1 Introduction

The idea of self-synchronization vibration system comes from the phenomena of synchronization. Blekhman et al.^[1] and Blekhman and Yaroshevich^[2] proposed the self-synchronization theory of vibration system driven by two induction motors. When the structure parameters of the two motors meet the requirements,the system can be operated as synchronously. Blekhman et al.^[3] also found that even if the revolving speed and the angular phase difference of the two eccentric rotors were disturbed or one of the motors was powered off,the system could still be operated as synchronously.

Sperling^{[4, 5]} described a well suited method for the derivation of the conditions for the existence and the stability of synchronization motion applying harmonic influence coefficients. Ragulskis et al.^{[6, 7]} analyzed the self-synchronization of nonlinear dynamic system. The small parameter methods were used to a number of problems by Nagaev^[8],which lead to better understanding of the self-synchronization theory.

Some scientists investigated the self-synchronization of motors having a limited power supplies (non-ideal system). Kononenko^[9] analyzed the model of non-ideal system based on the pure mechanical model of the motor and by considering the stationary characteristic of the energy source. However,this analysis considering mechanical interactions between the oscillating system and the energy source is limited by an assumed straight line. Balthazar et al.^{[10, 11]} studied the self-synchronization of four motors having limited power supply and mounted on a flexible structural frame support,where the velocity and phase differences in the responses were also analyzed graphically.

Wen^[12] established a branch of the vibration utilization engineering with the selfsynchronization theory in engineering,and numerous self-synchronous vibration machines were invented. The structure of the synchronous machines can be significantly simplified by applying the self-synchronization theory. This theory also makes the maintenance easier and more convenient,and increases the reliability of the machines. Zhao et al.^{[13, 14, 15]} developed the self-synchronization of the dual-motor and four-motor driven vibration system by the modified average method. He further explained the coupled dynamic characters and dynamic symmetry of the vibration system,and several methods were used to analyze the self-synchronization motion and to verify the self-synchronization of two motors.

The vibration from the operation of machine can affect the normal work,break the structure and the components of the machine,lower the rate of productivity,and harm human’s health. Thus,it is essential to implement the vibration isolation for reducing the dynamic load transmitted to the foundation. This paper focuses on the conditions of self-synchronization and stability of a dual-motor driven vibration system with a two-stage isolation vibration frame. A sufficiently large oscillation amplitude of the material box can be ensured on the vibration system in order to screen materials,and a reduction of the dynamic load transmitted to the foundation can be also achieved on the vibration system.

The structure of this paper is as follows. Section 2 establishes the motion equations of the vibration system. The conditions of implementing self-synchronization and motion stability are deduced in Section 3. Section 4 carries out numerical simulations to verify the selfsynchronization motion,and Section 5 provides the experimental results. Finally,the conclusions are given in Section 6. 2 Motion differential equations of vibration system

The dynamic model of a vibration system with a two-stage vibration isolation frame is shown in Fig. 1,consisting of a material box m₁,a supporting rigid body m₂,a vibration isolation frame m_r,and two eccentric rotors m₀₁ and m₀₂. m₁ is connected with m₂ with springs,and k_x and k_y are the stiffness coefficients of the springs in the x- and y-directions,respectively. m₂ is connected with m_r with springs k_z in the z-axis direction,and is fixed along the x- and y-directions. m_r is supported by the elastic foundation k_zr. O₁ and O₂ are rotational centers of m₀₁ and m₀₂,respectively. m₀₁ and m₀₂ are driven by two induction motors,which are installed symmetrically about the horizontal plane (Oxy) of the centroid O of m₁. The rotational plane is δ (resonance excitation angle) against Oxy. The rotational centers of two eccentric rotors and the material box’s center of mass are along the same vertical axis. The two motors rotate in the same direction from the top view.

The motion differential equations of the vibration system can be established by the Lagrange equation as follows:

where M₁ is the mass of the vibration system in the x- and y-directions,M₁ = m₁+m₀₁+m₀₂, M₂ is the mass of the vibration system in the z-axis direction,M₂ = m₁ + m₂ + m₀₁ + m₀₂, m_r is the mass of isolation vibration frame,m_0i are the masses of exciters i (i = 1,2),J_0i are the moments of inertia of the motor rotors i (i = 1,2),k_x,k_y,k_z,and k_zr are the stiffness coefficients of the springs in the x-,y-,z-,and z_r-directions,respectively,f_x,f_y,f_z,and f_zr are the damping coefficients in the x-,y-,z-,and z_r-directions,respectively,f_d1 and f_d2 are the damping coefficients of the two motors,φ_i are the phases of exciters i (i = 1,2),δ is the resonance excitation angle,r is the eccentric radius of each exciter,and T_ei are the electromagnetic torques of the induction motors i (i = 1,2).

