# Synchronous Stability of Four Homodromy Vibrators in a Vibrating System with Double Resonant Types

Authors: Xueliang Zhang; Chao Li; Zhihui Wang; Shiju Cui
Journal: Shock and Vibration
 (2018)

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## Abstract

This paper aims at studying the synchronous stability of four homodromy vibrators in subresonant and superresonant states. The motion differential equations are established firstly. The simplified form of analytical expressions is yielded, and the stability criterion of synchronous states satisfies Routh–Hurwitz criterion. The coupling dynamic characteristics of the system are analyzed in detail numerically, such as the maximum of coupling torque, coefficients of ability of synchronization and stability, and phase differences. Based on the ratio of operating frequencies to natural frequencies of the system, the resonant regions are divided into two areas: subresonant and superresonant. It is shown that the phase differences among four vibrators in the subresonant state are stabilized about zero, and exciting forces of four vibrators are positively superposed, while in the superresonant one, the phenomenon of the diversity of the nonlinear system occurs, i.e., two groups of synchronous and stable solutions of the phase differences (pi and pi/2) are found, and in this case, the exciting forces of four vibrators are counteracted, the rigid frame embodies no vibration, and the minimum of dynamic load transferring to foundation is realized. The correctness of theoretical results is verified by numerical characteristic analysis and simulations. This paper can provide a theoretical reference for designing a type of new high-frequency vibrating mill.


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## Body

## 1. Introduction

Synchronization is a natural phenomenon and often occurs in daily life. For example, two organ pipes close to each other can produce synchronous sounds [1], synchronization phenomenon is observed in nonlinear circuits [2], and so on. The earliest scientific research on synchronization was given by Huygens [3], which attracted more and more researchers to engage in research on it. In 1990, Pecora studied synchronization in chaotic systems and proposed its application of communication security [4]. Osipov et al. introduced synchronization in networks of Oscillatory [5]. Teufel et al. and Kapitaniak et al. investigated synchronization of pendula [6, 7]. Scholars Lu et al. analyzed synchronization on neuronal electrical activities [8]. Speed synchronization with the new hybrid system being used to dynamic shift coordinated control was given by Yan et al. [9]. With the rapid development of science and technology, the synchronization problems in other fields are also investigated by many scholars [10–13].Vibration phenomena are also often encountered in science and have a great influence in engineering. For a long time, some scholars found synchronization in the field of vibration. In 1960s, theoretical research studies about synchronization of two vibrators were given firstly by Blekhman by using the method of direct separation of motions [14–16]. In 1980, Inoue and Araki studied triple frequency synchronization of vibrators driven by double motors [17]. Scholar Quinn et al. described the behavior of synchronization of resonance in a weakly coupled multi-degree-of-freedom system [18]. Wen et al. investigated numerous types of synchronization problems of double identical vibrators and proposed the concept of vibratory synchronization transmission [19, 20]. In modern times, many theories on synchronization of vibrators were also studied by other scholars, for example, Fang and Hou investigated synchronization characteristics of a rotor-pendula system [21], and some research studies on self-synchronization of nonideal sources or exciters were given by Balthazar et al. [22, 23]. Zhang et al. [24–26] and Zhao et al. [27], studied problems of vibratory synchronization transmission on a cylindrical roller and synchronization of two or three vibrators in a nonresonant vibrating system.The above research studies are mainly devoted to synchronization problems of double or multiple vibrators in superresonant region, while synchronous and stable states of the subresonant region are mentioned rarely. This paper focuses on studying synchronous stability of four vibrators in the different resonant states (subresonant and superresonant). The dynamical model consists of four vibrators and a rigid frame. The vibrator is a kind of mechanism which generates exciting forces through eccentric rotors driven by AC motor. The present model is not complicated in structure, but the research results on synchronous and stable states of four vibrators are of great significance to engineering application.In this paper, the figure of the model is given, and the motion differential equations are established in Section2; the synchronization criterion of the four vibrators and the stability criterion are obtained in Section 3; characteristic analysis is discussed numerically and resonant regions are separated in Section 4; computer simulations are analyzed in Section 5; finally, conclusion is drawn.


## 2. Dynamic Model Description

In Figure1(a), the dynamical model consisting of four vibrators and a rigid frame is given, thereof a vibrator is defined as eccentric rotor driven by an induction motor, which can motivate a certain exciting force to make the rigid frame vibrate with a certain type of movement. Four vibrators which rotate in the same directions are driven separately by four induction motors and are fixed on a rigid frame. The rigid frame and a foundation are connected by four springs. Figure 1(b) illustrates three reference coordinates of the vibrating system, which includes the fixed frame oxy; the frame o′x′y′ is parallel to oxy, and the frame o″x″y″ rotates around o′. The displacement vectors of the system in x and y directions are denoted by x and y, respectively. The swing angle of the rigid frame is ψ. The rotational angles of four eccentric rotors are denoted by φi, i = 1, 2, 3, 4. The angles between four vibrators and the x-axis are β1, β2, β3, β4, respectively, with β1=0°, β2=90°, β3=180°, and β4=270°.Figure 1
Model and coordinate systems. (a) Dynamic model of the vibrating system; (b) three reference coordinate systems.
(a)
(b)Based on Lagrange’s equations and referring to [26], the motion differential equations of the vibrating system are established as the following form:(1)Mx¨+fxx˙+kxx=∑i=14mirφ˙i2cosφi+φ¨isinφi,My¨+fyy˙+kyy=∑i=14mirφ˙i2sinφi−φ¨icosφi,Jψ¨+fψψ˙+kψψ=lor∑i=14miφ˙i2sinφi−βi−φ¨icosφi−βi,Joi+mir2φ¨i+fiφ˙i˙=Tei−miry¨cosφi−x¨sinφi+l0ψ¨cosφi−βi+l0ψ˙2sinφi−βi,i=1,2,3,4,where M=m+m1+m2+m3+m4, fψ=fxly2+fylx2/2, J=Jm+m1+m2+m3+m4l02, J=Mle2, and kψ=kxly2+kylx2/2.Herem is the mass of the rigid frame; mi is the mass of the exciter i, i = 1, 2, 3, 4; M is the total mass of the vibrating system, which includes the mass of the rigid frame m and the mass of the four exciters; fi is damping coefficients of the axes of induction motor i, i = 1, 2, 3, 4; fx,fy,fψ are damping constants of the vibrating system in x, y, and ψ directions, respectively; J is the moment of inertia of the vibrating system about its mass center and Jm is that of the rigid frame about its mass center; le is equivalent rotational radius of the vibrating system about the mass center of the rigid frame; l0 denotes the distance between the rotational center of the exciter i and the mass center of the rigid frame; kx,ky,kψ are respectively stiffnesses of the vibrating system in x, y, and ψ directions; Tei [27] is electromagnetic torque of the motor i, i = 1, 2, 3, 4; and r is the eccentric radius of the exciter.


