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Shock and Vibration 18 (2011) 73–90 73DOI 10.3233/SAV-2010-0591IOS Press

Synchronization of two self-synchronousvibrating machines on an isolation frame

Chunyu Zhaoa,∗, Qinghua Zhaob, Zhaomin Gonga and Bangchun WenaaSchool of Mechanical Engineering and Automation, Northeastern University, Shenyang 110004, ChinabHebei State-owned Minerals Development & Investment Co., Ltd., Shijiazhuang, 050021, China

Received 10 February 2010

Revised 5 May 2010

Abstract. This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame.Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of thedisturbance parameters for the angular velocities of the four unbalanced rotors. Then the stability problem of synchronizationfor the four unbalanced rotors is converted into the stability problems of two generalized systems. One is the generalized systemof the angular velocity disturbance parameters for the fourunbalanced rotors, and the other is the generalized system of threephase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture betweeneach pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torquedifference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greaterthan the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibratingmachines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrixis definitepositive, and that all the eigenvalues for the generalized system of three phase disturbance parameters have negative real parts.Computer simulations are carried out to verify the results of the theoretical investigation.

1. Introduction

In the last few decades, much effort has been devoted to mathematically explain the mechanism of synchronization.One of the early and most widely used approaches has been the method of direct separation of motions, i.e., thatof two-timing (a special case of the method of multiple scales). Here the dynamics is divided into two parts: onecorresponding to motion on a fast time scale, and the other ona slow time scale. Using the average Lagrange functionof the vibrating system, Blekhman has applied this approachto successfully deal with a number of problems includingproblems of self-synchronization [2–8]. In vibrating systems with two identical unbalanced rotors, the approach isgreatly simplified by combining the differential equationsof the two unbalanced rotors into the differential equationof the phase difference between the two unbalanced rotors [14,15]. Using the Hill equations and the Floquet theory,Yamapi and Woafo derived instability and the complete synchronization in the ring of four coupled self-sustainedelectromechanical devices [16,17]. Taking the two disturbance parameters of the average angular velocity of the twounbalanced rotors in a vibrating system as the small parameters, the authors deduced the non-dimensional couplingequations of the rotors. The stability for synchronizationof two unbalanced rotors is converted into the problem ofstability for a system of the three first order differential equations and the stability condition is derived by meansof the Routh-Hurwitz criterion [18,19] or a general Lyapunov function [20]. On the other hands, several numericmethods have been developed to tackle the problem of synchronization [1,9,10,12,13]. But when the number of theunbalanced rotors is more than two, investigation of the stability is very difficult with the above methods.

∗Corresponding author. E-mail: [email protected].

ISSN 1070-9622/11/$27.50 2011 – IOS Press and the authors. All rights reserved

74 C. Zhao et al. / Synchronization of two self-synchronous vibrating machines on an isolation frame

Fig. 1. Dynamic model of a vibration isolation system.

Taking two self-synchronous vibrating machines on an isolation rigid frame for example, this paper extends ourprevious works on the synchronization of two unbalanced rotors into the synchronization of multiple unbalancedrotors in a vibrating system. Herein, the problem of synchronization stability for multiple unbalanced rotors isdivided into that of two generalized systems. One is the generalized system for the disturbance parameters ofmultiple angular velocities, and the other is the first orderdifferential equations for the disturbance parameters ofphase differences whose number is less than the number of unbalanced rotors by one. In the next section, theequations of motion of the system are described. The condition of implementing synchronization and that of stabilityof synchronization are deduced in Section 3. Computer simulations are carried out to verify the theoretical resultsin Section 4. Finally, conclusions are provided in Section 5.

2. Equations of motion

Figure 1 shows the dynamic model of a vibration isolation system, which consists of two vibrating machines,denoted by V1 and V2, and an isolation rigid frame. The rigid frame is supported by an elastic foundation, whichis composed of the two groups of springs installed symmetrically. Each vibrating machine is supported on the rigidframe by two groups of springs installed symmetrically and excited by two unbalanced rotors, which are separatelydriven by two induction motors rotating in opposite directions.

As illustrated in Fig. 1, the projectiono of the mass centerG of the rigid frame onto they-axis is fixed and thatof the mass centerGi of the mass center of the vibrating machine Vi onto theyi-axis is also fixed. For the rigidframe, three reference frames are assigned: the fixed frameoxy with the y-axis vertical and coinciding with thecenter line of the rigid frame; the nonrotating moving frameGx′y′z′, which undergoes translation motions whileremaining parallel tooxy, and the moving frameGx′′y′′ that is fixed to the rigid frame, as shown in Fig. 2(a). Foreach vibrating machine, three reference frames also are assigned: the fixed frameoixiyi, the nonrotating movingframeGx′iy

′

iz′

i and the moving frameGix′′i y′′

i , i = 1, 2, as shown in Fig. 2(b).Because the rigid frame is supported by the elastic foundation, it exhibits three degrees of freedoms. Mass center

coordinates,x andy, and an angular rotationψ are set as independent coordinates. In like manner, the masscentercoordinates of vibrating machine Vi, xi andyi, and its angular rotationψi also are set as independent coordinates.

C. Zhao et al. / Synchronization of two self-synchronous vibrating machines on an isolation frame 75

Fig. 2. Reference frames: the fixed frameoxyandoixiyi; Gy′z′ andGiy′

iz′i

nonrotating moving frame;Gy′′z′′, Giy′′

iz′′i

and moving framefixed to the rigid framei.

