Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

Shock and Vibration, Sep 2019

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 the disturbance parameters for the angular velocities of the four unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stability problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, 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.

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Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

Shock and Vibration 1070-9622 Synchronization of two self-synchronous vibrating machines on an isolation frame Chunyu Zhao 0 Qinghua Zhao 1 Zhaomin Gong 0 Bangchun Wen 0 0 School of Mechanical Engineering and Automation, Northeastern University , Shenyang 110004 , China 1 Hebei State-owned Minerals Development & Investment Co., Ltd. , Shijiazhuang, 050021 , China 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 the disturbance parameters for the angular velocities of the four unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stability problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, 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., that of two-timing (a special case of the method of multiple scales). Here the dynamics is divided into two parts: one corresponding to motion on a fast time scale, and the other on a slow time scale. Using the average Lagrange function of the vibrating system, Blekhman has applied this approach to successfully deal with a number of problems including problems of self-synchronization [2–8]. In vibrating systems with two identical unbalanced rotors, the approach is greatly simplified by combining the differential equations of the two unbalanced rotors into the differential equation of 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-sustained electromechanical devices [16,17]. Taking the two disturbance parameters of the average angular velocity of the two unbalanced rotors in a vibrating system as the small parameters, the authors deduced the non-dimensional coupling equations of the rotors. The stability for synchronization of two unbalanced rotors is converted into the problem of stability for a system of the three first order differential equations and the stability condition is derived by means of the Routh-Hurwitz criterion [18,19] or a general Lyapunov function [20]. On the other hands, several numeric methods have been developed to tackle the problem of synchronization [1,9,10,12,13]. But when the number of the unbalanced rotors is more than two, investigation of the stability is very difficult with the above methods. Taking two self-synchronous vibrating machines on an isolation rigid frame for example, this paper extends our previous works on the synchronization of two unbalanced rotors into the synchronization of multiple unbalanced rotors in a vibrating system. Herein, the problem of synchronization stability for multiple unbalanced rotors is divided into that of two generalized systems. One is the generalized system for the disturbance parameters of multiple angular velocities, and the other is the