two motors to overcome the differences of output torque for the two rotors by adjusting the phase difference between two rotors. The differences of output torque ΔM between the motors can be expressed as(29)ΔM=Me1Re1Me2Re2=η1η2mr2ωm2rmσμ11sinασμ12cosα+βμ21cosα+σβσμ12cosα+β+μ21cosα+σβ
As the parameter of the two asynchronous motors is identical, the differences of output torque should be zero. Therefore, to implement synchronous operation, the values of the parameters in the system must satisfy the following equation:(30)σμ11sinα+σμ12cosα+β+μ21cosα+σβμ23sinα+σ+1β=0
which is called balance equation of synchronous state. Obviously, the phase difference is influenced by the rotation direction of the motors, the coupling coefficients between rotor and pendula, and the installation location of the motors. According to (30), we can determine phase difference α with numerical computations. With trigonometric function, the equation above can be expanded as (31)cotα=σμ11+σμ12sinβ+σμ21sinβ+μ23cosσ+1βσμ12cosβ+μ21cosβμ23sinσ+1β
If some solutions of phase difference α exist in (31), the denominator of this equality in the right should be nonzero. Define the denominator of this equality with a symbol D; i.e.,(32)D=σμ12cosβ+μ21cosβμ23sinσ+1β
It can be known that denominator D is a function related to variables σ,μ12,μ21,μ23, and β, but parameter σ indicates the rotation directions of the motors independent of nonzero solution of denominator D. Therefore, we will discuss zero solution of parameter D related to installation angle and coupling coefficients. There are two cases causing the zero solution of denominator D.
(1) Parameter β represents installation angle of the motors. Thus, parameter β with a particular value leads to D being zero. The first "critical point" is (33)β=kππ,k=1,2,3….
When the installation angles of the motors are approximated or equal to this point, the absentsynchronization of the system will be implemented. In this situation, the vibrating characteristics of the oscillating body are unascertainable.
(2) According to (20), coupling coefficients μ12, μ21, and μ23 are fractions; thus parameters nx and nφ with particular values lead to D being zero. The second "critical point" is (34)nx=1ornφ=1.
The second "critical point" means that the excitation frequency of the motors is identical to the natural frequency of the springs. In this case, strong resonance exists in such value of the frequency ratios. However, frequency ratios nx and nφ are the function related to stiffness coefficients kx and kφ of the springs, which indicates that stiffness coefficients are the key parameters to determine phase difference α.
Clearly, to ensure the synchronization and stability of the system, the value of frequency ratios nx and nφ should be far away from 1. In light of different value of nφ and nφ, the coupling type of the system can be defined as follows:
Type 1: system of beforeresonance coupled beforeresonance (0.1<nx<1 and 0.1<nφ<1).
Type 2: system of afterresonance coupled beforeresonance (0.1<nx<1 and 1.0<nφ<7, or 1.0<nx<7 and 0.1<nφ<1).
Type 3: system of afterresonance coupled afterresonance (1.0<nφ<7 and 1.0<nx<7).
Eliminating the two kinds of "critical point", the phase difference between the two rotors can be calculated: (35)α=arccotσμ11+σμ12sinβ+σμ21sinβ+μ23cosσ+1βσμ12cosβ+μ21cosβμ23sinσ+1β.
Obviously, the value of the phase difference is related to the coupling coefficients (μ11, μ12, μ21, and μ23) and the installation angle. Coupling coefficients represent the mutual coupling ability among the oscillating body, rotors, and pendula through the springs. The larger the coupling coefficients are, the stronger the coupling ability of the system is. Obviously, coupling coefficients μ12, μ21, and μ23 contain sinβ; thus the absence of the coupling ability appears when β=0. In this situation, the rotors cannot implement synchronous operation. It should be noted that coupling coefficients are the functions of parameters nx, nφ, β, and rm, and these parameters may influence the value of phase difference α.