In the reference frameGix′′i y′′

i , the coordinates of mass center of each eccentric lump can beexpressed as

x′′

mi1 =

lai cosβ + ri1 cosϕi1lai sinβ + ri1 sinϕi1

, x′′

mi2 =

−lai cosβ − ri1 cosϕi2lai sinβ + ri1 sinϕi2

(i = 1, 2.) (1)

wherelai is the distance between the mass center of vibrating machineVi and the rotating center of the eccentriclump; andβ is the angle between the lineoioij and thex-axis.

In the reference frameGixiyi, the coordinates of the eccentric lump can be expressed as the following

xmij = xGi + Rixmij ,Ri =

[

cosψi − sinψisinψi cosψi

]

(i = 1, 2, ; j = 1, 2.) (2)

wherexGi is the coordinate vector of the mass centerGi of the vibrating machine Vi,xGi = xi, yiT .

Then the kinetic energyT of the system is expressed as

T =1

2xTGmRxG +

1

2JRψ

2 +1

2

2∑

i=1

xGimixGi +1

2

2∑

i=1

Jiψ2i +

(3)1

2

2∑

i=1

2∑

j=1

xTmijmijxmij +1

2

2∑

i=1

2∑

j=1

J0ijϕ2ij

wheremR is the mass matrix of the isolation frame,mR = diag(mR,mR), andJR is its moment of inertia aboutits mass center;xG is the displacement vector of the isolation frame,xG = x, yT ; mi is the mass matrix of thevibrating machine Vi,mi = diag(mV ,mV ), andJi is its moment of inertia about its mass center;mij is the massmatrix of the eccentric lump in the vibrating machine Vi andJ0ij is the moment of inertia of motorj in the vibratingmachine Vi;(•) denotes d• /dt.

When the vibrating system is running, the coordinates of thepoints that the supporting springs of the rigid frameare connected to the rigid frame on the referenceoxycan be expressed as

xri = xo + Rxr0i, R =

[

cosψ − sinψsinψ cosψ

]

, i = 1, 2. (4)

wherexo is the displacement vector of the mass center of the rigid frame,xo = x, yT ; xr01 andxr02 are thecoordinates of the corresponding points on the reference frameoxy when the system is in the equilibrium state,xr01 ≈ lk, 0

Tandxr02 ≈ −lk, 0T .

Then, the vector of transformation of the corresponding spring can be expressed as

∆xri = xri − xr0i, i = 1, 2. (5)


The coordinates of point of the supporting spring of the vibrating machine Vi, Xij , connected to itself on thereference frameoixiyi, arexij , then we can obtain its coordinate onoixiyi-frame as the following

x′

ij = xoi + Rixij , Ri =

[

cosψi − sinψisinψi cosψi

]

(i = 1, 2; j = 1, 2.) (6)

Assuming that the coordinates of point of the supporting spring of vibrating machine Vi, Ki, connected to the mainrigid frame isxrij on the reference frameoxy, then, during the operation of the system, its coordinates onthereference frameoxycan be expressed as

x′

rij = xo + Rxrij (i = 1, 2; j = 1, 2.) (7)

and the transformation vector of spring Kij is expressed as

∆xij = (xoi − xo) + (Ri − I)xij − (R − I)xrij (8)

Then the potential energyV of the system is expressed as

V =1

2

2∑

i=1

∆xTriKi∆xri +1

2

2∑

i=1

2∑

j=1

∆xTijKij∆xij (9)

whereKi is the stiffness matrix of the supporting springi of the rigid frame,Ki = diag(Kx/2,Ky/2); andKij isthe stiffness matrix of the supporting springj of vibrating machine Vi, Kij = diag(kx/2, ky/2).

The viscous dissipation functionD of the system can be described as the following:

D =1

2

2∑

i=1

∆xTi F i∆xi +1

2

2∑

i=1

2∑

j=1

∆xTijF ij∆xij +1

2

2∑

i=1

2∑

j=1

fjiϕ2ij (10)

whereF i is the damping matrix of the supporting springi of the rigid frame,F i = diag(Fx/2, Fy/2); andF ij isthe stiffness damping of the supporting springj of vibrating machine Vi, F ij = diag(fx/2, fy/2).

The equations of motion are set up by using Lagrange’s equations

ddt∂(T − V )

∂qi−∂(T − V )

∂qi+∂D

∂qi= Qi (11)

whereqi is the generalized coordinates of the considered system andQi is the system generalized force.If q = [x, y, ψ, x1, y1, ψ1, x2, y2, ψ2, ϕ11, ϕ12, ϕ21, ϕ22]

T is chosen as the generalized coordinates of the system,the generalized forcesQϕij = Teij and the others are zero, in whichTeij is the electromagnetic torque of motorijon vibrating machinei.

Substituting the coordinates of the correspondingpoints into Eqs (3), (9) and (10) and applying Lagranges’ Eq. (11)to them lead to the differential equations of motion of the system. Usually,mij << mi, ψ << 1 andψi << 1.Hence, the inertia coupling resulting from the asymmetry ofthe eccentric lump can be neglected during the runningprocess of the system. Therefore, the differential equations of motion of the system can be simplified as follows:

mRx+ (fx + 4fx0)x − fx0

2∑

i=1

xi + (kx + 4kx0)x− kx0

2∑

i=1

xi = 0

mRy + (fy + 4fy0)x− fy0

2∑

i=1

yi + (ky + 4ky0)y − ky0

2∑

i=1

yi = 0

JRψ + fψψ −

2∑

j=1

fψyj yj − fy0l20

2∑

j=1

ψj + kψψ −

2∑

j=1

kψyjyj − ky0l20

2∑

j=1

ψj = 0

mixi + fx0xi − fx0x+ kx0xi − kx0x =

2∑

j=1

(−1)j−1mijrij(ϕ2ij cosϕij + ϕij sinϕij) (12)


miyi + fy0yi − fy0y − fyiψψ + ky0yi − ky0y − kyiψψ =

2∑

j=1

mijrij(ϕ2ij sinϕij − ϕij cosϕij)