first order differential equations for the disturbance parameters of phase differences whose number is less than the number of unbalanced rotors by one. In the next section, the equations of motion of the system are described. The condition of implementing synchronization and that of stability of synchronization are deduced in Section 3. Computer simulations are carried out to verify the theoretical results in 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, which is composed of the two groups of springs installed symmetrically. Each vibrating machine is supported on the rigid frame by two groups of springs installed symmetrically and excited by two unbalanced rotors, which are separately driven by two induction motors rotating in opposite directions. As illustrated in Fig. 1, the projection o of the mass center G of the rigid frame onto the y-axis is fixed and that of the mass center Gi of the mass center of the vibrating machine Vi onto the yi-axis is also fixed. For the rigid frame, three reference frames are assigned: the fixed frame oxy with the y-axis vertical and coinciding with the center line of the rigid frame; the nonrotating moving frame Gx′ y′ z′ , which undergoes translation motions while remaining parallel to oxy, and the moving frame Gx′ y′ that is fixed to the rigid frame, as shown in Fig. 2(a). For each vibrating machine, three reference frames also are assigned: the fixed frame oixiyi, the nonrotating moving frame Gx′iyi′zi′ and the moving frame Gix′i yi′ , 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 and y, and an angular rotation ψ are set as independent coordinates. In like manner, the mass center coordinates of vibrating machine Vi, xi and yi, and its angular rotation ψi also are set as independent coordinates. In the reference frame Gix′i yi′ , the coordinates of mass center of each eccentric lump can be expressed as x′ mi1 = lai cos β + ri1 cos ϕi1 lai sin β + ri1 sin ϕi1 , x′ mi2 = −lai cos β − ri1 cos ϕi2 lai sin β + ri1 sin ϕi2 (i = 1, 2.) where lai is the distance between the mass center of vibrating machine Vi and the rotating center of the eccentric lump; and β is the angle between the line oioij and the x-axis. In the reference frame Gixiyi, the coordinates of the eccentric lump can be expressed as the following xmij = xGi + Rixmij , Ri = cos ψi − sin ψi (i = 1, 2, ; j = 1, 2.) sin ψi cos ψi where xGi is the coordinate vector of the mass center Gi of the vibrating machine Vi, xGi = {xi, yi}T . Then the kinetic energy T of the system is expressed as 1 T = 2 x˙ TGmR x˙G + 21 Xi=21 Xj=2 1 xTmij mij xmij + 12 JRψ˙ 2 + 21 Xi=21 x˙ Gimix˙ Gi + 1 X2 X2 J0ij ϕ˙ i2j 2 i=1 j=1 1 2 X Jiψ˙i2 + 2 i=1 ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) where mR is the mass matrix of the isolation frame, mR = diag(mR, mR), and JR is its moment of inertia about its mass center; xG is the displacement vector of the isolation frame, xG = {x, y}T ; mi is the mass matrix of the vibrating machine Vi, mi = diag(mV , mV ), and Ji is its moment of inertia about its mass center; mij is the mass matrix of the eccentric lump in the