## 3. Theoretical Derivation

It is assumed that the average phase of four vibrators isφ; the phase differences between vibrators 1 and 2, 2 and 3, and 3 and 4 are 2α1, 2α2, and 2α3, respectively, that is,(2)φ=14∑i=14φi,φ1−φ2=2α1,φ2−φ3=2α2,φ3−φ4=2α3.The change of average angular velocityφ˙ of four exciters is periodic because the vibrating system vibrates regularly. We can assume that the least common multiple period of four exciters is T, the average value ωm0 of the average angular velocity φ˙ during time T must be a constant, i.e., ωm0=1/T∫tt+Tφ˙dt=constant. ωm0 denotes the average angular velocity of four exciters and is also referred to as the synchronous operating frequency in the steady state.According to the modified average method of small parameters [25, 26], we can introduce small parameters ς0,ςi (ς0,ςi are functions of time, i = 1, 2, 3). It is assumed that the coefficients of the instantaneous change of φ˙ around ωm0, i.e., φ˙=1+ς0ωm0, α˙i=ςiωm0, i = 1, 2, 3.Thus, the rearrangement of Equation (2) is expressed as(3)φ1=φ+32α1+α2+12α3=φ+υ1,φ2=φ−12α1+α2+12α3=φ+υ2,φ3=φ−12α1−α2+12α3=φ+υ3,φ4=φ−12α1−α2−32α3=φ+υ4,φ˙1=1+ς0+32ς1+ς2+12ς3ωm0=1+ε1ωm0,φ˙2=1+ς0−12ς1+ς2+12ς3ωm0=1+ε2ωm0,φ˙3=1+ς0−12ς1−ς2+12ς3ωm0=1+ε3ωm0,φ˙4=1+ς0−12ς1−ς2−32ς3ωm0=1+ε4ωm0,φ¨1=ε˙1ωm0,φ¨2=ε˙2ωm0,φ¨3=ε˙3ωm0,φ¨4=ε˙4ωm0.If the average values ofεi i=1,2,3,4 over the single period are zero, i.e., ε¯i=0, i=1,2,3,4, we have φ˙1=φ˙2=φ˙3=φ˙4=ωm0, thus the four exciters can operate synchronously. At the moment, the angular accelerations φ¨i can be neglected in formulae of Equation (1). For the solutions of the first three formulae in Equation (1), according to [26], the responses of the system in x, y, and ψ directions can be obtained directly as(4)x=−rmrμx∑i=14ηicosφ+υi+γx,y=−rmrμy∑i=14ηisinφ+υi+γy,ψ=−rmrrlμψle∑i=14ηisinφ+υi−βi+γψ,where γi is the difference between π and the phase angle of the vibrating system in i-direction. γi=arctan2ξniωni/ωm0/1−ωni/ωm02. i = x, y, ψ; m0 is the mass of the standard exciter; ηi are the mass ratio of the exciter i to the standard one, ηi=mi/m0i=1,2,3,4; μi=1−ωni/ωm02, i = x, y, ψ; rm is mass ratio of the standard exciter to the total vibrating system, i.e., rm=m0/M.Assuming that the derivatives of Equation (4) are x¨,y¨,ψ˙, and ψ¨, we put them into the last three formulae of Equation (1) and then integrate them with φ=0∼2π, and the average differential equation of four vibrators is obtained as(5)Tei−T¯Li=0,i=1,2,3,4,T¯Li=m0r2ωm0χfi+χai,i=1,2,3,4,χf1=ωm0η12Ws0+η1η2Wcs12cos2α¯1+θs12+η1η3Wcs13cos2α¯1+2α¯2+θs13+η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs142,χa1=ωm0η1η2Wcc12sin2α1+θc12+η1η3Wcc13sin2α1+2α2+θc13+η1η4Wcc14sin2α1+2α2+2α3+θc142,χf2=ωm0η22Ws0+η1η2Wcs12cos2α¯1+θs12+η2η3Wcs23cos2α¯2+θs23+η2η4Wcs24cos2α¯2+2α¯3+θs242,χa2=ωm0−η1η2Wcc12sin2α1+θc12+η2η3Wcc23sin2α2+θc23+η2η4Wcc24sin2α2+2α3+θc242,χf3=ωm0η32Ws0+η2η3Wcs23cos2α¯2+θs23+η1η3Wcs13cos2α¯1+2α¯2+θs13+η3η4Wcs34cos2α¯3+θs342,χa3=ωm0−η2η3Wcc23sin2α2+θc23−η1η3Wcc13sin2α1+2α2+θc13+η3η4Wcc34sin2α3+θc342,χf4=ωm0η42Ws0+η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14+η2η4Wcs24cos2α¯2+2α¯3+θs24+η3η4Wcs34cos2α¯3+θs342,χa4=ωm0−η1η4Wcc14sin2α1+2α2+2α3+θc14−η2η4Wcc24sin2α2+2α3+θc24−η3η4Wcc34sin2α3+θc342,Ws0=rmrl02sinγψμψ+sinγxμx+sinγyμy,Wc0=rmrl02cosγψμψ+cosγxμx+cosγyμy,acsij=sinγxμx+sinγyμy+rl2sinγψμψcosβi−βj,i=1,2,3;i<j≤4,bcsij=rl2sinγψμψsinβi−βj,i=1,2,3;i<j≤4,accij=−cosγxμx−cosγyμy−rl2cosγψμψcosβi−βj,i=1,2,3;i<j≤4,bccij=rl2cosγψμψsinβi−βj,i=1,2,3;i<j≤4,Wcsij=rmacsij2+bcsij2,i=1,2,3;i<j≤4,Wccij=rmaccij2+bccij2,i=1,2,3;i<j≤4,θsij=arctan−bcsij/acsij,acsij≥0;π+arctan−bcsij/acsij,acsij<0;i=1,2,3;i<j≤4,θcij=arctanbccij/accij,accij≥0;π+arctanbccij/accij,accij<0;i=1,2,3;i<j≤4.


### 3.1. Synchronization Criterion

Based on the formula of electromagnetic torque, the frequency capture equation is obtained. When four vibrators can operate synchronously, we can obtain the output electromagnetic torqueT0i as follows:(6)T0i=Te0i−fiωm0=m0r2ωm0χfi+χai,i=1,2,3,4,where T0i indicates the differences between the electromagnetic torque of the motor i and the damping torque of its rotor.From Equation (6), the differences of output torque between the motors 1 and 2 (ΔT012), 2 and 3 (ΔT023), and 3 and 4 (ΔT034) can be expressed as(7)ΔT012=T01−T02=Tuη12−η22Ws0+η1η3Wcs13cos2α¯1+2α¯2+θs13−η2η3Wcs23cos2α¯2+θs23+η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η4Wcs24cos2α¯2+2α¯3+θs24+2η1η2Wcc12sin2α¯1+θc12+η1η3Wcc13sin2α¯1+2α¯2+θc13−η2η3Wcc23sin2α¯2+θc23+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14−η2η4Wcc24sin2α¯2+2α¯3+θc24,ΔT023=T02−T03=Tuη22−η32Ws0+η1η2Wcs12cos2α¯1+θs12−η1η3Wcs13cos2α¯1+2α¯2+θs13+η2η4Wcs24cos2α¯2+2α¯3+θs24−η3η4Wcs34cos2α¯3+θs34+2η2η3Wcc23sin2α¯2+θc23+η1η3Wcc13sin2α¯1+2α¯2+θc13−η1η2Wcc12sin2α¯1+θcc12+η2η4Wcc24sin2α¯2+2α¯3+θc24−η3η4Wcc34sin2α¯3+θc34,ΔT034=T03−T04=Tuη32−η42Ws0+η2η3Wcs23cos2α¯2+θs23−η2η4Wcs24cos2α¯2+2α¯3+θs24+η1η3Wcs13cos2α¯1+2α¯2+θs13−η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η3Wcc23sin2α¯2+θc23−η1η3Wcc13sin2α¯1+2α¯2+θc13+2η3η4Wcc34sin2α¯3+θc34+η2η4Wcc24sin2α¯2+2α¯3+θc24+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14,where Tsu=m0r2ωm02/2 indicates the kinetic energy of the standard vibrator.Through rearrangement, Equation (7) is expressed as(8)ΔT012TSU−η12−η22WS0=τc12α¯1,α¯2,α¯3,(9)ΔT023TSU−η22−η32WS0=τc23α¯1,α¯2,α¯3,(10)ΔT034TSU−η32−η42WS0=τc34α¯1,α¯2,α¯3.The coefficientsτc12α¯1,α¯2,α¯3,τc23α¯1,α¯2,α¯3,τc34α¯1,α¯2,α¯3 of Equations (8)–(10) are given by(11)τc12α¯1,α¯2,α¯3=η1η3Wcs13cos2α¯1+2α¯2+θs13−η2η3Wcs23cos2α¯2+θs13+η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η4Wcs24cos2α¯2+2α¯3+θs24+2η1η2Wcc12sin2α¯1+θc12+η1η3Wcc13sin2α¯1+2α¯2+θc13−η2η3Wcc23sin2α¯2+θc23+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14−η2η4Wcc24sin2α¯2+2α¯3+θc24,τc23α¯1,α¯2,α¯3=η1η2Wcs12cos2α¯1+θs12−η1η3Wcs13cos2α¯1+2α¯2+θs13+η2η4Wcs24cos2α¯2+2α¯3+θs24−η3η4Wcs34cos2α¯3+θs34+2η2η3Wcc23sin2α¯2+θc23+η1η3Wcc13sin2α¯1+2α¯2+θc13−η1η2Wcc12sin2α¯1+θc12+η2η4Wcc24sin2α¯2+2α¯3+θc24−η3η4Wcc34sin2α¯3+θc24,τc34α¯1,α¯2,α¯3=η2η3Wcs23cos2α¯2+θs23−η2η4Wcs24cos2α¯2+2α¯3+θs24+η1η3Wcs13cos2α¯1+2α¯2+θs13−η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η3Wcc23sin2α¯2+θc23−η1η3Wcc13sin2α¯1+2α¯2+θc13+2η3η4Wcc34sin2α¯3+θc34+η2η4Wcc24sin2α¯2+2α¯3+θc24+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14.τ
c
i
j
α
¯
1
,
α
¯
2
,
α
¯
3
i
j
=
12,23,34 denote the dimensionless coupling torque between vibrators i and j, which are limited functions of α¯1, α¯2, and α¯3, that is,(12)τc12α¯1,α¯2,α¯3≤τc12max,(13)τc23α¯1,α¯2,α¯3≤τc23max,(14)τc34α¯1,α¯2,α¯3≤τc34max.Considering Equations (8)–(10) and (12)–(14), we can obtain(15)ΔT012Tsu−η12−η22Ws0≤τc12max,(16)ΔT023Tsu−η22−η32Ws0≤τc23max,(17)ΔT034Tsu−η32−η42Ws0≤τc34max.Equations (15)–(17) denote the synchronization criterion of four vibrators. The left sides of inequalities show the absolute value of dimensionless residual torque differences between arbitrary two motors, and their right sides represent the maximum of dimensionless coupling torque.The sum of Equation (6) leads to(18)τaα¯1,α¯2,α¯3=14Tsu∑i=14Toi=14∑i=14ηi2Ws0+2η1η2Wcs12cos2α¯1+θs12+2η2η3Wcs23cos2α¯2+θs23+2η3η4Wcs34cos2α¯3+θs34+2η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14+2η1η3Wcs13cos2α¯1+2α¯2+θs13+2η2η4Wcs24cos2α¯2+2α¯3+θs24,where τaα¯1,α¯2,α¯3 indicates the average dimensionless loading torque of four motors. As a limited function, it satisfies the following formula:(19)τaα¯1,α¯2,α¯3≤τamax,where τamax denotes the maximum of the average dimensionless loading torque of four motors.We define thatζij denotes the coefficient of synchronization ability, that is,(20)ζ12=τc12maxτamax,ζ23=τc23maxτamax,ζ34=τc34maxτamax,ζ41=τc41maxτamax,with(21)τc41α¯1,α¯2,α¯3≤τc41max,τc41α¯1,α¯2,α¯3=−τc12α¯1,α¯2,α¯3+τc23α¯1,α¯2,α¯3+τc34α¯1,α¯2,α¯3.The larger the above coefficients, the stronger the coupling torques among vibrators. In other words, the larger these coefficientsζ12,ζ23,ζ34,andζ41, the more easily vibrators can achieve synchronization.τ
c
i
j
α
¯
1
,
α
¯
2
,
α
¯
3
i
j
=
12,23,34 and τaα¯1,α¯2,α¯3 are decided by parameters of the system, and their maximum will be discussed numerically in Section 4.