J0ψi + fy0l20ψi − fy0l

20ψ + ky0l

20ψi − ky0l

20ψ =

2∑

j=1

(−1)j−1mijrij lai(ϕ2ij sin(ϕij − β) − ϕij cos(ϕij − β))

(J0i1 +mi1r2i1)ϕi1 + fi1ϕi1 = Tei1 −mi1ri1(yi cosϕi1 − xi sinϕi1 − laiψi cos(ϕi1 − β))

(J0i2 +mi2r2i2)ϕi2 + fi2ϕi2 = Tei2 −mi2ri2(yi cosϕi2 + xi sinϕi2 + l0ψi cos(ϕi2 − β))

i = 1, 2.

with

fψ = fyl2a + 2fy0(l

21 + l22 + 2l20), fψy1 = fy0l1, fψy2 = fy0l2, fψy3 = −fy0l2, fψy4 = −fy0l1,

kψ = kyl2a + 2ky0(l

21 + l22 + 2l20), kψy1 = ky0l1, kψy2 = ky0l2, kψy3 = −ky0l2, kψy4 = −ky0l1

fy1ψ = fy0l1, fy2ψ = fy0l2, fy3ψ = −fy0l2, fy4ψ = −fy0l1, ky1ψ = ky0l1, ky2ψ = ky0l2,

ky2ψ = −ky0l2, ky4ψ = −ky0l1.

3. Synchronization of the vibrating system

When the system operates in the steady-state, the instantaneous average phase of the four unbalanced rotors isassumed to beϕ, and their instantaneous average angular velocity isϕ. If the average value ofϕ is assumed to beωm and the coefficient of the instantaneous change ofϕ is assumed to beν0, ϕ can be expressed as [18]

ϕ = (1 + ν0)ωm (13)

Assuming

α1 =1

2(ϕ11 − ϕ22), α2 =

1

2(ϕ21 − ϕ22), α3 =

1

4(ϕ11 + ϕ12 − ϕ21 − ϕ22), αi = νiωm(i = 1, 2, 3),

ϕ11 = (1 + ε1)ωm, ϕ12 = (1 + ε2)ωm, ϕ21 = (1 + ε3)ωm, ϕ22 = (1 + ε4)ωm,

we have

ϕ11 = ϕ+ α1 + α3, ϕ12 = ϕ− α1 + α3, ϕ21 = ϕ+ α2 − α3, ϕ21 = ϕ− α2 − α3

ϕ11 = (1 + ε1)ωm, ε1 = ν0 + ν1 + ν3, ϕ12 = (1 + ε2)ωm, ε2 = ν0 − ν1 + ν3, (14)

ϕ21 = (1 + ε3)ωm, ε3 = ν0 + ν2 − ν3, ϕ22 = (1 + ε4)ωm, ε4 = ν0 − ν2 − ν3.

When the vibrating system operates in the steady-state, theslip of an induction motor is very small. Hence, theeffect of the angular accelerations of the unbalance rotorson the response of the vibrating system can be neglected,i.e., ϕij ≈ 0 [18–20]. On the other hand, the natural frequency of the vibrating system is far less than its operatingone and its damping is very small [18–20]. In this case, the effect of the fluctuation of the angular velocities on theamplitude and phase angle of the response can also be neglected, i.e., the amplitude and phase angles of responseof the vibrating system can be expressed by usingωm and neglectingνi. Therefore, the responses of the vibratingsystem can be expressed as follows:

x = rmar0µx

2∑

i=1

(cos(ϕi1 − γxe) − cos(ϕi2 − γex))


y = rmar0µy

2∑

i=1

2∑

j=1

sin(ϕij − γey)

ψ =rmar0µeψ

le

ςy0

2∑

i=1

rei

2∑

j=1

sin(ϕij − γeψ) + ςψ0rea

2∑

i=1

(sin(ϕi1 − β − γeψ) − sin(ϕi2 − β − γeψ))

xi = rmr0µx0

−(cos(ϕi1 + γx0) − cos(ϕi2 + γx0) + ςx0ςex

2∑

j=1

(cos(ϕjl − γex0) − cos(ϕjl − γex0))

yi = rmr0µy0

−

2∑

j=1

sin(ϕij + γy0) +

2∑

j=1

2∑

l=1

ςy0(ςey + reirelςeψ0) sin(ϕjl − γey0)+

(15)

reireaςeψ0ςψ0

4∑

j=1

(sin(ϕj1 − β − γeyψ0) − sin(ϕj2 − β − γeyψ0)

ψi =rmr0µψ0

le0

[

−rea2∑

j=1

(sin(ϕi1 + γψ0) − sin(ϕi2 + γψ0)) + ςy0ςψ0

2∑

j=1

re0j2∑

l=1

sin(ϕjl − γeψy0)+

ςψ0ςeψrea2

∑

j=1

(sin(ϕj1 − γeψ0) − sin(ϕj2 − γeψ0))

]

whereγex, γey, γeψ, γex0, γey0, γeψ0, γeyψ0, π − γx0, π − γy0andπ − γψ0 are the phase angles.