vibrating machine Vi and J0ij is the moment of inertia of motor j in the vibrating machine Vi; ( •˙) denotes d • /dt. When the vibrating system is running, the coordinates of the points that the supporting springs of the rigid frame are connected to the rigid frame on the reference oxy can be expressed as xri = xo + Rxr0i, R = scionsψψ c−ossiψn ψ , where xo is the displacement vector of the mass center of the rigid frame, xo = {x, y}T ; xr01 and xr02 are the coordinates of the corresponding points on the reference frame oxy when the system is in the equilibrium state, xr01 ≈ {lk, 0}T and xr02 ≈ {−lk, 0}T . Then, the vector of transformation of the corresponding spring can be expressed as Δxri = xri − xr0i, The coordinates of point of the supporting spring of the vibrating machine Vi, Xij , connected to itself on the reference frame oixiyi, are xij , then we can obtain its coordinate on oixiyi-frame as the following ( 6 ) ( 8 ) ( 9 ) (10) (11) (12) 1 2 X Δx˙ iT F iΔx˙ i + 2 i=1 1 X2 X2 Δx˙ iTj F ij Δx˙ ij + 2 i=1 j=1 1 X2 X2 fjiϕ˙ i2j 2 i=1 j=1 where qi is the generalized coordinates of the considered system and Qi 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 forces Qϕij = Teij and the others are zero, in which Teij is the electromagnetic torque of motor ij on vibrating machine i. Substituting the coordinates of the corresponding points 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 and ψi << 1. Hence, the inertia coupling resulting from the asymmetry of the eccentric lump can be neglected during the running process of the system. Therefore, the differential equations of motion of the system can be simplified as follows: 2 miy¨i + fy0y˙i − fy0y˙ − fyiψψ˙ + ky0yi − ky0y − kyiψψ = X mij rij (ϕ˙ i2j sin ϕij − ϕ¨ij cos ϕij ) j=1 J0ψ¨i + fy0l02ψ˙i − fy0l02ψ˙ + ky0l02ψi − ky0l02ψ = 2 P (−1)j−1mij rij lai(ϕ˙ i2j sin(ϕij − β) − ϕ¨ij cos(ϕij − β)) j=1 (J0i1 + mi1ri21)ϕ¨i1 + fi1ϕ˙ i1 = Tei1 − mi1ri1(y¨i cos ϕi1 − x¨i sin ϕi1 − laiψ¨i cos(ϕi1 − β)) (J0i2 + mi2ri22)ϕ¨i2 + fi2ϕ˙ i2 = Tei2 − mi2ri2(y¨i cos ϕi2 + x¨i sin ϕi2 + l0ψ¨i cos(ϕi2 − β)) fψ = fyla2 + 2fy0(l12 + l22 + 2l02), kψ = kyla2 + 2ky0(l12 + l22 + 2l02), fψy1 = fy0l1, fψy2 = fy0l2, fψy3 = −fy0l2, fψy4 = −fy0l1, kψy1 = ky0l1, kψy2 = ky0l2, kψy3 = −ky0l2, kψy4 = −ky0l1 fy1ψ = fy0l1, fy2ψ = fy0l2, fy3ψ = −fy0l2, fy4ψ = −fy0l1, ky1ψ = ky0l1, ky2ψ = ky0l2, with 3. Synchronization of the vibrating system When the system operates in the steady-state, the instantaneous average phase of the four unbalanced rotors is assumed to be ϕ, and their instantaneous average angular velocity is ϕ˙ . If the average value of ϕ˙ is assumed to be ωm and the coefficient of the instantaneous change of ϕ˙ is assumed to be ν0, ϕ˙ can be expressed as [18] 1 α1 = 2 (ϕ11 − ϕ22), ϕ˙ 11 = (1 + ε1)ωm, 1 α2 = 2 (ϕ21 − ϕ22), ϕ˙ 12 = (1 + ε2)ωm, 1 α3 = 4 (ϕ11 + ϕ12 − ϕ21 − ϕ22), ϕ˙ 21 = (1 + ε3)ωm, ϕ˙ 22 = (1 + ε4)ωm, α˙ i = νiωm(i = 1, 2, 3), ϕ11 = ϕ + α1 + α3, ϕ12 = ϕ − α1 + α3, ϕ21 = ϕ + α2 − α3, ϕ21 = ϕ − α2 − α3 ϕ˙ 11 = (1 + ε1)ωm, ε1 = ν0 + ν1 + ν3, ϕ˙ 12 = (1 + ε2)ωm, ε2 = ν0 − ν1 + ν3, ϕ˙ 21 = (1 + ε3)ωm, ε3 = ν0 + ν2 − ν3, ϕ˙ 