### 3.2. Stability Criterion of the Synchronous States

According to Equation (3), the following formulae are obtained:(22)υ1=32α1+α2+12α3,υ2=−12α1+α2+12α3,υ3=−12α1−α2+12α3,υ4=−12α1−α2−32α3.AssumingΔυi=υi−υi0 and linearizing Δυi′ around α¯10,α¯20,α¯30, we can obtain(23)Δυ1′=−∑i=13∂χa1∂αi0Δαi,Δυ2′=−∑i=13∂χa2∂αi0Δαi,Δυ3′=−∑i=13∂χa3∂αi0Δαi,Δυ4′=−∑i=13∂χa4∂αi0Δαi,where ⋅0 denotes the values for α¯1=α¯10,α¯2=α¯20,α¯3=α¯30,andΔαi=α¯i−α¯i0,i=1,2,3.Through rearrangement, Equation (23) is expressed as(24)Δα˙=DΔα,with(25)D=dij3×3d11=−ωm0∗2η1η21/k11+1/k22Wcc12cos2α¯10+θc12+η1η3Wcc13cos2α¯10+2α¯20+θc13k11+η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k11,d12=ωm0∗2η2η3Wcc23cos2α¯20+θc23k22+η2η4Wcc24cos2α¯20+2α¯30+θc24k22−η1η3Wcc13cos2α¯10+2α¯20+θc13k11−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k11,d13=ωm0∗2η2η4Wcc24cos2α¯20+2α¯30+θc24k22−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k11,d21=ωm0∗2η1η2Wcc12cos2α¯10+θc12k22−η1η3Wcc13cos2α¯10+2α¯20+θc13k33,d22=−ωm0∗2η2η31/k22+1/k33Wcc23cos2α¯20+θc23+η2η4Wcc24cos2α¯20+2α¯30+θc24k22+η1η3Wcc13cos2α¯10+2α¯20+θc13k33,d31=ωm0∗2η1η3Wcc13cos2α¯10+2α¯20+θc13k33−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k44,d32=ωm0∗2η2η3Wcc23cos2α¯20+θc23k33+η1η3Wcc13cos2α¯10+2α¯20+θc13k33−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k44−η2η4Wcc24cos2α¯20+2α¯30+θc24k44,d33=−ωm0∗2η3η41/k33+1/k44Wcc34cos2α¯30+θc23+η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k44+η2η4Wcc24cos2α¯20+2α¯30+θc24k44.We assumeΔα=vexpλt and insert it into Equation (24) and then solve the equation detD−λI; the characteristic equation for eigenvalue λ is obtained as follows:(26)λ3−d11+d22+d33λ2+d11d33+d22d33+d11d22−d13d31−d23d32−d12d21λ−d11d22d33+d12d23d31+d13d21d32−d13d22d31−d11d23d32−d12d21d33=0.Considering to introduce additional parameters(27)a=−d11−d22−d33,b=d11d33+d22d33+d11d22−d13d31−d23d32−d12d21,e=−d11d22d33+d12d23d31+d13d21d32−d13d22d31−d11d23d32−d12d21d33.After the rearrangement, we can deduce(28)λ3+aλ2+bλ+e=0.Applying the Routh–Hurwitz criterion, we deduce the criterion of stability of the synchronous states in the following equation:(29)a>0,ab−e>0,e>0.Substituting the parametersa, b, and e into Equation (29) yields(30)H1=−d11−d22−d33>0,H2=d11d33+d22d33+d11d22−d13d31−d23d32−d12d21>0,H3=−d11d22d33+d12d23d31+d13d21d32−d13d22d31−d11d23d32−d12d21d33>0,or(31)H1>0,H2>0,H3>0,where Hii=1,2,3 denote the coefficients of ability of synchronization stability. Only when the structural parameters of the system meet Equation (31), can the synchronous states of the system be stable.Generally in engineering, the parameters of induction motors are selected as the same, that is,(32)kii≈k0,Te1≈Te2≈Te3≈Te4i=1,2,3,4.Introducing the following new symbols,(33)p12=η1η2Wcc12cos2α¯10+θc12,p23=η2η3Wcc23cos2α¯20+θc23,p34=η3η4Wcc34cos2α¯30+θc34,p14=η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14,p13=η1η3Wcc13cos2α¯10+2α¯20+θc13,p24=η2η4Wcc24cos2α¯20+2α¯30+θc24.Considering Equation (30), we have(34)P12>0,P23>0,P34>0,P14>0,P13>0,P24>0.Inserting Equation (32) into Equation (29) and considering Equation (34), the following formulae are deduced:(35)a=−d11−d22−d33=ωm0∗/k0p12+p23+p13+p24+p14+p34>0,(36)e=ωm0∗32k03p12p23p14+p12p23p24+p12p24p34+p12p13p34+p12p13p14+p12p13p24+p13p23p34+p13p23p14+p13p23p24+p13p24p34+p13p24p14+p14p23p34+p14p23p24+p14p24p34+p12p23p34+p12p34p14>0,(37)ab−e=ωm0∗34k033p14p242+4p23p142+3p12p242+4p13p242+4p24p132+3p34p132+3p14p132+3p34p142+3p12p142+3p13p142+3p24p142+8p12p23p34+8p12p23p14+9p12p24p14+8p12p13p34+7p12p13p14+8p12p13p24+7p13p23p34+8p13p23p14+8p13p23p24+8p13p24p34+8p13p24p14+8p14p23p34+8p14p23p24+7p14p24p34+9p14p13p34+8p12p34p14+3p232p34+4p232p14+3p232p24+3p232p13+3p232p12+3p242p23+3p242p34+4p342p12+3p342p13+3p342p14+3p342p23+3p342p24+4p122p14+3p122p14+3p122p24+3p122p13+3p122p23+3p132p12+3p132p23+9p23p24p34+9p12p13p23+7p12p23p24+8p12p24p34>0.Considering the stability criterion (Equation (31)) and Equations (35)–(37), we can achieve stable intervals of phase differences among four vibrators as follows:(38)2α¯10+θc12∈−π2,π2,2α¯20+θc23∈−π2,π2,2α¯30+θc34∈−π2,π2,2α¯10+2α¯20+2α¯30+θc14∈−π2,π2,2α¯10+2α¯20+θc13∈−π2,π2,2α¯20+2α¯30+θc24∈−π2,π2.