ςx0 =η2x0

η2x0 − 1

, kex = kx − 4ςx0kx0, ωex =

√

kex

mR

, ηex =ωex

ωm, µx =

ςx0η2ex − 1

,

µx0 =1

η2x0 − 1

, le0 =

√

J0

m, ςex =

η2ex0

η2ex − 1

, ηex0 =

√

kx0/mR

ωm, ςy0 =

η2y0

η2y0 − 1

,

key = ky − 4ςy0ky0, ωey =

√

key

mR

, µy =ςy0

η2ey − 1

, µy0 =1

η2y0 − 1

, ςey =η2ey0

η2ey − 1

,

ηey0 =

√

ky0/mR

ωm, ςψ0 =

η2ψ0

η2ψ0 − 1

, ηψ0 =

√

ky0l20/J0

ωm, re0 =

l0le0, rma =

m0

mR

, rm =m0

m,

m1 = m2 = m,m11 = m12 = m21 = m22 = m0,

keψ = kψ − 2ky0ςy0(l21 + l22) − 4ky0l

20ςψ0, re1 = −re2 =

l1le, rea =

lale, le =

√

JRmR

,

ωeψ =

√

keψ

JR, ηeψ =

ωeψ

ωm, ςyy0 =

η2cyη

2y0

(η2y − 1)(η2

y0 − 1) − 4η2cyη

2y0

, µeψ =1

η2eψ − 1

,

ςeψ =η2eψ0

η2eψ − 1

, ηeψ0 =

√

ky0l20/J0

ωm.

We differentiatexi, yi andψi in Eq. (15) with respect to timet by the chain rule (applied to each component ofα1,α2, α3 andϕ) to obtainxi, yi andψi, respectively. Then substitutingxi, yi andψi into the differential equationsof motion for the four unbalanced rotors in Eq. (12) and integrating them overϕ = 0 ∼ 2π and neglecting thehigh order terms ofεij , considering Eq. (14), we obtain the average differential equations of the four parameters as


follows:

(Jri +m0r20)ωm ˙εi + fdiωm(1 + εi) = Tei −mr2ωm

4∑

j=1

χ′

ij˙εj + ωm

4∑

j=1

χij εj

− χfi − χai

(16)i = 1, 2, 3, 4.

whereJr1 = J011, Jr2 = J012, Jr3 = J021, Jr4 = J022, fd1 = f11, fd2 = f12, fd3 = f21, fd4 = f22, Te1 = Te11,Te2 = Te12, Te3 = Te21, Te4 = Te22.

Compared with the change ofϕ(ϕ = ωm) with respect to timet, that ofεi andεi are very small, soεi andεi areconsidered to be slow-changing parameters in this study. During the aforementioned integration,εi, εi andαi areassumed to be the middle values of their integrationεi, ˙εi andαi, respectively. The coefficients ofεi, ˙εi in Eq. (16)are listed in Appendix A. In engineering, the damping of the vibrating system is very small, hence the sine of phaseangles inχ′

ij andχij are neglected [18–20].When a asynchronous motor operates in the vicinity of the angular velocity ofωm, the electromagnetic torque can

be expressed as the following [18]:

Tei = Te0i − keiεi (17)

whereTe0i andke0i are the electromagnetic torque and the stiffness coefficient of angular velocity of the motor whenits angular velocity isωm, respectively.

Because the moment of inertia of each motor’s rotor is much smaller than that of the unbalanced rotor, it can beneglected in Eq. (16). Introducing the following non-dimensional parameters

ρi = 1 −Wc0/2, i = 1, 2, 3, 4;

κ1 =ke01

m0r20ω2m

+fd1

m0r20+Ws1, κ2 =

ke02

m0r20ω2m

+fd2

m0r20+Ws2,

κ3 =ke03

m0r20ω2m

+fd3

m0r20+Ws2, κ4 =

ke04

m0r20ω2m

+fd4

m0r20+Ws2.

into Eq. (16), dividing each formula bym0r20ωm, rearranging them and writing it in the matrix form, we obtain

A ˙ε = Bε + u (18)

where

A =

ρ1 χ′

12 χ′

13 χ′

14

χ′

21 ρ2 χ′

23 χ′

24

χ′

31 χ′

32 ρ3 χ′

34

χ′

41 χ′

42 χ′

43 ρ4

, B =

κ1 χ12 χ13 χ14

χ21 κ2 χ23 χ24

χ31 χ32 κ3 χ34

χ41 χ42 χ43 κ4

,

ε = ε1 ε2 ε3 ε4 T , u = u1 u2 u3 u4 .

u1 =Te01

m0r20ωm−

fd1

m0r20−χf1 + χa1

m0r20ωm, u2 =

Te02

m0r20ωm−

fd2

m0r20−χf2 + χa2

m0r20ωm,

u3 =Te03

m0r20ωm−

fd3

m0r20−χf3 + χa3

m0r20ωm, u2 =

Te04

m0r20ωm−

fd4

m0r20−χf4 + χa4

m0r20ωm.

If the trial solution of Eq. (18) exists and is stable, the four unbalanced rotors can implement self-synchronization.Equation (18) is the average differential equations of the angular velocity disturbances of the four unbalanced rotorsover their average period and called the non-dimensional coupling equation of the four unbalanced rotors. Theanalytical approach used in this paper converts the problemof synchronization of the multiple unbalanced rotors ina vibrating system into that of existence and stability of trial solution for their non-dimensional coupling equation.


3.1. Conditions of implementing synchronization

Insertingu = into Eq. (18) yields

Toi = χfi + χai, i = 1, 2, 3, 4. (19)

whereToi is the differnece between the electromagnetic torque of themotori and the frictional torque of its rotorand called the output torque of the motori, Toi = Te0i − fdiωm.