22 = (1 + ε4)ωm, ε4 = ν0 − ν2 − ν3. When the vibrating system operates in the steady-state, the slip of an induction motor is very small. Hence, the effect of the angular accelerations of the unbalance rotors on the response of the vibrating system can be neglected, i.e., ϕ¨ij ≈ 0 [18–20]. On the other hand, the natural frequency of the vibrating system is far less than its operating one and its damping is very small [18–20]. In this case, the effect of the fluctuation of the angular velocities on the amplitude and phase angle of the response can also be neglected, i.e., the amplitude and phase angles of response of the vibrating system can be expressed by using ωm and neglecting νi. Therefore, the responses of the vibrating system can be expressed as follows: 2 x = rmar0μx X (cos(ϕi1 − γxe) − cos(ϕi2 − γex)) i=1 ψ = where γex, γey, γeψ, γex0, γey0, γeψ0, γeyψ0, π − γx0, π − γy0and π − γψ0 are the phase angles.   " ςeψ = ηe2ηψe2ψ−0 1 , ηeψ0 = pkyω0ml02/J0 . We differentiate xi, yi and ψi in Eq. (15) with respect to time t by the chain rule (applied to each component of α1, α2, α3 and ϕ) to obtain x¨i, y¨i and ψ¨i, respectively. Then substituting x¨i, y¨i and ψ¨i into the differential equations of motion for the four unbalanced rotors in Eq. (12) and integrating them over ϕ = 0 ∼ 2π and neglecting the high order terms of εij , considering Eq. (14), we obtain the average differential equations of the four parameters as # μx = , ςy0 = 1 ηy20 − 1 l0 le0 ςx0 ηe2x − 1 , 2 ηey0 , ηe2y − 1 m0 , rm = m0 , mR m (15) ςx0 = μx0 = 2 ηx0 ηx20 − 1 , 1 ηx20 − 1 ηey0 = pky0/mR ωm kex = kx − 4ςx0kx0, ωex = r J0 m , le0 = , ςex = r kex , mR ηex0 = ηex = ωex , ωm pkx0/mR ωm r key , mR 2 ηψ0 2 ηψ0 − 1 2 ηex0 , ηe2x − 1 μy = ςy0 ηe2y − 1 , pky0l02/J0 ωm key = ky − 4ςy0ky0, ωey = μy0 = , ςey = , ςψ0 = , ηψ0 = re1 = −re2 = ll1e , rea = llae ωeψ = r keψ , JR ηeψ = ωeψ , ςyy0 = ωm ηc2yηy20 2 , (ηy2 − 1)(ηy20 − 1) − 4ηc2yηy0 , le = μeψ = r JR mR 1 ηe2ψ − 1 , follows:  4 (Jri + m0r02)ωmε¯˙i + fdiωm(1 + ε¯i) = T¯ei − mr2ωm X χ′ij ε¯˙j + ωm j=1 4  X χij ε¯j  − χfi − χai j=1 i = 1, 2, 3, 4. where Jr1 = J011, Jr2 = J012, Jr3 = J021, Jr4 = J022, fd1 = f11, fd2 = f12, fd3 = f21, fd4 = f22, T¯e1 = T¯e11, Te2 = T¯e12, T¯e3 = T¯e21, T¯e4 = T¯e22. ¯ Compared with the change of ϕ(ϕ˙ = ωm) with respect to time t, that of εi and ε˙i are very small, so εi and ε˙i are considered to be slow-changing parameters in this study. During the aforementioned integration, εi, ε˙i and αi are assumed to be the middle values of their integration ε¯i, ε¯˙i and α¯i, respectively. The coefficients of ε¯i, ε¯˙i in Eq. ( 16 ) are listed in Appendix A. In engineering, the damping of the vibrating system is very small, hence the sine of phase angles in χ′ij and χij are neglected [18–20]. When a asynchronous motor operates in the vicinity of the angular velocity of ωm, the electromagnetic torque can be expressed as the following [18]: Tei = T¯e0i − k¯eiε¯i ¯ where Te0i and ke0i are the electromagnetic torque and the stiffness coefficient of angular velocity of the motor when its angular velocity is ωm, respectively. Because