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

-is paper aims at studying the synchronous stability of four homodromy vibrators in subresonant and superresonant states. -e motion differential equations are established firstly. -e simplified form of analytical expressions is yielded, and the stability criterion of synchronous states satisfies Routh–Hurwitz criterion.-e 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. -e correctness of theoretical results is verified by numerical characteristic analysis and simulations.-is paper can provide a theoretical reference for designing a type of new high-frequency vibrating mill.


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.e 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][11][12][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][15][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 multidegree-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 rotorpendula system [21], and some research studies on selfsynchronization of nonideal sources or exciters were given by Balthazar et al. [22,23].Zhang et al. [24][25][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.
e 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.
is paper focuses on studying synchronous stability of four vibrators in the different resonant states (subresonant and superresonant).e dynamical model consists of four vibrators and a rigid frame.e vibrator is a kind of mechanism which generates exciting forces through eccentric rotors driven by AC motor.e 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 Section 2; 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.

Dynamic Model Description
In Figure 1(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.
e 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 ′ .e displacement vectors of the system in x and y directions are denoted by x and y, respectively.e swing angle of the rigid frame is ψ. e rotational angles of four eccentric rotors are denoted by φ i , i � 1, 2, 3, 4. e angles between four vibrators and the x-axis are β 1 , β 2 , β 3 , β 4 , respectively, with β 1 � 0 °, β 2 � 90 °, β 3 � 180 °, and Based on Lagrange's equations and referring to [26], the motion differential equations of the vibrating system are established as the following form: where Here m is the mass of the rigid frame; m i 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; f i is damping coefficients of the axes of induction motor i, i � 1, 2, 3, 4; f x , f y , f ψ 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 J m is that of the rigid frame about its mass center; l e is equivalent rotational radius of the vibrating system about the mass center of the rigid frame; l 0 denotes the distance between the rotational center of the exciter i and the mass center of the rigid frame; k x , k y , k ψ are respectively stiffnesses of the vibrating system in x, y, and ψ directions; T ei [27] is electromagnetic torque of the motor i, i � 1, 2, 3, 4; and r is the eccentric radius of the exciter.

Theoretical Derivation
It is assumed that the average phase of four vibrators is φ; the phase differences between vibrators 1 and 2, 2 and 3, and 3 and 4 are 2α 1 , 2α 2 , and 2α 3 , respectively, that is, ( e change of average angular velocity _ φ 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 ω m0 of the average angular velocity _ φ during time T must be a constant, i.e., ω m0 � (1/T)  t+T t _ φ dt � constant.ω m0 denotes the average angular velocity of four exciters and is also referred to as the synchronous operating frequency in the steady state.

Shock and Vibration
According to the modi ed average method of small parameters [25,26], we can introduce small parameters ς 0 , ς i (ς 0 , ς i are functions of time, i 1, 2, 3).It is assumed that the coe cients of the instantaneous change of _ φ around ω m0 , i.e., _ us, the rearrangement of Equation ( 2) is expressed as ( If the average values of ε i (i 1, 2, 3, 4) over the single period are zero, i.e., ε i 0, (i 1, 2, 3, 4), we have , 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 rst three formulae in Equation (1), according to [26], the responses of the system in x, y, and ψ directions can be obtained directly as where c i is the di erence between π and the phase angle of the vibrating system in i-direction.c i arctan(2ξ ni (ω ni /ω m0 )/1 − (ω ni /ω m0 ) 2 ).i x, y, ψ; m 0 is the mass of the standard exciter; η i are the mass ratio of the exciter i to the standard one, η i m i /m 0 (i 1, 2, 3, 4); x, y, ψ; r m is mass ratio of the standard exciter to the total vibrating system, i.e., r m m 0 /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 di erential equation of four vibrators is obtained as Shock and Vibration 3 4 Shock and Vibration