## 3.1. Synchronization Criterion

Based on the formula of electromagnetic torque, the frequency capture equation is obtained. When four vibrators can operate synchronously, we can obtain the output electromagnetic torqueT0i as follows:(6)T0i=Te0i−fiωm0=m0r2ωm0χfi+χai,i=1,2,3,4,where T0i indicates the differences between the electromagnetic torque of the motor i and the damping torque of its rotor.From Equation (6), the differences of output torque between the motors 1 and 2 (ΔT012), 2 and 3 (ΔT023), and 3 and 4 (ΔT034) can be expressed as(7)ΔT012=T01−T02=Tuη12−η22Ws0+η1η3Wcs13cos2α¯1+2α¯2+θs13−η2η3Wcs23cos2α¯2+θs23+η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η4Wcs24cos2α¯2+2α¯3+θs24+2η1η2Wcc12sin2α¯1+θc12+η1η3Wcc13sin2α¯1+2α¯2+θc13−η2η3Wcc23sin2α¯2+θc23+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14−η2η4Wcc24sin2α¯2+2α¯3+θc24,ΔT023=T02−T03=Tuη22−η32Ws0+η1η2Wcs12cos2α¯1+θs12−η1η3Wcs13cos2α¯1+2α¯2+θs13+η2η4Wcs24cos2α¯2+2α¯3+θs24−η3η4Wcs34cos2α¯3+θs34+2η2η3Wcc23sin2α¯2+θc23+η1η3Wcc13sin2α¯1+2α¯2+θc13−η1η2Wcc12sin2α¯1+θcc12+η2η4Wcc24sin2α¯2+2α¯3+θc24−η3η4Wcc34sin2α¯3+θc34,ΔT034=T03−T04=Tuη32−η42Ws0+η2η3Wcs23cos2α¯2+θs23−η2η4Wcs24cos2α¯2+2α¯3+θs24+η1η3Wcs13cos2α¯1+2α¯2+θs13−η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η3Wcc23sin2α¯2+θc23−η1η3Wcc13sin2α¯1+2α¯2+θc13+2η3η4Wcc34sin2α¯3+θc34+η2η4Wcc24sin2α¯2+2α¯3+θc24+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14,where Tsu=m0r2ωm02/2 indicates the kinetic energy of the standard vibrator.Through rearrangement, Equation (7) is expressed as(8)ΔT012TSU−η12−η22WS0=τc12α¯1,α¯2,α¯3,(9)ΔT023TSU−η22−η32WS0=τc23α¯1,α¯2,α¯3,(10)ΔT034TSU−η32−η42WS0=τc34α¯1,α¯2,α¯3.The coefficientsτc12α¯1,α¯2,α¯3,τc23α¯1,α¯2,α¯3,τc34α¯1,α¯2,α¯3 of Equations (8)–(10) are given by(11)τc12α¯1,α¯2,α¯3=η1η3Wcs13cos2α¯1+2α¯2+θs13−η2η3Wcs23cos2α¯2+θs13+η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η4Wcs24cos2α¯2+2α¯3+θs24+2η1η2Wcc12sin2α¯1+θc12+η1η3Wcc13sin2α¯1+2α¯2+θc13−η2η3Wcc23sin2α¯2+θc23+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14−η2η4Wcc24sin2α¯2+2α¯3+θc24,τc23α¯1,α¯2,α¯3=η1η2Wcs12cos2α¯1+θs12−η1η3Wcs13cos2α¯1+2α¯2+θs13+η2η4Wcs24cos2α¯2+2α¯3+θs24−η3η4Wcs34cos2α¯3+θs34+2η2η3Wcc23sin2α¯2+θc23+η1η3Wcc13sin2α¯1+2α¯2+θc13−η1η2Wcc12sin2α¯1+θc12+η2η4Wcc24sin2α¯2+2α¯3+θc24−η3η4Wcc34sin2α¯3+θc24,τc34α¯1,α¯2,α¯3=η2η3Wcs23cos2α¯2+θs23−η2η4Wcs24cos2α¯2+2α¯3+θs24+η1η3Wcs13cos2α¯1+2α¯2+θs13−η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14−η2η3Wcc23sin2α¯2+θc23−η1η3Wcc13sin2α¯1+2α¯2+θc13+2η3η4Wcc34sin2α¯3+θc34+η2η4Wcc24sin2α¯2+2α¯3+θc24+η1η4Wcc14sin2α¯1+2α¯2+2α¯3+θc14.τ
c
i
j
α
¯
1
,
α
¯
2
,
α
¯
3
i
j
=
12,23,34 denote the dimensionless coupling torque between vibrators i and j, which are limited functions of α¯1, α¯2, and α¯3, that is,(12)τc12α¯1,α¯2,α¯3≤τc12max,(13)τc23α¯1,α¯2,α¯3≤τc23max,(14)τc34α¯1,α¯2,α¯3≤τc34max.Considering Equations (8)–(10) and (12)–(14), we can obtain(15)ΔT012Tsu−η12−η22Ws0≤τc12max,(16)ΔT023Tsu−η22−η32Ws0≤τc23max,(17)ΔT034Tsu−η32−η42Ws0≤τc34max.Equations (15)–(17) denote the synchronization criterion of four vibrators. The left sides of inequalities show the absolute value of dimensionless residual torque differences between arbitrary two motors, and their right sides represent the maximum of dimensionless coupling torque.The sum of Equation (6) leads to(18)τaα¯1,α¯2,α¯3=14Tsu∑i=14Toi=14∑i=14ηi2Ws0+2η1η2Wcs12cos2α¯1+θs12+2η2η3Wcs23cos2α¯2+θs23+2η3η4Wcs34cos2α¯3+θs34+2η1η4Wcs14cos2α¯1+2α¯2+2α¯3+θs14+2η1η3Wcs13cos2α¯1+2α¯2+θs13+2η2η4Wcs24cos2α¯2+2α¯3+θs24,where τaα¯1,α¯2,α¯3 indicates the average dimensionless loading torque of four motors. As a limited function, it satisfies the following formula:(19)τaα¯1,α¯2,α¯3≤τamax,where τamax denotes the maximum of the average dimensionless loading torque of four motors.We define thatζij denotes the coefficient of synchronization ability, that is,(20)ζ12=τc12maxτamax,ζ23=τc23maxτamax,ζ34=τc34maxτamax,ζ41=τc41maxτamax,with(21)τc41α¯1,α¯2,α¯3≤τc41max,τc41α¯1,α¯2,α¯3=−τc12α¯1,α¯2,α¯3+τc23α¯1,α¯2,α¯3+τc34α¯1,α¯2,α¯3.The larger the above coefficients, the stronger the coupling torques among vibrators. In other words, the larger these coefficientsζ12,ζ23,ζ34,andζ41, the more easily vibrators can achieve synchronization.τ
c
i
j
α
¯
1
,
α
¯
2
,
α
¯
3
i
j
=
12,23,34 and τaα¯1,α¯2,α¯3 are decided by parameters of the system, and their maximum will be discussed numerically in Section 4.