5. Synchronous Stability
From (35), some values of phase difference can be obtained; however, the stability of these values should be determined as I.I.Blekhman’s method [1, 2]. If a certain constant α=α∗ satisfies equation (36)∂P1P2∂αχ=0,
phase difference α∗ is stable under the condition that the value of χ is negative.
Thus, the residual torque difference can be obtained by (37)P1P2=μΦ1=Me1Re1Me2Re2+η1η2mr2ωm2rmσμ11sinα+σμ12cosα+β+μ21cosα+σβμ23sinα+σ+1β
As Mei,Reii=1,2 are irrelevant to phase difference α, in light of (33), the criterion of synchronous stability is expressed(38)χ=∂P1P2∂α=η1η2mr2ωm2rmσμ11cosασμ12sinα+βμ21sinα+σβμ23cosα+σ+1β<0.
Rearranging the equation above, the criterion of synchronous stability is simplified by (39)σμ11cosα+σμ12sinα+β+μ21sinα+σβ+μ23cosα+σ+1β>0
The formula above shows that synchronous stability of the system is also determined by rotation direction of the rotors, coupling coefficients, and installation location of the motors. Only should parameters of the system satisfy the balance equation (30) and the synchronous stability criterion (39), the synchronous operation of the rotors can be implemented. In this case, the phase difference between the rotors is called the stable phase difference.
6. Numerical Analysis
The abovementioned sections have given some theoretical discussions in the simplified form on synchronization problem for the vibration system that the unbalanced rotors are coupled with pendulum. In this section, we will employ some numerical analysis to discuss the stable phase difference, which can be calculated according to (30) and (39). The phase difference is determined by rotation direction of the rotors, coupling coefficients, and installation location of the motors. However, the coupling coefficients are the function of frequency ratios (nxnφ), mass ratios (rm, ηi), and motor installation angle (β). Therefore, it can be included that the stable phase difference may be influenced by the parameters above. The parameter values corresponding to general engineering application are as given in Table 1.
Parameter values.
Parameter values for system equation (6)
Unbalanced rotor for i=1,2
Vibroplatform
Pendulum rod
Asynchronous motor
mi=2[kg]
m0=100[kg]
l=0.3[m]
m3=m4=14[kg]
r=0.05[m]
kx=24649000050307[N/m]
kφ=498024402510[Nm/rad]
Lmi=0.13[H]
ωm=152157[rad/s]
fx=1064[Ns/m]
fφ=15[Nm/(rad/s)]
Lsi=0.1[H]


β=05π/12[rad]
np=2



Rri=0.54[Ω]



US0=220[V]
Parameter values according to dimensionless equations (11a), (11b), (11c), (11d), (11e)
η1=0.2,1
η2=1
η3=η4=12
σ=1,1
rm=0.02
nx=0.1~7
nφ=0.1~7
6.1. The Rotors Operated in Opposite Direction
The stable phase difference is determined by considering the different values of parameters β, ηii=1,2, nx, and nφ. Moreover, in light of (35) and (39), the rotation direction of the two rotors is opposite, and the stable phase difference between the two rotors is shown in Figure 2. According to (20) and (30), we have μ11+μ23sinα=0 when σ=1, and the synchronous balance condition as in (17a), (17b), (17c) is simply expressed as sinα=0. Similarly, the synchronous stability criterion can be rewritten by μ11+μ23sinα>0. It then follows that the value of the phase difference is stabilized at 0 [rad] in the blue area, which describes that the synphase motion is stable and the antiphase motion is unstable ([1] describes that the motion, as the existence of α∈(π/2,π/2), is called synphase synchronization; and the motion, as the existence of α∈(π/2,3π/2), is called antiphase synchronization). On the contrary, the brown area represents that the synphase synchronization is unstable and the antiphase synchronization is stable. Firstly, installation angle β and mass ratio rm are fixed at π/3 and 0.02, respectively, and the mass ratios η1,η2 are equal to 1.0. In such values of the parameters, the stable phase difference is shown in Figure 2(a). It is indicated that the stable phase difference is related to frequency ratios nx and nφ; in other words, the phase difference is dependent on the stiffness of the torque spring in the pendulum and the pressure spring in the vibrating body. Secondly, mass ratio rm is fixed at 0.02, and the frequency ratios nx and nφ are considered identical. Figure 2(b) describes that the stable phase difference is changed with the installation angle β. It can be seen that