Subtracting the second formula in Eq. (19) from the first one yields

∆To12 = To1 − To2(20)

=m0r20ω

2m[(Wc0 −Wy −Wψ) sin 2α1 + 2(Wψ −Wy) cos 2α3 cos α2 sin α1]

whereWy = rmµy0ςy0ςey cos γey0 andWψ = rmµψ0ςy0ςeψ0r2e1 cos γeψy0.

Subtracting the fourth formula in Eq. (19) from the third oneyields

∆To34 = To3 − To4(21)

=m0r20ω

2m[(Wc0 −Wy −Wψ) sin 2α2 + 2(Wψ −Wy) cos 2α3 cos α1 sin α2]

Subtracting the sum of the last two formulae in Eq. (19) from that of the first two ones yields

∆To = To1 + To2 − To3 − To4(22)

= 4m0r20ω

2m(Wψ −Wy) cos α1 cos α2 sin 2α3

In order to obtain the greater torque of frequency capture and the stable magnitude of a vibrating frame, a vibratingmachine is always designed to be the over resonant type, i.e., its frequency of operation is greater than its natural one.Usually, the frequency of operation is in the range of 4 to 5 times its natural one [14]. In this case, the amplitudes ofresponse of the vibrating system are almost independent of the exciting frequency and constants [14]. Assuming

ωm > 4ωn (23)

whereωn represents the natural frequency of the vibrating systems,then we have

ςx0 >1

15, ςy0 >

1

15, ςψ0 >

1

15(24)

Hence, in order to increase the torques of frequency capturebetween two of the four unbalanced rotors that arenot installed on the same vibrating rigid frame, the naturalfrequency of the isolation frame is designed to be muchhigher than the operation frequency of the vibrating system. Assuming

ωex > 4ωm, ωey > 4ωm, ωeψ > 4ωm (25)

we have

ςex >16

15≈ 1, ςey >

16

15≈ 1, ςeψ >

16

15≈ 1 (26)

As shown in Eqs (24) and (26), it can be seen thatWc0 is much greater thanWy andWψ . Hence, Equations (20)and (22) can be simplified as follows:

∆To12 = TC1 sin 2α1(27)

∆To34 = TC1 sin 2α2

whereTC1 = m0r20ω

2mWc0 is called the first torque of frequency capture.

Then the phase differences between each pair of the unbalanced rotors on the same rigid frame can be expressedas

2α1 = arcsin∆To12

TC1 (28)2α2 = arcsin

∆To34

TC1


As shown in Eq. (80), when the parameters of the vibrating system satisfy

|TC1| > max|∆To12| , |∆To34| (29)

the solutions of2α1 and2α2 exist.Equation (29) demonstrates that the condition that each pair of the unbalanced rotors on the same rigid frame is

that the first torque of frequency capture is equal to or greater than the difference in the output torque between eachpair of motors on the same rigid frame. When the two unbalanced rotors in a vibrating system with small dampingrotate in opposite directions, the first torque of frequencycapture is always much greater than the absolute valueof the difference of output torque for each pair of the motors. Hence,2α1 and2α2 are close to0 or π [14,15].Therefore,2α3 can be approximately expressed as

2α3 = arcsin∆ToTC2

(30)

where TC2 is called the second torque of frequency capture for the considered vibrating system,TC2 =4m0r

20ω

2m(Wψ −Wy).

Equation (30) demonstrates that the condition that the two vibrating machines can implement synchronizationis that the second torque of frequency capture is equal to or greater than the absolute value of the output torquedifference between the two pairs of the motors. i.e.,

|TC2| > |∆To| (31)

When the parameters of the vibrating system satisfy Eqs (29)and (31), the solutions of Eq. (19), denoted byα10, α20,α30 andωm0, can be determined by a numeric method.

3.2. Condition of the synchronization stability

Whenu = , Eq. (18) is a generalized system [22]

A0ε = B0ε (32)

whereA0 andB0 denote the values ofA andB for α1 = α10, α2 = α20, α3 = α30 andωm = ωm0.As shown in the expressions ofχ′

ij andχij (i = 1, 2, 3, 4;j = 1, 2, 3, 4) in Appendix A, when the parameters ofthe vibrating system satisfies the follow condition:

aij > 0, det(A2) > 0, det(A3) > 0, det(A0) > 0, (33)

the matricesA0 andB0 satisfy the generalized Lyapunov equations [22]:

ITB0 + BT0 I = −2ωmdiagκ1, κ2, κ3, κ4 (34)

AT0 I = IA0 > 0 (35)

whereI is the unit matrix.As shown in Appendix, whenα10, α20 andα30 satisfy

−π/2 < 2α10 < π/2, −π/2 < 2α20 < π/2, −π/2 < 2α30 < π/2,

π/2 < 2α30 − α10 − α20 < π/2, π/2 < 2α30 + α10 + α20 < π/2, (36)

π/2 < 2α30 − α10 + α20 < π/2, π/2 < 2α30 + α10 − α20 < π/2.