## 3.2. Stability Criterion of the Synchronous States

According to Equation (3), the following formulae are obtained:(22)υ1=32α1+α2+12α3,υ2=−12α1+α2+12α3,υ3=−12α1−α2+12α3,υ4=−12α1−α2−32α3.AssumingΔυi=υi−υi0 and linearizing Δυi′ around α¯10,α¯20,α¯30, we can obtain(23)Δυ1′=−∑i=13∂χa1∂αi0Δαi,Δυ2′=−∑i=13∂χa2∂αi0Δαi,Δυ3′=−∑i=13∂χa3∂αi0Δαi,Δυ4′=−∑i=13∂χa4∂αi0Δαi,where ⋅0 denotes the values for α¯1=α¯10,α¯2=α¯20,α¯3=α¯30,andΔαi=α¯i−α¯i0,i=1,2,3.Through rearrangement, Equation (23) is expressed as(24)Δα˙=DΔα,with(25)D=dij3×3d11=−ωm0∗2η1η21/k11+1/k22Wcc12cos2α¯10+θc12+η1η3Wcc13cos2α¯10+2α¯20+θc13k11+η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k11,d12=ωm0∗2η2η3Wcc23cos2α¯20+θc23k22+η2η4Wcc24cos2α¯20+2α¯30+θc24k22−η1η3Wcc13cos2α¯10+2α¯20+θc13k11−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k11,d13=ωm0∗2η2η4Wcc24cos2α¯20+2α¯30+θc24k22−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k11,d21=ωm0∗2η1η2Wcc12cos2α¯10+θc12k22−η1η3Wcc13cos2α¯10+2α¯20+θc13k33,d22=−ωm0∗2η2η31/k22+1/k33Wcc23cos2α¯20+θc23+η2η4Wcc24cos2α¯20+2α¯30+θc24k22+η1η3Wcc13cos2α¯10+2α¯20+θc13k33,d31=ωm0∗2η1η3Wcc13cos2α¯10+2α¯20+θc13k33−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k44,d32=ωm0∗2η2η3Wcc23cos2α¯20+θc23k33+η1η3Wcc13cos2α¯10+2α¯20+θc13k33−η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k44−η2η4Wcc24cos2α¯20+2α¯30+θc24k44,d33=−ωm0∗2η3η41/k33+1/k44Wcc34cos2α¯30+θc23+η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14k44+η2η4Wcc24cos2α¯20+2α¯30+θc24k44.We assumeΔα=vexpλt and insert it into Equation (24) and then solve the equation detD−λI; the characteristic equation for eigenvalue λ is obtained as follows:(26)λ3−d11+d22+d33λ2+d11d33+d22d33+d11d22−d13d31−d23d32−d12d21λ−d11d22d33+d12d23d31+d13d21d32−d13d22d31−d11d23d32−d12d21d33=0.Considering to introduce additional parameters(27)a=−d11−d22−d33,b=d11d33+d22d33+d11d22−d13d31−d23d32−d12d21,e=−d11d22d33+d12d23d31+d13d21d32−d13d22d31−d11d23d32−d12d21d33.After the rearrangement, we can deduce(28)λ3+aλ2+bλ+e=0.Applying the Routh–Hurwitz criterion, we deduce the criterion of stability of the synchronous states in the following equation:(29)a>0,ab−e>0,e>0.Substituting the parametersa, b, and e into Equation (29) yields(30)H1=−d11−d22−d33>0,H2=d11d33+d22d33+d11d22−d13d31−d23d32−d12d21>0,H3=−d11d22d33+d12d23d31+d13d21d32−d13d22d31−d11d23d32−d12d21d33>0,or(31)H1>0,H2>0,H3>0,where Hii=1,2,3 denote the coefficients of ability of synchronization stability. Only when the structural parameters of the system meet Equation (31), can the synchronous states of the system be stable.Generally in engineering, the parameters of induction motors are selected as the same, that is,(32)kii≈k0,Te1≈Te2≈Te3≈Te4i=1,2,3,4.Introducing the following new symbols,(33)p12=η1η2Wcc12cos2α¯10+θc12,p23=η2η3Wcc23cos2α¯20+θc23,p34=η3η4Wcc34cos2α¯30+θc34,p14=η1η4Wcc14cos2α¯10+2α¯20+2α¯30+θc14,p13=η1η3Wcc13cos2α¯10+2α¯20+θc13,p24=η2η4Wcc24cos2α¯20+2α¯30+θc24.Considering Equation (30), we have(34)P12>0,P23>0,P34>0,P14>0,P13>0,P24>0.Inserting Equation (32) into Equation (29) and considering Equation (34), the following formulae are deduced:(35)a=−d11−d22−d33=ωm0∗/k0p12+p23+p13+p24+p14+p34>0,(36)e=ωm0∗32k03p12p23p14+p12p23p24+p12p24p34+p12p13p34+p12p13p14+p12p13p24+p13p23p34+p13p23p14+p13p23p24+p13p24p34+p13p24p14+p14p23p34+p14p23p24+p14p24p34+p12p23p34+p12p34p14>0,(37)ab−e=ωm0∗34k033p14p242+4p23p142+3p12p242+4p13p242+4p24p132+3p34p132+3p14p132+3p34p142+3p12p142+3p13p142+3p24p142+8p12p23p34+8p12p23p14+9p12p24p14+8p12p13p34+7p12p13p14+8p12p13p24+7p13p23p34+8p13p23p14+8p13p23p24+8p13p24p34+8p13p24p14+8p14p23p34+8p14p23p24+7p14p24p34+9p14p13p34+8p12p34p14+3p232p34+4p232p14+3p232p24+3p232p13+3p232p12+3p242p23+3p242p34+4p342p12+3p342p13+3p342p14+3p342p23+3p342p24+4p122p14+3p122p14+3p122p24+3p122p13+3p122p23+3p132p12+3p132p23+9p23p24p34+9p12p13p23+7p12p23p24+8p12p24p34>0.Considering the stability criterion (Equation (31)) and Equations (35)–(37), we can achieve stable intervals of phase differences among four vibrators as follows:(38)2α¯10+θc12∈−π2,π2,2α¯20+θc23∈−π2,π2,2α¯30+θc34∈−π2,π2,2α¯10+2α¯20+2α¯30+θc14∈−π2,π2,2α¯10+2α¯20+θc13∈−π2,π2,2α¯20+2α¯30+θc24∈−π2,π2.


## 4. Numerical Characteristics Analyses

Based on the above deduced theoretical analytical results on criteria of synchronization and stability of four vibrators, we go a step further to analyze the coupling characteristics of the vibrating system in this section by numeric.It is assumed that four motors are the same and the corresponding parameters are as follows:M=1430kG, m0=10kG, J=1750kG⋅m2, kψ=20000kN/mμψ=0.95, fψ=3.37kNs/radξnx=ξny=ξnψ=0.07, kx=ky=23000kN/mμx=μy=0.95, and fx=fy=3.83kNs/m. The type of four motors is three-phase squirrel-cage (380 V, 50 Hz, 6-pole, and Δ-connected); the parameters (0.3 kW, rated speed 980 r/min) are the stator resistance Rs=3.35Ω, the rotor resistance Rr=3.4Ω, the stator inductance Ls=170mH, the mutual inductance Ls=164mH, and f1=f2=f3=f4=0.05. The natural frequencies are defined as(39)ωx=kxM,ωy=kyM,ωψ=kψJ.Because the stiffnesses of springs inx and y directions are identical, the natural frequencies in these two directions are the same, i.e., ωx=ωy=126.8≈127rad/s, and ωψ=106.9≈107rad/s means the natural frequency in ψ direction. Thus resonant regions of the vibrating system are divided into three ranges: regions Ιωm0<ωψ, ΙΙωψ<ωm0<ωx, and ΙΙΙωm0>ωx. It should be noted that in the following numeric characteristic analyses, we obtain the rounding values of ωx,ωy,andωψ in order to only discuss and describe conveniently, i.e., ωx=ωy≈127rad/s and ωψ≈107rad/s, which cannot influence the synchronization of the four vibrators and also not change the physical configuration of the system.


### 4.1. Synchronization Criterion

As shown in Section3, τcijα¯1,α¯2,α¯3 is the function of α1, α2, and α3. Figure 2 shows that values of τcijmax for the different ηi (i = 1, 2, 3, 4), in which maxima of τcijmax are achieved by ωx=127rad/s, and τcijmax under the condition of η1=η2=η3=η4=1 is larger than the others, and in this case, vibrators are easiest to implement synchronization.Figure 2
τ
c
i
j
max in different ηii=1,2,3,4. (a) η1=η2=η3=η4=1; (b) η1=η2=η3=1,η4=0.5; (c) η1=η2=1,η3=0.75,η4=0.5.
(a)
(b)
(c)The parameterζij denotes the coefficient of synchronization ability (CSA) and their expressions are given in Equation (20). Figure 3 shows the CSAs among four vibrators for the different ηi. In Figure 3(a), we can see that ζ12=ζ23=ζ34=ζ41>ζ13=ζ24 due to complete symmetry of the system; Figures 3(b) and 3(c) have ζ12=ζ23>ζ34>ζ41>ζ13>ζ24 and ζ12>ζ23>ζ41>ζ24>ζ13>ζ34, respectively, which indicate the important influence of ηi to synchronization. It should be noted here that the derivation method of τcijmax and ζij (ij = 13, 24) is similar to that of Equations (8)–(10), and hence, it is not dwelt on again. Additionally, CSAs in all conditions decrease as ωm0 increases in region ΙΙΙωm0>ωx.Figure 3
CGDSs among four vibrators in differentηii=1,2,3,4. (a) η1=η2=η3=η4=1; (b) η1=η2=η3=1,η4=0.5; (c) η1=η2=1,η3=0.75,η4=0.5.
(a)
(b)
(c)


### 4.2. Stable Phase Differences

From Figure4(a), we can find that the phase differences in region Ιωm0<ωψ are the same as that in region ΙΙωψ<ωm0<ωx. Due to the fact of the small difference value between ωx and ωψ in engineering, we may combine regions Ι and ΙΙ into one common region, which is defined as the subresonant region with respect to ωx, while region ΙΙΙ is defined as the superresonant region for ωx. Here L1 means the subresonant region and L2 refers to the superresonant region. Dashed line A means the simulation point of the subresonant region in next section, and 74 means ωm0=74rad/s. As such dashed line B means the simulation point of the superresonant region and 198 for ωm0=198rad/s. The validity of these calculated values will be verified in next section of computer simulations.Figure 4
Stable difference-values of phase in the differentηii=1,2,3,4. (a) η1=η2=η3=η4=1; (b) η1=η2=η3=1,η4=0.5; (c) η1=η2=1,η3=0.75,η4=0.5.
(a)
(b)
(c)It can be seen that from Figure4(a), in the case of stabilization, the phase differences are all close to 0° in the subresonance region, while in the superresonance region, the phase differences are near 90° and 180°, which denotes the existence of diversity of nonlinear system due to the existence of double equilibrium points [28].The vibrators 1, 2, and 3 are the same, and the mass of the vibrator 4 is changed, and the stable phase difference curves of Figure4(b) are obtained. The phase differences are near to 0°in the subresonance region; while in the superresonance region, there are two groups of stable phase differences among vibrators: 2α1−1=104°, 2α1−2=151°, 2α2−1=108°, 2α2−2=158°, 2α3−1=72°, 2α3−2=−158°. While in the near-resonance region, the transition value of the phase difference appears.The vibrators 1 and 2 are the same, and masses of the vibrators 3 and 4 are changed; the stable phase difference curves of Figure4(c) are obtained; the phase differences are in the vicinity of 0° in the subresonance region, while in the superresonance region, there are two stable phase differences: 2α1−1=112°, 2α1−2=153°, 2α2−1=104°, 2α2−2=162°, 2α3−1=54°, 2α3−2=−144°. In this case, the transition value of the phase differences also appears in the near-resonance region.