Making

and using the mode superposition method,we can get the vibration response of the first four equations of (1) as follows: where

3 Conditions of self-synchronization and motion stability 3.1 Dimensionless coupled equation of two eccentric rotors

It is assumed that the average phase of the two eccentric rotors is ψ during steady operation, and the phase difference between the two eccentric rotors is 2α. It is given by

Thus,each phase of the two eccentric rotors is Meanwhile,the angular velocity of eccentric rotors is assumed as = ω_m(t). Since the motion of vibration system changes periodically,i.e.,the load of each motor changes periodically,the average angular velocity ω_m0 of the two eccentric rotors is constant in period. The instantaneous variation coefficients of and are assumed to be ε₁ and ε₂,respectively,which are the functions of time t. and are given by

Substituting (5) into (4),and making v₁= ε₁ +ε₂ and v₂= ε₁ −ε₂,we can get the angular velocity and angular acceleration of two eccentric rotors as follows:

If two motors are operated with the same angular velocity,the average values of ε₁ and ε₂ over a single period must be zero,i.e.,ε₁ = 0 and ε₂ = 0.

Substituting (4) and (6) into (2) and introducing the following dimensionless parameters:

we can get the vibration responses of the system in the x_r-direction as follows:

Differentiate (7) with respect to time t to obtain ,and substitute them into the last two equations of (1). Then,by integrating them over ψ = 0-2π and neglecting the high order terms of v₁ and v₂,we get the average balanced equations of the two eccentric rotors as follows:

where Here,

Compared with the change of ψ ( = ω_m0) with respect to time t,those of α,v₁,v₂,₁, and ₂ are very small. Therefore,these above five parameters are slow-changing parameters. During the aforementioned integration over ψ = 0-2π,they can be assumed to be the middle values of their integration,i.e.,α,v₁,v₂,₁,and ₂,respectively.

If the angular velocity of the two motors is near ω_m0,their electromagnetic torques T_e1 and T_e2 can be expressed as

where T_e01 and T_e02 are the electromagnetic torques of the two motors when they are running in ω_m0,i.e.,they are in steady operation,and k_e01 and k_e02 are the coefficients of stiffness of the two motors when they are running in ω_m0.

In (8),since the inertia coefficients of the shafts of the two motors,J₀₁ and J₀₂,are far less than m₀₁r² and m₀₂r²,they can be neglected. Substituting (10) into (8) yields

Substituting (9) into (11),introducing the following dimensionless parameters:

and rearranging (11),we obtain the dimensionless coupled equation of the two eccentric rotors which describes the coupled relation between the two eccentric rotors as follows:

where

3.2 Condition of achieving self-synchronization motion

. If self-synchronization motion is realized on the system,ε₁ = 0,and ε₂ = 0. Substituting ε₁ = 0 and ε₂ = 0 into (12),we have u₁ = 0 and u₂ = 0. Rearranging (12),we get

where T_u is the kinetic energy of the eccentric rotor m₀₁,and T_u = m₀₁r²ω_m0²/2. T_R1 and T_R2 are the residual electromagnetic torques of motors 1 and 2,respectively,T_R1 = T_e01−f_d1ω_m0− T_uW_s0,and T_R2 = T_e02 − f_d2ω_m0 − T_uη²W_s0.

Subtracting two parts of (13),

where T_S is the torque of frequency capture,T_S = m₀₁r²ω_M0²|W_cc|,ΔT_R is the difference between the residual electromagnetic torques of two motors,i.e.,

Since |sin 2α| ≤ 1,the condition of achieving self-synchronization of the two eccentric rotors

i.e.,the torque of frequency capture is equal to or greater than the difference between the residual electromagnetic torques of the two motors. 3.3 Stability condition of self-synchronization motion