Equations (33), (34) and (35) can be satisfied.Therefore, the generalized system (Eq. (32)) is concessional and without pulse. If Aε

lim t→∞

= , Eq. (32) is

stable [22]. εlim t→∞

= means that the electromagnetic torques of the four motors are stably balanced with the load

torques that the vibrating system acts on them.Linearizing Eq. (19) aroundα10, α20, α30 andωm0, and neglectingWsc andWsc0, as well asfd1, fd2, fd3 and

fd4, we obtain


ke01(ν0 + ν1 + ν2) = −

3∑

i=1

(

∂χa1

∂αi

)

0

∆αi, (37)

ke02(ν0 + ν1 − ν2) = −

3∑

i=1

(

∂χa2

∂αi

)

0

∆αi, (38)

ke03(ν0 − ν1 + ν3) = −3

∑

i=1

(

∂χa3

∂αi

)

0

∆αi, (39)

ke04(ν0 − ν1 − ν3) = −

3∑

i=1

(

∂χa4

∂αi

)

0

∆αi, (40)

where(•)0 denotes the values forα1 = α10, α2 = α20 andα3 = α30; and∆αi = αi − αi0, i = 1, 2, 3.Summing Eqs (37)–(40), and rearranging them, we obtain

v0 = δ1ν1 + δ2ν2 + δ3ν3, (41)

where

δ1 = −ke01 + ke02 − ke03 − ke02

ke01 + ke02 + ke03 + ke02

,

δ2 = −ke01 − ke02

ke01 + ke02 + ke03 + ke04

,

δ3 = −ke03 − ke04

ke01 + ke02 + ke03 + ke04

.

It should be noticed that∆α =

∆α1 ∆α2 ∆α2

T= ν1 ν2 ν3 T . Substituting Eq. (41) into Eqs (37)–(40),

and writing them into the generalized system of∆α =

∆α1 ∆α2 ∆α2

Tin the following manner: subtracting

Eq. (38) from Eq. (37) as the first row, subtracting Eq. (39) from Eq. (40) as the second row, subtracting the sum ofEqs (39) and (40) from that of Eqs (37) and (38) as the third row, we obtain

E∆α = D∆α, (42)

whereE = [eij ]3×3, andD = [dij ]3×3.Equation (42) can be rewritten as:

∆α = C∆α,C = E−1D. (43)

Herein, Eq. (43) is called the generalized system for the disturbance parameters of phase differences. Exponentialtime-dependence of the form∆α = v exp(λt) is now assumed, and inserted into Eq. (43), then solving thedeterminant equationdet(C − λI) = 0, we obtain the characteristic equation for the eigenvalueλ as the following:

λ3 + c1λ2 + c2λ+ c3 = 0. (44)

The zero solutions of Eq. (43) are stable only if all the rootsof λin Eq. (44) have negative real parts. Using theRouth-Hurwitz criterion, Equation (45) satisfies the aboverequirements [11]:

c1 > 0, c3 > 0 and c1c2 > c3. (45)

∆αlim t→+∞

= 0 means νilim t→+∞

= 0, i = 0, 1, 2, 3.Using Eq. (14), we have εlim t→+∞

= , i.e., Aεlim t→+∞

= .

In engineering, the parameters of the four induction motorsare usually chosen to be similar [14,15], i.e.,

ke11 ≈ ke12 ≈ ke21 ≈ ke22 ≈ ke0,(46)

Te11 ≈ Te12 ≈ Te21 ≈ Te22.


In this case, the matrixE is approximately expressed as

E = diag4ke0, 2ke0, 2ke0. (47)

From Eqs (28) and (30), we havesin α10 ≈ 0, sin α20 ≈ 0 andsin α30 ≈ 0. Then the matricesD andE can bealso approximately simplified as two diagonal matrices. Hence, Eq. (43) can be simplified as follows:

∆α1 = −m0r20ωm0Wc0 cos 2α10∆α1

∆α2 = −m0r20ωm0Wc0 cos 2α20∆α2 (48)

∆α3 = −4m0r20ω

2m(Wψ −Wy) cos α10 cos α20 cos 2α30∆α3

In a vibrating system with dual-motor drivers rotating in opposite directions,Wc0 is always greater than 0, hence2α10 and2α20 are stabilized in the vicinity of 0 [14,15,18]. IfWψ > Wy , 2α30 is stabilized in the vicinity of 0; ifWψ < Wy, 2α30 is in the vicinity ofπ.

4. Computer simulations

In this section, computer simulations are carried out to verify the above theoretic results. The numeric algorithm isdeveloped from that of the vibrating system with two unbalanced rotors [19]. The parameters of the present systemare listed in Appendix B.

Figures 3 and 4 show the results of computer simulations for the parameterl1 = 0.8 (Wψ < Wy) and 2.5(Wψ > Wy), respectively. Because the damping coefficient of the rotor 11 is smaller than that of the rotor 21 and theparameters for the two pairs of motors are the same, the difference in the rotational speed between the two unbalancedrotors on V1 is greater than that on V2 during the starting process of the system, as shown in Fig. 3d. Hence, thephase difference between the two unbalanced rotors on V1 is greater than that between the two unbalanced rotorson V2, as shown in Figs 3a and b. But with the increase of the rotational speeds of the four motors, the vibrations ofthe two vibrating frames are excited and the first torque of frequency capture plays the role of synchronizing eachpair of the unbalanced rotors. Then the phase difference between each pair of unbalanced rotors is stabilized in thevicinity of 0, 2α10 = 0.04 and2α20 = 0.03, as illustrated in Figs 3a and b. At the same time, the second torque offrequency capture plays the role of synchronizing the rotational speeds of the unbalanced rotors on the two vibratingframes. Finally, the phase difference2α30 is stabilized in the vicinity of2α30 (2α30 = 3.19). When the rotationalspeed of each pair of unbalanced rotors passes through the natural frequency of each vibrating frame, the resonantresponses are excited. But when the system operates in the steady-state, the vibrations of each vibrating frame inx- andψ-directions are rapidly attenuated due to the damping of thesystem and each vibrating frame undergoes anonly vibration iny-direction, as shown in Figs 3e, f and g (vibrations of V2 are not shown in Fig. 3). Because thephase difference between the two pairs of unbalanced rotorsis in the vicinity ofπ and phase difference betweeneach pair of unbalanced rotors is in the vicinity of 0 when thesystem operates in the steady-state, the exciting forcesacting on the isolation inx- andy-directions are cancelled mutually and the exciting torques in ψ-direction aresuperposed. Therefore, when the system operates in the steady-state, the isolation frame undergoes an only vibrationin ψ-direction, as shown Fig. 3h, i and j. Forl1 = 2.5, the phase difference between the two pairs of unbalancedrotors is in the vicinity of 0 (2α30 = 0.25 rad) and the isolation frame undergoes a only vibrationin y-direction whenthe system operates in the steady-state, as shown in Fig. 4. These facts are consistent with the above theoreticalresults in Section 3.