the moment of inertia of each motor’s rotor is much smaller than that of the unbalanced rotor, it can be neglected in Eq. ( 16 ). Introducing the following non-dimensional parameters ( 16 ) (17) (18) ρi = 1 − Wc0/2, i = 1, 2, 3, 4; κ1 = κ3 = ke01 m0r02ωm2 ke03 m0r02ωm2 + mf0dr102 + Ws1, + mf0dr302 + Ws2, κ2 = κ4 = into Eq. ( 16 ), dividing each formula by m0r02ωm, rearranging them and writing it in the matrix form, we obtain Aε¯˙ = Bε¯ + u where  ρ 1′ χ′12 χ′′13 χ′′14  A =  χχ′′3211 ρχ2′′32 ρχ3′23 χχ′3244  , χ41 χ42 χ43 ρ4  κ1 χ12 χ13 χ14  B =  χχ3211 κχ232 κ3 χ34  χ23 χ24  , χ41 χ42 χ43 κ4 ε¯ = { ε¯1 ε¯2 ε¯3 ε¯4 }T , u = { u1 u2 u3 u4 }. u1 = u3 = ¯ Te01 m0r02ωm ¯ Te03 m0r02ωm − mf0dr102 − χf1 + χa1 , m0r02ωm − fd3 m0r02 − χf3 + χa3 , m0r02ωm u2 = u2 = ¯ Te02 m0r02ωm ¯ Te04 m0r02ωm − mf0dr202 − χf2 + χa2 , m0r02ωm − fd4 m0r02 − χf4 + χa4 . m0r02ωm 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 rotors over their average period and called the non-dimensional coupling equation of the four unbalanced rotors. The analytical approach used in this paper converts the problem of synchronization of the multiple unbalanced rotors in a vibrating system into that of existence and stability of trial solution for their non-dimensional coupling equation. 3.1. Conditions of implementing synchronization Inserting u =  into Eq. (18) yields Toi = χfi + χai, i = 1, 2, 3, 4. where Toi is the differnece between the electromagnetic torque of the motor i and the frictional torque of its rotor and called the output torque of the motor i, Toi = T¯e0i − fdiωm. Subtracting the second formula in Eq. (19) from the first one yields = m0r02ωm2[(Wc0 − Wy − Wψ) sin 2α¯1 + 2(Wψ − Wy ) cos 2α¯3 cos α¯2 sin α¯1] where Wy = rmμy0ςy0ςey cos γey0 and Wψ = rmμψ0ςy0ςeψ0re21 cos γeψy0. Subtracting the fourth formula in Eq. (19) from the third one yields ΔTo12 = To1 − To2 ΔTo34 = To3 − To4 where TC1 = m0r02ωm2Wc0 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 expressed (19) (20) (21) (22) (23) (25) (27) (28) = m0r02ωm2[(Wc0 − Wy − Wψ) sin 2α¯2 + 2(Wψ − Wy ) cos 2α¯3 cos α¯1 sin α¯2] Subtracting the sum of the last two formulae in Eq. (19) from that of the first two ones yields ΔTo = To1 + To2 − To3 − To4 = 4m0r02ωm2(Wψ − Wy) cos α¯1 cos α¯2 sin 2α¯3 In order to obtain the greater torque of frequency capture and the stable magnitude of a vibrating frame, a vibrating machine 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 of response of the vibrating system are almost independent of the exciting frequency and constants [14]. Assuming where ωn represents the natural frequency of the vibrating systems, then we have 1 1 1 ςx0 > , ςy0 > , ςψ0 > (24) 15 15 15 Hence, in order to increase the torques of frequency capture between two of the four unbalanced rotors that are not installed on the same vibrating rigid frame, the natural frequency of the isolation frame is designed to be much higher than the operation frequency of the vibrating system. Assuming ωex > 4ωm, ωey > 4ωm, ωeψ > 4ωm (29) (30) (31) (32) (33) (34) (35) (36) where TC2 is called the second torque of frequency capture