When the parameters of this system meet the condition of achieving the synchronization, the numerical solution of ω_m0 and α can be solved by (13),which are expressed as ω*_m0 and α₀^[14],respectively. When the vibration system operates synchronously,(12) is linearized at α = α₀. Using the notation z = {ε₁ ε₂ Δα}^T and considering in (13),we get^[15]

where

C is determined by C = A_'−1B',and the characteristic equation of C is determined by det(C − λI) = 0,which can be expressed as where According to the Routh-Hurwitz criterion,when (17) meets the solution z = 0 is stable. Then,(19) can be given by It is obvious that the working frequency of the vibration system is four times greater than the natural frequency in the non-resonant direction,and the damping ratio is very small (ξ ≤ 0.07), therefore,the vibration system meets H0 > 0^[15]. Thus,the condition of self-synchronization motion stability is 4 Numerical simulation of self-synchronization of vibration system 4.1 Selection of vibration isolation parameters

A reduction of the dynamic load transmitted to the foundation should be ensured on the vibration system in order to achieve vibration isolation. Therefore,the ratio of the mass m_r of the isolation vibration frame to the mass M₂ of the vibration system in the z-direction should be considered first during the parameters selection.

The parameters are selected as follows:

In general,when M₂ is twice as large as m_r,i.e.,the isolation mass ratio τ = 2,the two stage vibration isolation effect will be the best. Thus,the mass of vibration isolation frame m_r should be chosen as 1 250 kg.

According to (7),it is known that the system amplitude in the z-direction is closely related to the value of μ_z. The greater value of μ_z tends to result in a greater value of the system amplitude in the z-direction. The amplitude ratio of isolation vibration frame to the system in the z-direction is τμ_zr/μ_z. With a smaller value of τμ_zr/μ_z,the vibration isolation ability will get stronger. When τ = 2,μ_z and τμ_zr/μ_z are closely related to the frequency ratio in the zand zr-directions. The relation curves are shown in Fig. 2.

As seen in Fig. 2(a),it is only when n_zr is small enough,the whole system is possible to achieve resonance in the z-direction. When the resonance happens,the material box suffers a big amplitude. Each value of n_zr corresponds to a value of n_z,and it is obvious that when n_zr = 0.3,n_z = 0.9,the smaller value of n_zr is,the bigger value of n_z is. But the resonance amplitude slightly decreases.

From Fig. 2(b),when n_zr ≥ 1,it can be seen that the amplitude of isolation vibration frame is larger than the amplitude in the z-direction of the whole system. Therefore,the vibration isolation frame has no effect on the system. When n_zr < 1,the value of n_zr will become smaller,which means that the vibration isolation frame has better effect on the vibration system. When the value of n_zr is designated,with the increasing value of n_z,τμ_zr/μ_z decreases continuously and the vibration isolation effect gets better.

It can be seen in Fig. 2 that on the basis of a relatively small value of n_zr,a proper value of n_z is selected in order to make resonance in the z-direction realized on the system. This proper value of n_z can make the amplitude of the system in the z-direction big enough so as to meet the need of screening the materials. At the same time,this relatively small value of n_zr can make the system achieve vibration isolation in the z_z-direction. To sum up,the parameters should be selected as follows:

4.2 Self-synchronization motion of vibration system

The mass parameters of the vibration system are designated as follows:

Therefore,the system structure parameters are

The motor 1 and motor 2 are two three-phase squirrel-cage motors (380V,50Hz,6-pole,and Δ-connected). The parameters of the motor 1 and motor 2 are 3.7 kW,980 r/min and 0.75kW, 980 r/min,respectively. In order to ensure sufficient oscillation amplitude of the material box and reduce the dynamic load transmitted to the foundation,the relative parameters are chosen as τ = 2.064,n_z = 0.9,and n_zr = 0.3. Then,choose the other structure parameters as n_x = n_y = 4.0,ξ_x = ξ_y = ξ_z = ξ_zr = 0.07,δ = 15◦,and r = 0.2m. The numerical simulations of the vibration system are carried out,which are shown in Fig. 3. The power supply of motor 2 is cut off when t = 6 s,and a 10◦ phase disturbance is added to eccentric rotor 2 when t = 9s.

As shown in Fig. 3,when t = 4 s,the revolving speed of two motors reaches the same value (about 993.1 r/min),and the phase difference between the two eccentric rotors stabilizes at about 181.4◦. The displacement in every direction tends to be stable. At the moment,the π- phase self-synchronization and linear motion in the z-direction can be realized on the vibration system. Furthermore,the oscillation amplitude of the material box in the z-direction remains stable at about 1.19mm,and the oscillation amplitude of the isolation frame in the z_r-direction is about 0.04mm. Therefore,a good vibration isolation effect is achieved on the system.