5. Conclusions

This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame.The mathematical model is set up by using Lagrange’s equations. Using the modified average method of smallparameters, we deduce the non-dimensional differential equation of the disturbance parameters for the angularvelocities of the four unbalanced rotors, which includes the inertia coupling of the unbalanced rotors, the stiffness


Fig. 3. Results of computer simulation forl1 = 0.8m: a-phase difference between the two unbalanced rotors on V1; b-phase difference betweenthe two unbalanced rotors on V2; c-phase difference betweenthe unbalanced rotors 11 and 21; d-rotational velocities ofthe four unbalanced rotors;e-displacement of V1 inx1-direction, f-displacement of V1 iny1-direction; g-angular displacement of V1 inψ1-direction; h-displacement of theisolation frame inx-direction; i-displacement of the isolation frame iny-direction; j-angular displacement of the isolation framein ψ-direction.


Fig. 4. Results of computer simulation forl1 = 2.5m: a-phase differences; b-rotational speeds of the four unbalanced rotors; c-displacement ofthe isolation iny-direction; c-displacement of the isolation frame inψ–direction.

coupling of angular velocity of the four motors, and the loading coupling of the four motors. This analytical approachconverts the problem of synchronization for the four unbalanced rotors into the stability problem of a generalizedsystem and a system of three first order differential equations for the three phase differences. In a vibrating systemwith small damping, the inertia coupling matrix of the four unbalanced rotors is symmetric, the stiffness couplingmatrix is antisymmetrical and its diagonal elements are allnegative. These facts make the generalized systemsatisfy the generalized Lyapunov equations when the inertia coupling matrix is positive definite. The condition ofimplementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotorson a vibrating machine is greater than the absolute values ofthe output electromagnetic torque difference betweeneach pair of motors, and that the torque of frequency capturebetween the two vibrating machines is greater than theabsolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibratingmachines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrixof the generalized system is definite positive, and that all the eigenvalues for the generalized system of three phasedisturbance parameters have negative real parts.

Acknowledgement

This research is support by the National Science Foundationof China (Grant No: 51075063) and Program forChangjiang Scholars and Innovative Research Team in University.

Appendix A Non-diagonal elements of matrices A and B

χf1 =1

2ωm[Ws +Wsc12 cos 2α1 +Wsc13 cos(2α3 + α1 − α2) +Wsc14 cos(2α3 + α1 + α2)] (A1)

χf2 =1

2ωm[Wcs12 cos 2α1 +Ws0 +Wcs23 cos(2α3 − α1 − α2) +Wcs24 cos(2α3 − α1 + α2)] (A2)


χf3 =1

2ωm[Wcs13 cos(2α3 + α1 − α2) +Wcs23 cos(2α3 − α1 + α2)Ws0 +Wcs34 cos 2α2] (A3)

χf4 =1

2ωm[Wcs14 cos(2α3 + α1 + α2) +Wcs24 cos(2α3 − α1 + α2) +Wcs0 cos 2α2 +Ws0] (A4)

χa1 =1

2ωm[Wcc12 sin 2α1 +Wcc13 sin(2α3 + α1 − α2) +Wcc24 sin(2α3 + α1 + α2)] (A5)

χa2 =1

2ωm[−Wcc12 sin 2α1 +Wcc23 sin(2α3 − α1 − α2) +Wcc24 sin(2α3 − α1 + α2)] (A6)

χa3 =1

2ωm[−Wcc13 sin(2α3 + α1 − α2) −Wcc23 sin(2α3 − α1 − α2) +Wcc0 cos 2α2] (A7)

χa4 = −1

2ωm[Wcc14 sin(2α3 + α1 + ~α2) +Wcc24 cos(2α3 − α1 + α2) +Wcc0 cos 2α2] (A8)

χ′

11 = −1

2Wc0 (A9)

χ′

12 =1

2Wcc12 cos 2α1 (A10)

χ′

13 =1

2Wcc13 cos(2α3 + α1 − α2)) (A11)

χ′

14 =1

2Wcc14 cos(2α3 + α1 + α2)) (A12)

χ′

21 =1


χ′

22 = −1

2Wc0 (A14)

χ′

23 =1

2Wcc cos(2α3 − α1 − α2) (A15)

χ′

24 =1

2Wcc cos(2α3 − α1 + α2) (A16)

χ′

31 =1

2Wcc13 cos(2α3 + α1 − α2) (A17)

χ′

32 =1

2rm12Wcc23 cos(2α3 − α1 − α2) (A18)


χ′

33 = −1

2Wc0 (A19)