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 torque T 0i as follows: where T 0i indicates the differences between the electromagnetic torque of the motor i and the damping torque of its rotor.
rough rearrangement, Equation ( 7) is expressed as 8)-( 10) are given by 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, Shock and Vibration Considering Equations ( 8)-( 10) and ( 12)-( 14), we can obtain Equations ( 15)-( 17) denote the synchronization criterion of four vibrators.e 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.
e sum of Equation ( 6) leads to where τ a (α 1 , α 2 , α 3 ) indicates the average dimensionless loading torque of four motors.As a limited function, it satisfies the following formula: where τ a max denotes the maximum of the average dimensionless loading torque of four motors.We define that ζ ij denotes the coefficient of synchronization ability, that is, with e larger the above coefficients, the stronger the coupling torques among vibrators.In other words, the larger these coefficients ζ 12 , ζ 23 , ζ 34 , and ζ 41 , the more easily vibrators can achieve synchronization.
) and τ a (α 1 , α 2 , α 3 ) are decided by parameters of the system, and their maximum will be discussed numerically in Section 4.

Shock and Vibration 7
After the rearrangement, we can deduce Applying the Routh-Hurwitz criterion, we deduce the criterion of stability of the synchronous states in the following equation: or where H i (i � 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, Introducing the following new symbols, (37) Considering the stability criterion (Equation (31)) and Equations ( 35)-(37), we can achieve stable intervals of phase differences among four vibrators as follows: (38)

Numerical Characteristics Analyses
Based on the above deduced theoretical analytical results on criteria of synchronization and stability of four vibrators, we Shock and Vibration 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 � 1430 kG, m 0 � 10 kG, J � 1750 kG • m 2 , k ψ � 20000 kN/m(μ ψ � 0.95), f ψ � 3.37 kNs/rad(ξ nx � ξ ny � ξ nψ � 0.07), k x � k y � 23000 kN/m(μ x � μ y � 0.95), and f x � f y � 3.83 kNs/m.e 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 R s � 3.35 Ω, the rotor resistance R r � 3.4 Ω, the stator inductance L s � 170 mH, the mutual inductance L s � 164 mH, and 05. e natural frequencies are defined as Because the stiffnesses of springs in x and y directions are identical, the natural frequencies in these two directions are the same, i.e., ω x � ω y � 126.8 ≈ 127 rad/s, and ω ψ � 106.9 ≈ 107 rad/s means the natural frequency in ψ direction.us resonant regions of the vibrating system are divided into three ranges: regions Ι(ω m0 < ω ψ ), ΙΙ(ω ψ < ω m0 < ω x ), and ΙΙΙ(ω m0 > ω x ).It should be noted that in the following numeric characteristic analyses, we obtain the rounding values of ω x , ω y , and ω ψ in order to only discuss and describe conveniently, i.e., ω x � ω y ≈ 127 rad/s and ω ψ ≈ 107 rad/s, which cannot influence the synchronization of the four vibrators and also not change the physical configuration of the system.
e parameter ζ ij 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 η i .In Figure 3 8)- (10), and hence, it is not dwelt on again.Additionally, CSAs in all conditions decrease as ω m0 increases in region ΙΙΙ(ω m0 > ω x ).

Stable Phase Differences.
From Figure 4(a), we can find that the phase differences in region Ι(ω m0 < ω ψ ) are the same as that in region ΙΙ(ω ψ < ω m0 < ω x ).Due to the fact of the small difference value between ω x and ω ψ in engineering, we may combine regions Ι and ΙΙ into one common region, which is defined as the subresonant region with respect to ω x , while region ΙΙΙ is defined as the superresonant region for ω x .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 ω m0 � 74 rad/s.As such dashed line B means the simulation point of the superresonant region and 198 for ω m0 � 198 rad/s.e validity of these calculated values will be verified in next section of computer simulations.
It can be seen that from Figure 4(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].
e vibrators 1, 2, and 3 are the same, and the mass of the vibrator 4 is changed, and the stable phase difference curves of Figure 4(b) are obtained.e 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: While in the nearresonance region, the transition value of the phase difference appears.
e vibrators 1 and 2 are the same, and masses of the vibrators 3 and 4 are changed; the stable phase difference curves of Figure 4(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: In this case, the transition value of the phase differences also appears in the near-resonance region.