When t = 6 s,the power supply of motor 2 is cut off. During this process,the revolving speed of two motors decreases to about 992.1 r/min and the phase difference increases to about 181.9◦. But the system’s vibration in every direction is still stable. The self-synchronization motion of two motors continues. When t = 9s,a 10◦ phase disturbance is added to the motor 2. The phase difference turns to 191.9◦. Meanwhile,the revolving speed of two motors and the system vibration in every direction start to fluctuate. After about 1.5 s,the system vibration returns to the steady state. So the system can automatically adjust the phase difference and recovers the steady state. To sum up,the π-phase self-synchronization,vibratory synchronization transmission,and stability of self-synchronization operation of the vibration system can be realized. It can also meet the needs to screen the materials and reduce dynamic load to the foundation. 5 Experimental verification

The testing and analyzing system of the self-synchronous vibration system is shown in Fig. 4. The displacement is obtained by the accelerometer,and the revolving speed and the phase difference are obtained by the Laser Tacho Probe. The B&K data acquisition hardware is used to analyze the sensor signal. The experimental results are described in Table 1 and Fig. 5. As shown in Table 1 and Fig. 5,speed synchronization is realized on the two motors when the synchronous speed is 989 r/min. The phase difference between two eccentric rotors stands at 170 degree,which means that π-phase self-synchronization is nearly achieved on the system. Stable periodic motions are achieved on the vibration system in all directions,i.e., the x-,y-,and z-,and z_r-directions. The vibration amplitudes in x- and y-directions stabilize at 0.099 6mm and 0.092 1mm,respectively,and they are small enough to be neglected. The vibration amplitudes in the z- and z_r-directions remain stable at 1.52mm and 0.465mm,respectively, which demonstrates that the effect of the isolation vibration is significant. Then, the power supply of motor 2 is cut off. As shown in the Figs. 5(g) and 5(h),the phase difference between two eccentric rotors stands at 169 degree,which is sight away from π-phase self-synchronization. Even if the power of one of the motors is cut off,the speed synchronization at 988 r/min can be also achieved on the two motors.

As seen from the comparison between the experimental results and numerical simulation results,the parameter values are generally the same. The simulation results are accurate and reliable. 6 Conclusions

In this paper,the conditions of self-synchronization and motion stability of a dual-motor driven vibration system with a two-stage vibration isolation frame are studied. By the Lagrange equations,the differential equations of the vibration system are obtained,and the dimensionless coupled equation of the eccentric rotors is obtained by a modified average small parameter method. Then,the conditions of achieving self-synchronization and motion stability are deduced. At last,the numerical simulations and experiments are carried out to verify the results of the theoretical analysis. The conclusions are summarized as follows:

(i) If T_S ≥ |ΔT_R| is satisfied,i.e.,the torque of frequency capture is equal to or greater than the difference between the residual electromagnetic torques of two motors,and selfsynchronization motion can be achieved on the vibration system.

(ii) According to the Routh-Hurwitz criterion,it is found that the condition of selfsynchronization motion stability is H₁ > 0,H₃ > 0,4H₁H₂ − H₀H₃ > 0.

(iii) The system amplitude in the z-direction is closely related to the value of μ_z. The greater value of μ_z is,the greater value of the system amplitude in the z-direction is. And the amplitude ratio of isolation vibration frame to the system in the z-direction is τμ_zr /μ_z . With a smaller value of τμ_zr /μ_z ,the vibration isolation ability will get stronger. μ_z and τμ_zr /μ_z are closely related to the frequency ratio in the z-direction n_z ,the z_r -direction n_zr ,and the isolation mass ratio τ. For obtaining the better isolation effect,τ = 2,n_zr = 0.3,and n_z = 0.9 are chosen. Under the condition,the vibration system can meet the needs of screening the materials and get strong vibration isolation ability to reduce the dynamic load to the foundation. (iv) The numerical simulations prove that when the vibration system meet the conditions of self-synchronization and motion stability,the stable π-phase self-synchronization,vibratory synchronization transmission,and strong vibration isolation ability can be achieved on the vibration system.

(v) The self-synchronization experiments of the vibration system are conducted. As seen from the comparison between the experimental results and numerical simulation results,the parameter values are basically the same. The simulation results are accurate and reliable.