## 4.1. Synchronization Criterion

As shown in Section3, τcijα¯1,α¯2,α¯3 is the function of α1, α2, and α3. Figure 2 shows that values of τcijmax for the different ηi (i = 1, 2, 3, 4), in which maxima of τcijmax are achieved by ωx=127rad/s, and τcijmax under the condition of η1=η2=η3=η4=1 is larger than the others, and in this case, vibrators are easiest to implement synchronization.Figure 2
τ
c
i
j
max in different ηii=1,2,3,4. (a) η1=η2=η3=η4=1; (b) η1=η2=η3=1,η4=0.5; (c) η1=η2=1,η3=0.75,η4=0.5.
(a)
(b)
(c)The parameterζij denotes the coefficient of synchronization ability (CSA) and their expressions are given in Equation (20). Figure 3 shows the CSAs among four vibrators for the different ηi. In Figure 3(a), we can see that ζ12=ζ23=ζ34=ζ41>ζ13=ζ24 due to complete symmetry of the system; Figures 3(b) and 3(c) have ζ12=ζ23>ζ34>ζ41>ζ13>ζ24 and ζ12>ζ23>ζ41>ζ24>ζ13>ζ34, respectively, which indicate the important influence of ηi to synchronization. It should be noted here that the derivation method of τcijmax and ζij (ij = 13, 24) is similar to that of Equations (8)–(10), and hence, it is not dwelt on again. Additionally, CSAs in all conditions decrease as ωm0 increases in region ΙΙΙωm0>ωx.Figure 3
CGDSs among four vibrators in differentηii=1,2,3,4. (a) η1=η2=η3=η4=1; (b) η1=η2=η3=1,η4=0.5; (c) η1=η2=1,η3=0.75,η4=0.5.
(a)
(b)
(c)


## 4.2. Stable Phase Differences

From Figure4(a), we can find that the phase differences in region Ιωm0<ωψ are the same as that in region ΙΙωψ<ωm0<ωx. Due to the fact of the small difference value between ωx and ωψ in engineering, we may combine regions Ι and ΙΙ into one common region, which is defined as the subresonant region with respect to ωx, while region ΙΙΙ is defined as the superresonant region for ωx. Here L1 means the subresonant region and L2 refers to the superresonant region. Dashed line A means the simulation point of the subresonant region in next section, and 74 means ωm0=74rad/s. As such dashed line B means the simulation point of the superresonant region and 198 for ωm0=198rad/s. The validity of these calculated values will be verified in next section of computer simulations.Figure 4
Stable difference-values of phase in the differentηii=1,2,3,4. (a) η1=η2=η3=η4=1; (b) η1=η2=η3=1,η4=0.5; (c) η1=η2=1,η3=0.75,η4=0.5.
(a)
(b)
(c)It can be seen that from Figure4(a), in the case of stabilization, the phase differences are all close to 0° in the subresonance region, while in the superresonance region, the phase differences are near 90° and 180°, which denotes the existence of diversity of nonlinear system due to the existence of double equilibrium points [28].The vibrators 1, 2, and 3 are the same, and the mass of the vibrator 4 is changed, and the stable phase difference curves of Figure4(b) are obtained. The phase differences are near to 0°in the subresonance region; while in the superresonance region, there are two groups of stable phase differences among vibrators: 2α1−1=104°, 2α1−2=151°, 2α2−1=108°, 2α2−2=158°, 2α3−1=72°, 2α3−2=−158°. While in the near-resonance region, the transition value of the phase difference appears.The vibrators 1 and 2 are the same, and masses of the vibrators 3 and 4 are changed; the stable phase difference curves of Figure4(c) are obtained; the phase differences are in the vicinity of 0° in the subresonance region, while in the superresonance region, there are two stable phase differences: 2α1−1=112°, 2α1−2=153°, 2α2−1=104°, 2α2−2=162°, 2α3−1=54°, 2α3−2=−144°. In this case, the transition value of the phase differences also appears in the near-resonance region.


## 5. Computer Simulations

In this section, the vibrating system is simulated by using the fourth-order Runge–Kutta routine. Here, it is assumed that the mass of the standard vibrator ism0 = 10 kG, and the eccentric radii of four vibrators are same. Meanwhile, in the subresonant region, kx = 23000 kN/m and in the superresonant region, kx = 6240 kN/m; other parameters are the same as those in the previous section. Using quantitative analysis of simulations to verify qualitative analysis of characteristic analysis, we change spring stiffness to implement simulations of two regions. Thus, we need to introduce a new variable; it is the frequency ratio z, which can facilitate to derive ωm0 of the simulation point through it.Equation (40) denotes the frequency ratio; we substitute kx = 23000 kN/m into Equation (39) and obtain a new natural frequency of the point A, that is ωn∗=178rad/s. Simultaneously, based on the rotational speed in simulation (ωSA=103rad/s) and Equation (40), the frequency ratio of the simulation point A is obtained, that is z=0.58. Through the frequency ratio, we can deduce that the operating frequency of the simulation point A is ωm0=74rad/s. In the same way, the frequency ratio and operating frequency of the simulation point of B are z=1.56 and ωm0=198rad/s, when we consider kx = 6240 kN/m.(40)z=ωm0ωn.In other words, the simulation of subresonant region is in the condition ofz=0.58, corresponding to the simulation point A of Figure 4 and that of superresonant region is selected for z=1.56 corresponding to the simulation point B of Figure 4.


### 5.1. Superresonant Region (Simulation Point B) Simulations

#### 5.1.1. Simulation forη1=η2=η3=η4=1

Here, masses of four vibrators are identical, i.e.,mi = m0 = 10 kG (i = 1, 2, 3, 4). In Figure 5, we can see the following results: the angular acceleration of motors is identical when the four motors are in the starting process; therefore, the phase differences are near to 0°. But the responses of the system are aroused when the angular velocity of motors reaches resonance regions of the vibrating system. Until the angular velocity reaches stable and four vibrators can operate synchronously, the high frequency vibration is aroused, and in this moment, the synchronous rotational velocity of motors is 983 r/min, and the phase differences are near to 180°. In Figures 5(b)–5(d), at 20 s, a disturbance of π phase applies to the vibrator 2; 2α1 transforms from 180° to 90°, as well as 2α2 and 2α3, which indicates that there are two equilibrium points in the system, and these results agree well with that of simulation point B in Figure 4(a).Figure 5
Results of computer simulations forη1=η2=η3=η4=1. (a) Rotational velocities of the four motors; (b) phase difference between vibrator 1 and 2; (c) phase difference between vibrator 2 and 3; (d) phase difference between vibrator 3 and 4; (e) displacements in x and y directions; (f) displacement in ψ direction.In Figure5(a), at 40 s, we know that the system is back to the steady state and the synchronous rotational velocity of motors changes to 977 r/min. Meanwhile, in Figures 5(b)–5(d), at 40 s, the electricity of the system is switched off, and then 2α1 transforms from 90° to 107.23°, 2α2 from 90° to 70.15°, and 2α3 from 90° to 94.86°, which means that the phenomenon of vibratory synchronization transmission occurs in the system.In Figures5(e) and 5(f), the resonant responses of the rigid frame in x, y, and ψ directions are close to 0, when the system is back to the steady state, which indicates that the exciting forces of four vibrators are canceled with each other, and the system can still operate synchronously and stably in the superresonant region.


#### 5.1.2. Simulation forη1=η2=η3=1,η4=0.5

The analysis method in this section is the same as the previous section. Figure6 shows that the synchronous rotational velocity is 983 r/min, and the phase differences are 2α1=105°, 2α2=106°, 2α3=70.6°, when four vibrators operate synchronously. A disturbance of π phase applies to the vibrator 2, when time passes through 20 s and the system reach another steady state; in this moment, the phase differences change to 2α1=153°, 2α2=152°, 2α3=−155°. Contrasting the results with simulation point B in Figure 4(b), we can find that they are alike. At 40 s, the electricity of the system is switched off, and we see that the system is back to the steady state from Figure 6(a); in this case, the phase differences are 2α1=115°, 2α2=96°, 2α3=64°, and the synchronous rotational velocity changes to 978 r/min. These facts show that the vibrating system still can operate by the way of vibratory synchronization transmission.Figure 6
Results of computer simulations forη1=η2=η3=1,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)


#### 5.1.3. Simulation forη1=η2=1,η3=0.75,η4=0.5

Figure7 shows the following results. The four vibrators operate synchronously, when time reaches about 5 s. At this time, the phase differences are 2α1=155°, 2α2=158°, 2α3=−144°and the synchronous rotational velocity is 983 r/min. Similarly, a disturbance of π phase applies to the vibrator 2, and the phase differences transform to 2α1=112°, 2α2=103°, 2α3=51°. Reviewing characteristic analysis of Figure 4(c), we can know that the simulation results are line with it. When the electricity of the system is switched off, the system is back to the steady state and still keeps operating synchronously. In this moment, the phase differences are 2α1=125°, 2α2=89.85°, 2α3=42.99° and the synchronous rotational velocity is 978 r/min.Figure 7
Results of computer simulations forη1=η2=1,η3=0.75,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)


### 5.2. Subresonant Region (Simulation Point A) Simulations

The analysis method in this section is the same as the previous section, and the results will be presented in the table form.