χ′

34 =1


χ′

41 =1

2Wcc14 cos(2α3 + α1 + α2) (A21)

χ′

42 =1

2Wcc24 cos(2α3 − α1 + α2) (A22)

χ′

43 =1


χ′

44 = −Wc0 (A24)

χ11 = ωmWs0 (A25)

χ12 = ωmWcc0 sin 2α1 (A26)

χ13 = ωmWcc13 sin(2α3 + α1 − α2) (A27)

χ14 = ωmrmWcc14 sin(2α3 + α1 + α2) (A28)

χ21 = −ωmWcc12 sin 2α1 (A29)

χ22 = ωmrmWs0 (A30)

χ23 = ωmWcc23 sin(2α3 − α1 − α2) (A31)

χ24 = ωmWcc sin(2α3 − α1 + α2) (A32)

χ31 = −ωmWcc13 sin(2α3 + α1 − α2) (A33)

χ32 = −ωmWcc23 sin(2α3 − α1 + α2) (A34)

χ33 = ωmWs0 (A35)


χ34 = ωmWcc34 sin 2α2 (A36)

χ41 = −ωmWcc14 sin(2α3 + α1 + α2) (A37)

χ42 = −ωmWcc24 sin(2α3 − α1 + α2) (A38)

χ43 = −ωmWcc34 sin 2α2 (A39)

χ44 = Ws0 (A40)

Ws0 = rm[µx0(sin γx0 + ςx0ςex sinγex0) + µy0(sin γy0 + ςy0ςey sin γey0 +(A41)

r2e1ςy0ςeψ0 sin γψy0) + µψ0(r2ea sinγψ0 + ςeψ0ςψ0r

2ea sin γeψ0)]

Wc0 = rm[µx0(cos γx0 + ςx0ςex cos γex0) + µy0(cos γy0 + ςy0ςey cos γey0 +(A42)

r2e1ςy0ςeψ0 cos γψy0) + µψ0(r2ea cos γψ0 + ςeψ0ςψ0r

2ea cos γeψ0)]

Wcc12 =Wcc34 = Wcc0 + rm[µx0ςx0ςex cos γex0 − µy0(ςy0ςey cos γey0 +(A43)

r2e1ςy0ςeψ0 cos γeψy0) + µψ0r2eaςeψ0ςψ0 cos γeψ0]

Wcc0 = rm(µx0 cos γx0 − µy0 cos γy0 + r2eaµψ0 cos γψ0) (A44)

Wcs12 =Wcs34 = Wcs1 + rm[−µx0ςx0ςex sinγex0 + µy0(ςy0ςey sin γey0 +(A45)

r2e1ςy0ςeψ0 sin γeψy0) − µψ0r2eaςeψ0ςψ0 sinγeψ0]

Wcs1 = rm(µx0 cos γx0 − µy0 cos γy0 + r2eaµψ0 cos γψ0) (A46)

Wcc13 =Wcc24 = rm[−µx0ςx0ςex cos γex0 −(A47)

µy0(ςy0ςey cos γey0 − r2e1ςy0ςeψ0 cos γeψy0) − µψ0ςeψ0ςψ0r2ea cos γeψ0]

Wcs13 =Wcs25 = rm[µx0ςx0ςex sin γex0 +(A48)

µy0(ςy0ςey sin γey0 − r2e1ςy0ςeψ0 sinγeψy0) + µψ0ςeψ0ςψ0r2ea sin γeψ0]

Wcc14 =Wcc23 = rm[µx0ςx0ςex cos γex0 −(A49)

µy0(ςy0ςey cos γey0 − r2e1ςy0ςeψ0 cos γeψy0) + µψ0ςeψ0ςψ0r2ea cos γeψ0]

Wcs14 =Wcs23 = rm[−µx0ςx0ςex sinγex0 +(A50)

µy0(ςy0ςey sin γey0 − r2e1ςy0ςeψ0 sinγeψy0) − µψ0ςeψ0ςψ0r2ea sin γeψ0]


Appendix B: Parameters of the vibrating system

Table A1

Parameters of the four induction motors

Parameters Motor 11 Motor 12 Motor 21 Motor 22

Rated power (kW) 3.7 0.75 3.7 0.75Poles 6 6 6 6Rated frequency (Hz) 50 50 50 50Rated voltage (V) 220 220 220 220Rated rotational speed (r/min) 980 980 980 980Stator resistance (Ω) 0.54 3.35 0.54 3.35Rotor resistance referred to stator (Ω) 0.56 3.40 0.56 3.40Stator inductance (H) 0.141 0.170 0.141 0.170Rotor inductance referred to stator (H) 0.143 0.170 0.143 0.170Mutual inductance (H) 0.138 0.164 0.138 0.164Rotor damping coefficient (N.m.s/rad) 0.02 0.005 0.04 0.005

Table A2

Parameters of the vibrating system

Parameters V1 V2 Isolation frame

Mass of the vibrating frame (kg) 2400 2400 1000Moment of inertia about its centriod (kg·m2) 2190 2190 1600Mass of the eccentric lump (kg) 30 30Eccentric radius (m) 0.2 0.2Stiffness of springs inx-direction (N/m) 630000 630000 800000Stiffness of springs iny-direction (N/m) 630000 630000 800000Stiffness of springs inψ–direction (N·m/rad) 550000 550000 600000Damping constant inx-direction (N/(m/s)) 3850 3850 18000Damping constant iny-direction (N/(m/s)) 3850 3850 18000Damping constant inψ-direction (N·m/(rad/s)) 3400 3400 20000

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