the synchronous state of the system is locked in the synphase synchronization when nx and nφ>1.5, and the variation of installation angle β has little influence on the stable phase difference. However, the synchronous state is changed in region of nx and nφ<1.5. Finally, the effect of mass ratios rm, η1, and η2 on the phase difference needs to be discussed. Thus, the parameter values in Figure 2(c) are identical to Figure 2(a) except for rm. And the parameter values in Figure 2(d) are identical to Figure 2(b) except for η1. Comparing Figures 2(c) and 2(d) with Figures 2(a) and 2(b), respectively, it can be found that the values of the stable phase difference are independent of the mass ratios. Therefore, the mass variations of rotors and vibration body would not affect the synchronous motion when the two rotors operate in the opposite direction.
Stable phase difference when σ=1.
η1=η2=1,η3=η4=12,rm=0.02,β=π/3
η1=η2=1,η3=η4=12,rm=0.02,nx≈nφ
η1=η2=1,η3=η4=12,rm=0.04,β=π/3
η1=0.2,η2=1,η3=η4=12,rm=0.02,nx≈nφ
6.2. The Rotors Operated in the Same Direction
In the following calculations, the two rotors rotating in the same direction are considered according to (35) and (39). Firstly, the parameter values in Figure 3(a) are the same as Figure 2(a) except for rotation direction σ. From Figure 3(a), it follows that two rotors may synchronize either synphase or antiphase, also depending on the values of nx and nφ. The obtained results reveal that, in the interval of 0.1<nx<1 and 0.1<nφ<1, synphase synchronization is implemented between the rotors; in the interval of 0.1<nx<1 and 1<nφ<7, antiphase synchronization is implemented between the rotors; in the interval of 1<nx<7, 0.1<nφ<1 and 1<nx<7, 1<nφ<7, both antiphase and synphase synchronization collectively remained; in the interval of 1<nx<7 and for 1<nφ<7, the two rotors will be synchronously operated in synphase state. Figure 3(b) shows the variation of the stable phase difference with the installation angle. It is indicated that the synchronous state of the system is obviously influenced by β. Finally, the effect of mass ratios rm, η1, and η2 on the phase difference is discussed. Thus, the parameters in Figure 3(c) are identical with Figure 3(a) except for rm, and the parameters in Figure 3(d) are identical with Figure 3(b) except for η1. Comparing Figures 3(c) and 3(d) with Figures 3(a) and 3(b), respectively, it can be found that the stable phase difference is also independent of the mass ratios. In other words, synchronous state is less influenced by mass of the rotors when the two rotors operate in the same direction.
This subsection refers to the case that the system of afterresonance is coupled with the afterresonance; i.e., nx=6, nφ=6. The mass of the two rotors operated in opposite direction is identical; i.e., η1=η2=1. In the simulation model, the coefficients of the spring stiffness are kx=65969[N/m] and kφ=1053[N·m/rad], and the other parameter is identical with Table 1(a). The numerical results of this sample can be calculated through (9a), (9b), (9c), (9d), (9e). When the two motors are supplied by the electric source at the same time, the velocities of the two rotors are compatible (in Figure 4(a)). When the angular velocities of the motors reach the nominal velocity, then the motors oscillate steadily with the identical responses. At this moment, the coupling torques (in Figure 4(c)), making the phase difference α stabilized nearby 0 [rad], are fluctuated. In this case, the two motors rotate stably in synphase synchronization. From Figure 4(c), it follows the displacements of the vibration body and the two motors. It is clear that when the rotation velocity of the two rotors passes through the resonant region of the coupling system, the resonant responses of the system in the φ3 and φ4directions appear in the starting process. In the synchronous state, the displacements of the vibrating body and the motors are stable, and the amplitudes of them are 0[m], 0.01[rad], and 0.01[rad], respectively. Comparing simulation results with Figures 2(a) and 2(b), the stable phase difference obtained by the computer simulation is consistent with the theoretical computation (i.e., the stable phase difference in Figures 3(a) and 3(b) is equal to 0 [rad]; here, the stable phase difference is fluctuated near 0 [rad]).