#### 5.2.1. Simulation forη1=η2=η3=η4=1

The stable phase differences for Figures8(b)–8(d) are shown in Table 1.Figure 8
Results of computer simulations forη1=η2=η3=η4=1. (a) Rotational velocities of the four motors; (b) phase difference between vibrator 1 and 2; (c) phase difference between vibrator 2 and 3; (d) phase difference between vibrator 3 and 4; (e) displacements in x and y directions; (f) displacement in ψ direction.
(a)
(b)
(c)
(d)
(e)
(f)Table 1
The stable phase differences forη1=η2=η3=η4=1.
Simulation of point A
Characteristic analysis∗
Stable phase differences before disturbance
Stable phase differences after disturbance
Stable phase differences after outage
Stable phase differences
2
α
1
0
°
0
°
12.3
°
0
°
2
α
2
0
°
0
°
−
12.3
°
0
°
2
α
3
0
°
0
°
0.5
°
0
°
∗The results of characteristic analysis data from Figure 4(a).The stable phase differences of simulation and characteristic analysis are in good agreement with each other. In Figures8(e) and 8(f), the resonant responses of the rigid frame in x and y directions are larger and stabilize eventually in the range of −15∼+15, while the resonant response in ψ direction is zero. The above results indicate that the exciting forces superimpose positively, and a strong vibration occurs in the rigid frame, when the system operates in the subresonant region.


#### 5.2.2. Simulation forη1=η2=η3=1,η4=0.5

As shown in Figure9, the results are given in Table 2.Figure 9
Results of computer simulations forη1=η2=η3=1,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)Table 2
The stable phase differences forη1=η2=η3=1,η4=0.5.
Simulation of point A
Characteristic analysis∗
Stable phase differences before disturbance
Stable phase differences after disturbance
Stable phase differences after outage
Stable phase differences
2
α
1
0.8
°
0.8
°
13.3
°
5
°
2
α
2
1
°
1
°
−
11.1
°
4
°
2
α
3
−
11
°
−
11
°
−
12.65
°
−
3
°
∗The results of characteristic analysis come from Figure 4(b).


#### 5.2.3. Simulation forη1=η2=1,η3=0.75,η4=0.5

From Figure10, the results can be obtained and are presented in Table 3.Figure 10
Results of computer simulations forη1=η2=1,η3=0.75,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)Table 3
The stable phase differences forη1=η2=1,η3=0.75,η4=0.5.
Simulation of point A
Characteristic analysis∗
Phase differences before disturbance
Phase differences after disturbance
Phase differences after outage
Phase differences
2
α
1
1.37
°
1.31
°
13.9
°
6
°
2
α
2
−
2.71
°
−
2.78
°
−
15.5
°
3
°
2
α
3
−
8.11
°
−
8.13
°
−
9.18
°
0
°
∗The results of characteristic analysis can be observed in Figure 4(c).We can observe that the phase differences before disturbance are approximately equal to those after disturbance. The phenomenon accords with that of the characteristic analysis.


## 5.1. Superresonant Region (Simulation Point B) Simulations

### 5.1.1. Simulation forη1=η2=η3=η4=1

Here, masses of four vibrators are identical, i.e.,mi = m0 = 10 kG (i = 1, 2, 3, 4). In Figure 5, we can see the following results: the angular acceleration of motors is identical when the four motors are in the starting process; therefore, the phase differences are near to 0°. But the responses of the system are aroused when the angular velocity of motors reaches resonance regions of the vibrating system. Until the angular velocity reaches stable and four vibrators can operate synchronously, the high frequency vibration is aroused, and in this moment, the synchronous rotational velocity of motors is 983 r/min, and the phase differences are near to 180°. In Figures 5(b)–5(d), at 20 s, a disturbance of π phase applies to the vibrator 2; 2α1 transforms from 180° to 90°, as well as 2α2 and 2α3, which indicates that there are two equilibrium points in the system, and these results agree well with that of simulation point B in Figure 4(a).Figure 5
Results of computer simulations forη1=η2=η3=η4=1. (a) Rotational velocities of the four motors; (b) phase difference between vibrator 1 and 2; (c) phase difference between vibrator 2 and 3; (d) phase difference between vibrator 3 and 4; (e) displacements in x and y directions; (f) displacement in ψ direction.In Figure5(a), at 40 s, we know that the system is back to the steady state and the synchronous rotational velocity of motors changes to 977 r/min. Meanwhile, in Figures 5(b)–5(d), at 40 s, the electricity of the system is switched off, and then 2α1 transforms from 90° to 107.23°, 2α2 from 90° to 70.15°, and 2α3 from 90° to 94.86°, which means that the phenomenon of vibratory synchronization transmission occurs in the system.In Figures5(e) and 5(f), the resonant responses of the rigid frame in x, y, and ψ directions are close to 0, when the system is back to the steady state, which indicates that the exciting forces of four vibrators are canceled with each other, and the system can still operate synchronously and stably in the superresonant region.


### 5.1.2. Simulation forη1=η2=η3=1,η4=0.5

The analysis method in this section is the same as the previous section. Figure6 shows that the synchronous rotational velocity is 983 r/min, and the phase differences are 2α1=105°, 2α2=106°, 2α3=70.6°, when four vibrators operate synchronously. A disturbance of π phase applies to the vibrator 2, when time passes through 20 s and the system reach another steady state; in this moment, the phase differences change to 2α1=153°, 2α2=152°, 2α3=−155°. Contrasting the results with simulation point B in Figure 4(b), we can find that they are alike. At 40 s, the electricity of the system is switched off, and we see that the system is back to the steady state from Figure 6(a); in this case, the phase differences are 2α1=115°, 2α2=96°, 2α3=64°, and the synchronous rotational velocity changes to 978 r/min. These facts show that the vibrating system still can operate by the way of vibratory synchronization transmission.Figure 6
Results of computer simulations forη1=η2=η3=1,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)


### 5.1.3. Simulation forη1=η2=1,η3=0.75,η4=0.5

Figure7 shows the following results. The four vibrators operate synchronously, when time reaches about 5 s. At this time, the phase differences are 2α1=155°, 2α2=158°, 2α3=−144°and the synchronous rotational velocity is 983 r/min. Similarly, a disturbance of π phase applies to the vibrator 2, and the phase differences transform to 2α1=112°, 2α2=103°, 2α3=51°. Reviewing characteristic analysis of Figure 4(c), we can know that the simulation results are line with it. When the electricity of the system is switched off, the system is back to the steady state and still keeps operating synchronously. In this moment, the phase differences are 2α1=125°, 2α2=89.85°, 2α3=42.99° and the synchronous rotational velocity is 978 r/min.Figure 7
Results of computer simulations forη1=η2=1,η3=0.75,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)


## 5.1.1. Simulation forη1=η2=η3=η4=1

Here, masses of four vibrators are identical, i.e.,mi = m0 = 10 kG (i = 1, 2, 3, 4). In Figure 5, we can see the following results: the angular acceleration of motors is identical when the four motors are in the starting process; therefore, the phase differences are near to 0°. But the responses of the system are aroused when the angular velocity of motors reaches resonance regions of the vibrating system. Until the angular velocity reaches stable and four vibrators can operate synchronously, the high frequency vibration is aroused, and in this moment, the synchronous rotational velocity of motors is 983 r/min, and the phase differences are near to 180°. In Figures 5(b)–5(d), at 20 s, a disturbance of π phase applies to the vibrator 2; 2α1 transforms from 180° to 90°, as well as 2α2 and 2α3, which indicates that there are two equilibrium points in the system, and these results agree well with that of simulation point B in Figure 4(a).Figure 5
Results of computer simulations forη1=η2=η3=η4=1. (a) Rotational velocities of the four motors; (b) phase difference between vibrator 1 and 2; (c) phase difference between vibrator 2 and 3; (d) phase difference between vibrator 3 and 4; (e) displacements in x and y directions; (f) displacement in ψ direction.In Figure5(a), at 40 s, we know that the system is back to the steady state and the synchronous rotational velocity of motors changes to 977 r/min. Meanwhile, in Figures 5(b)–5(d), at 40 s, the electricity of the system is switched off, and then 2α1 transforms from 90° to 107.23°, 2α2 from 90° to 70.15°, and 2α3 from 90° to 94.86°, which means that the phenomenon of vibratory synchronization transmission occurs in the system.In Figures5(e) and 5(f), the resonant responses of the rigid frame in x, y, and ψ directions are close to 0, when the system is back to the steady state, which indicates that the exciting forces of four vibrators are canceled with each other, and the system can still operate synchronously and stably in the superresonant region.