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 is m 0 � 10 kG, and the eccentric radii of four vibrators are same.Meanwhile, in the subresonant region, k x � 23000 kN/m and in the superresonant region, k x � 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.us, we need to introduce a new variable; it is the frequency ratio z, which can facilitate to derive ω m0 of the simulation point through it.
Equation (40) denotes the frequency ratio; we substitute k x � 23000 kN/m into Equation (39) and obtain a new natural frequency of the point A, that is ω * n � 178 rad/s.Simultaneously, based on the rotational speed in simulation (ω SA � 103 rad/s) and Equation (40), the frequency ratio of the simulation point A is obtained, that is z � 0.58.

Shock and Vibration
rough the frequency ratio, we can deduce that the operating frequency of the simulation point A is ω m0 74 rad/s.In the same way, the frequency ratio and operating frequency of the simulation point of B are z 1.56 and ω m0 198 rad/s, when we consider k x 6240 kN/m.
In other words, the simulation of subresonant region is in the condition of z 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.

Superresonant Region (Simulation Point B) Simulations
5.1.1.Simulation for η 1 η 2 η 3 η 4 1.Here, masses of four vibrators are identical, i.e., m i m 0 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 di erences 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 10 Shock and Vibration the phase di erences are near to 180 °.In Figures 5(b)-5(d), at 20 s, a disturbance of π phase applies to the vibrator 2; 2α 1 transforms from 180 °to 90 °, as well as 2α 2 and 2α 3 , 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).
In Figure 5(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 o , and then 2α 1 transforms from 90 °to 107.23 °, 2α 2 from 90 °to 70.15 °, and 2α 3 from 90 °to 94.86 °, which means that the phenomenon of vibratory synchronization transmission occurs in the system.
In Figures 5(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.

Simulation for η
e analysis method in this section is the same as the previous section.Figure 6 shows that the synchronous rotational velocity is 983 r/min, and the phase di erences are 2α 1 105 °, 2α 2 106 °, 2α 3 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 di erences change to 2α 1 153 °, 2α 2 152 °, 2α 3 −155 °.Contrasting the results with simulation point B in Figure 4(b), we can nd that they are alike.At 40 s, the electricity of the system is switched o , and we see that the system is back to the steady state from Figure 6(a); in this case, the phase di erences are 2α 1 115 °, 2α 2 96 °, 2α 3 64 °, and the synchronous rotational velocity changes to 978 r/min.ese facts show that the vibrating system still can operate by the way of vibratory synchronization transmission.
5.1.3.Simulation for η 1 η 2 1, η 3 0.75, η 4 0.5.Figure 7 shows the following results.e four vibrators operate synchronously, when time reaches about 5 s.At this time, the phase di erences are 2α 1 155 °, 2α 2 158 °, 2α 3 −144 °and the synchronous rotational velocity is 983 r/min.Similarly, a disturbance of π phase applies to the vibrator 2, and the phase di erences transform to 2α 1 112 °, 2α 2 103 °, 2α 3 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 o , the system is back to the steady state and still keeps operating synchronously.In this moment, the phase di erences are 2α 1 125 °, 2α 2 89.85 °, 2α 3 42.99 °and the synchronous rotational velocity is 978 r/min.Shock and Vibration

Subresonant Region (Simulation Point A)
Simulations.e analysis method in this section is the same as the previous section, and the results will be presented in the table form.1.
e stable phase di erences of simulation and characteristic analysis are in good agreement with each other.In Figures 8(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.e 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.
We can observe that the phase di erences before disturbance are approximately equal to those after disturbance.e phenomenon accords with that of the characteristic analysis.

Conclusions
According to the above theoretical derivation, numerical characteristic analysis, and simulations, the following conclusions are given: (1) rough contrasting simulations of two regions in di erent η, we can observe the phase di erences in η 1 η 2 η 3 η 4 1 are more stable, which indicates that structure parameters have a large e ect 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 di erences in the subresonant maintain around 0 °and those in the superresonant maintain around 90 °and 180 °. e resonant responses of the rigid frame with double resonant types in x, y, and ψ directions are also discussed.
ese 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  18 Shock and Vibration 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.

Figure 1 :
Figure 1: Model and coordinate systems.(a) Dynamic model of the vibrating system; (b) three reference coordinate systems.