The dynamic characteristics when the rotors rotate in the opposite direction.
Simulation results obtained by (9a), (9b), (9c), (9d), (9e) are shown in Figure 5 whenσ=1,nx=6,nφ=0.3,η1=η2=1,η3=η4=12,rm=0.04, and β=π/3. Here, the coefficients of spring stiffness are kx=65969[N/m] and kφ=976100[N·m/rad], and the rotors are operated in the same direction. The coupling type of the system belongs to type 2. When the two motors are supplied by the electric source at the same time, the angular accelerations of the two rotors are compatible (in Figure 5(a)). The reason is that the inertia moments of the rotors are identical and the spring stiffness is extremely stronger. During the staring process of the system, the velocity difference exists between the two rotors, which leads to the phase difference instability, shown as in Figure 5(b). However, when the angular velocities of the motors reach the rated speed and the motor oscillate steadily, the synchronization phenomenon occurs. It should be noted that the velocity fluctuation of the motors is smaller than that of Section 7.1, because the vibration of the motors is feebler. At this moment, the average coupling torques (in Figure 5(c)), making the phase difference α stabilized at 0.8984 [rad], are approximated to 4.08[N·m]. In this case, the two motors rotate stably in the synphase synchronization, and the synchronous velocity is 157 [rad/s]. From Figure 5(d) it follows the displacements of the vibrating body and the two motors. It can be seen that the displacement responses of the vibrating body and the two motors are stable, and the amplitudes of them are 1.5×103[m], 4.1×104[rad] and 7.3×104[rad], respectively. Comparing simulation results with Figure 3(b), it should be noted that the value of the stable phase difference is in agreement with the results obtained for the case of the theoretical solutions (i.e., the stable phase difference of such parameters in Figure 3(a) is equal to 1.08 [rad]; here, the stable phase difference is equal to 0.8984 [rad]).
The dynamic characteristics when the rotors rotate in the same direction.
Velocity
Phase difference
Electromagnetic torque
Vibration response
8. Conclusions
In this paper, the dualpendulumrotor system is concerned. The research results show that the spring stiffness, the installation angular of the motor, and rotation direction of the rotors have a large influence on the existence and stability of the synchronization state in the coupling system, and the mass ratios of the system are irrelevant to the synchronous state of the system. It should be noted that to ensure the implementation of the synchronization of the system, the values of the parameters of the system should be far away to the two "critical points". To verify the correctness of the theoretical computations, some example simulations are preformed, and the results of theoretical computation are in accordance with that of example simulations.
The dualpendulumrotor system widely appears in aeropower plant, mining screening machines, parallel robots, and the like of the other rotation equipment. However, the dynamic characteristics and performance accuracy of the dualpendulumrotor system are mainly influenced by synchronous behavior between the rotors. In the early stage, for the developing and understanding the internal characteristics of the system, we only consider the vibrating body under the assumption of horizontal displacement. What is synchronization state of the system that the vibrating body simultaneously vibrates in the pitch and vertical directions? We believe that finding the answer to this question is the next step in challenging task of getting a complete understanding of synchronization in such system.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study is supported by National Natural Science Foundation of China (Grant no. 51705437), scientific research starting project of SWPU (no. 2017QHZ009), and the plan of science and technology of Sichuan province (no. 2016RZ0059).
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