## 5.1.2. Simulation forη1=η2=η3=1,η4=0.5

The analysis method in this section is the same as the previous section. Figure6 shows that the synchronous rotational velocity is 983 r/min, and the phase differences are 2α1=105°, 2α2=106°, 2α3=70.6°, when four vibrators operate synchronously. A disturbance of π phase applies to the vibrator 2, when time passes through 20 s and the system reach another steady state; in this moment, the phase differences change to 2α1=153°, 2α2=152°, 2α3=−155°. Contrasting the results with simulation point B in Figure 4(b), we can find that they are alike. At 40 s, the electricity of the system is switched off, and we see that the system is back to the steady state from Figure 6(a); in this case, the phase differences are 2α1=115°, 2α2=96°, 2α3=64°, and the synchronous rotational velocity changes to 978 r/min. These facts show that the vibrating system still can operate by the way of vibratory synchronization transmission.Figure 6
Results of computer simulations forη1=η2=η3=1,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)


## 5.1.3. Simulation forη1=η2=1,η3=0.75,η4=0.5

Figure7 shows the following results. The four vibrators operate synchronously, when time reaches about 5 s. At this time, the phase differences are 2α1=155°, 2α2=158°, 2α3=−144°and the synchronous rotational velocity is 983 r/min. Similarly, a disturbance of π phase applies to the vibrator 2, and the phase differences transform to 2α1=112°, 2α2=103°, 2α3=51°. Reviewing characteristic analysis of Figure 4(c), we can know that the simulation results are line with it. When the electricity of the system is switched off, the system is back to the steady state and still keeps operating synchronously. In this moment, the phase differences are 2α1=125°, 2α2=89.85°, 2α3=42.99° and the synchronous rotational velocity is 978 r/min.Figure 7
Results of computer simulations forη1=η2=1,η3=0.75,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)


## 5.2. Subresonant Region (Simulation Point A) Simulations

The analysis method in this section is the same as the previous section, and the results will be presented in the table form.


### 5.2.1. Simulation forη1=η2=η3=η4=1

The stable phase differences for Figures8(b)–8(d) are shown in Table 1.Figure 8
Results of computer simulations forη1=η2=η3=η4=1. (a) Rotational velocities of the four motors; (b) phase difference between vibrator 1 and 2; (c) phase difference between vibrator 2 and 3; (d) phase difference between vibrator 3 and 4; (e) displacements in x and y directions; (f) displacement in ψ direction.
(a)
(b)
(c)
(d)
(e)
(f)Table 1
The stable phase differences forη1=η2=η3=η4=1.
Simulation of point A
Characteristic analysis∗
Stable phase differences before disturbance
Stable phase differences after disturbance
Stable phase differences after outage
Stable phase differences
2
α
1
0
°
0
°
12.3
°
0
°
2
α
2
0
°
0
°
−
12.3
°
0
°
2
α
3
0
°
0
°
0.5
°
0
°
∗The results of characteristic analysis data from Figure 4(a).The stable phase differences of simulation and characteristic analysis are in good agreement with each other. In Figures8(e) and 8(f), the resonant responses of the rigid frame in x and y directions are larger and stabilize eventually in the range of −15∼+15, while the resonant response in ψ direction is zero. The above results indicate that the exciting forces superimpose positively, and a strong vibration occurs in the rigid frame, when the system operates in the subresonant region.


### 5.2.2. Simulation forη1=η2=η3=1,η4=0.5

As shown in Figure9, the results are given in Table 2.Figure 9
Results of computer simulations forη1=η2=η3=1,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)Table 2
The stable phase differences forη1=η2=η3=1,η4=0.5.
Simulation of point A
Characteristic analysis∗
Stable phase differences before disturbance
Stable phase differences after disturbance
Stable phase differences after outage
Stable phase differences
2
α
1
0.8
°
0.8
°
13.3
°
5
°
2
α
2
1
°
1
°
−
11.1
°
4
°
2
α
3
−
11
°
−
11
°
−
12.65
°
−
3
°
∗The results of characteristic analysis come from Figure 4(b).


### 5.2.3. Simulation forη1=η2=1,η3=0.75,η4=0.5

From Figure10, the results can be obtained and are presented in Table 3.Figure 10
Results of computer simulations forη1=η2=1,η3=0.75,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)Table 3
The stable phase differences forη1=η2=1,η3=0.75,η4=0.5.
Simulation of point A
Characteristic analysis∗
Phase differences before disturbance
Phase differences after disturbance
Phase differences after outage
Phase differences
2
α
1
1.37
°
1.31
°
13.9
°
6
°
2
α
2
−
2.71
°
−
2.78
°
−
15.5
°
3
°
2
α
3
−
8.11
°
−
8.13
°
−
9.18
°
0
°
∗The results of characteristic analysis can be observed in Figure 4(c).We can observe that the phase differences before disturbance are approximately equal to those after disturbance. The phenomenon accords with that of the characteristic analysis.


## 5.2.1. Simulation forη1=η2=η3=η4=1

The stable phase differences for Figures8(b)–8(d) are shown in Table 1.Figure 8
Results of computer simulations forη1=η2=η3=η4=1. (a) Rotational velocities of the four motors; (b) phase difference between vibrator 1 and 2; (c) phase difference between vibrator 2 and 3; (d) phase difference between vibrator 3 and 4; (e) displacements in x and y directions; (f) displacement in ψ direction.
(a)
(b)
(c)
(d)
(e)
(f)Table 1
The stable phase differences forη1=η2=η3=η4=1.
Simulation of point A
Characteristic analysis∗
Stable phase differences before disturbance
Stable phase differences after disturbance
Stable phase differences after outage
Stable phase differences
2
α
1
0
°
0
°
12.3
°
0
°
2
α
2
0
°
0
°
−
12.3
°
0
°
2
α
3
0
°
0
°
0.5
°
0
°
∗The results of characteristic analysis data from Figure 4(a).The stable phase differences of simulation and characteristic analysis are in good agreement with each other. In Figures8(e) and 8(f), the resonant responses of the rigid frame in x and y directions are larger and stabilize eventually in the range of −15∼+15, while the resonant response in ψ direction is zero. The above results indicate that the exciting forces superimpose positively, and a strong vibration occurs in the rigid frame, when the system operates in the subresonant region.


## 5.2.2. Simulation forη1=η2=η3=1,η4=0.5

As shown in Figure9, the results are given in Table 2.Figure 9
Results of computer simulations forη1=η2=η3=1,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)Table 2
The stable phase differences forη1=η2=η3=1,η4=0.5.
Simulation of point A
Characteristic analysis∗
Stable phase differences before disturbance
Stable phase differences after disturbance
Stable phase differences after outage
Stable phase differences
2
α
1
0.8
°
0.8
°
13.3
°
5
°
2
α
2
1
°
1
°
−
11.1
°
4
°
2
α
3
−
11
°
−
11
°
−
12.65
°
−
3
°
∗The results of characteristic analysis come from Figure 4(b).


## 5.2.3. Simulation forη1=η2=1,η3=0.75,η4=0.5

From Figure10, the results can be obtained and are presented in Table 3.Figure 10
Results of computer simulations forη1=η2=1,η3=0.75,η4=0.5. (a) Rotational velocities of the four motors; (b) phase difference between vibrators 1 and 2; (c) phase difference between vibrators 2 and 3; (d) phase difference between vibrators 3 and 4.
(a)
(b)
(c)
(d)Table 3
The stable phase differences forη1=η2=1,η3=0.75,η4=0.5.
Simulation of point A
Characteristic analysis∗
Phase differences before disturbance
Phase differences after disturbance
Phase differences after outage
Phase differences
2
α
1
1.37
°
1.31
°
13.9
°
6
°
2
α
2
−
2.71
°
−
2.78
°
−
15.5
°
3
°
2
α
3
−
8.11
°
−
8.13
°
−
9.18
°
0
°
∗The results of characteristic analysis can be observed in Figure 4(c).We can observe that the phase differences before disturbance are approximately equal to those after disturbance. The phenomenon accords with that of the characteristic analysis.


## 6. Conclusions

According to the above theoretical derivation, numerical characteristic analysis, and simulations, the following conclusions are given:(1)
Through contrasting simulations of two regions in differentη, we can observe the phase differences in η1=η2=η3=η4=1 are more stable, which indicates that structure parameters have a large effect on synchronization, i.e., the better the symmetry of the system is, the more easily the vibrators achieve synchronous stability.(2)
Resonant regions of the vibrating system are eventually divided to two regions: subresonant and superresonant; the phase differences in the subresonant maintain around0° and those in the superresonant maintain around 90° and 180°. The resonant responses of the rigid frame with double resonant types in x, y, and ψ directions are also discussed. These results indicate that the exciting forces of the system can be superposed positively, and the rigid frame has a large vibration amplitude, when the working region point of the vibrating system is in the subresonant. While in the superresonant, the phase differences among vibrators cannot be stabilized in the vicinity of zero, so it causes exciting forces to counteract each other, and the rigid frame produces no vibration.(3)
Utilizing the present study results of the synchronization and stability in the superresonant states, we can design a new type of high-frequency vibrating miller in engineering. For example, using the model and parameters of this paper, a new vibrating ball mill with four rollers can be designed, and the working region of the system is selected to be the superresonant region. Under the condition of ensuring good vibrating ball mill technology, the system can also reach the effect of vibration isolation automatically and has little harmful effect on the surrounding area.



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*Source: 9641231-2018-12-04.xml*