for the considered vibrating system, TC2 = 4m0r02ωm2(Wψ − Wy). Equation (30) demonstrates that the condition that the two vibrating machines can implement synchronization is that the second torque of frequency capture is equal to or greater than the absolute value of the output torque difference between the two pairs of the motors. i.e., When the parameters of the vibrating system satisfy Eqs (29) and (31), the solutions of Eq. (19), denoted by α¯10, α¯20, α¯30 and ωm0, can be determined by a numeric method. 3.2. Condition of the synchronization stability When u = , Eq. (18) is a generalized system [22] where A0 and B0 denote the values of A and B for α¯1 = α¯10, α¯2 = α¯20, α¯3 = α¯30 and ωm = ωm0. As shown in the expressions of χ′ij and χij (i = 1, 2, 3, 4; j = 1, 2, 3, 4) in Appendix A, when the parameters of the vibrating system satisfies the follow condition: As shown in Eq. (80), when the parameters of the vibrating system satisfy |TC1| > max{|ΔTo12| , |ΔTo34|} the solutions of 2α¯1 and 2α¯2 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 each pair of motors on the same rigid frame. When the two unbalanced rotors in a vibrating system with small damping rotate in opposite directions, the first torque of frequency capture is always much greater than the absolute value of the difference of output torque for each pair of the motors. Hence, 2α¯1 and 2α¯2 are close to 0 or π [14,15]. Therefore, 2α¯3 can be approximately expressed as aij > 0, det(A2) > 0, det(A3) > 0, det(A0) > 0, the matrices A0 and B0 satisfy the generalized Lyapunov equations [22]: IT B0 + B0T I = −2ωmdiag{κ1, κ2, κ3, κ4} A0T I = IA0 > 0 where I is the unit matrix. As shown in Appendix, when α¯10, α¯20 and α¯30 satisfy −π/2 < 2α¯10 < π/2, −π/2 < 2α¯20 < π/2, −π/2 < 2α¯30 < π/2, π/2 < 2α¯30 − α¯10 − α¯20 < π/2, π/2 < 2α¯30 + α¯10 + α¯20 < π/2, π/2 < 2α¯30 − α¯10 + α¯20 < π/2, π/2 < 2α¯30 + α¯10 − α¯20 < π/2. Equations (33), (34) and (35) can be satisfied. Therefore, the generalized system (Eq. (32)) is concessional and without pulse. If Aε = , Eq. (32) is lim t→∞ stable [22]. ε =  means that the electromagnetic torques of the four motors are stably balanced with the load lim t→∞ torques that the vibrating system acts on them. Linearizing Eq. (19) around α¯10, α¯20, α¯30 and ωm0, and neglecting Wsc and Wsc0, as well as fd1, fd2, fd3 and fd4, we obtain 3 ke01(ν0 + ν1 + ν2) = − X i=1 3 ke02(ν0 + ν1 − ν2) = − X i=1 3 ke03(ν0 − ν1 + ν3) = − X i=1 3 ke04(ν0 − ν1 − ν3) = − X i=1 ∂χa1 ∂αi ∂χa2 ∂αi ∂χa3 ∂αi ∂χa4 ∂αi 0 0 0 0 Δαi, Δαi, Δαi, Δαi, where v0 = δ1ν1 + δ2ν2 + δ3ν3, δ1 = − ke01 + ke02 − ke03 − ke02 , ke01 + ke02 + ke03 + ke02 δ2 = − δ3 = − ke01 − ke02 ke01 + ke02 + ke03 + ke04 ke03 − ke04 ke01 + ke02 + ke03 + ke04 , . where (•)0 denotes the values for α¯1 = α¯10, α¯2 = α¯20 and α¯3 = α¯30; and Δαi = α¯i − α¯i0, i = 1, 2, 3. Summing Eqs (37)–(40), and rearranging them, we obtain It should be noticed that Δ α˙ = Δα˙ 1 Δα˙ 2 Δα˙ 2 T = { ν1 ν2 ν3 }T . Substituting Eq. (41) into Eqs (37)–(40), and writing them into the generalized system of Δα = Δα1 Δα2 Δα2 T in 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 of Eqs (39) and (40) from that of Eqs (37) and (38) as the third row, we obtain EΔ α˙ = DΔα, where E = [eij ]3×3, and D = [dij ]3×3. Equation (42) can be rewritten as: Δ α˙ = CΔα, C = E−1D. λ3 + c1λ2 + c2λ + c3 = 0. c1 > 0, c3 > 0 and c1c2 > c3. Herein, Eq. (43) is called the generalized system for the disturbance parameters of phase differences. Exponential time-dependence of the form Δα = v exp(λt) is now assumed, and inserted into Eq. (43), then solving the determinant equation det(C − λI) = 0, we obtain the characteristic equation for the eigenvalue λ as the following: The zero solutions of Eq. (43) are stable only if all the roots of λin Eq. (44) have negative real parts. Using the Routh-Hurwitz criterion, Equation (45) satisfies the above requirements [11]: lim Δt→α+∞ = 0 means lim tν→i +∞ = 0, i = 0, 1, 2, 3. Using Eq. (14), we have lim t→ε+∞ = , i.e., lim At→ε+∞ = . In engineering, the parameters of the four induction motors are usually chosen to be similar [14,15], i.e., ke11 ≈ ke12 ≈ ke21 ≈ ke22 ≈ ke0, Te11 ≈ Te12 ≈ Te21 ≈ Te22. (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) In this case, the matrix E is approximately expressed as E = diag{4ke0, 2ke0, 2ke0}. From Eqs (28) and (30), we have sin α¯10 ≈ 0, sin α¯20 ≈ 0 and sin α¯30 ≈ 0. Then the matrices D and E can be also approximately simplified as two diagonal matrices. Hence, Eq. (43) can be simplified as follows: (47) (48) 2 Δα˙ 1 = −m0r0 ωm0Wc0 cos 2α¯10Δα1 2 Δα˙ 2 = −m0r0 ωm0Wc0 cos 2α¯20Δα2 Δα˙ 3 = −4m0r02ωm2(Wψ − Wy ) cos α¯10 cos α¯20 cos 2α¯30Δα3 In a vibrating system with dual-motor drivers rotating in opposite directions, Wc0 is always greater than 0, hence 2α¯10 and 2α¯20 are stabilized in the vicinity of 0 [14,15,18]. If Wψ > Wy , 2α¯30 is stabilized in the vicinity of 0; if Wψ < Wy , 2α¯30 is in the vicinity of π. 4. Computer simulations 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 small parameters, we deduce the non-dimensional differential equation of the disturbance parameters for the angular velocities of the four unbalanced rotors, which includes the inertia coupling of the unbalanced rotors, the stiffness coupling of angular velocity of the four motors, and the loading coupling of the four motors. This analytical approach converts the problem of synchronization for the four unbalanced rotors into the stability problem of a generalized system and a system of three first order differential equations for the three phase differences. In a vibrating system with small damping, the inertia coupling matrix of the four unbalanced rotors is symmetric, the stiffness coupling matrix is antisymmetrical and its diagonal elements are all negative. These facts make the generalized system satisfy the generalized Lyapunov equations when the inertia coupling matrix is positive definite. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix of the generalized system is definite positive, and that all the eigenvalues for the generalized system of three phase disturbance parameters have negative real parts. Acknowledgement This research is support by the National Science Foundation of China (Grant No: 51075063) and Program for Changjiang Scholars and Innovative Research Team in University. Appendix A Non-diagonal elements of matrices A and B 1 2 ωm[Ws + Wsc12 cos 2α¯1 + Wsc13 cos(2α¯3 + α¯1 − α¯2) + Wsc14 cos(2α¯3 + α¯1 + α¯2)] χf2 = ωm[Wcs12 cos 2α¯1 + Ws0 + Wcs23 cos(2α¯3 − α¯1 − α¯2) + Wcs24 cos(2α¯3 − α¯1 + α¯2)] (A1) χf3 = 21ωm[Wcs13 cos(2α¯3 + α¯1 − α¯2) + Wcs23 cos(2α¯3 − α¯1 + α¯2)Ws0 + Wcs34 cos2α¯2] χf4 = 12ωm[Wcs14 cos(2α¯3 + α¯1 + α¯2) + Wcs24 cos(2α¯3 − α¯1 + α¯2) + Wcs0 cos2α¯2 + Ws0] χa1 = 12ωm[Wcc12 sin2α¯1 + Wcc13 sin(2α¯3 + α¯1 − α¯2) + Wcc24 sin(2α¯3 + α¯1 + α¯2)] χa2 = 21ωm[−Wcc12 sin2α1 + Wcc23 sin(2α¯3 − α¯1 − α¯2) + Wcc24 sin(2α¯3 − α¯1 + α¯2)] χa3 = 12ωm[−Wcc13 sin(2α¯3 + α¯1 − α¯2) − Wcc23 sin(2α¯3 − α¯1 − α¯2) + Wcc0 cos2α¯2] χa4 = −21ωm[Wcc14 sin(2α¯3 + α¯1 + α~2) + Wcc24 cos(2α¯3 − α¯1 + α¯2) + Wcc0 cos2α¯2] χ′11 = −21Wc0 χ′12 = 12Wcc12 cos2α¯1 χ′13 = 12Wcc13 cos(2α¯3 + α¯1 − α¯2)) χ′14 = 21Wcc14 cos(2α¯3 + α¯1 + α¯2)) χ′21 = 21Wcc12 cos2α¯1 χ′22 = −21Wc0 χ′23 = 12Wcc cos(2α¯3 − α¯1 − α¯2) χ′24 = 12Wcc cos(2α¯3 − α¯1 + α¯2) χ′31 = 12Wcc13 cos(2α¯3 + α¯1 − α¯2) χ′32 = 21rm12Wcc23 cos(2α¯3 − α¯1 − α¯2) χ34 = ωmWcc34 sin 2α¯2 χ41 = −ωmWcc14 sin(2α¯3 + α¯1 + α¯2) χ42 = −ωmWcc24 sin(2α¯3 − α¯1 + α¯2) χ43 = −ωmWcc34 sin 2α¯2 χ44 = Ws0 Ws0 = rm[μx0(sin γx0 + ςx0ςex sin γex0) + μy0(sin γy0 + ςy0ςey sin γey0 + re21ςy0ςeψ0 sin γψy0) + μψ0(re2a sin γψ0 + ςeψ0ςψ0rea sin γeψ0)] 2 Wc0 = rm[μx0(cos γx0 + ςx0ςex cos γex0) + μy0(cos γy0 + ςy0ςey cos γey0 + re21ςy0ςeψ0 cos γψy0) + μψ0(re2a cos γψ0 + ςeψ0ςψ0rea cos γeψ0)] 2 Wcc12 = Wcc34 = Wcc0 + rm[μx0ςx0ςex cos γex0 − μy0(ςy0ςey cos γey0 + re21ςy0ςeψ0 cos γeψy0) + μψ0re2aςeψ0ςψ0 cos γeψ0] Wcc0 = rm(μx0 cos γx0 − μy0 cos γy0 + re2aμψ0 cos γψ0) Wcs12 = Wcs34 = Wcs1 + rm[−μx0ςx0ςex sin γex0 + μy0(ςy0ςey sin γey0 + re21ςy0ςeψ0 sin γeψy0) − μψ0re2aςeψ0ςψ0 sin γeψ0] 2 Wcs1 = rm(μx0 cos γx0 − μy0 cos γy0 + reaμψ0 cos γψ0) Wcc13 = Wcc24 = rm[−μx0ςx0ςex cos γex0 − μy0(ςy0ςey cos γey0 − re21ςy0ςeψ0 cos γeψy0) − μψ0ςeψ0ςψ0rea cos γeψ0] 2 Wcs13 = Wcs25 = rm[μx0ςx0ςex sin γex0 + Wcc14 = Wcc23 = rm[μx0ςx0ςex cos γex0 − Wcs14 = Wcs23 = rm[−μx0ςx0ςex sin γex0 + μy0(ςy0ςey sin γey0 − re21ςy0ςeψ0 sin γeψy0) + μψ0ςeψ0ςψ0rea sin γeψ0] 2 μy0(ςy0ςey cos γey0 − re21ςy0ςeψ0 cos γeψy0) + μψ0ςeψ0ςψ0rea cos γeψ0] 2 μy0(ςy0ςey sin γey0 − re21ςy0ςeψ0 sin γeψy0) − μψ0ςeψ0ςψ0rea sin γeψ0] 2 (A36) (A37) (A38) (A39) (A40) (A41) (A42) (A43) (A44) (A45) (A46) (A47) (A48) (A49) (A50) Appendix B: Parameters of the vibrating system Table A1 Parameters of the four induction motors Parameters Rated power (kW) Poles Rated frequency (Hz) Rated voltage (V) Rated rotational speed (r/min) Stator resistance (Ω) Rotor resistance referred to stator (Ω) Stator inductance (H) Rotor inductance referred to stator (H) Mutual inductance (H) Rotor damping coefficient (N.m.s/rad) [22] R. 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Chunyu Zhao, Qinghua Zhao, Zhaomin Gong, Bangchun Wen. Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame, Shock and Vibration, DOI: 10.3233/SAV-2010-0591